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Fluctuations and phase transitions in quantum Ising systems McKenzie, Ryan 2016

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Fluctuations and Phase Transitions in Quantum IsingSystemsbyRyan McKenzieB.Sc. Math and Physics Honours, The University of British Columbia, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics and Astronomy)The University of British Columbia(Vancouver)August 2016c© Ryan McKenzie, 2016AbstractThe quantum Ising model is perhaps the simplest possible model of a quantum magnetic mate-rial. Despite its simplicity, its versatility and wide range of applications, from quantum compu-tation, to combinatorial optimization, to biophysics, make it one of the most important modelsof modern physics. In this thesis, we develop a general framework for studying quantum Isingsystems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantumfluctuations, and the quantum phase transition between paramagnetic and ferromagnetic statesthat occurs when a magnetic field is applied transverse to the easy axis of the system.The magnetic insulating crystal LiHoF4 is a physical realization of the quantum Isingmodel, with the additional features that the dominant coupling between spins is the long rangedipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom.These nuclear degrees of freedom constitute a spin bath environment acting on the system. Inthis thesis, we present an effective low temperature Hamiltonian for LiHoF4 that incorporatesboth these features, and we analyze the effects of the nuclear spin bath on the system. We findthe lowest energy crystal field excitation in the system is gapped at the quantum critical pointby the presence of the nuclear spins, with spectral weight being transferred down to a lowerenergy electronuclear mode that fully softens to zero at the quantum critical point. Further-more, we present a toy model, the spin half spin half model, that illustrates the effects of ananisotropic hyperfine interaction on a quantum Ising system. We find the critical transversefield is increased when the longitudinal hyperfine coupling is dominant, as well as an enhance-ment of both the longitudinal electronic susceptibility and an applied longitudinal field.In addition, we present a field theoretic formalism for incorporating the effects of fluctua-tions beyond the random phase approximation in general quantum Ising systems. We find thatany regular on site interaction, such as a nuclear spin bath, does not fundamentally alter thecritical properties of a quantum Ising system. This formalism is used to calculate correctionsto the magnetization of LiHoF4.iiPrefaceIn this thesis, we study fluctuations and phase transitions in quantum Ising systems, of whichthe fascinating magnetic material LiHoF4 is a particular example. The study of LiHoF4, inparticular, the study of the effect the nuclear degrees of freedom have on the system, was sug-gested by Dr. Philip Stamp. All the research carried out in this thesis, and the tools developedfor that purpose, are the work of the author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The LiHoF4 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.1 The Crystal Field and Exchange Interaction . . . . . . . . . . . . . . . 232.1.2 The Hyperfine Interactions . . . . . . . . . . . . . . . . . . . . . . . . 272.1.3 Domains in LiHoF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 The Effective Low Temperature Hamiltonian . . . . . . . . . . . . . . . . . . . 352.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Dipolar Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1 Dipolar Interaction in the Continuum Limit . . . . . . . . . . . . . . . . . . . 413.1.1 Finite Sized Spherical Sample . . . . . . . . . . . . . . . . . . . . . . 443.1.2 Finite Sized Cylindrical Sample . . . . . . . . . . . . . . . . . . . . . 463.2 Direct Summation of the Dipolar Interaction in LiHoF4 . . . . . . . . . . . . . 483.3 Ewald Summation in LiHoF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52iv3.3.1 Ewald Summation in a Cylindrical Sample . . . . . . . . . . . . . . . . 573.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Spin Half Spin Half Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1 Perturbation Theory in the Ordered Phase . . . . . . . . . . . . . . . . . . . . 664.2 Mean Field Electronic Spin Magnetization . . . . . . . . . . . . . . . . . . . . 684.3 Mean Field Nuclear Spin Magnetization . . . . . . . . . . . . . . . . . . . . . 754.4 Longitudinal Electronic Correlation Function . . . . . . . . . . . . . . . . . . 804.4.1 Paramagnetic Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.2 Ferromagnetic Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5 Electronic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 LiHoF4 in the Random Phase Approximation . . . . . . . . . . . . . . . . . . . . 935.1 Electronic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Field Theoretic Treatment of Quantum Ising Systems . . . . . . . . . . . . . . . 1066.1 Field Theory: From Heisenberg to Ising Systems . . . . . . . . . . . . . . . . 1146.1.1 Paramagnetic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1.2 Ferromagnetic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 First Order Phase Transitions in an Ising System . . . . . . . . . . . . . . . . . 1186.3 Quantum Ising Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3.1 Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.3.2 Gaussian Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3.3 The Interacting Field Theory . . . . . . . . . . . . . . . . . . . . . . . 1296.4 The Renormalized Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377 Corrections to Mean Field Magnetization . . . . . . . . . . . . . . . . . . . . . . 1387.1 Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2 Spin Half Transverse Field Ising Model . . . . . . . . . . . . . . . . . . . . . 1477.3 Spin Half Spin Half Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.4 LiHoF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156vBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A Mean Field Basis Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170B Transverse Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176B.1 Mean Field Operator Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.3 Susceptibility and Correlation Functions . . . . . . . . . . . . . . . . . . . . . 178C A Diagramatic Expansion for Spin Systems . . . . . . . . . . . . . . . . . . . . . 180C.1 Cumulants and the Generating Function . . . . . . . . . . . . . . . . . . . . . 180C.2 The Diagramatic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182D Susceptibilities and Correlation Functions . . . . . . . . . . . . . . . . . . . . . . 186D.1 Ising Model: Structure of the Green’s Function . . . . . . . . . . . . . . . . . 189D.2 RPA Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190E Spin Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196E.1 Two Spin Cumulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197E.2 Three Spin Cumulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198E.3 Four Spin Cumulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201viList of TablesTable 2.1 In this table, we list various estimates of the crystal field parameters ofLiHoF4 in units of Kelvin. References [10, 76] are based on the suscep-tibility measurements of Hansen et al. and Beauvillain et al., respectively.References [77, 78] are, respectively, the optical light scattering experimentsof Gifeisman et al. and Christensen. Reference [79] contains the estimatesbased on Shakurov et al.’s EPR experiments. In [67], we have the numericalestimates of Rønnow et al., and in [80], we have the estimates of Babkevichet al. based on neutron scattering experiments. . . . . . . . . . . . . . . . . 25Table 7.1 In this table, we show corrections to the mean field magnetization of LiHoF4calculated using the high density approximation. Each column correspondsto a different value of the applied transverse magnetic field Bx, and each rowcorresponds to a finer division of the Brillouin zone (a larger value of N inequation (7.40)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152viiList of FiguresFigure 1.1 The above figure, taken from the paper of Rønnow et al., illustrates manyaspects of the magnetic insulating crystal LiHoF4. In A, we see the phasediagram as determined by magnet susceptibility (circles) and neutron scat-tering (squares). At low temperatures, we see that the ferromagnetic phaseis stabilized by the hyperfine interaction. In B, we see the gap in the spec-trum at what would be the electronic soft mode in the absence of hyperfineinteractions. The dashed line shows the soft mode in the absence of thehyperfine interaction. Diagram C is a schematic illustration of the elec-tronic eigenstates split into multiplets by the nuclear degrees of freedom.Finally, in D, we see the ratio of the energy gap Ec shown in diagram B, tothe splitting of the two lowest MF electronic energy levels (∆), plotted as afunction of temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.1 The above figure shows the phase diagram of LiHoF4, measured by Bitkoet al. via magnetic susceptibility (solid circles). The vertical axis is theapplied transverse field, and the horizontal axis is the temperature. Thedashed line is the phase diagram calculated using mean field theory, ne-glecting the hyperfine interaction. The solid line is calculated using meanfield theory, with the hyperfine interaction included. There are several freeparameters in the MF calculations which lead to the apparently good agree-ment between the experimental data and the MF results. In reality, fluctua-tions in the MF have a significant impact on the phase diagram of LiHoF4due to the dominant long range dipolar coupling between electronic spins. . 18Figure 2.2 The figure above, taken from the thesis of Kraemer [16], shows the struc-ture of materials in the LiReF4 series, where Re is a rare earth atom. . . . . 22viiiFigure 2.3 In this figure, we show the energy levels of the crystal field component ofthe LiHoF4 Hamiltonian (in Kelvin) as a function of an applied transversefield Bx (in Tesla). The inset shows the lowest three energy levels with theground state taken as zero energy. We see that the second excited state isseperated from the low energy doublet by a gap of over 10K. . . . . . . . . 27Figure 2.4 In this figure, we show the mixing of the even electronic eignestates of theJz operator by the crystal field as a function of an applied transverse fieldfor the ground state (solid line) and first excited state (dashed line) of theelectronic component of the single ion Hamiltonian for LiHoF4. Each α jcorresponds to the electronic eigenstate such that Jz| j〉= j| j〉 . . . . . . . 28Figure 2.5 In this figure, we show the mixing of the odd electronic eignestates of theJz operator by the crystal field as a function of an applied transverse fieldfor the ground state (solid line) and first excited state (dashed line) of theelectronic component of the single ion Hamiltonian for LiHoF4. Each α jcorresponds to the electronic eigenstate such that Jz| j〉= j| j〉 . . . . . . . . 29Figure 2.6 In the figures above, we plot the non-zero matrix elements of the effectivespin half operators, Jµ = Cµ(Bx) +∑ν=x,y,zCµν(Bx)τν , for the truncatedLiHoF4 Hamiltonian. The plot on the left shows the larger matrix elements,with the upper most matrix element being Czz. Below Czz, in descendingorder, we have Cx, Cxx and Cyy. The matrix elements in the right hand plotare much smaller than those on the left. In descending order, we have Cy,Cxy, and Cyx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.7 In this figure, we plot the effective transverse field, ∆ (in Kelvin), actingon the effective Ising spins in LiHoF4, as a function of the physical trans-verse field Bx (in Tesla). The inset shows the next largest parameters inthe LiHoF4 Hamiltonian, these being the effective transverse field actingdirectly on the nuclear spins, ∆xn, and the longitudinal hyperfine coupling, Az. 38Figure 2.8 In this figure, we plot the magnitudes of the transverse hyperfine parame-ters, A⊥ being the uppermost line, and A++ being the middle line, in theeffective low temperature Hamiltonian for LiHoF4 as a function the appliedtransverse magnetic field Bx. The lowest line is the stray field, ∆yn, actingon the nuclear spins in the direction transverse to the easy axis and the di-rection of the applied transverse field. All these parameters are about anorder of magnitude smaller than the other parameters in the model in thevicinity of the critical transverse field Bx = 4.9T . . . . . . . . . . . . . . . 39ixFigure 3.1 In the figure above we see the Fourier transform of the dipolar interactionin a long thin cylindrical sample of LiHoF4 as a function of transversemomenta at kz = 0 (left), and kz = pi3c (right). . . . . . . . . . . . . . . . . . 60Figure 3.2 In the figure above we see the Fourier transform of the dipolar interactionin a long thin cylindrical sample of LiHoF4 as a function of transversemomenta at kz = 2pi3c (left), and kz =pic (right). . . . . . . . . . . . . . . . . 60Figure 4.1 The plot above shows z (solid line) and x (dashed line) ground state com-ponents of the electronic spin operators (in the MF approximation) of thespin half transverse field Ising model with an isotropic hyperfine interac-tion, as a function of the applied transverse field ∆. We work in units ofthe exchange interaction strength J. We find the z component of the mag-netization is uniformly reduced with increasing transverse field strength.We obtain these results from the MF Hamiltonian of the spin half spin halfmodel, given in equation (4.28). . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.2 The plot above shows z (solid line) and x (dashed line) ground state com-ponents of the electronic spin operators (in the MF approximation) of thespin half transverse field Ising model with an anisotropic hyperfine inter-action, as a function of the applied transverse field ∆. We work in units ofthe exchange interaction strength J. The plot on the left shows the magne-tization when the longitudinal hyperfine interaction (Az = 0.8) is dominant,whereas the plot on the right is for a dominant transverse hyperfine interac-tion (A⊥ = 0.8). With the longitudinal hyperfine interaction dominant, wesee that the critical transverse field is driven to larger values with a decreas-ing transverse hyperfine interaction A⊥. With A⊥ dominant, decreasing Azreduces the critical transverse field. We obtain these results from the MFHamiltonian of the spin half spin half model, given in equation (4.28). . . . 70xFigure 4.3 The plot above shows z (solid line) and x (dashed line) ground state compo-nents of the electronic spin operators (in the MF approximation) of the spinhalf transverse field Ising model with an anisotropic hyperfine interactionand a transverse field acting directly on the nuclear spins, as a function oftransverse field ∆. We work in units of the exchange interaction strength J.The longitudinal hyperfine interaction (Az = 0.8) is dominant, and the ap-plied transverse field acting directly on the nuclear spins is ∆n= ∆50 . We ob-tain these results from the MF Hamiltonian of the spin half spin half model,given in equation (4.28), with an additional termHn=−∆n2 (I++I−) to ac-count for the transverse field acting on the nuclear spins. This additionaltransverse field leads to a reduction in the critical transverse field of thesystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 4.4 The plot above shows the z (solid lines) and x (dashed lines) componentsof the ground state expectation values of the nuclear spin operators (in theMF approximation) of the spin half transverse field Ising model with anisotropic hyperfine interaction, as a function of transverse field ∆. Wework in units of the exchange interaction strength J. We obtain these re-sults from the MF Hamiltonian of the spin half spin half model, given inequation (4.28). We plot the absolute value of the expectation values of thenuclear operators, noting that they are equal and opposite their electroniccounterparts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.5 The plot above shows the z (solid lines) and x (dashed lines) componentsof the ground state expectation values of the nuclear spin operators (in theMF approximation) of the spin half transverse field Ising model with ananisotropic hyperfine interaction, as a function of transverse field ∆. Wework in units of the exchange interaction strength J. We obtain these re-sults from the MF Hamiltonian of the spin half spin half model, given inequation (4.28). We plot the absolute value of the expectation values of thenuclear operators, noting that they are equal and opposite their electroniccounterparts. The plot on the left shows the magnetization while the longi-tudinal hyperfine interaction (Az = 0.8) is dominant, while the plot on theright is for a dominant transverse hyperfine interaction (A⊥ = 0.8) . . . . . 77xiFigure 4.6 The plot above shows the z (solid lines) and x (dashed lines) componentsof the ground state expectation values of the nuclear spin operators (in theMF approximation) of the spin half transverse field Ising model with anisotropic hyperfine interaction, as a function of transverse field ∆. Wework in units of the exchange interaction strength J. We obtain these re-sults from the MF Hamiltonian of the spin half spin half model, given inequation (4.28), with an additional transverse field acting directly on thenuclear spins.. We plot the absolute value of the expectation values of thenuclear operators, noting that they are equal and opposite their electroniccounterparts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 4.7 The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin half model with an anisotropic hyper-fine interaction, calculated from the Green’s function, equation (4.74), atzero wavevector~k= 0. We work in units of J, the strength of the exchangecoupling between spins. Here, the longitudinal hypefine interaction has thesame strength as the exchange interaction, and the transverse hyperfine in-teraction is a factor of ten smaller. We see the lower mode softens to zeroin a quantum phase transition and the associated spectral weight diverges.The middle mode carries most of the spectral weight throughout the rest ofthe diagram, except in weak transverse fields where the upper mode maycarry some of the spectral weight. . . . . . . . . . . . . . . . . . . . . . . 86Figure 4.8 The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin half model with an anisotropic hyper-fine interaction, calculated from the Green’s function, equation (4.74), atzero wavevector~k= 0. We work in units of J, the strength of the exchangecoupling between spins. Here, the longitudinal hypefine interaction has thesame strength as the exchange interaction, and the transverse hyperfine in-teraction is a factor of two smaller. We see the lower mode softens to zeroin a quantum phase transition and the associated spectral weight diverges.The middle mode carries most of the spectral weight throughout the rest ofthe diagram, except in weak transverse fields where the upper mode maycarry some of the spectral weight. . . . . . . . . . . . . . . . . . . . . . . 87xiiFigure 4.9 The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin spin half model with an anisotropic hy-perfine interaction, calculated from the Green’s function, equation (4.74),at zero wavevector~k = 0. We work in units of J, the strength of the ex-change coupling between spins. We take the strength of the hyperfine in-teraction to be the same as the strength of the exchange interaction. Wesee the lower mode softens to zero in a quantum phase transition and theassociated spectral weight diverges. The middle mode carries most of thespectral weight throughout the rest of the diagram. . . . . . . . . . . . . . 87Figure 4.10 The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin spin half model with an anisotropic hy-perfine interaction, calculated from the Green’s function, equation (4.74),at zero wavevector~k = 0. We work in units of J, the strength of the ex-change coupling between spins. Here, the transverse hypefine interactionhas the same strength as the exchange interaction, and the longitudinal hy-perfine interaction is a factor of two smaller. We see the lower mode softensto zero in a quantum phase transition and the associated spectral weight di-verges. The middle mode carries most of the spectral weight throughoutthe rest of the diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 4.11 RPA Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.12 Spectral Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.13 The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin spin half model with an anisotropic hy-perfine interaction, calculated from the Green’s function, equation (4.74),at zero wavevector~k = 0. We work in units of J, the strength of the ex-change coupling between spins. Here, the transverse hypefine interactionhas the same strength as the exchange interaction, and the longitudinal hy-perfine interaction is a factor of ten smaller. The RPA modes and theirspectral weight are colour coordinated. We see the lower mode softens tozero in a quantum phase transition and the associated spectral weight di-verges. The middle mode carries most of the spectral weight throughoutthe rest of the diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89xiiiFigure 5.1 The plot above shows the energy levels of the effective low temperaturemean field Hamiltonian of LiHoF4, given in equation (5.1), as a functionof the physical transverse magnetic field, Bx. We consider a long thin cylin-drical, or needle shaped, sample. . . . . . . . . . . . . . . . . . . . . . . . 95Figure 5.2 The plot above shows the transverse and longitudinal electronic (〈Jx〉 and〈Jz〉), and nuclear (|〈Ix〉| and |〈Iz〉|), magnetizations of LiHoF4. The mag-netizations are calculated self consistently from the mean field Hamilto-nian given in equation (5.1). We consider a long thin cylindrical, or needleshaped, sample. We see the nuclear magnetizations, |〈Ix〉| and |〈Iz〉|, sat-urate near the quantum phase transition and in the zero field limit, respec-tively. The electronic magnetizations, 〈Jx〉 and 〈Jz〉, fail to saturate due tothe disordering effect of the crystal field. . . . . . . . . . . . . . . . . . . . 96Figure 5.3 The plots above show the matrix elements of the electronic spin operatorτz, with ci j = 〈i|τz| j〉, for the low temperature effective Hamiltonian forLiHoF4, given in equation (5.1), in the MF basis. We plot the matrix ele-ments as a function of the physical transverse field Bx (in Tesla). Note thescale of the vertical axis in the plot to the left is three orders of magnitudesmaller than that of the plot to the right. . . . . . . . . . . . . . . . . . . . 99Figure 5.4 The plot above shows the zero wavevector longitudinal susceptibility ofLiHoF4 in the random phase approximation, along with the lowest energymode (at zero wavevector, k = 0) in the electronic (RPA) spectrum, as afunction of the applied transverse magnetic field Bx (in Tesla). We considera long thin cylindrical sample. We see that the susceptibility diverges, andthe lowest energy mode softens all the way to zero, at the quantum criticalpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 5.5 The plot above shows the RPA spectrum (in Kelvin) of the low temperatureeffective Hamiltonian for LiHoF4 given in (5.1), at zero wavevector, as afunction of the transverse magnetic field Bx (in Tesla). . . . . . . . . . . . . 101Figure 5.6 The plot above shows the spectral weight of the RPA modes (E1 and E8 toE15) of LiHoF4, calculated from the longitudinal Green’s function (5.15),at zero momentum~k= 0, as a function of the physical transverse magneticfield field Bx (in Tesla). We see the spectral weight of the lowest energymode diverges at the quantum critical point. Above Bx ≈ 3T , most of thespectral weight is carried by E8, except near the phase transition. Themodes not shown in the figure carry no spectral weight. . . . . . . . . . . 102xivFigure 5.7 The plot above shows the spectral weight of the upper RPA modes (E8 toE15) of LiHoF4, calculated from the longitudinal Green’s function (5.15),at zero momentum~k = 0, as a function of the physical transverse field Bxbetween 1T and 3.5T . Above Bx ≈ 3T , E8 is the dominant mode. Thepeaks in the intensities, in order of decreasing amplitudes, correspond toE9, E10, ..., E15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 5.8 The plot above shows the RPA spectrum of LiHoF4 in momentum space, atthe critical mean field ∆0c = 5.33T , calculated from the longitudinal Green’sfunction given in equation (5.15). The central mode, separated from therest, carries the spectral weight away from k = 0. Near k = 0, spectralweight is transferred to the lowest energy mode. . . . . . . . . . . . . . . . 104Figure 5.9 The plot above shows the spectral weight of the RPA modes of LiHoF4 inmomentum space, calculated from the longitudinal Green’s function givenin equation (5.15), at the critical mean field, ∆0c = 5.33T . . . . . . . . . . . 104Figure 6.1 The figure above shows the one loop diagrams that contribute to the lead-ing order correction (order 1z , z being the coordination number) to the con-nected two point longitudinal correlation function of a quantum Ising sys-tem given in equation (6.107). The left most diagram is the balloon contri-bution gB, the center diagram is the loop contribution gL, and the rightmostdiagram corresponds to gu. . . . . . . . . . . . . . . . . . . . . . . . . . . 135Figure 7.1 The diagram above corresponds to the leading order correction to the mag-netization of a quantum Ising system in the high density approximation (anexpansion in the inverse coordination number). . . . . . . . . . . . . . . . 139Figure 7.2 In this figure, we plot the Landau exponents, r˜0, g˜0, u˜0 given in equations7.23 and 7.24, of the effective field theory for the transverse field Isingmodel at zero temperature, as a function of the transverse field ∆. Thefigure on the left shows the system in the absence of a longitudinal field,whereas the figure on the right shows the exponents in the presence of alongitudinal field of h = Jnn, where Jnn is the nearest neighbour exchangeinteraction between the spins. . . . . . . . . . . . . . . . . . . . . . . . . . 145xvFigure 7.3 In this figure, we plot the zero temperature Landau exponents, r˜0, g˜0, u˜0given in equations 7.23 and 7.24, of the effective field theory for the spinhalf spin half model (left) and LiHoF4 (right), as a function of the trans-verse magnetic field. For the spin half spin half model, given in equa-tion (7.37), we assume a nearest neighbour exchange interaction Jnn be-tween spins, and we assume there is no effective field acting directly on thenuclear spins (ε = 0). We assume a weak isotropic hyperfine interactionAz = A⊥ = 0.01Jnn. The LiHoF4 Hamiltonian is given in equation (7.39). . 146Figure 7.4 In this figure, we plot the mean field longitudinal magnetization of thetransverse field Ising model (dashed line), along with the leading order cor-rection in the high density approximation calculated from equation (7.36),as a function of the transverse field ∆. We consider a simple cubic crystalwith exchange interaction strength J. We see the theory breaks down inthe vicinity of the critical transverse field where fluctuations become moreimportant. The point at which the corrected magnetization reaches zerogives the leading order correction to the critical transverse field. We find∆c ≈ 2.84J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure 7.5 In this figure, we plot the MF magnetization of the spin half spin half modelgiven in equation (7.37), along with its leading order correction in the highdensity approximation calculated from equation (7.17). We set the dipolarinteraction to zero, and assume no transverse field acting directly on thenuclear spins. We assume an isotropic hyperfine interaction, with Az =A⊥ = 0.01Jnn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Figure 7.6 The figure above shows the MF longitudinal magnetization of LiHoF4(dashed line), along with the leading order correction calculated from equa-tion (7.17), as a function of the transverse field Bx. The point at which thecorrected magnetization reaches zero gives the leading order correction tothe critical transverse field. We find Bcx ≈ 4.4T. The experimental value ofthe critical transverse field is Bcx = 4.9T; hence, with our choice of crys-tal field parameters, and nearest neighbour exchange interaction, the lead-ing order correction underestimates the critical transverse field by about asmuch as it is overestimated by MF theory. . . . . . . . . . . . . . . . . . . 153xviAcknowledgmentsI would like to acknowledge the support of my thesis supervisor, Dr. Philip Stamp. His breadthof knowledge is remarkable. I would also like to thank the friends and family, too numerous toname, who have helped and supported me over the years.xviiDedicationThis thesis is dedicated to my mother, whose love and support have made my life possible.xviiiChapter 1IntroductionMagnetic materials have a venerable history. They were of interest to Thales, often consideredthe first known philosopher, in the 6th century BC. However, according to Aristotle, Thalesbelieved a magnet’s ability to attract iron filings was due to its soul [1]. Although magnetswere put to practical use as compass needles as early as 1000AD, there was little improvementin the understanding of magnetism for over two millennia. This changed during the scientificrevolution, starting in 1600 with the publication of William Gilbert’s book, De Magnete. Thisbook is remarkable for its reliance on empiricism, and for Gilbert’s contempt for some of theestablished beliefs of his time. Gilbert looked for truth ”in things themselves,” rather thanrelying on the beliefs of his predecessors. The magnetic field of the earth is perhaps the bestknown of Gilbert’s discoveries [2].Over the following centuries, many of the world’s best scientists developed the theory ofelectricity and magnetism, and shed light on the connection between the two, relying on ex-periment to be their guide. This culminated with the publication of the electromagnetic fieldequations of James Clerk Maxwell in 1873. These equations explained light as an electro-magnetic phenomenon, a discovery that was fully illuminated by Einstein’s special theory ofrelativity at the start of the 20th century. Despite the success of Maxwell’s electromagnetictheory, an understanding of magnetic materials remained elusive.In the early part of the 19th Century, Andre´-Marie Ampere, building on Hans Christian Oer-sted’s discovery in 1820 that a current carrying wire could deflect a compass needle, proposedthat the magnetic properties of materials may be due to continually flowing microscopic elec-tric currents inside the material [2]. In the latter part of the 19th century, through experiments,Pierre Curie deduced that each atom in certain magnetic materials (paramagnets) that lack aspontaneous magnetization, behaves like a little magnet whose orientation may be altered byan applied magnetic field, or, the collective behaviour of which, may be altered by a change intemperature. This led Pierre Weiss to propose that in certain materials with a permanent mag-1netization (ferromagnets) there was an internal field responsible for aligning each little magnet.However, at the time, there was no known mechanism that could produce the field necessaryin most materials. In 1926, the quantum mechanical exchange interaction was discovered byHeisenberg and Dirac [3, 4]. This interaction proved to be typically responsible for the dom-inant interactions between the atomic magnetic moments in magnetic materials, and is strongenough to account for the fields observed in a ferromagnet. Although the exchange interactionis the dominant interaction in most magnetic materials, there exist exceptions. In magneticmaterials where the electrons are tightly bound to their host atoms, the exchange interactionwill be weak. In this case, the dominant interaction may be the dipole-dipole interaction be-tween the magnetic moments at each atomic site. This thesis will deal with a material in whichdipolar interactions are dominant, viz., the magnetic insulating crystal LiHoF4.Weiss’ mean field (MF) approximation, in which each atomic magnet, or spin (the magneticmoment of a particle is proportional to its intrinsic quantum property of spin), responds tothe average field produced by all other spins in the material, is of primary importance in thestudy of magnetic materials because, at a qualitative level, it produces many of the featuresof a magnetic substance, such as the phase diagram, the magnetization, and the susceptibility.In order to obtain quantitative agreement between theory and experiment, or to understandthe behaviour of a magnetic system as it undergoes a phase transition, it is often necessaryto go beyond MF theory. Much of this thesis will be devoted to systematically determiningcorrections to the MF approximation.The simplest correction to MF theory is known as the Gaussian approximation, or equiva-lently, the random phase approximation (RPA). As discussed in many textbooks, for example[5], phenomenological Landau theory involves writing down an energy functional for the mag-netization of a systemL=∫d3~r[L [M(~r)]+γ2(∇M(~r))2], (1.1)whereL [M(~r)] is a function of the magnetization, and the gradient term represents the energycost of the spatial variation of the magnetization. In the MF approximation, we assume M(~r) =M is constant, and minimizeL (M) to find the mean field. In the Gaussian approximation, weexpandL [M(~r)] for small M(~r), and truncate at quadratic order. The free energy of the systemmay then be calculated fromF =−β ln(∫DM(~r) e−βL), (1.2)where β is the inverse temperature, and∫DM(~r) denotes a functional integral is to be per-2formed over every configuration of the magnetization. In this thesis we will use∫Dx to denotea functional integral, and we will use∫dx to denote ordinary integration. The Gaussian approx-imation allows for a system to fluctuate around its MF; however, these fluctuations are treatedas non-interacting. In this thesis, we will rigorously derive equations such as (1.2), startingfrom a microscopic model, and we will systematically determine corrections to the Gaussianapproximation.The Ising model, in its simplest guise, is a microscopic model of a magnetic material inwhich the spins are arranged on a lattice, and each spin may only point up or down. Thereis an energy associated with every configuration of the spins depending on the interactionenergy between neighbours. This is perhaps the simplest possible microscopic model of amagnetic material. The model was given to Ising, by Lenz, in an attempt to better understandferromagnetism and phase transitions. Ising solved the model in one dimension in his 1924thesis, and found no evidence of a phase transition to a ferromagnetic state. He concluded thatthis held true in higher dimensions; however, this was shown to be false by an argument madeby Peierls [6]. Peierls divided a two dimensional square lattice into postive and negative regionscorresponding to spin up and spin down. He then showed that at a low enough temperature thearea encapsulated by one region may be smaller than the area encapsulated by the oppositeregion, which leads to the conclusion there will be an excess of one spin or another. In 1944,a rigorous solution to a two dimensional Ising model was published by Onsager [7]. Thissolution clearly showed a transition from a paramagnetic to a ferromagnetic state at a criticaltemperature Tc, and showed the specific heat of the system is divergent at the critical point inthe thermodynamic limit, i.e., there is a continuous phase transition. The deceptively simpleIsing model is able to exhibit complex behaviour.The Ising model will be a central topic of this thesis. However, we will not be concernedwith the classical Ising model discussed above, in which each spin points up or down Sz ∈{1,−1}. Instead, we will be concerned with the quantum Ising model in which we promote Szto a quantum spin operator. In, for example, a transverse magnetic field, this allows for spins tobe in a superposition of states. At low temperatures (T ∼ 1K), the magnetic insulating crystalLiHoF4 is a physical realization of the quantum Ising model. The anisotropy of the system isa result of the crystal electric field, due to the surrounding ions (the ligands), that mixes andsplits the J = 8 electronic multiplet of the holmium ion. In the absence of an applied magneticfield, spectroscopy and susceptibility experiments find a degenerate ground state separatedfrom the first excited state by a gap of around 11K [8–10]. LiHoF4 has the additional featurethat the observed ferromagnetic order stems from the dominant dipolar interaction between theelectronic spins, rather than the exchange interaction. Furthermore, each Ho3+ ion is stronglycoupled to its spin I = 72 nucleus [11]. The interaction between the electronic and nuclear3degrees of freedom is known as the hyperfine interaction, and is of primary interest in thisthesis.In this thesis, we will focus on LiHoF4. However, we note that many of the procedurescarried out on the LiHoF4 Hamiltonian, and the tools developed for the study of the system,are applicable to the rest of the LiReF4 series (Re=rare earth), and to other quantum Isingsystems. Large single crystals of materials in the LiReF4 series (of very high quality in thecase of LiHoF4) can be grown from a melt, making them well suited for research [12]. TheLiReF4 series displays a range of interesting behaviour. Both LiHoF4 and LiTbF4 are dipolarcoupled Ising ferromagnets; however, in LiTbF4 the lowest two MF energy levels are gappedeven in zero transverse field and the hyperfine interaction is roughly half as strong as in LiHoF4[13, 14]. The material LiErF4 is a dipolar coupled XY antiferromagnet [15]. The rare earthelements in LiReF4 compounds may be mixed, or replaced by non-magnetic yttrium, leadingto interesting spin glass behaviour [16], in which the magnetic moments freeze in a randommanner.The dipolar coupling between spins caused a great deal of early interest in LiHoF4, asthe upper critical dimension (the dimension above which there is MF critical behaviour) of adipolar coupled Ising system is three, whereas in an exchange coupled system the upper criticaldimension is four. The properties of a magnetic system near a phase transition obey power lawsbecause of the scale invariance of the underlying theory. For example, the magnetization of anIsing system varies as m∼ (Tc−T )β for T < Tc. Above the critical dimension, this behaviourmay be understood using Landau MF theory, in which case we find β = 12 ; below the criticaldimension, fluctuations play a key role. The renormalization group provides a method forsystematically determining the effect of fluctuations, and can be used to determine criticalexponents such as β [5]. A system at its critical dimension is known as marginal.In 1969, Larkin and Khmel’nitzkii showed that a dipolar coupled Ising system in three di-mensions is expected to have MF critical exponents with logarithmic corrections [17]. Renor-malization group calculations later established this as the marginal critical behaviour of suchsystems [18–20]. In the thermodynamic limit, the Fourier transform of the longitudinal dipolarinteraction (the dipole wave sum) in the long wavelength limit has the formDk = D0+a1k2zk2+a2k2+a3k2z + · · · , (1.3)where the coefficients depend on details of the underlying lattice, and, due to the nature of thelong range dipolar interaction, the zero momentum summation depends on the sample shape.The a1 term in the dipole wave sum is not well defined in the zero momentum limit. Thisambiguity is removed if a finite sized sample is considered, in which case the dipole wave sum4becomesDk = D0+ a˜2k2+ a˜3k2z + · · · (1.4)It is the a1 term in equation (1.3) that leads to the dipolar fixed point, with logarithmic cor-rections to MF critical behaviour, in a renormalization group treatment of a dipolar coupledsystem. Carrying out renormalization on a term that is not well defined in the long wave-length limit seems like a questionable procedure, nevertheless, the dipolar fixed point has beenobserved in experiments [21].In 1978, Beauvillain et al. investigated the critical behaviour of the dipolar coupled materialLiHoF4 using magnetic susceptibility measurements; however, they were unable to differenti-ate between MF critical behaviour and the logarithmic corrections expected at the marginal di-mension [22]. Magnetization measurements made by Griffin et al. in 1980 definitively showedevidence for the logarithmic corrections to MF behaviour [23]. The marginality of the criticalbehaviour of LiHoF4 is reviewed in a 2001 paper of Nikkel and Ellman, in which they presentfurther evidence for marginality based on specific heat measurements [21].It is the marginal critical behaviour that sparked early interest in LiHoF4; however, in thisthesis, our focus won’t be on the marginal critical behaviour due to the dipolar interaction,rather, we will concern ourselves with LiHoF4’s other main feature, the coupling to the nucleardegrees of freedom. Furthermore, we will be primarily interested in LiHoF4 in an appliedtransverse magnetic field that splits the degenerate ground state.The degenerate ground state doublet of LiHoF4 may be split by the application of a mag-netic field transverse to the easy axis of the crystal, making LiHoF4 a physical realization ofthe transverse field quantum Ising model (TFIM). This model was introduced by de Gennes in1963 to account for protons tunneling between two different states in the ferroelectric phase ofKH2PO4, and is discussed in a somewhat more general context in a 1966 paper of Brout et al.[24, 25]. The TFIM has a wide range of applicability beyond ferroelectric and ferromagneticmaterials. For example, in a Jahn-Teller system spin up and spin down states may correspondto different distortions of the crystal lattice. The coupling between ”spins”, and the effectivetransverse field, are due to the crystal electric field of the system. For a list of real systemsto which the TFIM model can be applied see [26]. A transverse field applied to a quantumIsing system is able to destroy the ordered phase of the system at any temperature. At zerotemperature, a transverse magnetic field of about 4.9T applied to LiHoF4 is able to destroy theferromagnetic order of the system in a quantum phase transition [27].A quantum phase transition (QPT) is a zero temperature phase transitions driven by quan-tum fluctuations between competing quantum ground states, whose energy depends on some5control parameter such as magnetic field, pressure, or doping. These differ from classicalphase transitions, which are driven by thermal fluctuations. In a classical system the kineticand potential parts of the Hamiltonian, H(p,q) = Hkin(p)+Hpot(q), commute (we use p andq to represent the three dimensional momenta and positions of all N particles in the system).This means that static thermodynamic properties of a classical system, which follow from thepartition functionZclassical =1N!∫dpe−βHkin(p)∫dqe−βHpot(q), (1.5)may be studied without considering the dynamics. In a quantum system the kinetic and poten-tial parts of the Hamiltonian are operators that might not commute. Therefore, the dynamicalbehaviour of the system must be included when studying its static thermodynamic properties.This is an essential part of the study of QPTs.Phase transitions are generally classified as either first order, or continuous. In a first ordertransition, such as the transition between water and ice, the two phases may coexist at the tran-sition point. In a continuous phase transition, this ceases to be the case. The ferromagnetic toparamagnetic transition of the TFIM is an example of such a transition. The study of continu-ous QPTs is interesting, as they are prevalent in nature, occurring in materials such as high Tcsuperconductors, and heavy fermion systems, and the effects of a QPT on the properties of amaterial extend to regions of the phase diagram away from the quantum critical point [28–30].The TFIM is perhaps the simplest system that undergoes a continuous QPT.A system that undergoes a continuous phase transition can often be characterized by anorder parameter. An order parameter is a thermodynamic quantity that is non-zero in onephase, and zero in the other. For example, the longitudinal magnetization serves as an orderparameter in the ferromagnetic to paramagnetic transition of the TFIM. Close to a criticalpoint, spatial correlations between fluctuations of the order parameter are characterized by thecorrelation length ξ , which diverges at the critical point. Likewise, temporal fluctuations arecharacterized by a correlation time ξτ . The correlation time is related to the correlation lengthby ξτ = ξ z. The dynamic critical exponent z follows from an anisotropy in the scaling of spaceand time under scale transformations (renormalization) of a system. It is model dependent, forexample, the TFIM has z= 1, meaning that time effectively acts as an extra spatial dimensionin the quantum critical regime, whereas for an itinerant antiferromagnet z= 2. In general, theeffective dimensionality of a system near a quantum critical point will be de f f = d+z, with theadditional dimensions stemming from temporal fluctuations. For more details on the dynamiccritical exponent and the effective dimensionality of a system near a quantum critical point, seethe original work of Hertz [31].6The partition function of the TFIM at zero temperature has been mapped to a classicalmodel in one higher dimension by Suzuki [32]. This is an explicit example of the shift ineffective dimensionality of a quantum system due to temporal fluctuations. More generally, wemay write the partition function of a quantum system asZ =∫DM(~r,τ)exp(−∫d~r∫ h¯β0dτ L [M(~r,τ)]), (1.6)where L [M(~r,τ)] is the Lagrangian of the (fluctuating) order parameter of the system. Wework with imaginary time τ = it. Written in this way, it appears that time manifests itselfin the quantum partition function as an additional dimension with a domain determined byβ , the inverse temperature. This is misleading because under renormalization space and timemight not scale isotropically leading to a dynamical critical exponent z 6= 1. Suzuki’s mappingclearly demonstrates that for the TFIM the dynamic critical exponent is z = 1. The transverseIsing system LiHoF4 also has z= 1; however, the complicated single ion Hamiltonian make anexplicit mapping to a higher dimensional classical system impracticable, and it is necessary todeal with the quantum Hamiltonian.Despite the fact that temporal fluctuations in a quantum system effectively manifest them-selves as additional spatial dimensions, with de f f = d+ z, QPTs are of fundamental interestbecause their behaviour can be qualitatively different from that of their classical counterparts.There are many reasons for this. For example, in a system with random disorder in space, thedisorder will be infinitely correlated in time. Typically, this causes the effects of disorder to bestronger in quantum systems. See the book of Sachdev for further discussion of this point [28].Another example, discussed in the reviews of Vojta, Belitz and Kirkpatrick [29, 33, 34], is thatthe presence of soft modes in a quantum system (not including the order parameter fluctuations)may fundamentally alter its critical behaviour. This is due to long range dynamical interactionsbetween the order parameter fluctuations not necessarily present in classical systems.In LiHoF4, where the upper critical dimension of the classical phase transition is three, weexpect MF critical behaviour in the quantum regime where, due to the quantum to classicalcorrespondence, the effective dimensionality is four. Such behaviour is observed in the 1996susceptibility measurements of Bitko et al., where the quantum regime is accessed by varyingthe transverse field near zero temperature [27]. In this thesis, we will investigate the effect ofthe nuclear degrees of freedom on this QPT.Bitko et al. note that, due to the strong hyperfine interaction, the nuclear degrees of free-dom in LiHoF4 have a significant impact on the system’s phase diagram. At low temperatures,the hyperfine interaction stabilizes the ferromagntic phase against the effects of the transversemagnetic field [27]. The hyperfine interaction also manifests itself in specific heat measure-7ments. Each electronic eigenstate is split into eight electronuclear levels by the I = 72 nuclearspin, and, as discussed in a 1983 paper of Mennenga et al., these electronuclear degrees offreedom exhibit themselves as a Shottky type contribution (a significant increase) to the lowtemperature specific heat [35]. The electronic excitation spectrum of LiHoF4 is also stronglyaffected by the hyperfine interaction. In a continuous quantum phase transition, the energyrequired for order parameter fluctuations is expected to soften to zero at the quantum criticalpoint. The TFIM undergoes such a transition. In 2005, Rønnow et al. published neutron scat-tering experiments that show that what should have been the electronic soft mode in LiHoF4is gapped by the presence of the nuclear spins [36]. Although the soft mode is gapped, thelongitudinal magnetization of the system, which serves as an order parameter, goes to zerocontinuously at the critical point, which is characteristic of a continuous phase transition. Ananalysis of this behaviour based on an effective low temperature Hamiltonian for LiHoF4 is animportant part of this thesis. Much of the physics of LiHoF4 discussed so far is illustrated inFigure 1.1, taken from [36].Our analysis of LiHoF4 begins in Chapter 2, where we present the Hamiltonian of thesystem with a discussion of each term, and the expected domain formation in the material.We then derive a low temperature effective Hamiltonian for LiHoF4 that fully incorporates thenuclear degrees of freedom. No such model exists in the literature. We find that the effectivetransverse magnetic field acting directly on the nuclear spins is quite large, a fact that has beenoverlooked by past researchers. This field originates from a shift in the 4 f electron cloudsurrounding each holmium ion due to the applied transverse magnetic field. In Chapter 3, wetake the Fourier transform of the long range dipolar interaction present in LiHoF4. Althoughour focus is not on the dipolar coupling between the electronic spins, and the resultant criticalbehaviour, the dipolar interaction is an important aspect of the material, so the interaction isdealt with in detail.An alternative effective Hamiltonian for LiHoF4 has been developed by Schechter andStamp [37, 38]. In their work, a set of eight pseudospin operators are introduced to account fortransitions between low energy electronuclear eigenstates. When compared to the Hamiltonianderived in this thesis, this pseudospin Hamiltonian is rather cumbersome to work with. Fur-thermore, the pseudospin Hamiltonian does not make clear that the hyperfine interaction leadsto a significant effective transverse magnetic field that acts directly on the nuclear spins, as dis-cussed in the previous paragraph. This field plays a crucial role in determining the behaviourof the system near its quantum critical point.After discussing the Hamiltonian of real LiHoF4 in Chapters 2 and 3, we turn to a toy modelthat elucidates some of the physics of quantum Ising systems coupled to nuclear degrees offreedom in Chapter 4. The toy model consists of an exchange coupled spin half Ising system,8The Ho ions in LiHoF4are placed on atetragonal Scheelite lattice with parametersa 0 5.175 ) and c 0 10.75 ). The crystal-fieldground state is a G3,4doublet with only a ccomponent to the angular momentum andhence can be represented by the sz 0 T1 Isingstates. A transverse field in the a-b planemixes the higher lying states with the groundstate; this produces a splitting of the doublet,equivalent to an effective Ising model field.The phase diagram of LiHoF4(Fig. 1A) wasdetermined earlier by susceptibility measure-ments (10) and displays a zero-field Tcof 1.53K and a critical field of Hc0 49.5 kOe in thezero temperature limit. The same measure-ments confirmed the strong Ising anisotropy,with longitudinal and transverse g factors dif-fering by a factor of 18 (10). The suddenincrease in Hcbelow 400 mK was explained byalignment of the Ho nuclear moments throughthe hyperfine coupling. Corrections to phasediagrams as a result of hyperfine couplingshave a long history (18) and were noted for theLiREF4(RE 0 rare earth) series, of whichLiHoF4is a member, more than 20 years ago(19). What is new here is that the applicationof a transverse field and the use of high-resolution neutron scattering spectroscopy al-low us to carefully study the dynamics as wetune through the quantum critical point (QCP).We measured the magnetic excitationspectrum of LiHoF4with the use of theTAS7 neutron spectrometer at RisL NationalLaboratory, with an energy resolution (fullwidth at half maximum) of 0.06 to 0.18 meV(20). The transverse field was aligned to betterthan 0.35-, and the sample was cooled in adilution refrigerator. At the base temperature of0.31 K, giving a critical field of 42.4 kOe, theexcitation spectrum was mapped out below, at,and above the critical field (Fig. 2). For allfields, a single excitation branch dispersesupward from a minimum gap at (2,0,0) toward(1,0,0). From (1,0,0) to (1,0,1), the mode showslittle dispersion but appears to broaden. Thediscontinuity on approaching (1,0,1 – e) and(1 þ e,0,1) as e Y 0 reflects the anisotropyand long-range nature of the magnetic dipolecoupling. However, the most important ob-servation is that the (2,0,0) energy, which isalways lower than the calculated single-ionenergy (È0.39 meV at 42.4 kOe), shrinksupon increasing the field from 36 to 42.4 kOeand then hardens again at 60 kOe. At thisqualitative level, what we see agrees with themode softening predicted for the simple Isingmodel in a transverse field. However, it ap-pears that the mode softening is incomplete. Atthe critical field of 42.4 kOe, the mode retains afinite energy of 0.24 T 0.01 meV. This result isapparent in Fig. 1B, which shows the gapenergy as a function of the external field.To obtain a quantitative understanding ofour experiments, we consider the full rare-earthHamiltonian, which closely resembles that ofHoF3(21, 22). Each Ho ion is subject to thecrystal field, the Zeeman coupling, and thehyperfine coupling. The interaction betweenmoments is dominated by the long-rangedipole coupling, with a small nearest neighborexchange interaction J12:H 0XiEHCFðJiÞ þ AJi I Ii j gmBJi I H^j12XijXabJDDabðijÞJiaJjbj12Xn:n:ijJ 12 Ji I Jj ð2Þwhere J and I are the electronic and nuclearmoments, respectively, and for 165Ho3þ J 0 8and I 0 7/2. Hyperfine resonance (23) and heatcapacity measurements (24) show the hyper-fine coupling parameter A 0 3.36 meV as forthe isolated ion, with negligible nuclear-quadrupole coupling. The Zeeman term isreduced by the demagnetization field. Thenormalized dipole tensor Dab(ij) is directly cal-culable, and the dipole coupling strength JDissimply fixed by lattice constants and the mag-netic moments of the ions at JD0 (gmB)2N 01.1654 meV, where mBis the Bohr magneton.This leaves as free parameters various num-bers appearing in the crystal-field HamiltonianHCFand the exchange constant J12. The formerare determined (25) largely from electron spinresonance for dilute Ho atoms substituted forY in LiYF4, whereas the latter is constrainedby the phase diagram determined earlier (10)(Fig. 1A). We have used an effective mediumtheory (9) previously applied to HoF3(26) tofit the phase diagram, and we conclude that agood overall description—except for a modest(14%) overestimate of the zero-field transitiontemperature—is obtained for J120 –0.1 meV.On the basis of quantum Monte Carlo simu-lation data, others (27) have also concluded thatJ12is substantially smaller than JD.Having established a good parameterizationof the Hamiltonian, we model the dynamics,where expansion to order 1/z (where z is thenumber of nearest neighbors of an ion in thelattice) leads to an energy-dependent re-normalization E1 þ S(w)^–1 (on the order of10%) of the dynamic susceptibility calculatedin the random phase approximation, with theself energy S(w) evaluated as described in(26). For the three fields investigated in detail,the dispersion measured by neutron scatteringis closely reproduced throughout the Brillouinzone. As indicated by the solid lines in Fig. 2,the agreement becomes excellent if the calcu-lated excitation energies are multiplied by a re-normalization factor Z 0 1.15. The point is notthat the calculation is imperfect but rather thatit matches the data as closely as it does. Indeed,it also predicts a weak mode splitting of about0.08 meV at (1,0,1 – e), consistent with theincreased width in the measurements. Theagreement for the discontinuous jump between(1,0,1 – e) and (1 þ e,0,1) as a result of thelong-range nature of the dipole coupling showsthat this is indeed the dominant coupling.Fig. 1. (A) Phase diagram ofLiHoF4 as a function of transversemagnetic field and temperaturefrom susceptibility (10) (circles)and neutron scattering (squares)measurements. Lines are 1/z cal-culations with (solid) and without(dashed) hyperfine interaction.Horizontal dashed guide marksthe temperature 0.31 K at whichinelastic neutron measurementswere performed. (B) Field depen-dence of the lowest excitationenergy in LiHoF4 measured atQ 0 (1 þ e,0,1). Lines are calcu-lated energies scaled by Z 0 1.15with (solid) and without (dashed)hyperfine coupling. The dashedvertical guides show how in eithercase the minimum energy occursat the field of the transition[compare with (A)]. (C) Schematicof electronic (blue) and nuclear(red) levels as the transverse fieldis lowered toward the QCP.Neglecting the nuclear spins, the electronic transition (light blue arrow) would soften all the way tozero energy. Hyperfine coupling creates a nondegenerate multiplet around each electronic state. TheQCP now occurs when the excited-state multiplet through level repulsion squeezes the collective modeof the ground-state multiplet to zero energy, hence forestalling complete softening of the electronicmode. Of course, the true ground and excited states are collective modes of many Ho ions and shouldbe classified in momentum space. (D) Calculated ratio of the minimum excitation energy Ec to thesingle-ion splitting D at the critical field as a function of temperature. This measures how far theelectronic system is from the coherent limit, for which Ec/D 0 0.R E P O R T S15 APRIL 2005 VOL 308 SCIENCE www.sciencemag.org390 on August 30, 2010 www.sciencemag.orgDownloaded from Figure 1.1: The above figure, taken from the paper of Rønnow et al., illustra es many as-p cts of the magn tic insulating cryst l LiHoF4. In A, we see the phase diagram asdetermined by magnet susceptibility (circles) and neutron scattering (squares). Atlow temperatures, we see that the ferromagnetic phase is stabilized by the hyperfineinteraction. In B, we see the gap in the spectrum at what would be the electronicsoft mode in the absence of hyperfine interactions. The dashed line shows the softmode in the absence of the hyperfine interaction. Diagram C is a schematic illus-tration of the electronic eigenstates split into multiplets by the nuclear degrees offreedom. Finally, in D, we see the ratio of the energy gap Ec shown in diagram B, tothe splitting of the two lowest MF electronic energy levels (∆), plotted as a functionof temperature.9coupled to a spin half nuclear spin with an anisotropic hyperfine interaction, in the presence ofa transverse magnetic field. This spin half spin half model (SHSH) shows that an anisotropichyperfine interaction, with a dominant longitudinal component, leads to an enhancement ofthe longitudinal susceptibility of an Ising system, as well as an enhancement of an appliedlongitudinal field. Furthermore, this model clearly illustrates the gap in the electronic spectrumcaused by the nuclear spins. We see that spectral weight of the gapped mode is transferredto a low energy electronuclear mode that fully softens to zero at the quantum critical point.In Chapter 5, we apply what we have learned studying the toy model to the effective lowtemperature Hamiltonian of LiHoF4.Above, we have introduced the quantum Ising system LiHoF4, and discussed two of its keyfeatures: the dipolar interaction, and the strong coupling to the nuclear degrees of freedom.As mentioned, our primary interest in the system is due to the hyperfine interaction. Thisstems from the fact that the nuclear spins in LiHoF4 can be viewed as a spin bath environmentacting on the electronic degrees of freedom. The physics of such an environment has beenelucidated in the work Prokof’ev and Stamp [39]. Early work on quantum systems coupledto environmental degrees of freedom was carried out by Feynman and Vernon in their seminal1963 paper on influence functionals, in which they considered a quantum system coupled to anenvironment consisting of a bath of harmonic oscillators [40]. Performing an average over theoscillator bath degrees of freedom yields the influence functional, which encodes the effects ofthe environment on a quantum system. This approach was further developed by Caldeira andLeggett to make quantitative predictions about macroscopic systems of practical importancesuch as SQUIDs [41]. Caldeira and Leggett consider a single two state system coupled toa bath of harmonic oscillators. If all but one of the holmium ions in LiHoF4 are replacedwith non-magnetic yttrium ions, the resulting crystal is a physical realization of the Caldeira-Leggett model, with the oscillators being the phonons in the crystal. More accurately, as theelectronic holmium spin is coupled to a nuclear spin, the model consists of a pair of interactingspins coupled to an environmental sea (PISCES). The PISCES model was considered in a 1998paper by Dube´ and Stamp [42], although, because the coupling between the nuclear spin andthe phonons in a holmium ion embedded in LiYF4 is very weak, it is a poor realization of themodel studied in [42].We have discussed both a spin bath environment and an oscillator bath environment. Per-haps the best way of clarifying the difference between the two is with a representative modelH =−J2∑〈i j〉SziSzj−∆∑iSxi +∑i∑nAn~Ini ·~Si+ c∑iSzi (a†i +ai)+∑kεka†kak. (1.7)Here we consider an exchange coupled transverse Ising system, coupled at each site to addi-10tional degrees of freedom ~I. We consider a diagonal coupling, but this does not necessarilyhave to be the case. The ~Is could be nuclear spins, or they could equally well be a collectionof crystal defects, or other impurities, that couple to the Ising spins. The final two terms repre-sent a bath of harmonic oscillators (bosons) with a longitudinal coupling to each spin. Again,the coupling does not necessarily have to take this form. This bath of bosons could representphonons in a crystal. If a set of N environmental modes is weakly coupled to the system ofinterest, it is possible to map them to an oscillator bath type environment consisting of delocal-ized modes belonging to an infinite dimensional Hilbert space. As discussed by Prokof’ev andStamp [39], if the coupling to the environmental modes is strong (independent of the numberof modes in the system), no such mapping to an oscillator bath environment is possible. Theseenvironmental modes constitute the spin bath environment {Ini }, each element of which be-longs to a finite dimensional Hilbert space, and couples to the system in a manner independentof the number of spin bath modes.The result of an environment acting on a quantum system is decoherence. The state of aquantum system may be specified by its wavefunction |Ψ〉= ∑i ci|φi〉, with the correspondingdensity operator being ρ̂ = |Ψ〉〈Ψ|=∑i j cic∗j |φi〉〈φ j|. Performing an average over the environ-mental degrees of freedom in the density matrix yields the reduced density matrix of the systemalone. If the system is entangled with its environment (this means the wavefunction of the sys-tem plus the environment can’t be written as a simple product |Ψ〉= |φ〉env⊗|θ〉sys), averagingover environmental degrees of freedom causes phase relations between different componentsof the system to be lost. This is because the system’s phase information may be distributedamongst the environmental degrees of freedom. This causes the reduced density matrix to be-come a mixed state obeying the laws of classical probability rather than being a pure quantumstate. Decoherence lies at the heart of the quantum to classical transition and, possibly, themeasurement paradox in quantum mechanics, as well as being of great practical importancefor the development of quantum computation. In a quantum computation, entanglement be-tween quantum states must be maintained long enough to carry out meaningful manipulations.Decoherence destroys this entanglement, and it is necessary to mitigate, or control, the effectsof decoherence in order to carry out meaningful quantum computations.The spin bath environment is a particularly inimical source of decoherence because it per-sists down to zero temperature. At zero temperature, precession of the bath spins between spinflips in the system of interest, as well as exchange of phase information between the systemand the bath when a system spin flips, leads to dechorence [39]. Controlling decoherence dueto a spin bath environment is essential in the effort to build a quantum computer.Quantum annealing is a method of quantum computation that takes advantage of quantumtunneling to facilitate finding the (approximate) ground state of a system with an energy land-11scape consisting of many local minima separated by high and narrow energy barriers. In afrustrated quantum Ising model,H =−∑i jVi jSziSzj, (1.8)there may be no single ground state consisting of an orderly arrangement of the spins. Frustra-tion stems from Vi j assuming positive and negative values, or from certain underlying lattices,for example, a triangular lattice, in antiferromagnetic systems. A frustrated Ising system isoften referred to as the Ising spin glass model because it may have a spin glass ground state. Afrustrated quantum Ising system may have an energy landscape amenable to quantum anneal-ing. Quantum annealing involves applying a strong transverse field to a quantum Ising system,then weakening the field so that the system can attempt to settle into its ground state. A quan-tum system may tunnel out of local minima in the energy landscape allowing the system toachieve its ground state faster than if energy barriers are overcome by repeatedly heating andcooling the system, as in thermal annealing. Many problems with significant practical appli-cations, such as the traveling salesman, and the graph partition problem, can be mapped toa an Ising spin glass model [43]; however, the energy landscape of these problems does notnecessarily give quantum annealing an advantage when trying to find their ground state. Find-ing problems with industrial significance that map to the Ising spin glass model, and have anenergy landscape amenable to quantum annealing, is a major challenge facing those interestedin the commercial development of the field.Quantum annealing has received a great deal of media attention in the past several yearsdue to the commercial development of a quantum annealer by D-Wave Systems. D-Waveartificially engineers systems with an Ising spin glass Hamiltonian by making use of super-conducting flux qubits, in which the qubits (Ising spins) are superconducting loops in whichthe current can circulate clockwise, or counterclockwise. Recently, D-Wave has achieved theremarkable feat of linking over 1000 of these flux qubits, and the resulting quantum annealeris benchmarked in [44]. The D-Wave annealer clearly demonstrates that quantum tunnelingis being utilized to find the ground state of a certain realization of the Ising spin glass model.However, due to limited connectivity between the qubits, and rapid decoherence times, it isunclear whether or not the machine will offer a practical advantage over a classical computer.A superconducting flux qubit quantum annealer with longer coherence times, and better con-nectivity, than the D-Wave machine is under development by John Martinis’ group. Martinishopes to have a functioning quantum annealer consisting of about 100 qubits as soon as 2017.More information can be found in the following MIT technology review article [45].The development of a superconducting flux qubit quantum annealer is seriously hindered12by the problem of achieving high connectivity between the constituent qubits. One possibilityfor circumventing this difficulty is to engineer Ising spin glass systems by trapping atoms in anoptical lattice. Different states of the trapped atoms serve as the Ising spins, and the interactionsbetween the spins are mediated by phonons whose interactions can be tuned through the opticallattice [46]. Two dimensional lattices consisting of up to 350 trapped atoms have been achieved[47]. There have been recent proposals of how to make use of trapped atoms in an optical latticeas a quantum annealer to solve the number partition problem, which is a problem that mapsto the Ising spin glass model, and that may have an energy landscape amenable to quantumannealing [48].When LiHoF4 is doped with non-magnetic yttrium, the frustrated nature of the long rangedipolar interaction leads to an energy landscape amenable to quantum annealing. It was inLiHoxY1−xF4 that, in 1999, Brooke et al. first demonstrated the practicality of quantum an-nealing [49]. The study of LiHoxY1−xF4 is interesting in its own right as the system has manyinteresting features. For example, as discussed by Schechter, off diagonal dipolar couplings inthe doped material effectively induce a longitudinal random field [50]. The critical behaviourof a dipolar coupled Ising model in the presence of a random field is discussed for Mn12 ac-etates by Millis et al. [51]. The Mn12 acetate system is quite similar to the doped LiHoF4system; however, Mn12 is simpler because hyperfine interactions play no role in the thermody-namics. Also, LiHoxY1−xF4 will undergo a spin glass transition below a critical value of thedoping of at most xc = 0.46, as discussed by Reich et al. [52, 53]. A spin glass is a systemwhere the spins freeze in a randomly aligned manner with a transition that may be character-ized by the divergence of the non-linear susceptibility χ3, where, expanding the magnetizationabout small applied fields, χ3 is defined by M(H) = χH+χ3H3+χ5H5+ · · · , as discussed ina 1977 paper of Suzuki [54]. The theoretical analysis of spin glass in dilute magnetic mate-rials such as LiHoxY1−xF4 is complicated by the quenched nature of the disorder, that is, thenon-magnetic impurities are not in thermodynamic equilibrium, and it is necessary to considera set of replicas of the system, where a different realization of the disorder may be present ineach replica, in order to obtain meaningful results [43]. The spin glass phase of LiHoxY1−xF4has been of particular interest because the dynamic susceptibility of dilute samples appearedto exhibit anomalous (antiglass) behaviour [55]. Evidence for this anomalous behaviour hasnot been borne out in more recent experiments [56]; however, the role of the nuclear spins onthe dynamics of the doped system remains an interesting problem [37, 38]. A review of thephysics of LiHoxY1−xF4 is available in [57].Our focus in this thesis is on fluctuations and phase transitions in quantum Ising mod-els, rather than on decoherence and the other challenges facing the development of quantumcomputation, or the interesting behaviour of LiHoxY1−xF4. The prevalence of quantum Ising13systems in nature make this an interesting topic in its own right, particularly when the systemis coupled to additional degrees of freedom. In Chapters 2, 4, and 5, we have analyzed LiHoF4,and the spin half spin half model, within the random phase approximation (RPA), thus elucidat-ing the effects of the nuclear degrees of freedom. As mentioned in the opening paragraphs ofthis introduction, a goal of this thesis is to systematically determine corrections to the RPA. Wedo so by developing a field theoretic formalism for quantum Ising systems. The inclusion ofthe effects of fluctuations beyond the RPA is important for model determination, particularlyin dipolar coupled systems where, due to the frustrated long range nature of the interaction,the effects of fluctuations can be significant. Furthermore, the field theoretic formalism thatwe develop in Chapter 6 of this thesis has a great deal of versatility, and applications beyondmodel determination, such as the spin glass problem discussed above, and the behaviour ofsystems with time dependent parameters. Future applications of the field theoretic formalismare discussed in the closing chapter of this thesis.We develop the field theoretic formalism by making use of the well known Hubbard-Stratonovich transformation to decouple the interaction between spins in the partition functionfor the system. We then proceed to integrate over all the microscopic spin degrees of freedomleaving an effective theory for the Hubbard-Stratonovich field. This procedure was used byYoung in 1976 to discuss the quantum phase transition in the spin half transverse field Isingmodel [58]; however, Young does not calculate the coefficients of the resulting theory beyondquadratic order. In this thesis, we develop the formalism to all orders, and we provide explicitexpressions for the coefficients of the theory to quartic order for a system with an arbitrarysingle ion Hamiltonian. We note that this procedure has been extensively developed for itin-erant magnetic systems, beginning with the work of Hertz [31], and is discussed in the recentreview of Brando et al. [59]. Alternative methods for incorporating the effects of fluctuationsin quantum Ising systems that do not make use of an effective field theory are discussed in theintroduction to Chapter 6. We think that the development of the field theoretic formalism is im-portant because it allows for a treatment of a system’s critical behaviour via the renormalizationgroup.We use the field theoretic formalism of Chapter 6 to develop a diagrammatic perturbationtheory for including the effects of fluctuations in quantum Ising systems, and, in Section 6.3.3,we show that it is equivalent to the high density approximation (an expansion in the inversecoordination number) introduced by Brout [60, 61], and presented for the spin half transversefield Ising model by Stinchecombe [26, 62, 63]. In Chapter 7, we apply the field theoreticformalism to the calculation of corrections to the MF magnetization of various quantum Isingsystems.We close this introduction with a brief summary of what we think are the most significant14accomplishments of this thesis. These are:1. A derivation of a low temperature effective Hamiltonian for LiHoF4 in a transverse mag-netic field that fully incorporates the nuclear degrees of freedom in Chapter 2. No suchmodel exists in the literature. This Hamiltonian shows that the dominant mixing of thenuclear degrees of freedom is due to an effective transverse field acting directly on thenuclear spins due to a shift in each Ho3+ ions 4 f electron cloud. In Chapter 3, a detailedanalysis of the long range dipolar interaction between the electronic spins is provided.2. In Chapter 4 we introduce a toy model with exactly solvable single ion eigenstates thatelucidates the effect of an anisotropic hyperfine interaction on a quantum Ising system.We find that a dominant longitudinal hyperfine interaction stabilizes the system againstthe disordering effects of an applied transverse field, as well as enhancing both the lon-gitudinal susceptibility of the system, and the effect of a longitudinal field. The enhance-ment of a longitudinal field has not been previously noted.3. In Chapters 4 and 5, an RPA analysis of the toy model and LiHoF4 is provided. The RPAresults clearly show that the effect of the nuclear spins is to gap what would have beenthe electronic soft mode in a system with no hyperfine coupling, with spectral weightbeing transferred down to a low energy electronuclear mode that fully softens to zero atthe quantum critical point. This explains the neutron scattering experiments carried outon LiHoF4 by Rønnow et al. [36].4. In Chapter 6, a general field theoretic formalism for quantum Ising systems that allowsfor the effect of fluctuations beyond the RPA is developed. This field theory is usedto derive a diagrammatic perturbation theory, with the perturbation parameter being theinverse coordination number, equivalent to the theory introduced by Brout [60, 61], andapplied to the spin half transverse field Ising model by Stinchcombe [26, 62, 63]. We usethe theory to derive corrections to the magnetization of LiHoF4 in Chapter 7.5. We find that a regular nuclear spin bath, that is, a single species of nuclear spin coupled toeach electronic degree of freedom, will have no effect on the quantum critical behaviourof a quantum Ising system.Despite the fundamental importance of quantum Ising systems, and of the transverse fieldIsing model in particular, there has been no significant improvements in Stinchcombe’s worksince its publication in 1973 [26, 62, 63]. Thermodynamic research on quantum Ising systemshas been largely focused on critical behaviour [28], rather than on the more general problemof the inclusion of the effects of fluctuations beyond the random phase approximation in real15magnetic systems. In this thesis, we generalize Stinchcombe’s work to systems with an arbi-trary single ion Hamiltonian, and, by using the field theoretic approach, we provide a derivationof the diagrammatic perturbation theory that offers a great deal of clarity and simplicity whencompared to past approaches. If the work here becomes more widely appreciated, we hope tosee it applied to Ising systems doped with non-magnetic impurities, antiferromagnets, systemscoupled to an oscillator bath environment, and systems with time dependent parameters. Wethink that these topics, in particular the study of disordered systems and non-equilibrium dy-namics, are among the most important problems of 21st century condensed matter physics, andbelieve that the formalism presented here is well suited to the study of such problems.16Chapter 2The LiHoF4 HamiltonianThe rare earth magnetic insulating crystal LiHoF4 is a central topic of this thesis. At lowtemperatures, LiHoF4 is a physical realization of the magnetic dipole coupled quantum Isingmodel; however, there is the additional feature that each electronic spin is strongly coupled to anuclear degree of freedom. The system undergoes a phase transition from a paramagnetic to aferromagnetic state, induced by the long range dipolar interaction, below a critical temperatureof 1.53K [11]. The ferromagnetic order may be destroyed by the application of a magneticfield transverse to the easy axis of the system. At zero temperature, the ferromagnetic orderis destroyed in a quantum phase transition at a critical value of the transverse field of about4.9T. The experimental phase diagram, taken from Bitko et al. [27], is shown in Figure 2.1. Inthis chapter, we introduce the Hamiltonian of LiHoF4, and derive an effective low temperatureHamiltonian that includes the nuclear degrees of freedom. The truncation procedure used hereto obtain the low energy Hamiltonian was first employed by Chakraborty et al. [64]; however,this is the first time the nuclear spins have been fully incorporated. In addition to the derivationof the low temperature effective Hamiltonian, this chapter includes a discussion of domainformation in LiHoF4. We note that the Hamiltonian for LiHoF4 has many free parameters (theCF parameters, and the exchange interaction). As these parameters may be tuned in theoreticalcalculations, or numerical simulations, claims regarding quantitative agreement between theoryand experiment should be treated with some suspicion. A challenge facing experimentalists isto better determine the free parameters in LiHoF4.17VOLUME 77, NUMBER 5 P HY S I CA L REV I EW LE T T ER S 29 JULY 1996FIG. 2. Mean-field critical behavior of the magnetic suscep-tibility in the T ! 0 limit as functions of reduced tempera-ture (open circles, Tc ­ 0.114 K, Ht ­ 49.0 kOe) and reducedtransverse field (filled circles, Hct ­ 49.3 kOe, T ­ 0.100 K).Here J is a measure of the interaction between spins, k isBoltzmann’s constant, and G is dependent on the mixingby Ht of the ground-state doublet with the excited crystal-field states.The exact mean-field phase boundary can be calculatedby solving the Hamiltonian for a single Ho31 ion sJ ­8, I ­ 72 d self-consistently:H ­ Vc 2 g'mBHtJˆx 1 AsIˆ ? Jˆd 2 2J0kJˆzlJˆz , (2)where Vc represents the zero-field crystal-field operator[12], g' is the transverse g factor, A is the hyperfinecoupling strength, and J0 is an averaged spin-spin lon-FIG. 3. Experimental phase boundary (filled circles) for theferromagnetic transition in the transverse field-temperatureplane. Dashed line is a mean-field theory including only theelectronic spin degrees of freedom; solid line is a full mean-field theory incorporating the nuclear hyperfine interaction[Eq. (2)]. Both theories have the same two fitting parameters.gitudinal coupling constant. The hyperfine term arisesfrom the interaction of the Ho nuclear spins with the elec-tronic states through a core polarization effect [21], and itspractical relevance for the magnetic ordering of quantummagnets in effective transverse fields was first noted byAndres for PrCu2 [22]. For LiHoF4, both heat capacity[21] and hyperfine resonance [23] measurements at low Tgive A ­ 0.039 K ­ sAkdgygk, where Ak ­ 0.43 K, theLandé g-factor g ­ 1.25, and the ground-state longitudi-nal g-factor gk ­ 13.8.A solution for Tc as a function of Ht is found by fix-ing Ht and then calculating kJˆzl self-consistently, start-ing at a high temperature and then decreasing T in smallsteps until a nonzero (spontaneous) magnetization is ob-served. The hyperfine interaction effectively mixes thenuclear and electronic eigenstates together; therefore, thesolution proceeds by diagonalizing Eq. (2) in a s136 3136d eigenfunction space (17 crystal field states 38 nu-clear states). The solution is shown in Fig. 3 as the solidline, providing an excellent account of the experimentaldata. We find best fit values J0 ­ 0.0270 6 0.0005 Kand g' ­ 0.74 6 0.04. The value J0 ­ 0.0270 K >2hTcsHt ­ 0dj hgygkj2. The experimentally determinedvalue of g' is remarkably close to the single ion Landé gfactor given the large uncertainty in the matrix elements ofJx which connect the ground state and excited state crystal-field levels. These matrix elements are calculated from theeigenstates of Vc and depend on the measurements whichnot only contain statistical errors $25%, but are interpo-lated from the dilute limit (lightly doped LiYF4) [12].We can illuminate the underlying physics and recoverthe more conventional mean-field form of the phase dia-gram by fixing J0 and g' to their best-fit values and settingA ­ 0 in Eq. (2). Solving self-consistently for the mag-netization gives the dashed line in Fig 3. At high temper-ature, J is the only pertinent quantum number. At low T ,however, the eigenstates of Iˆ and Jˆ are slaved together,and an effective composite spin sI 1 Jd raises the trans-verse field scale required to destroy the ferromagnetic state.Hence, it is clear that the upturn in the phase boundary forT , 0.6 K results directly from the inclusion of the well-known Ho31 hyperfine term in the Hamiltonian. Given thespherical symmetry of the hyperfine interaction and the nu-clear eigenstates, the hyperfine interaction would not shiftthe axis of quantization if the electronic crystal-field statesalso possessed spherical symmetry. In LiHoF4, sphericalsymmetry is broken by the strongly Ising nature of thecrystal-field states and the effect of the hyperfine term islarge for T # Ak.As a further test that the full mean-field Hamiltonianof Eq. (2) is an accurate description of the physics, weuse it to calculate the susceptibility x 0sHtd of LiHoF4in the paramagnet. The calculation is performed byadding a small s,1023 Oed longitudinal field hz to theHamiltonian and solving self-consistently for the magne-tization Mz with no floating parameters The susceptibil-942Figure 2.1: The above figure shows the phase diagram of LiHoF4, measured by Bitko etal. via magnetic susceptibility (solid circles). The vertical axis is the applied trans-verse field, and the horizontal axis is the temperature. The dashed line is the ph sediagram calculated using mean field theory, neglecting the hyperfine interaction.The solid line is calculated using mean field theory, with the hyperfine interactionincluded. There are several free parameters in the MF calculations which lead tothe apparently good agreement between the experimental data and the MF results.In reality, fluctuations in the MF have a significant impact on the phase diagram ofLiHoF4 due to the dominant long range dipolar coupling between electronic spins.The HamiltonianThe magnetic properties of LiHoF4 are due to the partially filled 4 f shells of the Ho3+ ions.Trivalent rare earth ions shed their two outer 6s electrons, and a single 5d electron, leaving acore with the electronic configuration of xenon, and a partially filled 4 f shell. In the groundstate of Ho3+, there are ten 4 f electrons that fill the orbital angular momentum eigenstates,Lz ∈ {−3,−2,−2,0,1,2,3}, in accord with Hund’s rules. That is, the total spin of the electronsis maximized, thereby minimizing the exchange energy between electrons, leading to S = 2;the total orbital angular momentum is maximized, minimizing the coulomb interaction energybetween electrons, leading to L= 6; and the total angular momentum is given by J = L+S= 8,thereby minimizing the spin orbit interaction energy. Hund’s rules follow from the Russell-Saunders coupling scheme, which is known to be accurate for 4 f electrons. The term symbol,182S+1LJ , for the Ho3+ ion is 5I8. A discussion of the physics of rare earth atoms may be foundin [65].If we take U(ri) = −eV (ri) = − Ze24piε0ri to be the radially symmetric potential of the ith 4 felectron in a rare earth element, the spin orbit interaction due to the 4 f electrons isHso =∑i12m2ec21ri∂V (ri)∂ ri~li ·~si =±λ (LS)~S ·~L=±λ (LS)2 (~J2−~L2−~S2), (2.1)where~L= ∑i~li and ~S= ∑i~si, and the + (−) sign refers to a less (more) than half full 4 f shell.In order to obtain the spin orbit coupling λ , we express the spin orbit interaction in a basisof states of total orbital and spin angular momentum, L2,Lz and S2,Sz. It is a consequence ofthe Wigner-Eckart projection theorem that λ (LS) is a function of only ||L2|| = L(L+ 1) and||S2|| = S(S+1), and not of Lz and Sz. Holmium’s large atomic number leads to a significantspin orbit interaction; therefore, we work in a basis of states of the operators L2,S2,J2, and Jz.With the gyromagnetic ratio of the electron taken to be exactly two, the Zeeman energy of aholmium ion in a magnetic field will be given byHZ =−µB(~L+2~S) ·~B=−µBgL~J ·~B, (2.2)where the Bohr magneton is given by µB = 0.6717K/T , and we have made use of the Wigner-Eckart projection theorem in the form〈L S J Jz|~L+2~S|L S J Jz′〉= gL〈J Jz|~J|J Jz′〉 (2.3)to obtain the final expression. The Lande´ g factor, with L= 6 and S= 2, is given bygL = 1+J(J+1)−L(L+1)+S(S+1)2J(J+1)=54. (2.4)More intuitively, noting that the time average of the spin angular momentum will lie along thetotal angular momentum vector~Savg =~S · ~J||J2||~J, (2.5)we may write 〈~L+2~S〉= 〈~J+~Savg〉 which leads to the Lande´ g factor given in equation (2.4).After our brief discussion of the electronic configuration of a holmium ion, and the spinorbit interaction, we move on to the magnetic insulator LiHoF4. The following Hamiltonian19may be used to model LiHoF4,H =∑iVC(~Ji)−gLµB∑iBxJxi −12JD∑i6= jDµνi j Jµi Jνj (2.6)+12Jnn ∑<i j>~Ji ·~J j+A∑i~Ii ·~Ji,where VC(~Ji) is the crystal field, which will be discussed in Section 2.1.1, and Bx is an appliedtransverse magnetic field. The dipolar interaction between electronic spins,Dµνi j =3(rµi − rµj )(rνi − rνj )−|~ri−~r j|2δµν|~ri−~r j|5 , (2.7)has strength JD =µ04pi (gLµB)2. The summation is over a tetragonal Bravais lattice with fourHo3+ ions per unit cell. The lattice spacing in the xy plane is a= 5.175A˚, and the longitudinallattice spacing is c = 10.75A˚. The holmium ions have the Scheelite structure, with fractionalcoordinates (0,0, 12), (0,12 ,34), (12 ,12 ,0) and (12 ,0,14) [66]. In Figure 2.2, taken from the thesis ofKraemer [16], the structure of LiHoF4 is illustrated including all atoms in the unit cell. Usingthe in plane lattice spacing as a reference, we find the strength of the dipolar interaction to beJDa3 = 7mK. Estimates of the antiferromagnetic exchange interaction vary, as will be discussedin Section 2.1.1. Throughout this thesis, unless otherwise noted, we will use the estimate ofRønnow et al., Jnn = 1.16mK [67]. The net spin of the holmium nucleus is I = 72 , and thehyperfine interaction is A = 39mK. A discussion of the nuclear interactions is provided inSection 2.1.2.Due to the frustrated long range nature of the dipolar interaction, fluctuations have a sig-nificant impact on the properties of LiHoF4. The validity of the Hamiltonian given in equation(2.6), taking into consideration the effect of fluctuations, has been tested via Monte Carlo sim-ulations, and through the application of a high density approximation (an expansion in theinverse coordination number). The high density approximation was carried out by Rønnow etal. in [67]. They diagonalize the full 136× 136 electronic plus nuclear single ion Hamilto-nian in the presence of a transverse magnetic field, and make use of the formalism of Jensen,presented in [68], to include the effects of fluctuations. By tuning the crystal field parameters,and the antiferromagnetic exchange interaction, Rønnow et al. are able to obtain agreementbetween theory and experiment for most of the phase diagram; however, no single choice ofparameters was found to account for the entire phase diagram. At low temperatures, Rønnowet al. are also able to obtain a good fit to the (gapped) lowest energy crystal field excitation,apart from an overall scaling factor of 1.15. Magnetoelastic interactions are suggested as a20possible source of the discrepancy in the excitation energy; however, they appear insufficientto account for the discrepancies observed in the phase diagram.Quantum Monte Carlo simulations have been carried out on an effective low temperatureHamiltonian for LiHoF4 in a transverse magnetic field by Chakraborty et al. [64]. In theirpaper, Chakraborty et al. ignore the hyperfine interaction in their low temperature effectiveHamiltonian, choosing instead to incorporate its effects through a sytematic renormalization ofthe transverse magnetic field. At a fixed temperature, with T < Tc = 1.53K, the experimentalcritical transverse field is upto 30% larger than the value predicted by Monte Carlo simula-tions using the same parameters as the work of Rønnow et al. [67] discussed in the previousparagraph. Chakraborty et al. attribute the discrepancy to uncertainties in the crystal field pa-rameters. They find the results of their calculations become increasingly sensitive to the crystalfield parameters as the transverse field is raised. The work of Chakraborty et al. has been revis-ited by Tabei et al. in a paper in which classical Monte Carlo is used to analyze the Bx/Tc 1regime. Effects due to quantum fluctuations are included in their work perturbatively. Thedeviation between experimental and theoretical phase diagrams in increasing transverse fieldpersists in Tabei et al.’s work, and, by comparing several numerical techniques, they concludeit is most likely not of computational origin. Tabei et al. test a single set of alternative crys-tal field parameters, and, provided the antiferromagnetic exchange interaction is adjusted toobtain the correct zero transverse field critical temperature, they find little difference in theexperimental phase diagram. This leads them to tentatively conclude that in weak transversefields the crystal field parameters are not the source of the difference between theory and ex-periment. Tabei et al suggest anisotropic exchange, higher order multipolar interactions, ormagnetoelastic couplings as possible sources of the discrepancy.We note that in Chakraborty et al.’s work the hyperfine interaction is incorporated as arenormalization of the effective transverse field, and in Tabei et al.’s work the hyperfine inter-action is left out altogether because it is thought to be irrelevant in the regime of interest BxTc  1[64, 69]. As will be shown in Section 2.2, in zero transverse field the longitudinal hyperfineinteraction has a significant magnitude. One might expect that it will stabilize the LiHoF4 sys-tem against the effects of fluctuations, in a way not accounted for by a renormalization of theeffective transverse field. We suggest this as a possible source of the discrepancy between thetheoretical and experimental phase diagrams determined by Monte Carlo simulations.In Section 2.2, we derive an effective low temperature Hamiltonian for the LiHoF4 systemthat fully incorporates the effects of the nuclear spins. This Hamiltonian will be used to performan analysis of the sytem in the random phase approximation in Chapter 5. In Chapter 6, we willdevelop a field theoretic formalism that leads to a high density approximation (an expansionin the inverse coordination number) to include the effects of fluctuations in quantum Ising21systems such as LiHoF4. We will defer further discussion of the high density approximation,and a comparison to the formalism of Jensen used in the paper of Rønnow et al. [67, 68], tothat chapter.Chapter 2Materials and Methods2.1 The Compound LiReF4The lithium rare earth tetrafluorides LiReF4 crystalize in a tetragonal scheel-ite structure with space group I41/a. The Re-ion, four per unit cell, occupypositions with point symmetry S4. In figure 2.1 the unit cell is depicted.Further crystallographic specifications and the lattice constants for variouscompounds are listed in the appendix A.ReLiFFigure 2.1: Unit cell of LiReF4. To enhance visibility of the illustration thefluorine ions are only drawn around the (a/2, a/2, c/2) Re position.LiReF4 is an ideal system to investigate an entire chemical series of com-pound, because it crystalizes for almost every element Re of the rare earthfamily without any significant structural change. The replacement onlyslightly affects the position of the fluorine ions and the lattice constants.8Figure 2.2: The figure above, taken from the thesis of Kraemer [16], shows the structureof materials in the LiRe 4 series, where Re is a rare earth atom.22The Crystal Field and Exchange InteractionThe high electronegativity of the fluorine ions in LiHoF4 leads to a distortion of the of the4 f electron cloud surrounding each holmium ion, lifting the 17-fold degeneracy of the J = 8magnetic moment. This perturbation is accounted for by the inclusion of the crystal electricfieldVC(~Ji). The physics of the crystal field, which we will review here, is discussed in [70, 71].The crystal field Hamiltonian due to the ions surrounding each holmium ion (the ligands),is given byVC =− e24piε0∑k jZ j|~R j−~rk|, (2.8)where e is the electron charge, Z j is the effective charge of the jth ligand, and ~R j and~rk arethe positions of the jth ligand and kth electron in the 4 f cloud, respectively. We may expandequation (2.8) in spherical harmonics to obtainVC =∞∑n=0n∑m=−nAmn ∑krnkYmn (θk,φk). (2.9)This Hamiltonian may be put in a form more amenable to calculation by writing the sphericalharmonics in terms of their Cartesian coordinates, ∑kYmn (θk,φk) = fmn (x,y,z), and replacingthe functions fmn with spin operators sharing all the same symmetries. This is known as theStevens’ operator equivalents method [72–74], and leads to the following expression for thecrystal field HamiltonianVC(~J) =∞∑n=0n∑m=−nBmnOmn (~J). (2.10)The operator equivalents Omn corresponds to sums or differences of the spherical harmonicsY±mn . In practice, the crystal field parameters Bmn are determined by experiments. In rareearth materials, the summation over n is restricted to n = 2,4,6, and the possible values ofm are restricted by the point group of the crystal in question. For an extensive review of thecrystal field Hamiltonian in rare earth compounds, see [75]. For LiHoF4, the symmetry of thetetragonal crystal (scheelite, or space group C64h− I41/a) leads to a crystal field given byVC(~J) = B02O02+B04O04+B06O06+B44(C)O44(C)+B46(C)O46(C)+B44(S)O44(S)+B46(S)O46(S),(2.11)23where the Stevens’ operators areO02 = 3J2z − J(J+1) (2.12)O04 = 35J4z −30J(J+1)J2z +25J2z −6J(J+1)+3J2(J+1)2O06 = 231J6z −315J(J+1)J4z +735J4z +105J2(J+1)2J2z −525J(J+1)J2z +294J2z−5J3(J+1)3+40J2(J+1)2−60J(J+1)O44(C) =12(J4++ J4−)O46(C) =14(J4++ J4−)[11J2z − J(J+1)−38]+h.c.O44(S) =12i(J4+− J4−)O46(S) =14i(J4+− J4−)[11J2z − J(J+1)−38]+h.c.We use h.c. to denote the Hermitian conjugate.In LiHoF4, the crystal field induces strong anisotropy leading to the Ising nature of thematerial. Typically, in a crystal with tetragonal symmetry, the n = 2 terms are dominant. Itis easy to see how this leads to anisotropy by considering only the O02 term and the Zeemanenergy of the mean fieldH = B02O02−gLµB~J ·~B. (2.13)By diagonalizing the Hamiltonian above, we find that with B02 < 0 the ground state energyof the system is minimized if the mean field (MF) is in the z direction (the long axis of thetetragonal crystal), and if B02 > 0 the preferential direction for ordering is in the xy plane. Thesign of B02 depends on whether the 4 f electron cloud of the rare earth element is flattened inthe z direction (B02 < 0), or elongated in the z direction (B02 > 0). In LiHoF4, B02 < 0 and thereis strong Ising anisotropy.The crystal field parameters (CFPs) of LiHoF4 have been measured via a number of ex-perimental techniques, by many different groups; however, no consensus has been reachedregarding their values. An early estimate of the parameters, based on susceptibility measure-ments, was made by Hansen et al. in 1975 [10]. In 1980, the CFPs were again determined viamagnetic susceptibility measurements by Beauvillain et al. [76]; however, Beauvillain et al.make no reference to the work of Hansen et al. Optical spectroscopy was used to determinethe CFPs by Christensen, and by Gifeisman et al., in the late 1970s [77, 78]. Electron param-agnetic resonance was used by Shakurov et al. to estimate the CFPs in 2005 [79]. Around thesame time, numerical estimates of the CFPs, based on neutron scattering data, were made by24CFP (K) Ref. [10] Ref. [76] Ref. [78] Ref. [77] Ref. [79] Ref. [67] Ref. [80]B02 −0.754 −0.853 −0.609 −0.606 −0.609 −0.696 −0.672103B04 4.94 5.55 3.25 3.75 3.75 4.06 3.58106B06 1.16 1.16 8.41 6.06 6.05 4.64 6.26102B44(C) 5.26 5.44 4.29 4.16 3.15 4.18 4.07104B46(C) 9.92 9.99 8.17 7.95 6.78 8.12 7.32102B44(S) 0 0 0 0 2.72 0 0104B46(S) 1.96 1.37 0 0 4.14 1.14 1.98Table 2.1: In this table, we list various estimates of the crystal field parameters of LiHoF4in units of Kelvin. References [10, 76] are based on the susceptibility measurementsof Hansen et al. and Beauvillain et al., respectively. References [77, 78] are, respec-tively, the optical light scattering experiments of Gifeisman et al. and Christensen.Reference [79] contains the estimates based on Shakurov et al.’s EPR experiments.In [67], we have the numerical estimates of Rønnow et al., and in [80], we have theestimates of Babkevich et al. based on neutron scattering experiments.Rønnow et al. [67]. More recently, in 2015, neutron scattering was used by Babkevich et al.to determine the CFPs [80]. In Table 2.1, we list these values of the CFPs. In this thesis, weuse the crystal field parameters of Rønnow et al. [67] because, in the low temperature regimethat we will be primarily concerned with, they provide a good fit to the experimental phase di-agram of LiHoF4. LiHoF4 provides an arena for testing many aspects of fundamental physics,as discussed in the introduction to this thesis, therefore it is crucial that these parameters aredetermined with greater accuracy.The antiferromagnetic exchange interaction has not been directly determined. An estimateof its strength has been made by Rønnow et al [67], based on inelastic neutron scatteringdata. Using the crystal field parameters listed above, they find an exchange interaction ofJnn = 1.16mK provides a good fit to the experimental phase diagram, except in the vicinity ofthe zero transverse field critical temperature (T 0c = 1.53K), that is, their fit is good when whenTcgLµBBx > 1, where Tc is the critical temperature in an applied transverse field Bx. By takingB46(S) = 0.87×10−5K and Jnn = 3.13mK they are able to obtain a better fit to the experimentalphase diagram near T 0c = 1.53K; however, there are now significant discrepancies betweenthe experimental and calculated phase diagram at intermediate temperatures of about 0.4K to1.5K.Another estimate of the exchange interaction was made by Tabei et al., based on zero trans-verse field Monte Carlo simulations [69]. The exchange interaction is used to tune the criticaltemperature to the correct experimental value. These simulations neglect the effect of thehyperfine interaction; however, they note that prior Monte Carlo simulations [64], which incor-25porated the effect of the hyperfine interaction via a renormalization of the effective transversefield, indicate that the effect of the hyperfine interaction is unimportant when the transversefield is zero. We are skeptical of this result as a renormalization of the transverse field failsto account for any stabilizing effect the longitudinal hyperfine interaction might have againstthe disorder caused by fluctuations. Using the crystal field parameters of [67], Tabei et al.find the exchange interaction to be Jnn = 3.91mK. The parameters in Tabei et al.’s work aretuned to obtain the correct experimental value for the zero transverse field critical tempera-ture; however, the tuned parameters fail to produce the experimental phase diagram away fromBx = 0. They suggest that magnetoelastic couplings, higher order multipolar interactions, oranisotropic exchange, may be sources of the discrepancy [69].In this thesis, we will use the crystal field parameters of [67], and take Jnn = 1.16mKunless otherwise noted. These parameters have been demonstrated to provide a good fit to theexperimental phase diagram in the low temperature regime.The electronic energy levels in the crystal field are mixed and split by the applied transversemagnetic field. In the absence of the applied field, the system has a degenerate ground stateseparated from the first excited state by about 11K. The degenerate ground state is split by thetransverse field leading to the effective transverse Ising nature of the system. As the transversefield is increased towards the quantum critical point, there is also significant mixing with thehigher lying levels that cannot be included perturbatively. In Figure 2.3, we plot the energylevels of the electronic single ion Hamiltonian,He =∑iVC(~Ji)−gLµB∑iBxJxi , (2.14)as a function of the transverse field. The inset shows the splitting of the lowest two energylevels, relevant to the effective transverse field Ising Hamiltonian used to model the system,and the next highest excited state. This higher lying state is separated from the lower lyingdoublet by at least 10K.Group theoretic considerations determine the general structure of the ground state doubletto be a mixture of the odd electronic eigenstates. The applied transverse field breaks the sym-metry of the ground state, and mixes even eigenstates of the Jz operator into the ground stateHamiltonian. The mixing of the eigenstates of the Jz operator, Jz| j〉= j| j〉, in the ground stateand the first excited state of equation (2.14),|Ψ1,2〉=7∑j=−7α1,2j | j〉, (2.15)is illustrated in Figures 2.4 and 2.5.260 1 2 3 4 5 6Bx (T)-200-1000100200Energy (K)                      Energy Levels of the Single Ion Electronic Hamiltonian0 2 4 60102030Figure 2.3: In this figure, we show the energy levels of the crystal field component of theLiHoF4 Hamiltonian (in Kelvin) as a function of an applied transverse field Bx (inTesla). The inset shows the lowest three energy levels with the ground state taken aszero energy. We see that the second excited state is seperated from the low energydoublet by a gap of over 10K.The Hyperfine InteractionsThe Ho3+ ions in LiHoF4 consist of a single nuclear isotype with nuclear spin I = 72 . Asdiscussed by Mennenga et al. in their paper on the specific heat of LiHoF4 [35], the nuclearpart of the Hamiltonian for a Ho3+ ion may be written asHn = A~I · ~J+gnµn~I · ~Hn+HQ. (2.16)The first term, which we will refer to as the hyperfine interaction, is the most significant. It isdue primarily to the magnetic interaction between each holmium ion’s 4 f electron cloud andits nucleus. The final two terms have been dropped from the LiHoF4 Hamiltonian given inequation (2.6) because they are expected to be small. The second term includes any externallyapplied field, or mean field (MF), due to the neighbouring holmium ions. It also includes nu-clear dipole-dipole interactions between the holmium nucleus and neighbouring nuclear spins,and Fermi contact interactions other than core polarization effects. Any core polarization ef-270 2 4 6Bx (T)00.050.10.150.20.250.30.35Electronic Eigenstates                        Mixing of Electronic Eigenstates|α± 8 ||α± 6 ||α± 4 ||α± 2 ||α0 |Figure 2.4: In this figure, we show the mixing of the even electronic eignestates of theJz operator by the crystal field as a function of an applied transverse field for theground state (solid line) and first excited state (dashed line) of the electronic com-ponent of the single ion Hamiltonian for LiHoF4. Each α j corresponds to the elec-tronic eigenstate such that Jz| j〉= j| j〉fects are included in the first term. The nuclear magneton is µn = 3.66× 10−4K/T and thenuclear g factor for holmium is gn= 4.17 [81]. The final term is the nuclear electric quadrupoleinteraction at the holmium nucleus. In addition, there will be hyperfine terms involving onlythe lithium and fluorine nuclei Hn′(~IF ,~ILi). The most significant parts of Hn′(~IF ,~ILi) are thetransferred hyperfine interactions due to the Fermi contact interaction between the holmiumelectrons, and the lithium and fluorine nuclei. We will discuss each of these terms in turnbelow, beginning with the most significant, the hyperfine interaction. For a more detailed dis-cussion of hyperfine interactions in rare earth atoms see the review by Bleaney [82]. Here weprovide a brief review of what we deem to be most significant.The hyperfine interaction may be written asHA = A~I · ~J =−gnµn~I · ~H4 f , (2.17)where ~H4 f is the magnetic field at the nucleus of a Ho3+ ion due primarily to its 4 f electroncloud. We say primarily because we allow this term to include small corrections due to the280 2 4 6Bx (T)00.10.20.30.40.50.6Electronic EigenstatesMixing of Electronic Eigenstates|α± 7 ||α± 5 ||α± 3 ||α± 1 |Figure 2.5: In this figure, we show the mixing of the odd electronic eignestates of theJz operator by the crystal field as a function of an applied transverse field for theground state (solid line) and first excited state (dashed line) of the electronic com-ponent of the single ion Hamiltonian for LiHoF4. Each α j corresponds to the elec-tronic eigenstate such that Jz| j〉= j| j〉atom’s core electrons. The largest contribution to ~H4 f comes from the orbital angular momen-tum of the 4 f electrons, and is given by~HL =−2µB∑i〈r−3i 〉~li =−2µB〈r−3〉~L, (2.18)where we have assumed that each orbital has the same average radius to obtain the final ex-pression. The next largest contribution is due to the dipolar field of the electron spins, and maybe written as~Hs = 2µB∑i〈r−3i 〉[~si−3(~si ·~ri)~ri] = 2µB〈r−3〉ξ [L(L+1)~S−3(~L ·~S)~L]. (2.19)The final expression is obtained by using the Steven’s equivalent operator method, the detailsof which are found in [73]. For Ho3+ the numerical factor is ξ = − 1990 . The orbital and spinangular momentum vectors precess rapidly around the (conserved) total angular momentumvector ~J =~L+~S. Projecting onto ~J, we find the time averaged field felt by the nucleus to be29given by~H4 f =−2µB〈r−3〉[~L · ~J−ξ(L(L+1)~S · ~J−3(~L ·~S)(~L · ~J))]~JJ(J+1). (2.20)This expression is subject to several corrections, such as the Fermi contact interaction due tothe polarization of core s electrons. These corrections are discussed by Bleaney in [82], andwill not be discussed here. The hyperfine field ~H4 f is largest for Ho3+ and Er3+ with fieldsapproaching 800 Tesla. This, in conjunction with with holmium’s large (relative to the otherrare earths elements) nuclear g factor, gn = 4.17, and large nuclear spin, I = 72 , lead to thesignificant hyperfine coupling in LiHoF4.We now consider the second term in equation (2.16), Hext = gnµn~I · ~Hn, which we take tobe the energy of a holmium nuclear spin due to the field generated by all sources external tothe holmium ion itself. This term includes any externally applied magnetic field, as well asthe MF caused by the neighbouring holmium ions. It also includes the dipolar fields due tothe nuclear moments of neighbouring ions. All these interactions are suppressed by a factor(or factors) of µNµB relative to the interaction of these fields with a holmium electronic spin, orthe dipole-dipole interaction between holmium electronic spins. This term may also includeFermi contact interactions between the holmium nucleus and electrons belonging to neighbour-ing ions. Spectroscopy experiments carried out by Magarin˜o et al. on LiHoxY1−xF4, wheremost of the holmium has been replaced by non-magnetic yttrium, are presented in [9], andspectroscopy experiments in pure LiHoF4 are presented in [13]. They found a spacing betweenhyperfine resonance lines of A = 479G 1T104G ∗ gL ∗ µB = 40.2mK in the dilute sample, and inthe pure sample they found A= 39.8mK. This shows neighbouring holmium atoms in the puresample have little impact on the spacing of the lines in the hyperfine spectrum. Assuming thatFermi contact interactions with the lithium and fluorine electrons are small, we are justified indroppingHext = gnµn~I · ~Hn from the thermodynamic analysis of the LiHoF4 Hamiltonian.The most significant part of Hn′(~IF ,~ILi) is the transferred hyperfine interaction due to theFermi contact interaction between the holmium electrons and the lithium and fluorine nuclei.NMR experiments carried out by Hansen and Nevald have determined these interactions [83].They found that for fluorine, the transferred hyperfine interaction is of the same order of magni-tude as the dipole-dipole coupling between the fluorine nuclear spin and the holmium electonicspin. For lithium, the transferred hyperfine interaction is five times smaller than the lithium-holmium dipolar coupling. These terms may be safely dropped when performing a thermody-namic analysis of the LiHoF4 Hamiltonian; however, we note that, as pointed out by Schechterand Stamp [38], they may still have an impact on the relaxational dynamics, and decoherence,in LiHoF4.30The final term in equation (2.16) is the nuclear electric quadrupole interaction HQ. Re-call that the energy of a charge distribution, q =∫ρ(r)d~r, placed in an electric field may beexpanded as [84]Hmultipole = qV (0)−~P ·~E(0)− 16∑i jQi j∂E j∂ ri(0)+ · · · , (2.21)where ~E =−∇V is the electric field, and ~P and Qi j are, respectively, the dipole and quadrupolemoments of the charge distribution. We are interested in the charge distribution of eachholmium nucleus in the presence of the electric field created by its surroundings. The firstterm corresponds to a constant shift in the ground state energy and may be ignored, and thedipolar term is zero because the electric field is zero at the nucleus (otherwise the nucleuswould move). The final term, the quadrupolar term, may be non-zero because derivatives ofthe electric field may be non-zero at the nucleus. In terms of equivalent spin operators, we maywrite the quadrupolar operator asQi j =∫ρ(r)(3rir j− r2δi j)d~r = QI(2I−1)[32(IiI j+ I jIi)− I(I+1)δi j], (2.22)with Q ≡ Qzz. With an appropriate choice of axes Vi j = ∂ iE j = 0 for any i 6= j, and, for asystem with axial symmetry about the z axis such that Vxx = Vyy, the quadrupolar interactionwill take the formHQ = P[I2z −13I(I+1)]. (2.23)Such is the case in LiHoF4. The value of Q for a holmium atom is Q = 2.4 barns, whichis unremarkable when compared to the Q values of the neighbouring rare earth atoms in theperiodic table. Specific heat measurements have been used to find the value of P for LiHoF4 byMennenga et al. [35]. We note that in Mennenga et al. A‖ = Ag‖gL, where an effective g factorof g‖ = 13.5 has been introduced to account for the systems crystal field. They fit their data tothe Shottky anomaly in the specific heat due to the nuclear spins, and allow in their model acontribution from the hyperfine interaction, and a nuclear quadrupole field. For the hyperfineinteraction they find A= 38.8mK, and for the nuclear electric quadrupole interaction they findP= 2mK, which is very close to the value for the free ion which they state is P= 1.7mK. Thisindicates that the electric field gradient is due primarily to the 4 f electron cloud, and the effectof the crystal electric field is relatively small. We will find, after performing the truncationto obtain the low energy Hamiltonian, that the effective longitudinal hyperfine interaction isenhanced to Az ≈ 200mK, whereas the quadrupole coupling is left unchanged, so neglecting31the quadrupole term introduces errors on the order of about 1% to the Hamiltonian. Errors ofthe same order of magnitude are incurred when the transverse dipolar interactions are droppedfrom the effective theory. In the remainder of this thesis, the nuclear quadrupole interactionwill be neglected, and we will take A= 39mK.Domains in LiHoF4In order to discuss domains in LiHoF4, we begin by discussing the local field felt by eachholmium ion inside the crystal. We follow closely the discussion of Mennenga et al. in [35].The local field will be given by~hloc =~ha+4pi3~M+λdip ~M−~N · ~M+λex ~M, (2.24)where ~M is the magnetic moment per unit volume, which, for the sake of simplicity, we taketo be constant. This corresponds to assuming our sample is a uniformly magnetized ellipsoid,in which case the demagnetizing field, to be discussed shortly, is uniform [85]. There is anexternally applied magnetic field~ha, and~hex= λex ~M is the field due to the exchange interaction.The dipolar field is ~hdip = 4pi3 ~M+ λdip ~M−~N · ~M. The three terms in the dipolar field are,respectively, the Lorentz local field due to the exclusion of the origin in the dipolar sums,a contribution due strictly to the structure of the lattice, and the demagnetizing field, whichdepends on the external shape of the sample [8, 35]. The dipolar interaction will be dealt within detail in Chapter 3.From Maxwell’s equations, we know the total magnetostatic energy of our sample will begiven byE =−µ02∫~M ·~h d3~r, (2.25)where the integral is over the volume of our sample, and~h= ~ha−~N · ~M is the macroscopic fieldgiven by Maxwell’s equations. In the case of a uniformly magnetized ellipsoid considered sofar, the integrand is constant and we simply have E = −µ02 V ~M ·~h. In the ferromagnetic phaseof the system, the analysis is no longer as simple. The system may arrange itself into domainsin order to minimize its total energy, in which case the macroscopic field and magnetizationwill be non-homogenous.As discussed in [35], there is experimental evidence indicating that in the ferromagneticphase of LiHoF4, in a field applied along the easy axis of the crystal, the macroscopic average32field,hz =1V∫Vhz = hza−NzMz, (2.26)will be zero up to the saturation value of the magnetization, i.e. hza = NzMz, where Mz =Mz++Mz− is now the difference between the contributions of the spins in the up and downdomains. In zero applied field, we are led to the conclusion that the system forms long needlelike domains, in which case the demagnetization factor for each domain is Nz = 0, or that thenet magnetization of the system is zero (or both!). As discussed by Tabei et al. [69], accordingto Griffiths’ theorem [86] the magnetization of the sample must be zero in the thermodynamiclimit. Griffiths’ theorem states that spins on a lattice with magnetic dipole-dipole interactions,exchange, and anisotropy energy, possess a well defined bulk free energy, independent of sam-ple shape, in the thermodynamic limit, in the absence of an externally applied magnetic field.This implies the net magnetization of the system must be zero, otherwise, magnetic momentson the surface of the sample would couple to the dipolar moments in the sample, causing shapedependence [35]. Mennenga et al. note that their specific heat measurements in zero appliedfield are independent of sample shape, which can also be understood as a result of Griffiths’theorem. We note that because Griffiths’ theorem is only applicable in the thermodynamiclimit, the question remains as to how large a system must be in order to have zero net magne-tization, and, if there are impurities pinning domain walls, how long it will take the system toreach a state of zero magnetization.Mennenga et al. [35] provide an interesting argument for the vanishing of the macroscopicaverage field, hz, that we reproduce here. They begin by defining the measured susceptibilityMz = χzza hza, and the internal susceptibility Mz = χzzmedhz, where the internal susceptibility isthe response of the medium to the macroscopic average field. As the domain structure ismacroscopic, the field to which it will respond is hz, so χzzmed measures the susceptibility of thedomains. We find that the measured susceptibility may be written asχzza =Mzhza=1Nz+ hzMz=1Nz+(χzzmed)−1. (2.27)In order for the macroscopic average field to vanish in the ferromagnetic phase of the system,we must have χzza = 1Nz , which implies the internal susceptibility of the medium must diverge.This means that the domain walls must have high mobility. As Mennenga et al. put it [35] , inequilibrium, we have hz = 0. A change in hza will produce a non-zero hz; the domain structurewill immediately readjust itself so as to keep hz = 0. We note that this argument was presentedin an earlier paper of Cooke et al. [11], in which they verify that χzza = 1Nz in the ferromagnetic33phase of LiHoF4. They also note that no hysteresis effects are present in LiHoF4, which isanother effect of the mobile domain walls.Experimental evidence for the domain structure of LiHoF4 from optical light scatteringis presented in [8], where they found evidence of needle like domains with the extension ofone domain perpendicular to the easy axis of the crystal being about 5µm, at a temperatureof 0.92Tc (Tc = 1.54K), in cylindrical samples with the z-axis parallel to the easy axis of thematerial. In addition, Cooke et al. argue in favour of the formation of needle like domainsbased on energy considerations [11].Further evidence for the formation of needle shaped domains in LiHoF4 has been foundby Kjønsberg and Girvin using Monte Carlo simulations [87]. Considering a spherical sampleconsisting of 215 dipoles, they see evidence of needle shaped domains. Domains are alsoconsidered by Biltmo and Henelius in [88]. Based on energy considerations in a finite cylinderwith non-zero demagnetization factor, they predict the formation of parallel sheet domains inthe system’s ground state. Unfortunately, there is no experimental evidence to support thisprediction, so, in this thesis, we will consider a long thin uniformly magnetized cylinder withzero demagnetization field, or a system divided into needle like domains, as has been observvednear the zero transverse field phase transition [8].As noted by Chakraborty et al. in [64], in LiHoF4, at low temperatures, the transversedipolar interaction is negligible compared to the longitudinal dipolar interaction. This meansa spin pointing in a direction transverse to the easy axis is unaware of the orientation of itsneighbours. Hence, an applied transverse field polarizes spins uniformly in the x direction,or, as Chakraborty et al. put it [64], the magnetization Mx is unaware of the domain structureformation in LiHoF4.Within a single domain away from the domain wall, where the macroscopic field is hz = 0,the local field acting on a spin will be~hloc =(4pi3+λdip+λex)~M. (2.28)The anisotropy energy in LiHoF4 is large, leading to domain walls with a very narrow width;hence, most of the spins will experience the same local field. In what follows, we will considera long thin cylindrical sample with zero demagnetizing field. Such a sample should consist of asingle domain. It also reflects the local field felt by a spin in a sample of LiHoF4 that is dividedinto needle shaped domains, as discussed above.34The Effective Low Temperature HamiltonianWe now derive an effective low temperature Hamiltonian for the LiHoF4 system. Following[64], we diagonalize the electronic part of the single ion Hamiltonian H˜e =UHeU†, withHe =∑iVC(~Ji)−gLµB∑iBxJxi . (2.29)We apply the same rotation to the spin operators J˜µ =UJµU†, and truncate the operators downto the two by two subspace that mixes the lower two eigenstates of HeJµ =Cµ(Bx)+ ∑ν=x,y,zCµν(Bx)τν . (2.30)The lower two electronic eigenstates of He are well separated from the rest of the electroniceigenstates as illustrated in Figure 2.3. The hyperfine interaction, and the interaction energy be-tween holmium ions, is too weak to cause significant mixing with the higher lying eigenstates,which justifies the truncation procedure. We apply a second rotation in order to diagonalize theJz operator in the two by two subspace so that Jz=Czzτz. In terms of the two lowest eigenstatesof He, |α〉 and |β 〉, our basis is | ↑〉= 1√2 [|α〉+exp iθ |β 〉] and | ↓〉=1√2[|α〉−exp iθ |β 〉], wherethe phase is fixed such that the coefficient of the lowest eigenstate |α〉 is real and positive. Wenote that, contrary to the recent claim made in [89], the Ising nature of the electronic degreesof freedom is maintained in an applied transverse field, with the relevant Ising eigenstates, | ↑〉and | ↓〉, being a mixture of all the Jz eigenstates, given in Figures 2.4 and 2.5. In Figure 2.6,we plot the non-zero matrix elements of the effective spin half operators as a function of thetransverse field.In terms of the effective spin operators, the Hamiltonian He may be written in the two bytwo subspace asHe ≈∑iECM,i(Bx)− 12∆(Bx)∑iτxi (2.31)− 12JDC2zz(Bx)∑i6= jDzzi jτzi τzj +12JnnC2zz(Bx) ∑<i j>τzi τzj ,where ECM is the average of the two lowest electronic energy levels, and ∆ is their difference.The terms neglected in this approximation either vanish due to symmetry considerations, orthey are significantly smaller (∼ 1%) than the terms given in equation (2.31). For a discussionof these terms see [69]. The Ising nature of the system is apparent in the truncated Hamiltonian.We now reintroduce the nuclear spins by truncating the hyperfine interaction, Hhyp=A∑i~Ii ·350 2 4 6Bx(T)-10123456Matrix Elements                   Matrix Elements of the Effective Spin OperatorsCzzCxCxxCyy0 2 4 6Bx(T)-0.1-0.08-0.06-0.04-0.0200.02Matrix Elements                   Matrix Elements of the Effective Spin OperatorsCxyCyCyxFigure 2.6: In the figures above, we plot the non-zero matrix elements of the effectivespin half operators, Jµ = Cµ(Bx)+∑ν=x,y,zCµν(Bx)τν , for the truncated LiHoF4Hamiltonian. The plot on the left shows the larger matrix elements, with the uppermost matrix element being Czz. Below Czz, in descending order, we have Cx, Cxxand Cyy. The matrix elements in the right hand plot are much smaller than those onthe left. In descending order, we have Cy, Cxy, and Cyx.~Ji, down to the lowest two electronic levels (we replace ~Ji with the effective spin half operatorfor the two by two subspace). The result for the hyperfine interaction isHhyp = ACx∑iIxi +ACy∑iIyi (2.32)+ACzz∑iτzi Izi +ACxz∑iτzi Ixi +ACyz∑iτzi Iyi+ACxx+Cyy+ i(Cyx−Cxy)4 ∑iτ+i I−i +ACxx+Cyy− i(Cyx−Cxy)4 ∑iτ−i I+i+ACxx−Cyy+ i(Cyx+Cxy)4 ∑iτ−i I−i +ACxx−Cyy− i(Cyx+Cxy)4 ∑iτ+i I+i .Dropping the energy shift ECM, and keeping only nonzero terms in the hyperfine interaction,we find our effective low temperature Hamiltonian to be (suppressing the field dependence ofour operators)He f f =− ∆2∑iτxi −12JDC2zz∑i6= jDzzi jτzi τzj +12JnnC2zz ∑<i j>τzi τzj (2.33)+∑i~∆n ·~Ii+Az∑iτzi Izi +A⊥∑iτ+i I−i +A†⊥∑iτ−i I+i+A++∑iτ+i I+i +A†++∑iτ−i I−i ,36where~∆n = (ACx,ACy,0) (2.34)andAz = ACzz (2.35)A⊥ = ACxx+Cyy+ i(Cyx−Cxy)4A++ = ACxx−Cyy− i(Cyx+Cxy)4Incorporating the nuclear spins into the model by applying the trunctation procedure for theelectronic spins to the hyperfine interaction was suggested by Chakraborty et al. in [64]; how-ever, it was never carried out in their work. This is the first time such a procedure has beenused to analyze LiHoF4.In Figure 2.7, we show the effective transverse field acting on the electronic spins as afunction of the physical transverse field Bx (in Tesla). The inset shows the next two largestparameters in the effective Hamiltonian, the transverse field acting directly on the nuclear spins,and the longitudinal hyperfine interaction. We see that the effective transverse field acting onthe nuclear spins is rather large, and should not be neglected. The remaining parameters in ourmodel, the magnitudes of which are illustrated in Figure 2.8, are significantly smaller.The important point to take from the low energy effective Hamiltonian is the anisotropy ofthe hyperfine interaction, and the large effective transverse magnetic field acting directly on thenuclear spins. We see that the effective longitudinal hyperfine interaction is about Az= 200mK,and the transverse component, A⊥, is over ten times smaller. As for the electronic dipole-dipoleinteraction, the source of the anisotropy is the deformation of the electronic 4 f orbitals dueto the crystal electric field. The effective transverse field acting on the nuclear spins, ∆xn, isroughly 100mK when the real transverse field, Bx, is between 3T and 6T. It is this effectivetransverse field that is responsible for the dominant mixing of the nuclear spins, rather thanthe transverse hyperfine interaction. This effective field is a result of the strong hyperfineinteraction in LiHoF4, viz., the physical transverse field shifts the electronic 4 f orbitals leadinga significant effective field acting directly on the nuclear spins via the hyperfine coupling. Thedominant longitudinal hyperfine interaction is well known, and was considered by Mennengaet al. in their specific heat measurements in 1983 [35]; however, the large effective field actingon the nuclear spins has not been previously noted.370 2 4 6Bx (T)01234567∆ (K)Magnitude of Effective Parameters0 2 4 6Bx (T)00.10.20.3Energy (K)∆nxAzFigure 2.7: In this figure, we plot the effective transverse field, ∆ (in Kelvin), acting onthe effective Ising spins in LiHoF4, as a function of the physical transverse field Bx(in Tesla). The inset shows the next largest parameters in the LiHoF4 Hamiltonian,these being the effective transverse field acting directly on the nuclear spins, ∆xn,and the longitudinal hyperfine coupling, Az.SummaryIn this chapter, we discussed the rare earth insulating magnet LiHoF4, and its effective lowtemperature Hamiltonian. At low temperatures, this system is a physical realization of thedipolar coupled Ising model. In Section 2.1, we introduced the Hamiltonian thought to modelthe material and illustrated its crystal structure. The crystal field, which leads to the Isinganisotropy of the system, and the exchange interaction, are then discussed in Section 2.1.1. InLiHoF4, each electronic degree of freedom is strongly coupled to a nuclear spin. The physics ofthis hyperfine interaction is discussed in Section 2.1.2. Before turning to the low temperatureeffective Hamiltonian, we discussed domain formation in LiHoF4 in Section 2.1.3. We relayedthe fact that the material forms needle like domains (near its it critical point at least), and thatthe domain walls have high mobility, dominating the susceptibility in the ordered phase.In Section 2.2, we truncated the Hamiltonian obtaining a low temperature effective modelthat fully incorporates the nuclear degrees of freedom. The nuclear spins have not been in-380 2 4 6Bx (T)00.0050.010.0150.02Parameters (K)Magnitude of Effective Parameters∆nyA⊥A++Figure 2.8: In this figure, we plot the magnitudes of the transverse hyperfine parame-ters, A⊥ being the uppermost line, and A++ being the middle line, in the effectivelow temperature Hamiltonian for LiHoF4 as a function the applied transverse mag-netic field Bx. The lowest line is the stray field, ∆yn, acting on the nuclear spins inthe direction transverse to the easy axis and the direction of the applied transversefield. All these parameters are about an order of magnitude smaller than the otherparameters in the model in the vicinity of the critical transverse field Bx = 4.9T .cluded as part of the truncation procedure in previous work. We saw that this effective Hamil-tonian is essentially the Ising model, with a hyperfine interaction that is anisotropic, and aneffective transverse magnetic field acting on the nuclear spins with a magnitude that is com-parable to that of the longitudinal hyperfine interaction. Although the effective anisotropichyperfine interaction is well known, the effective transverse field acting on the nuclear spinshas not been pointed out previously. It is this field that is primarily responsible for the mixingof the nuclear degrees of freedom at low temperatures.39Chapter 3Dipolar InteractionIn LiHoF4, one must take into consideration the long range dipolar interaction between spins.Recall that in the low temperature effective hamiltonian for LiHoF4 the dipolar component isgiven byHdip =−12JDC2zz(Bx)∑i6= jDzzi jτzi τzj , (3.1)where JDa3 = 7mK, with a= 5.175A˚ being the transverse lattice spacing. The effective spin halfoperator is given by Jz =Czz(Bx)τz, withCzz(Bx) plotted in Figure 2.6. The spatial dependenceof the longitudinal component of the dipolar interaction strength is given byDzzi j =1r3i j(3z2i jr2i j−1), (3.2)where ri j = |~ri−~r j|, and zi j = zi− z j.An analysis of the long range dipolar interaction, taking into account the underlying crystalstructure of LiHoF4, is an essential part of understanding the system. This analysis has not beenpresented explicitly in the literature, so we do so here. We analyze the longitudinal componentof the dipolar interaction in Fourier space, performing a dipole wave sum to obtain each Fouriercomponent of the dipolar interaction.Dipole wave sums were calculated in the continuum limit by Holstein and Primakoff, in1940, in a paper in which second quantization was used to derive the spin wave spectrum ofa dipole-dipole coupled Heisenberg ferromagnet [90]. Working in the continuum limit corre-sponds to assuming the underlying lattice structure of the sample is simple cubic, and fails toproduce the correct result for crystals with a more complicated lattice. Furthermore, Holsteinand Primakoff neglect any boundary effects; boundary effects are important for momenta with40wavelengths on the order of the system size, and have a significant impact on the zero momen-tum summation. In Section 3.1, we reproduce Holstein and Primakoff’s results for a sphericalsample, and incorporate the effects of the system’s boundary. We then repeat the calculationfor a long, thin, finite cylinder, which corresponds to the dipolar field felt by a spin withina needle shaped domain. In Section 3.2, we perform the discrete dipole wave sum for smallmomenta by brute force numerically, and compare with the continuum results.Dipole wave sums in primitive cubic lattices (simple cubic, body centered cubic, and facecentered cubic) were carried out by Cohen and Keffer in a 1955 paper [91]. Cohen and Kefferconsider boundary effects in their work, and a key result is that the system’s boundary is onlysignificant for momenta such that kR < 10, with R being the system size. They also find thatthe dipole wave sum is independent of position unless kR < 10, with the exception of thepoint at k = 0. At k = 0, the shaped dependent dipole wave sum is completely independentof the choice of origin, except for origins immediately next to the sample surface. The resultsof Cohen and Keffer were obtained using the Ewald summation method, which we apply toLiHoF4 in Section 3.3.The Ewald summation method divides a sum into a short range part, and a long range part.Performing the long range part of the summation in Fourier space leads to rapid convergenceof what might otherwise be a slowly converging series. In LiHoF4, the Ewald summation iscomplicated by the underlying lattice. Ewald summation for a lattice with a basis (the setof atoms associated with each lattice point) has been considered by Bowden and Clark [92].Bowden and Clark sum over a set of sublattices to account for each atom in the crystal. Ratherthan sum over sublattices, we prefer to introduce a geometric factor to account for the basis.The calculation is essentially the same. In Section 3.3, we perform the dipole wave sum usingEwald summation in a spherical sample of LiHoF4. In Section 3.3.1, we redo the calculationin a long cylindrical sample of LiHoF4, relevant to a system with needle like domains. No suchcalculation appears to exist in the literature.Dipolar Interaction in the Continuum LimitWe begin by calculating the Fourier transform of the dipolar interaction in the continuum limit.This calculation follows the work published in 1940 by Holstein and Primakoff [90]. The41Fourier transform of the dipolar interaction is given byDk =1N ∑i 6= jDi jeik(ri−r j) =1N ∑i6= j1r3i j(3z2i jr2i j−1)eikri j (3.3)=NV∫r 6=01r3(3z2r2−1)ei~k·~rd3~r.We exclude the origin (∫r 6=0) from the integral on the final line because the summation excludesthe term i = j. Clearly, the dipole wave sum on a discrete lattice will depend on the structureof the underlying lattice. In the continuum limit, any information regarding the structure of thelattice is lost. Explicit calculations show that the continuum result is equivalent to performingthe discrete summation over a simple cubic lattice [91]. This may be understood as a reflectionof the fact that the simple cubic lattice contains a homogeneous distribution of points. In,for example, a tetragonal crystal, one of the spatial directions is stretched. This leads to thediscrepancy between the lattice summation over a tetragonal crystal and the continuum result.We may rewrite equation (3.3) asDk =NVD(0)+NV∫r 6=01r3(1− 3z2r2)(1− ei~k·~r)d3~r (3.4)=NVD(0)+NkzV∂∂kz∫ 1r3ei~k·~rd3~r+NV[zr3(1− ei~k·~r)]∂ zdxdy,where an integration by parts is performed on the z component of the integral, and D(0) =∫r 6=01r3 (3z2r2 −1)d3~r. The above manipulations separate the zero wavevector component of thesum, which contains the divergence at the origin, from the rest.The zero wavevector component of the sum is given byNVD(0) =−NV∫r 6=01r3(1− 3z2r2)d3~r (3.5)=−NV∫inzr3zˆ ·~ndΣ− NV∫outzR3zˆ ·~ndΣ,where in denotes a small sphere centered at the origin, and out denotes the outer surface of thespecimen. We exclude the origin and use Gauss’ theorem to obtain the second line from thefirst. The first term is a surface integral, oriented towards the origin, over a small sphere, whichgives us the Lorentz local field. The second term is an integral over the outer surface of thesample, oriented towards infinity, which gives us the demagnetizing fieldHD =−NV∫outzR3zˆ ·~ndΣ. (3.6)42Ignoring the demagnetizing field for the moment, we find the Lorentz local field to beNVD(0)∣∣∣∣in=−NV∫inzr3zˆ ·~ndΣ (3.7)=NV∫ 2pi0∫ pi0sin(θ)cos2(θ)dθdφ =NV4pi3.Given a spherical sample of a simple cubic crystal, the demagnetizing field is equal and oppo-site to the Lorentz local field, and the zero wavevector sum vanishes. With a more complicatedunderlying lattice, or a more complicated sample shape, the contributions will not cancel. Ina uniformly magnetized ellipsoid, the demagnetizing field is constant; however, the demagne-tizing field of a non-ellipsoidal sample may be a rather complicated function of the specimen’sshape that likely needs to be worked out numerically [85]. We will deal with lattices morecomplicated than simple cubic later in Sections 3.2 and 3.3.We now turn to the momentum dependent part of the dipole wave sum. We consider aspherically shaped sample with radius R, and perform the first integral in equation (3.4) inspherical coordinatesI1 =NkzV∂∂kz∫ 1r3ei~k·~rd3~r =4piNkzV∂∂kz∫ R0sin(kr)kr2dr. (3.8)In Section 3.1.2, we will consider a long cylindrical sample, relevant to materials that formneedle like domains. We now perform the ∂∂kz derivative and integrate to getI1 =−4piNVk2zk2[1− j0(kR)], (3.9)where j0 is a spherical Bessel function of the first kind. In an infinite sample j0(kR)→ 0, andwe’re left with only the term to the left of the square brackets.The boundary term in equation (3.4) from the integration by parts can be evaluated asfollows (assuming a spherical sample)I2 =NV∫ [ zr3(1− ei~k·~r)]∂ zdxdy (3.10)=NV2R3∫ R−R∫ √R2−y2−√R2−y2√R2− r2[1− ei~k⊥·~r⊥ cos(kz√R2− r2)]dxdy43where~r⊥ = (x,y) and r = |~r⊥|. In polar coordinates, the integral becomesI2 =NV2R3∫ R0∫ 2pi0r√R2− r2[1− eik⊥r cosθ cos(kz√R2− r2)]drdθ (3.11)=NV4pi∫ 10x2[1− J0(k⊥R√1− x2)cos(kzRx)]dx,where J0(x) is a Bessel function of the first kind. In the limit of infinite sample size (k⊥R 1),we findI2 =NV4pi3− NV4pi∫ 10y√1− y2J0(k⊥Ry)cos(kzR√1− y2)dy (3.12)Numerical integration indicates the integral on the right vanishes as we take the limit of aninfinite sized system. This is expected because as R→ ∞, J0 and the cosine become rapidlyvarying functions that average to zero.Our final result, for a spherical sample with cubic symmetry in the limit of infinite samplesize, isDk =NV4pi3 +HD : k = 0NV4pi3 +HD+NV4pi3(1−3 k2zk2): k 6= 0Recall that in a spherical sample the Lorentz local field and the demagnetizing field are equaland opposite and will sum to zero. We leave them in the expressions above as a reminderthat they may not cancel in a system with a more complicated shape, or a more complicatedunderlying lattice. In such a sample, the zero frequency dipole wave sum is given by D0 =NV(4pi3 +λdip)+HD, where λdip accounts for the lattice structure. Note that in the limit k→ 0,the term k2zk2 can take on any value in the interval [0,1]. This ambiguity is removed when wetake into consideration corrections due to the finite size of the spherical sample.Finite Sized Spherical SampleWe now consider the dipole wave sum for a finite sized spherical sample in the continuum limit.This calculation is performed in the 1955 work of Cohen and Keffer [91], in which they performdipole wave sums for primitive cubic lattices taking into account finite size effects. Cohen andKeffer find that boundary effects are negligible outside the region kR < 10. For momentainside this region, the dipole wave sum will have strong position and shape dependence, exceptat k= 0. At k= 0, the dipole wave sum is independent of the choice of origin; however, it willstill be dependent on the shape of the sample.The boundary contributes three additional terms to the dipolar sum, ∂Dk = HD+ ∂D2k +44∂D3k . The first term, which stems from the zero wavevector sum, is the demagnetization fieldHD =−NV∫outzR3zˆ ·~ndΣ=−NV4pi3. (3.13)In a spherical sample with cubic symmetry, this is equal and opposite to the Lorentz local fieldgiven by equation (3.7). The second term comes from the boundary of the integral in equation(3.9). We get an additional contribution to the dipole sum of∂D2k =NV4pik2zk2j0(kR). (3.14)In the limit kR 1, the Bessel function may be approximated as j0(kR)≈ 1− 16(kR)2, andI1 =−4piNVk2zk2[1− j0(kR)]≈−NV2pi3(kzR)2. (3.15)The third boundary term comes from (3.11), which we consider in the small k limit, whereJ0(k⊥R√1− x2)≈ 1− (k⊥R)24(1− x2) cos(kzRx)≈ 1− (kzRx)22. (3.16)We find∂D3k =−NV4pi∫ 10x2J0(k⊥R√1− x2)cos(kzRx)dx (3.17)≈−NV4pi[13− (kzR)210− (k⊥R)230].This gives usI2 ≈ NV4pi3[(kR)210+(kzR)25], (3.18)which leads to the following expression for the dipole wave sum at small wavevectors (λ  R)in a spherical sample with simple cubic symmetryDk =NV4pi3+HD+NV4pi3R210[k2⊥−2k2z ]. (3.19)In a spherical sample, the first two terms cancel, but we leave them in as a reminder that theremay be a zero frequency contribution to the sum for other sample shapes, or with an underlyinglattice that is not simple cubic. In a ferromagnet, a system will order at a wavevector such thatDk is a maximum. From the above expression, we see that it is energetically favourable for the45system to order at k = k⊥ 6= 0. It is energetically favourable for a spherically sample to orderat wavevectors slightly away from zero.In a sample that is large enough, the system will rearrange itself, forming domains, inorder to minimize its free energy. In such a system, it no longer makes sense to consider theshape of the sample, rather, it is the shape of the domains that is important. Stray fields fromneighbouring domains are negligible because the system forms domains in order to eliminatethese fields. In Section 3.1.2, we turn to a long thin cylindrical sample, consistent with a systemthat forms needle like domains.Finite Sized Cylindrical SampleWe now consider the dipole wave sum, in the continuum limit, for a long thin cylindricalsample. This sample shape has been considered by Cohen and Keffer [91]; however, theydid not publish their results. The expansion in plane waves presented here is straightforward;however, an expansion in cylindrical waves may prove useful for obtaining analytic results. Weleave the cylindrical wave calculation as a subject of future work. The calculation most likelyexists elsewhere in the literature, but the time required to track it down far exceeds the timerequired to carry out the calculation.We begin with the expressionDk =NVD(0)+NkzV∂∂kz∫ 1r3ei~k·~rd3~r+NV∫ [ zr3(1− ei~k·~r)]∂ zdxdy. (3.20)In a long thin cylinder, magnetized along its longitudinal axis, the demagnetizing field is ap-proximately zero. This is because the magnetic surface charge (unpaired dipoles) lie at the topand the bottom of the cylinder. As the length of the cylinder is increased, the interaction energyof these dipoles goes to zero [85]. This meansNVD(0) =NV4pi3. (3.21)We perform the first integral in cylindrical coordinatesI1 =NkzV∂∂kz∫ 1r3ei~k·~rd3~r (3.22)=NkzV∂∂kz∫ h−h∫ 2pi0∫ R0r(r2+ z2)32eik⊥r cosθeikzzdrdθdz=−N4pikzV∫ h0∫ R0rzJ0(k⊥r)(r2+ z2)32sin(kzz)drdz,46where, in the final line, we have performed the angular integral and reduced the z domain to[0,h]. In the long wavelength limit, kzh 1 and k⊥R 1, we findI1 ≈−N4pik2zV∫ h0∫ R0rz2(r2+ z2)32drdz (3.23)=−N2pik2zV[h2−h(R2+h2) 12 +R2 ln(h+(R2+h2) 12R)].Assuming a long thin cylinder, we findI1 ≈−NV 2pi(kzh)2[R2h2ln(2hR)− R22h2+ · · ·]. (3.24)For the second integral we haveI2 =NV∫ [ zr3(1− ei~k·~r)]∂ zdxdy (3.25)=NV∫ 2pi0∫ R0hr(r2+h2)32[2−2eik⊥r cosθ cos(kzh)]drdθ=NV4pih∫ R0r(r2+h2)32[1− J0(k⊥r)cos(kzh)]drExpanding in the long wavelength limit we findI2 ≈ NV 4pih[k2⊥4(R2+2h2(R2+h2)12−2h)+k2zh22(1h− 1(R2+h2)12)]. (3.26)For a long thin cylinder, R h, the expression becomesI2 ≈ NV 2pik2zh2 R22h2− NV2pik2⊥R2 R24h2+ · · · . (3.27)The dipole sum is then given byDk ≈ NV4pi3− NV2pik2zh2[R2h2ln(2hR)− R2h2]− NV2pik2⊥R2 R24h2+ · · · . (3.28)Recall that for a long thin cylinder we have HD ≈ 0. We see that in a cylindrical sample, thequadratic term will depend on the aspect ratio, with the longitudinal momentum componentbecoming more dominant as the length of the cylinder is increased. Unlike the spherical sam-ple, in the case of a long thin cylinder, we see that Dk has a maximum at k = 0. Hence, a longthin cylindrical sample will order at zero wavevector.47Direct Summation of the Dipolar Interaction in LiHoF4In this section, we resort to direct summation to obtain the dipole wave sums in LiHoF4. Thissummation converges at a reasonable rate in the long wavelength limit kR 1; however, ittakes an unreasonably long time to converge for momenta belonging to the rest of the Brillouinzone. In Section 3.3, we will perform perform the summation over the entire Brillouin zoneusing the Ewald summation method. The work here, which is valid at small momenta, can beused to verify the Ewald summation results.We will consider the cases of a spherical sample and a long cylindrical sample, or needleshaped domain. The summations are complicated by the fact that the underlying lattice of thesystem is not simple cubic. We will sum over four sublattices to account for each atom in theLiHoF4 crystal. The dipolar sum is given byDzz~k =1N ∑i6= jDi jei~k·(~ri−~r j) = ∑l 6=01r3l(3z2lr2l−1)ei~k·~rl , (3.29)where ~rl =~ri−~r j. The positive and negative components of the dipole wave sums are notindependent, they are related by Dzz~k = (Dzz−~k)∗. Upon summation over the Brillouin zone, thepositive and negative momentum components of the imaginary part of the dipole wave sumwill cancel, and may be neglected from the subsequent analysis. For the real component, wehave Re[Dzz~k ] = Re[Dzz−~k]; hence, all terms odd in~k will vanish.Note that Di j is a function of an atom’s distance from the origin. We will group the termsin the sum accordingly. We divide the sum into four parts, Dzz~k = D1k +D2k +D3k +D4k , wherethe four terms correspond to the following atoms:~r 1l = (ma,na, pc) ~r2l = (m+12,n+12, p+12)~r 3l = (±m,±(n+12), p+14) and ~r 3l = (±(n+12),±m,−p− 14)~r 4l = (±(m+12),±n, p+ 34) and ~r 4l = (±n,±(m+12),−p− 34). (3.30)We introduceD(m,n, p) =1[(m2+n2)a2+(pc)2]32[3(pc)2(m2+n2)a2+(pc)2−1](3.31)for notational compactness.First, we consider~r 1l = (ma,na, pc), and, to avoid overcounting lattice sites, we treat the48axes, planes, and bulk of the sample separately. Along the axes we findD1k |axes =∞∑m=1[cos(kxma)+ cos(kyma)]2D(m,0,0)+∞∑p=12cos(kzpc)D(0,0, p) (3.32)≈∞∑m=1[4D(m,0,0)− k2⊥(ma)2D(m,0,0)]+∞∑p=1[2D(0,0, p)− k2z (pc)2D(0,0, p)],where k2⊥ = k2x + k2y . We are careful to distinguish between the summation along the trans-verse axes and the summation along the easy axis, as these terms will contribute differentlydepending on whether we are considering a sphere or a cylinder. Along the planes we findD1k |planes =∞∑m,n=14cos(kxma)cos(kyna)D(m,n,0) (3.33)+∞∑m,p=1[cos(kxma)cos(kzpc)+ cos(kyma)cos(kzpc)]4D(m,0, p)≈∞∑m,n=1[4D(m,n,0)−2k2⊥(ma)2D(m,n,0)]+∞∑m,p=1[8D(m,0, p)−2k2⊥(ma)2D(m,0, p)−4k2z (pc)2D(m,0, p)],and in the bulk we findD1k |bulk =∞∑m,n,p=18cos(kxma)cos(kyna)cos(kzpc)D(m,n, p) (3.34)≈∞∑m,n,p=1[8D(m,n, p)−4k2⊥(ma)2D(m,n, p)−4k2z (pc)2D(m,n, p)].For compactness, we now introduce, for example, m+ 12 ≡ m 12 , or p+34 ≡ p 34 . For theatoms at~r 2l = (m 12 ,n 12 , p 12 ), we only need to consider the bulk. We findD2k =∞∑m,n,p=08cos(kxm 12a)cos(kyn 12a)cos(kzp 12c)D(m 12,n 12, p 12) (3.35)≈∞∑m,n,p=0[8D(m 12,n 12, p 12)−4k2⊥m212a2D(m 12,n 12, p 12)−4k2z p212c2D(m 12,n 12, p 12)].Next, we consider the atoms at~r 3l = (±m,±n 12 , p 14 ) and~r3l = (±n 12 ,±m,−p 14 ). We must49consider the planes and the bulk of the sample separately. In the planes, where m= 0, we findD3k |planes =∞∑n,p=02[cos(kxn 12a)exp(− ikzp 14c)+ cos(kyn 12a)exp(ikzp 14c)]D(0,n 12, p 14)(3.36)≈∞∑n,p=0[4D(0,n 12, p 14)− k2⊥n212a2D(0,n 12,p 14)−2k2z p214c2D(0,n 12, p 14)],and in the bulk we haveD3k |bulk =∞∑m=1n,p=04[cos(kxma)cos(kyn 12a)exp(ikzp 14c)(3.37)+ cos(kxn 12a)cos(kyma)exp(− ikzp 14c)]D(m,n 12, p 14)≈∞∑m=1n,p=0[8D(m,n 12, p 14)−4k2⊥[m2+n212]a2D(m,n 12, p 14)−4k2z p214c2D(m,n 12, p 14)].Finally, we consider the atoms at~r 4l = (±m 12 ,±n, p 34 ) and~r4l = (±n,±m 12 ,−p 34 ). Alongthe axis, where n= 0, we findD4k |planes =∞∑m,p=02[cos(kxm 12a)exp(ikzp 34c)(3.38)+ cos(kym 12a)exp(− ikzp 34c)]D(m 12,0, p 34)≈∞∑m,p=0[4D(m 12,0, p 34)− k2⊥m212a2D(m 12,0, p 34)−2k2z p234c2D(m 12,0, p 34)],and in the bulk we haveD4k |bulk =∞∑n=1m,p=04[cos(kxm 12a)cos(kyna)exp(ikzp 34c)+ (3.39)+ cos(kxna)cos(kym 12a)exp(− ikzp 34c)]D(m 12,n, p 34)≈∞∑n=1m,p=0[8D(m 12,n, p 34)−4k2⊥[n2+m212]a2D(m 12,n, p 34)−4k2z p234c2D(m 12,n, p 34)].We have divided the dipole wave sum into summations over four sublattices, and then50proceeded to expand the result in the limit kR 1. The result may be divided into three parts,Dzz~k = D0+Dk⊥R2(k⊥R)2+Dkzh2(kzh)2, (3.40)representing the zero wavevector contribution, the transverse contribution, and the longitudinalcontribution, respectively. We introduce the radial system size R, and the longitudinal systemsize h, to emphasize that the small parameter we are expanding in is kR, or kh. Collectingtogether like terms in the dipole wave summation we findD0 = 4∞∑m=1D(m,0,0)+2∞∑p=1D(0,0, p)+4∞∑m,n=1D(m,n,0) (3.41)+4∞∑m,p=0[2D(m1,0, p1)+D(0,m 12, p 14)+D(m 12,0, p 34)]+8∞∑m,n,p=0[D(m1,n1, p1)+D(m 12,n 12, p 12)+D(m1,n 12, p 14)+D(m 12,n1, p 34)]−Dk⊥ =∞∑m=1(ma)2D(m,0,0)+2∞∑m,n=1(ma)2D(m,n,0)+2∞∑m,p=0[((m+1)a)2D(m1,0, p1)+(m+12)2a2(D(0,m 12, p 14)+D(m 12,0, p 34))]+4∞∑m,n,p=0[((m+1)a)2D(m1,n1, p1)+(m+12)2a2D(m 12,n 12, p 12)+((m+1)2+(n+12)2)a2(D(m1,n 12, p 14)+D(m 12,n1, p 34))]−Dkz =∞∑p=1(pc)2D(0,0, p)+2∞∑m,p=0[2((p+1)c)2D(m1,0, p1)+((p+14)c)2D(0,m 12, p 14)+((p+34)c)2D(m 12,0, p 34)]+4∞∑m,n,p=0[((p+1)c)2D(m1,n1, p1)+((p+12)c)2D(m 12,n 12, p 12)++((p+14)c)2D(m1,n 12, p 14)+((p+34)c)2D(m 12,n1, p 34)]In a spherical sample (R= h) we find D0 ∗a3 = 3.205, where a= 5.175A˚ is the transverseunit cell length. Recall that in the continuum limit the zero momentum dipole wave sum is zero.The value obtained here is due strictly to the underlying lattice. We find D0 ∗a3 = caλdip, withλdip = 1.54. The transverse and longitudinal components of the dipole sum converge much51slower than the zero frequency contribution. If we take the system size to be R= 1000, we findDk⊥ ∗ a3R2 = 32.41 and Dz ∗ a3R2 =−43.22. In the continuum limit, the result is Dk⊥ =−12Dz.In a cylindrical sample, we find D0 ∗a3 = 11.272, while Dk⊥ and Dz depend on the aspectratio of the cylinder. This dependence on the aspect ratio is consistent with the result obtainedin Section 3.1.2 by treating the dipole sum in the continuum limit (valid for a simple cubiccrystal). The demagnetization field for a long thin cylinder is zero; hence, we may write thezero momentum dipole sum as D0 ∗a3 = ca(4pi3 +λdip), with the correction due to the latticebeing λdip = 1.24.Ewald Summation in LiHoF4We now calculate the Fourier transform of the dipolar interaction in LiHoF4 making use of theEwald summation method. Ewald summation leads to rapid convergence of the dipole wavesum over the entire Brillouin zone. We follow the papers of Aharony and Fisher, and of Bowdenand Clark [92, 93]. The paper of Bowden and Clark generalizes the Ewald summation methodto systems with more than one atom per unit cell, such as LiHoF4, by summing over a set ofsublattices. Here, rather than summing over sublattices, we prefer to introduce a geometricfactor to account for each of the atoms in the unit cell. We begin by performing the Ewaldsummation in a spherical sample. In Section 3.3.1, we go on to perform the Ewald summationin a long thin cylinder, relevant to a system with needle like domains. We have found no suchcalculation in the literature.The dipolar interaction may be written asDzz~k =1N ∑i 6= j1r3i j(3z2i jr2i j−1)ei~k·~ri j = ∂2∂ z2 ∑l 6=0ei~k·~rl|~rl−~r|∣∣∣∣~r=0, (3.42)where, in the last line, we have set ~ri−~r j = ~rl and the sum runs over all atoms in the latticeexcluding the atom at~r. We begin by making use of the Gaussian integral1|~rl−~r| =2√pi∫ ∞0dρe−ρ2|~rl−~r|2 (3.43)to obtainDzz~k (~r) =∂ 2∂ z2 ∑l 6=0ei~k·~rl 2√pi∫ ∞0dρe−ρ2|~rl−~r|2 . (3.44)Note that we are now treating the dipole wave sum as a function of~r. Rearranging, and adding52and subtracting the l = 0 contribution, we findDzz~k (~r) =∂ 2∂ z22√pi∫ ∞0dρ[∑le−ρ2|~rl−~r|2+i~k·(~rl−~r)]ei~k·~r− ∂2∂ z21r, (3.45)where r = |~r|. The function in square brackets is a periodic function of~r on the lattice; hence,we may consider its Fourier transformg(~K) =1Vc∑l∫celld3~re−ρ2|~rl−~r|2+i(~k+~K)·(~rl−~r)e−i~K·~rl . (3.46)At this point, we must take the basis of our lattice into consideration. Specializing to thecase of LiHoF4 , the fractional coordinates of the four holmium ions in the basis are given by(0,0,0),(1/2,1/2,1/2),(0,1/2,1/4) and (1/2,0,3/4). We consider each atom (labeled 1 to4) separately, breaking the sum up into four termsg1(~K) =1Vc∑l0∫celld3~re−ρ2|~rl−~r|2+i(~k+~K)·(~rl−~r) =1Vc∫d3~re−ρ2r2+i(~k+~K)·~r (3.47)g2(~K) =e−i~K·(a2 ,a2 ,c2 )Vc∫d3~re−ρ2r2+i(~k+~K)·~rg3(~K) =e−i~K·(0,a2 ,c4 )Vc∫d3~re−ρ2r2+i(~k+~K)·~rg4(~K) =e−i~K·(a2 ,0,3c4 )Vc∫d3~re−ρ2r2+i(~k+~K)·~r.The summation over l0 includes all Bravais lattice vectors, rather than being over each atomin the crystal. We replace the sum of integrals over unit cells with a single integral over allspace. At this point, we specialize to the case of a spherical sample, and perform the integralappearing in the gn(~K) functions in spherical coordinatesI =∫r2 sin(θ)drdθdφe−ρ2r2+i|~k+~K|r cos(θ) = 4pi∫r2drsin(|~k+~K|r)|~k+~K|r e−ρ2r2. (3.48)Taking y= |~k+~K|r, we may writegm(~K) =αmVc4pi|~k+~K|3∫ysin(y)exp(− ρ2y2|~k+~K|2 )dy, (3.49)where the αm are geometric factors from our four atomic sites. Summing over all four contri-53butions yieldsg(~K) =α(~K)Vc4pi|~k+~K|3∫ysin(y)exp(− ρ2y2|~k+~K|2 )dy, (3.50)where the geometric factor isα(~K) =∑mαm = 1+ e−i~K·( a2 , a2 , c2 )+ e−i~K·(0,a2 ,c4 )+ e−i~K·(a2 ,0,3c4 ). (3.51)Making use of equation (3.50), we return to the dipole wave sum, which is given bylim~r→0Dzz~k (~r), with Dzz~k(~r) given in equation (3.45). We separate the integral appearing inequation (3.45) into two parts to obtainDzz~k (~r) =∂ 2∂ z22√pi[∫ Λ0dρ∑~Kg(~K)ei(~k+~K)·~r+∫ ∞Λdρ∑lei~k·~rle−ρ2|~rl−~r|2]− ∂2∂ z21r(3.52)=− 1Vc∑~K(kz+Kz)2|~k+~K|22α(~K)√piei(~k+~K)·~r∫ Λ|~k+~K|0dzF|~k+~K|R(z)+∂ 2∂ z22√pi∑lei~k·~rlH(Λ|~rl−~r|)− ∂2∂ z21r,withFx(z) = 4pi∫ x0ysin(y)e−z2y2dy H(s) =Λs∫ ∞sdye−y2. (3.53)We have introduced a convergence factor Λ. When the integration variable ρ is large, weperform the summation in real space; when ρ is small, we perform the summation in Fourierspace. An appropriate choice of Λ ensures the rapid convergence of the dipole wave sum.Separating the ~K = 0 and ~rl = 0 components of the sums from the rest, we findDzz~k =−1Vc(kzk)2 8√pi∫ Λk0dzFkR(z)− 1Vc ∑~K 6=0(kz+Kz)2|~k+~K|22α(~K)√pi∫ Λ|~k+~K|0dzF|~k+~K|R(z) (3.54)+ limr→0∂ 2∂ z22√pi ∑~rl 6=0ei~k·~rlH(Λ|~rl−~r|)− limr→0(∂ 2∂ z21r− ∂2∂ z22√piH(Λr)).This expression is an extension of Aharony and Fisher’s work in [93] to a lattice with a basis.The above expression matches equation (14) of Bowden and Clark [92], except that in theabove expression we have summed over the four atoms in the basis introducing a geometricfactor, rather than summing over each sublattice separately. Following Bowden and Clark, we54consider each of the four terms in equation (3.54) separately.We begin withA=− 1Vc(kzk)2 8√pi∫ Λk0dzFkR(z) (3.55)In the limit kR 1, we may take the range of y= kr in the integral in Fy(z) to be zero to infin-ity. This neglects boundary effects; however, as discussed in the introduction to this chapter,boundary effects are only significant when kR< 10. Making use of the integral∫ ∞0ysin(y)e−z2y2dy=√pi4z3exp(− 14z2)(3.56)we findA=− 1Vc(kzk)28pi∫ Λk0e−14z2z3dz=− 1Vc(kzk)216pi exp(− k24Λ2). (3.57)We have dropped the boundary term∂A=1Vc(kzk)232√pi∫ Λk0dz∫ ∞kRdy ysin(y)e−z2y2 . (3.58)Similarly, we findB=− 1Vc∑~K 6=0(kz+Kz)2|~k+~K|22α(~K)√pi∫ Λ|~k+~K|0dzF∞(z) (3.59)=− 1Vc∑~K 6=0(kz+Kz)2|~k+~K|2 4piα(~K)exp(− |~k+~K|24Λ2),with a boundary contribution of∂B=1Vc∑~K 6=0(kz+Kz)2|~k+~K|2 8√piα(~K)∫ Λ|~k+~K|0dz∫ ∞|~k+~K|Rdy ysin(y)e−z2y2. (3.60)We may rewrite the third term asC = limr→0∂ 2∂ z22√pi ∑~rl 6=0ei~k·~rlH(Λ|~rl−~r|) = ∑~rl 6=0ei~k·~rl ∂2∂ z2l(erfc(Λ|~rl|)|~rl|), (3.61)where we have used the fact limr→0 ∂2∂ z2 f (|~rl −~r|) = ∂2∂ z2lf (|~rl|), and made use of the compli-55mentary error functionerfc(z) =2√pi∫ ∞ze−y2dy. (3.62)Taking the derivatives, we find C = ∑~rl 6=0 ei~k·~rlEΛ(~rl) withEΛ(~rl) =[2Λ√pie−(Λrl)2r2l(2Λ2z2l +3z2lr2l−1)+erfc(Λrl)r3l(3z2lr2l−1)]. (3.63)Finally, using the error function, we may rewrite the last term as followsD=− limr→0(∂ 2∂ z21r− ∂2∂ z22√piH(Λr))(3.64)=− limr→0(∂ 2∂ z2[1r(1− 2√pi∫ ∞Λrdye−y2)]=− limr→0(∂ 2∂ z2[1rerf(Λr)]).Making use of the series expansionerf(z) =2√pi∞∑n=0(−1)nz2n+1n!(2n+1)(3.65)we findD=4Λ33√pi. (3.66)Our result for the dipolar sum is thenDzz~k =4Λ33√pi+ ∑~rl 6=0ei~k·~rlEΛ(~rl)− 1Vc(kzk)216pi exp(− k24Λ2)(3.67)− 1Vc∑~K 6=0(kz+Kz)2|~k+~K|2 4piα(~K)exp(− |~k+~K|24Λ2).Note that in the limit Λ→ 0 we recover the original dipole sum, as we must. The first twoterms in the above expression are generally valid. The remaining pair of terms are valid fora spherical sample in the limit R λ . When considering the limit k→ 0, it is important toremember that we must include contributions from the boundary. In practice, it is easiest toperform the dipole sum directly for small values of k, and use the Ewald summation methodotherwise.56Ewald Summation in a Cylindrical SampleWe now consider the Ewald summation method for a cylindrical sample. Long thin cylindersare of particular interest because LiHoF4 forms needle like domains, as discussed in the intro-duction to this chapter. We perform our analysis using plane waves; however, an expansionin cylindrical waves may prove useful for obtaining analytic results. We leave the cylindri-cal wave calculation as a subject of future work. We begin by evaluating integral (3.48) incylindrical coordinatesI =∫d3~re−ρ2r2+i(~k+~K)·~r =∫ h−h∫ 2pi0∫ R0e−ρ2(r2+z2)+i|~k⊥+~K⊥|r cosθ+i(kz+Kz)zrdrdθdz. (3.68)Performing the angular integral givesI = 2pi∫ h−h∫ R0re−ρ2r2J0(|~k⊥+~K⊥|r)e−ρ2z2ei(kz+Kz)zdrdz. (3.69)The r and z integrals are separable so that I = IRIh withIR = 2pi∫ R0re−ρ2r2J0(|~k⊥+~K⊥|r)dr Ih =∫ h−he−ρ2z2ei(kz+Kz)zdz (3.70)We begin by evaluating Ih for a long thin cylinder. We findIh = 2∫ h=∞0e−ρ2z2 cos((kz+Kz)z)dz=√piρexp(−(kz+Kz)24ρ2). (3.71)Recall from equation (3.50) that g(~K) = 4piVc I. We findg(~K) =α(~K)Vc2pi32|~k⊥+~K⊥|2e− (kz+Kz)24ρ2ρ∫ |~k⊥+~K⊥|R0yJ0(y)exp( −ρ2y2|~k⊥+~K⊥|2)dy, (3.72)and, following the same steps as for the spherical sample, we find the dipole wave sum to beDzz~k∣∣∣∣k⊥ 6=0=− 4piVc∑~K 6=0(kz+Kz)2|~k⊥+~K⊥|2α(~K)∫ Λ|~k⊥+~K⊥|0e− (kz+Kz)24z2|~k⊥+~K⊥|2zG|~k⊥+~K⊥|R(z)dz (3.73)− 16piVc(kzk⊥)2 ∫ Λk⊥0e− (kz)24z2(~k⊥)2zGk⊥R(z)dz+4Λ33√pi+ ∑~rl 6=0ei~k·~rlEΛ(~rl),57whereGx(z) =∫ x0yJ0(y)e−z2y2dy=12z2e−14z2∣∣∣∣x=∞. (3.74)This result is not valid when k⊥= 0 as the expressions involve a division by zero. In the specialcase k⊥ = 0 and kz 6= 0, the result isDzz~k∣∣∣∣k⊥=0=− 4piVc∑~K⊥ 6=0(kz+Kz)2|~K⊥|2α(~K)∫ Λ|~K⊥|0e− (kz+Kz)24z2|~K⊥|2zG|~K⊥|R(z)dz (3.75)−∑Kzα(~K)piVc∫ ∞(kz+Kz)24Λ2e−z(1− e− [(kz+Kz)R]24z )dz+4Λ33√pi+ ∑~rl 6=0ei~k·~rlEΛ(~rl).The first summation is over all reciprocal lattice vectors with ~K⊥ 6= 0, and the second summa-tion is over all reciprocal lattice vectors such that ~K⊥ = 0. Note that the summation over Kzincludes the Kz = 0 term. If~k = 0, the above result is still correct, except for the fact we mustnow neglect the ~K = 0 term from the second summation. The discontinuity between the~k = 0result and the result at finite~k is due to the boundary terms neglected in taking the limit h→∞in equation (3.71).We now evaluate the dipole sum in the limit, k⊥R 1. Definingβ (~K) = 1+ γ(~K) γ(~K) =(kz+Kz)2|~k⊥+~K⊥|2(3.76)we find, in the limit k⊥R 1,Dzz~k =−2piVc∑~K 6=0γ(~K)α(~K)∫ Λ|~k⊥+~K⊥|0e−β (~K)4z2z3dz (3.77)− 8piVcγ(0)∫ Λk⊥0e−β (0)4z2z3dz+4Λ33√pi+ ∑~rl 6=0ei~k·~rlEΛ(~rl).Performing the integral over z yieldsDzz~k =−4piVc∑~K 6=0(kz+Kz)2(~k+~K)2α(~K)e−|~k+~K|24Λ2 − 16piVc(kzk)2e−k24Λ2 +4Λ33√pi+ ∑~rl 6=0ei~k·~rlEΛ(~rl).(3.78)This is identical to the result for the spherical sample, which is to be expected as the shape58dependence of the sum is important in the opposite limit k⊥R 1.In the limit x= k⊥R 1 we find for equation (3.74)Gx(z) =∫ x0yJ0(y)e−z2y2dy≈ 12z2(1− e−z2x2). (3.79)We may then write the ~K = 0 component of the momentum space summation in equation (3.73)as,I0 =−16piVc γ(0)∫ Λk⊥0e−γ(0)4z2zGk⊥R(z)dz=−4piVcγ(0)∫ ∞k2⊥Λ2e−γ(0)u4 (1− e− x2u )du. (3.80)Assuming Λ is chosen so that ΛR 1, which is a reasonable assumption because good con-vergence is usually achieved with Λ chosen to be approximately the inverse lattice spacing, wefindI0 =−4piVc∫ ∞k2z4Λ2e−z(1− exp(− (kzR)24z))dz≈−4piVc∫ ∞k2z4Λ2e−zdz=−4piVce−k2z4Λ2 . (3.81)We are left with the task of evaluating terms in equation (3.73) for which ~K 6= 0. We separateterms for which |~K⊥|= 0 (the Kz summation) and treat them in the same fashion as we treatedthe integral I0. For |~K⊥| 6= 0, we have |~k⊥+ ~K⊥|R 1, and we may make use of equation(3.74). The final result for the dipole sum in a finite sized cylindrical sample in the limitk⊥R 1 is given byDzz~k =−4piVc∑|~K⊥|6=0(kz+Kz)2(~k+~K)2α(~K)e−|~k+~K|24Λ2 − piVc∑Kzα(~K)e−(kz+Kz)24Λ2 +4Λ33√pi+ ∑~rl 6=0ei~k·~rlEΛ(~rl).(3.82)If we take k⊥= 0, we will obtain the same result as will be obtained by performing the integralsin (3.75).If~k = 0 then the result above is still valid as long as the Kz = 0 term is excluded from thesecond summation. We reiterate the fact that the apparent discontinuity at~k = 0 is due to theboundary term neglected when taking the limit of a long cylinder h→ ∞. In reality, there is nodiscontinuity, but the function varies extremely rapidly in the region near~k= 0. What equation(3.82) gives us is the small momentum behaviour (k⊥R 1) of the dipole wave sum in a longcylinder (kzh 1) of LiHoF4.In order to perform the dipolar sum in LiHoF4, we take Λ= 2a , and consider a system withsize R= 10000a, where a= 5.175A˚ is the transverse lattice spacing. This roughly corresponds59to a domain with radius 5µm. If k⊥R ≤ 1 we use (3.82), otherwise, we use (3.78). In thespecial case~k = 0, we exclude the Kz = 0 term from equation (3.82). In real space, we sumover a cylinder with a radius and height of 10 unit cells, and sum over a corresponding numberof reciprocal lattice vectors. This is more than enough to ensure convergence of the series∑~rl 6=0 ei~k·~rlEΛ(~rl). In Figures 3.1 and 3.2, we plot the dipole summation as a function of thetransverse momenta for various values of the longitudinal momenta. We clearly see the rapidvariation of the sum near~k = 0.1ky ∗pia0.5Dipolar Interaction Strength (kz = 0)240kx ∗pia6Dipolar Sum 80.5 010112-200-101Dipolar Sum 0Dipolar Interaction Strength (kz =pi3c)kx ∗pia100.5ky ∗pia0.51 0Figure 3.1: In the figure above we see the Fourier transform of the dipolar interaction ina long thin cylindrical sample of LiHoF4 as a function of transverse momenta atkz = 0 (left), and kz = pi3c (right).-201-101Dipolar Sum 0Dipolar Interaction Strength (kz =2pi3c)ky ∗pia0.5kx ∗pia0.5100 01kx ∗pia0.5Dipolar Interaction Strength (kz =pic)-151-10ky ∗pia-5Dipolar Sum0.50050Figure 3.2: In the figure above we see the Fourier transform of the dipolar interaction ina long thin cylindrical sample of LiHoF4 as a function of transverse momenta atkz = 2pi3c (left), and kz =pic (right).60SummaryIn this chapter, we considered the Fourier transform of the dipolar interaction, necessary for amomentum space analysis of systems such as LiHoF4. In Section 3.1, we performed the dipolewave sum in the continuum limit. This gave correct results when the underlying lattice of thesystem being dealt with is simple cubic. In an infinite spherical sample, we found there is anambiguity in the zero momentum limit of the dipole wave sum. This ambiguity is resolved byconsidering the effect of the boundary. In Section 3.1.2, we redid the previous calculation fora long thin cylindrical sample, and show that the small momentum behaviour will depend onthe aspect ratio of the cylinder.After performing the dipole wave sum in the continuum limit, we turned to the dipole wavesum for a sample of LiHoF4, where we must account for the underlying lattice. In Section3.2, we examined the small momentum behaviour of the dipole wave sum in LiHoF4 by directsummation. Direct summation becomes unreasonably slow away from k = 0. In order toobtain the dipole wave sum in LiHoF4 over the entire Brillouin zone, we made use of theEwald summation method. We performed the Ewald summation for a spherical sample, andfor a long thin cylindrical sample. The cylindrical sample reflects the dipolar field felt in asample of LiHoF4 that is divided into needle like domains. The results obtained via Ewaldsummation in a long thin cylindrical sample are used throughout the rest of this thesis.61Chapter 4Spin Half Spin Half ModelConsider a spin half Ising system in a transverse field coupled to a spin half nuclear spin bathwith HamiltonianH =−J2∑i6= jVi jSziSzj−∆∑iSxi +Az∑iIzi Szi +A⊥2 ∑i(I+i S−i + I−i S+i ). (4.1)The Ss represent electronic spin operators, and the Is represent nuclear spin operators. Weassume a longitudinal ferromagnetic interaction between electronic spins, and take ∆, and thehyperfine interaction, to be positive. Our choice of energy units is arbitrary. For simplicity,we choose energy units such that J = 1, and it is to be understood that the transverse field andhyperfine interaction are now in units of J. We introduce this relatively simple Hamiltonian asa toy model that illustrates the effects of an anistropic hyperfine interaction on the transversefield Ising model. With the inclusion of a transverse field acting directly on the nuclear spins,Hn = −∆n∑i Ixi , this serves as a toy model for LiHoF4. Many of the qualitative features ofLiHoF4 are easily illustrated by this model.In Section 4.4.1, we will show the spin half spin half (SHSH) model undergoes a ferromag-netic to paramagnetic phase transition at a critical temperature of∆c ≈ V02 +12(A2zA⊥−A⊥). (4.2)For ∆ > ∆c, there will be no longitudinal mean field (MF) acting on the system. In this case,62the single ion Hamiltonian,H0i =−∆Sxi +AzIzi Szi +A⊥2(I+i S−i + I−i S+i ) =Az4 −∆2 0 0−∆2 −Az4 A⊥2 00 A⊥2 −Az4 −∆20 0 −∆2 Az4 , (4.3)may be diagonalized analytically. We use as our basis |~Ψ〉= (| ⇑↑〉, | ⇓↑〉, | ⇑↓〉, | ⇓↓〉), wherethe double arrows denote electronic spins, and the single arrows denote nuclear spins. Theeigenvalues of the single ion Hamiltonian areE1 =−12√∆2+(Az+A⊥)24− A⊥4E2 =−12√∆2+(Az−A⊥)24+A⊥4(4.4)E3 =12√∆2+(Az+A⊥)24− A⊥4E4 =12√∆2+(Az−A⊥)24+A⊥4and their associated eigenvectors are|1〉= α1−∆2E1− Az4−E1+ Az4∆2 |2〉= α2∆2−E2+ Az4−E2+ Az4∆2 (4.5)|3〉= α3−∆2E3− Az4−E3+ Az4∆2 |4〉= α4∆2−E4+ Az4−E4+ Az4∆2 ,where the α’s are normalization constants. In the limit ∆ Az,A⊥, making use of these eigen-values and eigenvectors, simple analytic expressions for the magnetization, susceptibility, andother physically relevant quantities may be obtained in terms of the parameters of the Hamil-63tonian. We list here, for future reference, the normalization constants expanded in this limit,α1 =1∆[1− Az+A⊥4∆− (Az+A⊥)28∆2+ · · ·](4.6)α2 =1∆[1− Az−A⊥4∆− (Az−A⊥)28∆2+ · · ·]α3 =1∆[1+Az+A⊥4∆− (Az+A⊥)28∆2+ · · ·]α4 =1∆[1+Az−A⊥4∆− (Az−A⊥)28∆2+ · · ·].We also list the difference between energy levels, E ji = E j−Ei, to O((Az,⊥∆)3)E21 =A⊥2[1+Az2∆+ · · ·]E41 = ∆[1+A⊥2∆+A2z +A2⊥8∆2+ · · ·](4.7)E43 =A⊥2[1− Az2∆+ · · ·]E32 = ∆[1− A⊥2∆+A2z +A2⊥8∆2+ · · ·]E31 = ∆[1+(Az+A⊥)28∆2+ · · ·]E42 = ∆[1+(Az−A⊥)28∆2+ · · ·].We now introduce the single ion basis operatorsLmn = |m〉〈n|, (4.8)where m,n ∈ {1, ..,4} refer to the eigenstates introduced in (4.5). In this thesis, these operatorswill also be referred to as mean field (MF) operators. We prefer to use the term single ion here,as we are working in a regime where the MF is zero. Further discussion of these operators isavailable in Appendix A.In terms of the single ion operators, the z component of the electronic spin operator may bewritten asSz = c12[L12+L21]+ c34[L34+L43]+ c14[L14+L41]+ c23[L23+L32] (4.9)whereci j =12αiα j[A2z8− ∆22+2EiE j− Az2 (Ei+E j)](4.10)64In the limit ∆ Az,A⊥, we find the matrix elements to be given by (to O((Az,⊥∆)3))c12 =−c34 = Az4∆ c14 = c23 =−12[1− 5A2z +3A2⊥16∆2]. (4.11)The x component of the electronic spin operator is given bySx = a11L11+a22L22+a33L33+a44L44+a13[L13+L31]+a24[L24+L42], (4.12)whereaii =−α2i ∆(Ei− Az4)(4.13)anda13 =α1α34∆(Az+A⊥) a24 =α2α44∆(Az−A⊥). (4.14)We have listed the z and x components of the electronic spin operator in the paramagneticphase of the model. We now proceed to do so for the nuclear spin operators as well. The zcomponent of the nuclear spin operator is given byIz = d12[L12+L21]+d34[L34+L43]+d14[L14+L41]+d23[L23+L32] (4.15)where the matrix elements aredi j =−12αiα j[A2z8+∆22− Az2(Ei+E j)+2EiE j]. (4.16)In the limit ∆ Az,A⊥, we find the matrix elements to be given by (to O((Az,⊥∆)3))d12 = d34 =−12[1− 3A2z +5A2⊥16∆2]. d14 =−d23 = A⊥4∆ (4.17)The x component of the nuclear spin operator is given byIx = b11L11−b22L22+b33L33−b44L44+b13[L13+L31]+b24[L24+L42], (4.18)wherebii = α2i ∆(Ei− Az4)(4.19)65andb13 =−α1α34 ∆(Az+A⊥) b24 =α2α44∆(Az−A⊥). (4.20)These analytic expressions are listed here for reference, as they are useful for understandingthe behaviour of the electronic and nuclear spins. They will be used throughout the remainderof this chapter to analyze the magnetization, susceptibility, and excitation spectrum of the spinhalf spin half (SHSH) model .The SHSH Hamiltonian given in equation (4.1) may be divided into two parts,H =H0+H ′, where, in terms of the single ion operators,H0 =∑i∑nEnLinn (4.21)is the diagonalized single ion Hamiltonian, andH ′ =−12∑i 6= jVi jSziSzj (4.22)is the interaction, with the Sz operator given in terms of the single ion operators in equation(4.9).Perturbation Theory in the Ordered PhaseIn the presence of a longitudinal field, or in the ordered phase of the system, we are no longerable to diagonalize the single ion Hamiltonian of the spin half spin half (SHSH) model exactly;however, we can treat a longitudinal field that is much weaker than the transverse field pertur-batively. This allows us to study properties of the ordered phase of our model analytically inthe vicinity of the system’s quantum critical point. The Hamiltonian isH =HMF − 12∑i6= jVi jS˜zi S˜zj, (4.23)where S˜zi = Szi −〈Sz〉0, and the subscript 0 denotes the average is taken with respect to the MFHamiltonian. The MF Hamiltonian is given byHMF =−∆∑iSxi −H∑iSzi +Az∑iIzi Szi +A⊥2 ∑i(I+i S−i + I−i S+i ), (4.24)66with H = h+V0〈Sz〉 and V0 = ∑ jVi j. H includes an applied longitudinal field h as well as theMF of the neighbouring electronic spins, which we obtain by using the MF Hamiltonian tosolve for the magnetization self consistently. At each site, we divide the MF Hamiltonian intotwo parts,HMFi=H0i+H εi , whereH0i is the single ion Hamiltonian given in (4.3), andHεiis the longitudinal component of the MF Hamiltonian, which we assume is small. Working ina basis of eigenstates ofH0i we find,H εi =0 −Hc12 0 −Hc14−Hc12 0 −Hc23 00 −Hc23 0 −Hc34−Hc14 0 −Hc34 0 , (4.25)where the ci j are given in equation (4.10). In what follows, we will drop the subscript i, and itis to be assumed that we are referring to the MF Hamiltonian of a single ion.We now include the effects ofH ε perturbatively to second order in the ratio of the longi-tudinal field to the transverse field, H∆ . We find the perturbed energies to beE˜1 = E1− H2c212E21− H2c214E41E˜2 = E2+H2c212E21− H2c223E32(4.26)E˜3 = E3+H2c223E32− H2c234E43E˜4 = E4+H2c214E41+H2c234E43,and the normalized (to order H2∆2 ) wavefunctions to be˜|1〉=|1〉+ Hc12E21 |2〉+Hc14E41|4〉+ H2c23c12E31E21 |3〉+H2c34c14E31E41|3〉√1+ H2c212E221+H2c214E241(4.27)˜|2〉=|2〉− Hc12E21 |1〉+Hc23E32|3〉− H2c14c12E42E21 |4〉+H2c34c23E42E32|4〉√1+ H2c212E221+H2c223E232˜|3〉=|3〉− Hc23E32 |2〉+Hc34E43|4〉+ H2c12c23E31E32 |1〉−H2c14c34E31E43|1〉√1+ H2c223E232+H2c234E243˜|4〉=|4〉− Hc14E41 |1〉−Hc34E43|3〉+ H2c12c14E42E41 |2〉+H2c23c34E42E43|2〉√1+ H2c214E241+H2c234E243.These expressions are listed here for reference, and will be used subsequently to analyze the67magnetization and susceptibility of the SHSH model in the vicinity of its quantum criticalpoint.Mean Field Electronic Spin MagnetizationWe calculate the electronic spin magnetization of the SHSH model self consistently from theMF Hamiltonian given byHMF =−∆∑iSxi −H∑iSzi +Az∑iIzi Szi +A⊥2 ∑i(I+i S−i + I−i S+i ), (4.28)where H = h+V0〈Sz〉0 and V0 = ∑ jVi j. At each atomic site, the Hamiltonian is given byHMFi =Az4 − H2 −∆2 0 0−∆2 −Az4 + H2 A⊥2 00 A⊥2 −Az4 − H2 −∆20 0 −∆2 Az4 + H2 , (4.29)where we are using the same basis as in equation (4.3). In this basis, the electronic spinoperators are given bySz =12 0 0 00 −12 0 00 0 12 00 0 0 −12 Sx =0 12 0 012 0 0 00 0 0 120 0 12 0 . (4.30)In Figures 4.1 and 4.2, we show the ground state expectation values of the electronic spinoperators in the MF approximation for an isotropic and an anisotropic hyperfine interaction,solved for self consistently, plotted as a function of the transverse magnetic field ∆. For brevity,we will often refer to the expectation value of a spin operator as the magnetization, when, infact, they are proportional to each other. We consider a simple cubic crystal with a nearestneighbour exchange interaction so thatV0 = 6J, with J= 1 being the exchange coupling. Whenthe hyperfine interaction is isotropic, we find its only effect on the electronic magnetization is asmall overall reduction. The effect of an anisotropic hyperfine interaction is far more dramatic,as can be seen in Figure 4.2. We find that a dominant longitudinal hyperfine interaction tendsto increase the longitudinal electronic magnetization, stabilizing it against the effects of thetransverse field; whereas a dominant transverse hyperfine interaction has the opposite effect.680 1 2 3 4 5∆00.10.20.30.40.5〈 Sz 〉, 〈 Sx 〉                        Ground State Magnetization in MF ApproximationAz=A⊥=1Az=A⊥=0.5Az=A⊥=0.01Figure 4.1: The plot above shows z (solid line) and x (dashed line) ground state compo-nents of the electronic spin operators (in the MF approximation) of the spin halftransverse field Ising model with an isotropic hyperfine interaction, as a function ofthe applied transverse field ∆. We work in units of the exchange interaction strengthJ. We find the z component of the magnetization is uniformly reduced with increas-ing transverse field strength. We obtain these results from the MF Hamiltonian ofthe spin half spin half model, given in equation (4.28).In Figures 4.1 and 4.2, we neglect the effect of an applied transverse magnetic field actingon the nuclear spins, Hn = ∆n∑iI+i +I−i2 . When one considers an effective low temperatureHamiltonian for systems such as LiHoF4, such a field is present, and cannot be neglected. Thistransverse nuclear field is present because when we truncate an electronic spin down to a spinhalf subspace, the effective operator takes the formJµ =Cµ + ∑ν=x,y,zCµντν . (4.31)In the hyperfine interaction, theCµ terms lead to an effective field acting directly on the nuclearspins. This field may be understood as the effect of a shift in the electron cloud surroundingan ion due to the applied transverse field, that causes a transverse field to act on the nuclearspins via the hyperfine coupling. This transverse field due to an atom’s electron cloud maybe orders of magnitude larger than the applied field acting directly on the nuclear spins. In690 5 10 15∆00.10.20.30.40.5〈 Sz 〉, 〈 Sx 〉                        Ground State Magnetization in MF ApproximationA⊥ = 0.8A⊥ = 0.1A⊥ = 0.01A⊥ = 0.00010 1 2 3 4 5∆00.10.20.30.40.5〈 Sz 〉, 〈 Sx 〉                        Ground State Magnetization in MF ApproximationAz=0.8Az=0.4Az=0.01Figure 4.2: The plot above shows z (solid line) and x (dashed line) ground state compo-nents of the electronic spin operators (in the MF approximation) of the spin halftransverse field Ising model with an anisotropic hyperfine interaction, as a func-tion of the applied transverse field ∆. We work in units of the exchange interactionstrength J. The plot on the left shows the magnetization when the longitudinal hy-perfine interaction (Az = 0.8) is dominant, whereas the plot on the right is for adominant transverse hyperfine interaction (A⊥ = 0.8). With the longitudinal hyper-fine interaction dominant, we see that the critical transverse field is driven to largervalues with a decreasing transverse hyperfine interaction A⊥. With A⊥ dominant,decreasing Az reduces the critical transverse field. We obtain these results from theMF Hamiltonian of the spin half spin half model, given in equation (4.28).particular, in the Ho3+ ion, the field at the nucleus due to the 4 f electron cloud is close to 800Tesla [82]. The effect of the the applied field acting directly on the nuclear spins (4.9 Tesla atthe zero temperature critical transverse field in LiHoF4) is suppressed by a factor of the nuclearmagneton µN , which is about 1836 times smaller than the electronic Bohr magneton µB.We see in Figure 4.2, that as A⊥→ 0, with Az fixed, the critical transverse field is driven tohigher values. This is because with A⊥ = 0 there is no mechanism for flipping nuclear spinsin Iz eigenstates; the nuclear spin up and down subspaces are entirely disjoint, as is apparentin equation (4.2). Within each subspace, the nuclear spins constitute an effective longitudinalfield acting on the electronic system. The effect of the transverse field acting on the nuclearspins is a reduction of the critical field values, even when A⊥ = 0. This point is illustrated inFigure 4.3 where we have repeated the MF calculation for the anisotropic case with ∆n = ∆50 .This roughly corresponds to the ratio of the strength of the effective transverse field actingon the nuclear spins in LiHoF4, to the strength of the effective field acting on the electronicspins, when the system goes critical (∆(Bcx) ≈ 50∆n(Bcx), with Bcx being the physical criticaltransverse field of LiHoF4). Even with this modest value of the transverse field acting on thenuclear spins, we see by comparing Figure 4.2 and Figure 4.3 that it has a significant impact70on ∆c, the point at which ∆ drives the longitudinal magnetization to zero.0 1 2 3 4 5 6∆00.10.20.30.40.5〈 Sz 〉, 〈 Sx 〉                        Ground State Magnetization in MF ApproximationA⊥ = 0.8A⊥ = 0.1A⊥ = 0.01A⊥ = 0.0001Figure 4.3: The plot above shows z (solid line) and x (dashed line) ground state compo-nents of the electronic spin operators (in the MF approximation) of the spin halftransverse field Ising model with an anisotropic hyperfine interaction and a trans-verse field acting directly on the nuclear spins, as a function of transverse field ∆.We work in units of the exchange interaction strength J. The longitudinal hyperfineinteraction (Az = 0.8) is dominant, and the applied transverse field acting directlyon the nuclear spins is ∆n = ∆50 . We obtain these results from the MF Hamiltonianof the spin half spin half model, given in equation (4.28), with an additional termHn =−∆n2 (I++ I−) to account for the transverse field acting on the nuclear spins.This additional transverse field leads to a reduction in the critical transverse field ofthe system.In the high field limit, with ∆ > ∆c so that 〈Sz〉 = 0, and ∆ Az,A⊥, we find the groundstate electronic magnetization in the x direction to be〈Sx〉 ≈ 12[1− 5(Az+A⊥)216∆2+ · · ·]. (4.32)We see that the effect of the nuclear spins is to suppress the x component of the magnetizationin the paramagnetic phase.In the vicinity of the phase transition, where the longitudinal magnetization is small, we71may use our perturbation theory results given in equation (4.27), and the Sz operator given inequation (4.9), to write down an analytic expression for the longitudinal magnetization. Theground state longitudinal electronic magnetization of the system is given by (to order H2∆2 )˜〈1|Sz ˜|1〉=H[2c212E21+2c214E41+2H2c212c223E31E221+2H2c234c214E31E241+ 4H2c12c23c34c14E31E41E21][1+ H2c212E221+H2c214E241] . (4.33)This expression will be used in Section 4.4 when discussing the susceptibility of the system.For now, we note that ˜〈1|Sz ˜|1〉 = dm is the infinitesimal change in the longitudinal electronicmagnetization upon application of a small longitudinal field, H = dH. To leading order in H∆we havedm=[2c212E21+2c214E41]dh. (4.34)Recall that H = h+V0〈Sz〉0. We have used the fact that dH ≈ dh. Differentiating the MFmagnetization with respect to h yields the RPA expression for the susceptibility, as shown inAppendix B. We identifyχzz0 =[2c212E21+2c214E41]≈ 12∆[1+12∆(A2zA⊥[1− Az2∆+O(A2z ,A2⊥∆2)]−A⊥+O(A2z ,A2⊥∆2))](4.35)as the single ion longitudinal electronic susceptibility of our SHSH system, at zero temperature.In the final expression, we have expanded in the limit Az,A⊥ ∆. To leading order we seeχzz0 ≈12∆+14∆2[A2zA⊥−A⊥], (4.36)which matches the result derived in [64]. This result ceases to be valid when A⊥→ 0 becausethe lower two energy levels, and the upper two energy levels, of the SHSH system becomedegenerate; hence E21 in equation (4.33) is zero. As previously mentioned in this section, whenA⊥ = 0, in the absence of a transverse field acting on the nuclear spins, the nuclear spin up andnuclear spin down subspaces are disjoint. When calculating thermodynamic averages in theMF approximation, we may consider the two subspaces separately. Within each subspace, thenuclear spin constitutes a longitudinal field acting on the electronic degree of freedom, and thephysics is the same as that of the transverse field Ising model in a longitudinal field, discussedin Appendix B. We will comment further on the degenerate case in Section 4.4, in which we72calculate the single ion susceptibility making use of the MF operator formalism discussed inAppendix A. Degeneracies are easily dealt with using this formalism.In the non-degenerate case, equation (4.36) shows that an isotropic hyperfine interactionwill have little effect on the susceptibility of a magnetic system. However, with an anisotropichyperfine interaction, where the longitudinal component is dominant, the longitudinal suscep-tibility is enhanced. Conversely, if the transverse component of the hyperfine interaction isdominant the longitudinal susceptibility is suppressed.As a final note, we point out that the total magnetization of the system will have a nuclearcomponent, as well as the electronic component discussed above. In the MF Hamiltonian(4.28), we have neglected the effect of the longitudinal field acting directly on the nuclearspins because this term depends on the ratio of the nuclear to the Bohr magneton. The MFHamiltonian including this term is (including a transverse field acting directly on the nuclearspins)HMF =−∆∑iSxi −H∑iSzi −µNµBh∑iIzi −∆n2 ∑i(I+i + I−i ) (4.37)+Az∑iIzi Szi +A⊥2 ∑i(I+i S−i + I−i S+i ).From the free energy we find−∂F∂h=M = 〈Sz〉0+ µNµB 〈Iz〉0. (4.38)Performing a second derivative yields−∂ 2F∂h2=∂M∂h= β[〈S2z 〉0−〈Sz〉20+2µNµB(〈SzIz〉0−〈Sz〉0〈Iz〉0)+ µ2Nµ2B(〈I2z 〉0−〈Iz〉20)]. (4.39)The result above is the classical result for the relationship between the susceptibility and theelectronic and nuclear correlation functions. See the discussion surrounding equation (D.15)for further details. The susceptibility given by χzz0 in equation (4.36) is the zero temperatureVan Vleck contribution to the susceptibility of the system due solely to the electronic spins.For an explanation of the Van Vleck contribution to the susceptibility, see the discussion thatfollows equation (D.28). For convenience, we note thatχzz0 =−Re[g(ω = 0)], (4.40)where g(ω) is given in Appendix D. The total Van Vleck contribution to the single ion suscep-73tibility of the system may be written as−∂ 2F∂h2= χzz0,total = χzz0 +2µNµBχzz0,en+µ2Nµ2Bχzz0,n, (4.41)where χzz0,en is the mixed electronuclear contribution, and χzz0,n is the purely nuclear contribution.In the following section, we turn our attention to the nuclear spins, and perform a similarMF analysis.74Mean Field Nuclear Spin MagnetizationWe now calculate the nuclear spin magnetization of the SHSH model in the MF approximation.For brevity, we will often refer to the expectation value of a spin operator as the magnetization,when, in fact, they are proportional to each other. Using the same basis as for (4.3), the nuclearspin operators are given byIz =12 0 0 00 12 0 00 0 −12 00 0 0 −12 Ix =0 0 12 00 0 0 1212 0 0 00 12 0 0 . (4.42)In Figures 4.4 and 4.5, we show the expectation value of the nuclear spin operators as a functionof transverse field for an isotropic and an anisotropic hyperfine interaction. We see that thenuclear spins order at the same critical field as the electronic spins.750 1 2 3 4 5∆00.10.20.30.40.5|〈 Iz 〉|, |〈 Ix 〉|                       Ground State Magnetization in MF ApproximationAz=A⊥=1Az=A⊥=0.5Az=A⊥=0.01Figure 4.4: The plot above shows the z (solid lines) and x (dashed lines) components ofthe ground state expectation values of the nuclear spin operators (in the MF approx-imation) of the spin half transverse field Ising model with an isotropic hyperfineinteraction, as a function of transverse field ∆. We work in units of the exchangeinteraction strength J. We obtain these results from the MF Hamiltonian of the spinhalf spin half model, given in equation (4.28). We plot the absolute value of theexpectation values of the nuclear operators, noting that they are equal and oppositetheir electronic counterparts.When the hyperfine interaction is isotropic, the electronic and nuclear magnetizations aresimilar. However, with an anisotropic hyperfine interaction, we see in Figure 4.5 that the nu-clear magnetization is quite different from the electronic magnetization shown in Figure 4.2.When the longitudinal hyperfine interaction is dominant, the longitudinal nuclear spin magne-tization is far more resistant to the effects of the transverse field than its electronic counterpart.This can be understood by inspecting equation (4.28). We see that if we set the transversehyperfine interaction A⊥ to zero, there is no energy benefit associated with aligning the nuclearspins opposite the transverse field. The longitudinal hyperfine interaction holds nuclear spinsalong the easy axis of the system. When the transverse hyperfine interaction is dominant, wesee that the longitudinal nuclear magnetization is more easily reduced by the transverse fieldthan its electronic counterpart. Again, by inspection of equation (4.28), we see that if we takeAz = 0, it is energetically favourable for the nuclear spins to lie opposite the transverse field.760 5 10 15∆00.10.20.30.40.5|〈 Iz 〉|, |〈 Ix 〉|                       Ground State Magnetization in MF ApproximationA⊥ = 0.8A⊥ = 0.1A⊥ = 0.010 1 2 3 4 5∆00.10.20.30.40.5|〈 Iz 〉|, |〈 Ix 〉|                       Ground State Magnetization in MF ApproximationAz=0.8Az=0.4Az=0.01Figure 4.5: The plot above shows the z (solid lines) and x (dashed lines) componentsof the ground state expectation values of the nuclear spin operators (in the MFapproximation) of the spin half transverse field Ising model with an anisotropichyperfine interaction, as a function of transverse field ∆. We work in units of theexchange interaction strength J. We obtain these results from the MF Hamiltonianof the spin half spin half model, given in equation (4.28). We plot the absolute valueof the expectation values of the nuclear operators, noting that they are equal andopposite their electronic counterparts. The plot on the left shows the magnetizationwhile the longitudinal hyperfine interaction (Az = 0.8) is dominant, while the ploton the right is for a dominant transverse hyperfine interaction (A⊥ = 0.8).There is no energy benefit associated with spins aligned along the easy axis.We have ignored the effect of the applied transverse field on the nuclear spins in Figures 4.4and 4.5. As with the electronic spins, we find the effect of a transverse field acting directly onthe nuclear spins is a reduction in ∆c relative to the critical field when the effect of the transversefield on the nuclear spins is ignored. We also find that if the strength of the transverse fieldacting on the nuclear spins is comparable to the strength of the hyperfine interaction, there willbe qualitative changes in the magnetization of the nuclear spins, as can be seen in Figure 4.6. Inthis figure we include a transverse field acting directly on the nuclear spins, Hn = ∆n∑iI+i +I−i2 ,with ∆n = ∆100 . We see that with a hyperfine interaction of strength Az = A⊥ = 0.01 the nuclearmagnetization is affected by the transverse field.770 1 2 3 4 5∆00.10.20.30.40.5|〈 Iz 〉|, |〈 Ix 〉|                       Ground State Magnetization in MF ApproximationAz=A⊥=1Az=A⊥=0.5Az=A⊥=0.01Figure 4.6: The plot above shows the z (solid lines) and x (dashed lines) components ofthe ground state expectation values of the nuclear spin operators (in the MF approx-imation) of the spin half transverse field Ising model with an isotropic hyperfineinteraction, as a function of transverse field ∆. We work in units of the exchangeinteraction strength J. We obtain these results from the MF Hamiltonian of the spinhalf spin half model, given in equation (4.28), with an additional transverse fieldacting directly on the nuclear spins.. We plot the absolute value of the expecta-tion values of the nuclear operators, noting that they are equal and opposite theirelectronic counterparts.In the high field limit, with ∆ > ∆c so that 〈Iz〉 = 0, and ∆ Az,A⊥, we find the groundstate magnetization in the x direction to be equal and opposite its electronic counterpart,〈Ix〉 = −〈τx〉. When the longitudinal magnetization is small, we may use our perturbationtheory results (4.27), and the Iz operator (4.15), to write down an analytic expression for thelongitudinal component of the nuclear magnetization. The ground state expectation value ofthe longitudinal nuclear spin operator is given by˜〈1|Iz ˜|1〉=H[2c12d12E21+ 2c14d14E41 +2H2c212c23d23E31E221+2H2c214c34d34E31E241+ 2H2c12c34c14d23E31E41E21+ 2H2c12c23c14d34E31E41E21][1+ H2c212E221+H2c214E241] .(4.43)78To lowest order in H∆ we find˜〈1|Iz ˜|1〉 ≈[2c12d12E21+2c14d14E41]H. (4.44)We identify the term in brackets with the Van Vleck contribution to the electronuclear suscep-tibility given in (4.41). In the limit Az,A⊥ ∆, we findχzzo,en =−12∆[AzA⊥(1− Az2∆+O(A2z ,A2⊥∆2))+A⊥2∆+O(A2z ,A2⊥∆2)]. (4.45)Dropping the contribution to the Van Vleck susceptibility coming entirely from the nuclearspins, which is suppressed by a factor of µ2Nµ2B, we find from (4.41) that the zero temperature VanVleck contribution to the susceptibility of our system is given byχzz0,total ≈ χzz0 +2µNµBχzz0,en =12∆(1+2∣∣∣∣µNµB∣∣∣∣ AzA⊥)+14∆2(A2zA⊥−A⊥)(1−2∣∣∣∣µNµB∣∣∣∣), (4.46)We note that the overall contribution from the electronuclear term is positive. The minus signin χzz0,en is cancelled by a minus sign in the ratio of the nuclear to the Bohr magneton. As withthe electronic component of the susceptibility, we see that a dominant longitudinal hyperfineinteraction will lead to an enhancement of the electronuclear susceptibility. We also see thatthe expressions cease to be valid in the limit A⊥→ 0. This is because degeneracies in the MFHamiltonian lead to a division by zero in the calculations. We will consider the degenerate casein the following section.79Longitudinal Electronic Correlation FunctionWe now calculate the cumulant part of the longitudinal electronic spin spin correlation functionG(k,τ) =−〈Tτ S˜zk(τ)S˜z−k(0)〉 (4.47)where S˜z = Sz−〈Sz〉. This function is of primary importance when it comes to understandingmagnetic systems, as explained in Appendix D. It will be used here in the random phase ap-proximation (RPA) to obtain the RPA excitation modes, their spectral weight, and the criticaltransverse field of the SHSH model. The spectral weights associated with this function arerelevant to neutron scattering experiments.The RPA, or, equivalently, the Gaussian approximation, includes the effect of fluctuations,but treats the fluctuations as non-interacting. This approximation takes the fluctuations to bedistributed normally about their mean value. It is essentially the simplest possible correctionto MF theory. The RPA breaks down in the vicinity of a phase transition, where the effects offluctuations become important. Any good textbook, for example Goldenfeld [5], will discussthe validity of the RPA. A common estimate of the error implicit in the RPA involves calcu-lating how sharply peaked the normally distributed fluctuations are around their mean value.This is known as the Ginzburg criterion. In order for the Ginzburg criterion to be satisfied werequireEG =1N∑i j∣∣∣∣〈(Szi −〈Szi 〉)(Szj−〈Szj〉)〉∣∣∣∣〈Szi 〉〈Szj〉 1 (4.48)The numerator is the zero momentum component of the equal time connected correlation func-tion Szzk=0(t = 0) =∫ dω2pi Szzk=0(ω), which is discussed in Appendix D. This correlation functionmay be expressed in terms of the spectral density, Szzk (ω) =ρzzk (ω)1−e−βω , where the spectral densityis given byρzzk (ω) =−2Im[Gk(iω → ω+ iδ )]. (4.49)In terms of the spectral density, the Ginzburg criterion can be written asEG =1〈Sz〉20∫ dω2piρzzk=0(ω)1− e−βω  1, (4.50)where we have assumed a uniform magnetization 〈Szi 〉= 〈Sz〉0. In Chapter 6, we develop a fieldtheoretic formalism that allows for simpler estimates of the error involved in the RPA. A rough80estimate of the significance of the effect of fluctuations may be obtained by comparing theprefactors of the effective field theory. In Chapter 7, we will use the field theoretic formalismto make estimates of the significance of fluctuations in various magnetic models.Written in the Matsubara formalism, the Green’s function is given byG(k,τ) =−〈Tτ S˜zk(τ)exp(− ∫ β0 dτV (τ))S˜z−k(0)〉0〈Tτ exp(− ∫ β0 dτV (τ))〉0, (4.51)whereV (τ) =−12∑kVkS˜zk(τ)S˜z−k(τ), (4.52)and we define Vk = 1N ∑i jVi jeik(ri−r j).Making use of the MF basis operator formalism discussed in Appendices A and D, we findthe MF, or unperturbed, propagator to be (assuming no degeneracies)g(τ) =−〈Tτ S˜z(τ)S˜z(0)〉0(4.53)=−∑j>ic2i jDi j2E jiE2ji− (iωn)2−β4∑j=1c2j jD jδω0+β[ 4∑j=1c j jD j]2δω0.Note that the momentum index has been dropped as we are dealing with spins at a singlesite. All averages are with respect to the MF Hamiltonian. In the case where any of the MFenergy levels are degenerate, for example, if we take A⊥ = 0 in the SHSH model in which caseE21 = E43 = 0, we simply exclude the degenerate terms from the first summation in equation(4.53).Armed with the unperturbed magnon propagator, we may perform perturbation theory inthe fluctuations around the MF. It should be noted that Wick’s theorem does not apply to thespin operators in the usual way; hence, the fourth and higher order interactions cannot befactored into simple products of pairs of spin operators. This is because spins do not commuteor anticommute. The MF basis operator approach provides a general reduction scheme for spinoperators, and has been applied to products of up to four operators in Appendix E. In the RPA,the propagator for the system isGRPA(k, iωn) =g(iωn)1+g(iωn)Vk. (4.54)81The remainder of this chapter will be devoted to exploring some of the properties of the RPApropagator.Paramagnetic PhaseWe now examine the longitudinal electronic correlation function in the paramagnetic phaseof the system, extracting the low frequency spectrum and the critical transverse field. In theT → 0 limit, for low energy excitations, we find−g(iωn)≈−g12(iωn)−g14(iωn)≈ A+B(iωn)2, (4.55)wheregi j(iωn) = c2i jDi j2E jiE2ji− (iωn)2(4.56)andA=2c212D12E21+2c214D14E41B=2c212D12E321+2c214D14E341. (4.57)Note that the zero temperature single ion longitudinal electronic susceptibility is given by χzz0 =−g(0) = A. Recall from (4.36) that expanding in the high field limit, Az,A⊥ ∆, we haveχzz0 =2c214E41+2c212E21≈ 12∆+14∆2[A2zA⊥−A⊥]. (4.58)We are working in the low temperature limit, so we set the population factors to one, Di j = 1.The result (4.58) is consistent with the result derived perturbatively in [64]. This thesis isprimarily concerned with the zero temperature limit; however, we note that the formalism usedhere allows for easy generalization to finite temperatures. In the case A⊥→ 0, where E21 = 0,the result becomesχzz0∣∣∣∣A⊥=0=2c214E41≈ 12∆[1− 3A2z4∆2+ · · ·]. (4.59)This is similar to the result for the spin half transverse field Ising model discussed in AppendixB, with the longitudinal hyperfine interaction playing the same role as a longitudinal magneticfield.In terms of the parameters A and B given in equation (4.57), the RPA Green’s function is82given byG(k, iωn) =11g +Vk=−A2B[AB − (iωn)2− VkA2B ], (4.60)and the spectrum is given byωk =√AB√1−VkA. (4.61)Note that this is the spectrum in the low frequency limit. The full RPA spectrum will bediscussed in Section 4.5. The spectrum softens to zero, at k = 0, at a critical field given by1−V0χzz0 = 0. In the limit Az,A⊥ ∆, the critical field is given by∆c ≈ V02 +12(A2zA⊥−A⊥). (4.62)The RPA value of the critical field is subject to further corrections due to the nuclear spins.Recall from (4.41) that the total susceptibility contains an electronuclear contribution and acontribution solely from the nuclear spins. The nuclear contribution is suppressed by a factorof (µNµB )2, and may safely be ignored. The leading order correction, due to the electronuclearsusceptibility, follows from 1−V0(χzz0 +2µNµB χzzen) = 0. We find the following correction to thecritical field due to the electronuclear susceptibility∆c ≈ V02(1+2∣∣∣∣µNµB∣∣∣∣ AzA⊥)+12(A2zA⊥−A⊥)(1−2∣∣∣∣µNµB∣∣∣∣[1− AzA⊥]). (4.63)Ferromagnetic PhaseAlthough we can’t diagonalize the MF Hamiltonian of the SHSH model in the ordered phaseof the system analytically, we may include the effects of a weak longitudinal field, H = h+V0〈Sz〉  ∆, perturbatively, making use of the expressions derived in Section 4.1. We begin byexpressing the MF magnetization (4.33) as˜〈1|Sz ˜|1〉= m= χzz0 H[1−δH2+ γH2], (4.64)whereδ =c212E221+c214E241, (4.65)83andγ =(c12c23E41+ c14c34E21)2E21E31E41(c212E41+ c214E21). (4.66)The MF energy levels and matrix elements are given by (4.4) and (4.10) respectively. Dif-ferentiating the MF magnetization (4.64) with respect to h, we find the static susceptibility tobeχzz(H) =∂ 〈Sz〉0∂h=χzz0 (H)1−V0χzz0 (H), (4.67)where χzz0 (H) is the single ion susceptibility, in the presence of a longitudinal field H. Thisresult may also be obtained by taking the zero momentum and zero frequency component of(4.54). We choose here to differentiate the magnetism to show the equivalence of the twoapproaches. The single ion susceptibility, keeping terms up to quadratic order in H, is given byχzz0 (H) = χzz0[1−3δH2−3γH2]. (4.68)Expanding in the limit Az,A⊥ ∆ we find thatδ =14∆2[1− A⊥∆+O(A2z,⊥∆2)+A2zA2⊥(1− Az∆+O(A2z,⊥∆2))](4.69)≈ 14∆2[1+A2zA2⊥],andγ =14∆2A2zA2⊥[1+O(Az,⊥∆)][1+O(Az,⊥∆)+AzA⊥Az2∆(1+O(Az,⊥∆))]−1(4.70)≈ 14∆2A2zA2⊥[1+AzA⊥Az2∆]−1We include γ here for the sake of completeness; however, we note that it stems from the secondorder perturbative correction to the wavefunctions in (4.27). This correction is unreliable,and will be dropped from subsequent analysis. Taking only the leading order perturbativecorrection to the energies and wavefunctions, we find, in the limit Az,A⊥ 1,χzz0 (H) = χzz0[1− 3H24∆2(1+A2zA2⊥)]. (4.71)84If the hyperfine interaction is isotropic this reduces to the result for the transverse Ising modelgiven in equation (B.11) of Appendix B.We note that the effects of an anisotropic hyperfine interaction (Az > A⊥) on the suscep-tibility are twofold. First, the single ion susceptibility is enhanced by an additive factor of14∆2 [A2zA⊥ −A⊥], causing the system to magnetize more rapidly upon the application of a longitu-dinal field. Second, the effect of the longitudinal field is enhanced by a multiplicative factor of√12(1+ A2zA2⊥). We define the enhanced field to beH+ = H√12(1+A2zA2⊥), (4.72)which givesχzz0 (H) = χzz0[1− 3H2+2∆2]. (4.73)Electronic SpectrumIn Section 4.4.1, we analyzed the low frequency spectrum of the SHSH model in the paramag-netic phase of the system. In this section we use the full RPA longitudinal electronic Green’sfunction to find all the RPA modes of the system, and their associated spectral weights. Wewill then examine the low frequency spectrum in the ferromagnetic phase, and find how thespectral gap varies with the longitudinal field.We consider the full zero temperature RPA spectrum, which follows from the poles ofG(k,z) =g(iωn)1+g(iωn)Vk=∑4j=2 |c1 j|22E j1∏s 6= j(E2s1− z2)Vk∑4j=2 |c1 j|22E j1∏s 6= j(E2s1− z2)−∏4j=2(E2j1− z2). (4.74)We assume a simple cubic crystal with nearest neighbour exchange interactions between theelectronic spins, and calculate the zero wavevector gap in the modes as a function of transversefield, and the associated spectral weights as discussed in Appendix B. We see the upper bands,corresponding to a spin opposed to the MF, and the lower bands, corresponding to a spin in linewith the MF, are split by the hyperfine interaction. In the absence of the hyperfine interaction,the upper band would soften to zero at a critical value of the transverse field. We see in theplots that this band is now gapped; however, the spectral weight is transferred down to a lowerelectronuclear level which fully softens to zero at the quantum critical point.850 2 4 6 8∆0246810EnergyRPA Modes (Az =1, A⊥=0.1)0 2 4 6 8∆00.050.10.150.20.250.30.350.4Intensity        Spectral Weight of RPA modesFigure 4.7: The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin half model with an anisotropic hyperfine inter-action, calculated from the Green’s function, equation (4.74), at zero wavevector~k = 0. We work in units of J, the strength of the exchange coupling between spins.Here, the longitudinal hypefine interaction has the same strength as the exchangeinteraction, and the transverse hyperfine interaction is a factor of ten smaller. Wesee the lower mode softens to zero in a quantum phase transition and the associ-ated spectral weight diverges. The middle mode carries most of the spectral weightthroughout the rest of the diagram, except in weak transverse fields where the uppermode may carry some of the spectral weight.In general, the RPA modes of the system are obtained by factoring the denominator of(4.74) numerically. However, in the paramagnetic phase of the system the matrix elementc13 = 〈1|τz|3〉 is zero; hence, there is a common factor of E231− (iωn)2 in the numerator anddenominator. This MF mode carries no spectral weight. The remaining equation is quadraticand may be solved analytically. After removing the common factor, the RPA Green’s function(4.74) becomesG(k,z) =|c12|22E21(E241− z2)+ |c14|22E41(E221− z2)Vk[|c12|22E21(E241− z2)+ |c14|22E41(E221− z2)]− (E221− z2)(E241− z2)(4.75)=az2−χzz0bz4+ cz2+(1−Vkχzz0 ),860 1 2 3 4 5 6∆01234567EnergyRPA Modes (Az =1, A⊥=0.5)0 1 2 3 4 5 6∆00.10.20.30.40.50.60.70.80.9Intensity        Spectral Weight of RPA modesFigure 4.8: The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin half model with an anisotropic hyperfine inter-action, calculated from the Green’s function, equation (4.74), at zero wavevector~k = 0. We work in units of J, the strength of the exchange coupling between spins.Here, the longitudinal hypefine interaction has the same strength as the exchangeinteraction, and the transverse hyperfine interaction is a factor of two smaller. Wesee the lower mode softens to zero in a quantum phase transition and the associ-ated spectral weight diverges. The middle mode carries most of the spectral weightthroughout the rest of the diagram, except in weak transverse fields where the uppermode may carry some of the spectral weight.0 1 2 3 4 5∆0123456EnergyRPA Modes (Az =1, A⊥=1)0 1 2 3 4 5∆00.20.40.60.811.21.41.6Intensity        Spectral Weight of RPA modesFigure 4.9: The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin spin half model with an anisotropic hyperfineinteraction, calculated from the Green’s function, equation (4.74), at zero wavevec-tor~k = 0. We work in units of J, the strength of the exchange coupling betweenspins. We take the strength of the hyperfine interaction to be the same as the strengthof the exchange interaction. We see the lower mode softens to zero in a quantumphase transition and the associated spectral weight diverges. The middle mode car-ries most of the spectral weight throughout the rest of the diagram.870 1 2 3 4 5∆0123456EnergyRPA Modes (Az =0.5, A⊥=1)0 1 2 3 4 5∆00.20.40.60.811.21.41.61.82Intensity        Spectral Weight of RPA modesFigure 4.10: The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin spin half model with an anisotropic hyper-fine interaction, calculated from the Green’s function, equation (4.74), at zerowavevector~k = 0. We work in units of J, the strength of the exchange couplingbetween spins. Here, the transverse hypefine interaction has the same strength asthe exchange interaction, and the longitudinal hyperfine interaction is a factor oftwo smaller. We see the lower mode softens to zero in a quantum phase transitionand the associated spectral weight diverges. The middle mode carries most of thespectral weight throughout the rest of the diagram.where χzz0 is given in equation (4.58), anda=2|c12|2E21E241+2|c14|2E221E41≈ 2A2⊥∆[1− 2Az+A⊥2∆+AzA⊥2∆2− 2A2z +A2⊥2∆2+O(A3z,⊥∆3)](4.76)b=1E221E241≈ 4A2⊥∆2[1− Az+A⊥∆+AzA⊥∆2− 3(A2z +A2⊥)8∆2+O(A3z,⊥∆3)]c=Vkα− 1E221− 1E241≈− 4A2⊥[1− Az∆− A2z −A2⊥4∆2− Vk2∆(1− 2Az+A⊥2∆+AzA⊥2∆2− 2A2z +A2⊥2∆2)+O(A3z,⊥∆3)].Solving the quadratic equation obtained by setting the denominator of equation (4.75) to zero,we find the lowest energy RPA mode in Figures 4.7 - 4.13, in the paramagnetic phase of thesystem, to be given byE1k =A⊥2√1−Vkχzz01− Vk2∆+O(Az,⊥∆). (4.77)We see that this mode softens to zero at the critical field given by equation (4.62), and vanishes880 1 2 3 4 5∆0123456EnergyRPA Modes (Az =0.1, A⊥=1)Figure 4.11: RPA Modes0 1 2 3 4 5∆00.511.522.53Intensity        Spectral Weight of RPA modesFigure 4.12: Spectral WeightFigure 4.13: The plots above show the RPA modes (left), and their associated spectralweight (right), of the spin half spin spin half model with an anisotropic hyper-fine interaction, calculated from the Green’s function, equation (4.74), at zerowavevector~k = 0. We work in units of J, the strength of the exchange couplingbetween spins. Here, the transverse hypefine interaction has the same strengthas the exchange interaction, and the longitudinal hyperfine interaction is a factorof ten smaller. The RPA modes and their spectral weight are colour coordinated.We see the lower mode softens to zero in a quantum phase transition and the as-sociated spectral weight diverges. The middle mode carries most of the spectralweight throughout the rest of the diagram.altogether in the absence of the transverse hyperfine interaction. The second RPA mode isgiven byE2k = ∆[1+A⊥∆− Vk2∆(1+A⊥∆)+O(A2z,⊥∆2)] 12. (4.78)In the absence of the hyperfine interaction, we recover the RPA mode of the spin half transverseIsing model, given in equation (B.20) of Appendix B, in which case this mode will soften tozero at ∆ = V02 . With the nuclear spins present, this mode remains gapped at the critical fieldgiven by equation (4.62).We now make use of our perturbation theory results in Section 4.1 to analyze the low energyexcitation of the system, in the ferromagnetic phase, near the quantum phase transition. We89expand −g(iωn) = A+B(iωn)2 in the low frequency, low temperature, limit to obtainA=2c212E21+2c213E31+2c214E41(4.79)B=2c212E321+2c213E331+2c214E341.Note these expressions contain matrix elements between the first and third MF eigenstates,absent in the paramagnetic phase of the system. The matrix elements, and the differencesbetween the low energy eigenstates, are given by (to leading order in H,Az,A⊥∆ )c12 ≈ c012[1− H24∆2(1+2A2zA2⊥)]E21 ≈ E021[1+H22∆2A2zA2⊥](4.80)c13 ≈ HAz4∆2 E31 ≈ E031[1+H22∆2]c14 ≈ c014[1− H24∆2(1− A2zA2⊥)]E41 ≈ E041[1+H22∆2(1+A2z4A⊥∆)]The superscript zero is to indicate that these are the energy differences and the matrix elementsof the system in the absence of any longitudinal field. The E0jis and c0i js are given in equa-tions (4.7) and (4.11), respectively. The corrections above are calculated using the first orderperturbative corrections to the wave functions.As in the paramagnetic phase, the low frequency spectrum of the system is given byωk =√AB√1−VkA. (4.81)However, A and B are now given by equation (4.79), and we have A= χzz0 (H) given in equation(4.71). We now consider the gap in the spectrum at zero wavevector. At the critical field,defined by Vk=0χzz0 (H = 0) = 1, the expression above reduces toω0 =√32V0B|H=0H+∆c, (4.82)where we keep only terms linear in H. Recall from equation (4.72) that H+ is the longitudinalfield enhanced by an anisotropic hyperfine interaction. In the absence of the nuclear spins,the gap in the spectrum due to an applied longitudinal field at ∆c is given by ω0 =√32H.This mode in the transverse field Ising model corresponds to the middle mode in Figures 4.7- 4.13 of the SHSH model. When we include the hyperfine interaction there is a lower energy90electronuclear mode that softens to zero. It is the gap in this mode due to a longitudinal fieldthat is given in equation (4.82). Expanding in the limit Az,A⊥ 1, we findB|H=0 = A2z∆2A3⊥[1+12∆A3⊥A2z]+O(Az,⊥∆). (4.83)In the limit Az,A⊥ ∆, we find the gap in the spectrum at the critical field to be given byω0 ≈ H+√3A3⊥2A2zV0[1− A3⊥2∆cA2z]+O(H2∆2c). (4.84)We see that the gap in the spectrum between the ground state and the lowest electronuclearlevel varies linearly with our enhanced field H+, and the rate at which the gap opens dependson the nuclear spins. In the limit Az,A⊥→ 0, this mode vanishes and it is the middle mode inFigures 4.7 - 4.13 that will soften to zero at the quantum critical point.91SummaryIn this chapter, we introduced the spin half spin half model (SHSH)H =−J2∑i6= jVi jSziSzj−∆∑iSxi +Az∑iIzi Szi +A⊥2 ∑i(I+i S−i + I−i S+i ). (4.85)This relatively simple model demonstrates the effects of the hyperfine interaction on the trans-verse field Ising model, and, with the inclusion of a transverse field acting on the nuclear spins,may be used to illustrate many of the qualitative features of the magnetic material LiHoF4. Thesingle ion Hamiltonian of the SHSH model can be diagonalized exactly in the paramagneticphase of the system. We began by doing so, and gave the electronic and nuclear spin operatorsin this basis. In Section 4.1, we introduced a longitudinal field to the system and gave secondorder perturbation theory results suitable for analyzing the system in the vicinity of its quantumcritical point.In Sections 4.2 and 4.3, we illustrated the electronic and nuclear magnetizations of the sys-tem in the MF approximation. We proceeded to use our perturbation theory results to giveanalytic expressions for the magnetizations in the presence of a longitudinal field, and showedthe longitudinal susceptibility of the system is enhanced or suppressed by an anisotropic hy-perfine interaction.After exploring the MF results, we turned to the longitudinal electronic Green’s functionof the SHSH model in Section 4.4. Treating the Green’s function in the RPA, we derivedthe low energy paramagnetic spectrum and the critical value of the transverse field. We thenused perturbation theory to analyze the susceptibility of the system, and we found that boththe single ion susceptibility, and the longitudinal field itself, are enhanced by an anisotropichyperfine interaction with a dominant longitudinal component.Finally, in Section 4.5, we calculated all the RPA modes of the SHSH model, and theirassociated spectral weight, by factoring the RPA Green’s function numerically. We saw thatwhat would have been the soft mode in the absence of the nuclear spins is now gapped. Thismode carries most of the spectral weight except near the transition, where the spectral weightis transferred to a lower energy mode that softens completely at the critical transverse field.This is the primary result of this chapter. We closed the chapter by making use of the basis ofparamagnetic eigenstates to derive analytic expressions for the RPA modes.92Chapter 5LiHoF4 in the Random PhaseApproximationIn this chapter, we analyze the longitudinal electronic correlation function of LiHoF4 in therandom phase approximation (RPA), making use of the effective low temperature Hamiltonianderived in Section 2.2. The techniques developed in this thesis, and presented in Chapter 4 forthe simpler spin half spin half model, make such an analysis straightforward. The low energyexcitations in LiHoF4 have been measured via neutron scattering experiments by Rønnow etal. in [36]. Their data shows that the expected soft mode at the system’s critical point isgapped. This result was subsequently analyzed using a numerical approximation that includesthe leading order corrections to the RPA in a high density approximation, that is, an expansionin the inverse coordination number of the system, in [67]. The numerical calculation shows thegap in the crystal field spectrum caused by the hyperfine interaction, but their is no discussion ofthe other excitation modes of the system and their associated spectral weight. Most importantly,although there is a brief mention of a lower energy electronuclear mode that softens to zero atthe quantum critical point, there is no discussion of this mode. In this chapter, we calculate thelow energy mode that softens to zero, and we calculate the transfer of spectral weight from thegapped mode to the low energy mode that occurs near the phase transition. A previous RPAanalysis has been carried out on a toy model for LiHoF4 in a transverse field by Banerjee andDattagupta in [94]. Their work includes only a longitudinal hyperfine interaction, and fails tocapture the full complexity LiHoF4. Most importantly, they neglect the effect of the transversefield acting directly on the nuclear spins due to a shift in the electronic cloud of each holmiumion’s 4 f electronic cloud caused by an applied transverse magnetic field, and the mixing ofthe nuclear spins by the transverse component of the hyperfine interaction. These effects arecrucial in the vicinity of the quantum critical point, as the phase transition is dependent on themobility of the nuclear spins. The low temperature effective Hamiltonian used here does not93suffer from these shortcomings.We begin by self consistently solving for the expectation value of the τz operator in themean field (MF) Hamiltonian, HMF = ∑iH iMF , whereH iMF =−∆2τxi −V0〈τz〉τzi + ~∆n ·~Ii+Azτzi Izi+ (5.1)+A⊥τ+i I−i +A†⊥τ−i I+i +A++τ+i I+i +A†++τ−i I−i ,withV0 =C2zz[JD∑jDzzi j −4Jnn]. (5.2)Recall, that the exchange interaction involves a sum over each holmium ion’s four nearestneighbours, and in the two by two low energy subspace Jz = Czzτz. We assume a long thincylindrical sample, consistent with the domain structure of LiHoF4 near its phase transition asdiscussed in Section 2.1.3, in which case a3Dzz0 = a3∑ jDzzi j = 11.272, where a= 5.175A˚ is thetransverse lattice spacing. The strength of the dipolar interaction is JDa3 = 7mK, and the strengthof the antiferromagnetic exchange interaction is Jnn = 1.2mK. We see from this that at zerowave vector the strength of the dipolar interaction is more than 16 times that of the exchangeinteraction. The transverse fields, ∆ and ∆n, and the hyperfine couplings, Az, A⊥ and A++, arefunctions of the physical transverse field Bx. All these energies are given in Section 2.2.In Figure 5.1, we plot the energy levels of the MF Hamiltonian as a function of the physicaltransverse magnetic field Bx. The upper and lower sets of energy levels correspond to an elec-tronic spin in line with the MF of the system, and an electronic spin opposed to the MF of thesystem, respectively. The two clusters of energy levels in Figure 5.1 stem from the lower twoelectronic crystal field excitations plotted in Figure 2.3 of Chapter 2. Each of these electronicenergy levels is split into eight electronuclear levels by the hyperfine interaction, separated byAz ≈ 200mK. In Figure 5.2, we plot the transverse and longitudinal electronic magnetization,〈Jz〉=Czz〈τz〉 and 〈Jx〉=Cx+∑ν=x,y,zCxν〈τν〉, along with the transverse and longitudinal nu-clear spin magnetization. Note that at the phase transition, the transverse nuclear magnetizationnearly saturates at 〈Ix〉= 72 , whereas the transverse electronic magnetization, strongly effectedby the crystal field, is far from its saturation value of 〈Jx〉= 8.940 1 2 3 4 5 6Bx (T)-4-3-2-101234Energy (K)Mean Field EigenvaluesFigure 5.1: The plot above shows the energy levels of the effective low temperature meanfield Hamiltonian of LiHoF4, given in equation (5.1), as a function of the physicaltransverse magnetic field, Bx. We consider a long thin cylindrical, or needle shaped,sample.Electronic SpectrumIn this section, we calculate the connected imaginary time ordered longitudinal electronic spincorrelation function (Green’s function for short)Gk(τ) =−〈Tτ J˜zk(τ)J˜z−k(0)〉=−C2zz(Bx)〈Tτ τ˜zk(τ)τ˜z−k(0)〉, (5.3)where J˜z = Jz− 〈Jz〉 = Czz(τz− 〈τz〉), in order to determine the excitation spectrum of theLiHoF4 system. This function is relevant to the neutron scattering experiments of Rønnow etal. [36], with the spectral weight of each excitation mode corresponding to the intensity ofscattered neutrons. The electronic spin operator τz is given byτz =[1 00 −1]⊗ I8, (5.4)where I8 is the eight by eight identity matrix corresponding to the nuclear subspace. We men-tion this to remind the reader we are dealing with the sixteen lowest electronuclear levels of950 1 2 3 4 5 6Bx (T)0123456Mean Field Magnetization                  Electronic and Nuclear Mean Field Magnetization〈 Jz 〉〈 Jx  〉| 〈 Iz 〉 || 〈 Ix  〉 |Figure 5.2: The plot above shows the transverse and longitudinal electronic (〈Jx〉 and〈Jz〉), and nuclear (|〈Ix〉| and |〈Iz〉|), magnetizations of LiHoF4. The magnetizationsare calculated self consistently from the mean field Hamiltonian given in equation(5.1). We consider a long thin cylindrical, or needle shaped, sample. We see thenuclear magnetizations, |〈Ix〉| and |〈Iz〉|, saturate near the quantum phase transitionand in the zero field limit, respectively. The electronic magnetizations, 〈Jx〉 and〈Jz〉, fail to saturate due to the disordering effect of the crystal field.the Hamiltonian. Transforming to frequency space, in the random phase approximation (RPA),the Green’s function is given byGRPAk (iωn) =C2zzg(iωn)1+Vkg(iωn), (5.5)where the unperturbed propagator is given byg(τ) =−〈Tτ τ˜z(τ)τ˜z(0)〉0, (5.6)and the subscript 0 denotes the average is to be taken with respect to the MF Hamiltonian.In what follows, we will drop the superscript RPA from the Green’s function, as it is to beunderstood that we are working in the RPA throughout this chapter.The interaction is given by Vk =C2zz[JDDzzk − Jnnγk]. Note that Dzzk is the Fourier transform96of the dipolar interaction, discussed in Chapter 3, andγk = 2Re[cos(kxa2)e−ikzc4 + cos(kya2)eikzc4]= 2cos(kzc4)[cos(kxa2)+ cos(kya2)](5.7)is the Fourier transform of the exchange interaction taking into account the four nearest neigh-bour atoms at (±a2 ,0,− c4) and (0,±a2 , c4). We take only the real part of Vk, as the imaginarypart vanishes from the Hamiltonian upon summation.In terms of the MF eigenstates, we define ci j = 〈i|τz| j〉0 to be the MF matrix elements ofthe τz operator, E ji = E j−Ei to be the difference between the ith and jth energy levels, and thepopulation factor to be Di j = Di−D j, where Di = e−βEiZ0 and Z0 is the MF partition function.In terms of these parameters, the unperturbed propagator is given by g(iωn) = ∑i< j gi j(iωn)+gelδiωn,0, wheregi j(iωn 6= 0) =−|ci j|2Di j 2E jiE2ji− (iωn)2. (5.8)We will drop the elastic contribution to the unperturbed propagator gel that arises in the orderedphase of the system (〈τz〉 6= 0) at finite temperatures. This term is discussed in Appendix D. Itvanishes in the limit T → 0, and will not effect the calculation of the inelastic spectrum. In thelimit T → 0, we find only the lowest eigenstate is populated, hence our unperturbed propagatorbecomes−g(iωn)∣∣∣∣T=0= ∑j>1|c1 j|2 2E j1E2j1− (iωn)2≈ χzz0 +χzz2 (iωn)2, (5.9)whereχzz0 = ∑j>12|c1 j|2E j1χzz2 = ∑j>12|c1 j|2E3j1. (5.10)We expect this expansion to be valid for frequencies much less than the minimum gap in theMF spectrum, |ω|  min{E21(∆)} = E21(∆c) ≈ 80mK. Note that χzz0 is simply the electroniccomponent of the static single ion longitudinal susceptibility of the effective low temperatureHamiltonian given in equation (5.1) in the MF approximation. For LiHoF4, we have χzz0 |liho =C2zzχzz0 .97Using the identitylimε→0+1x± iε =P(1x)∓ ipiδ (x) (5.11)we may rewrite the unperturbed propagator as−g(iωn→ ω+ i0+)∣∣∣∣T=0= ∑j>1|c1 j|2(2P( 1E j1)+ ipi[δ (E j1−ω)−δ (E j1+ω)]). (5.12)The spectral weight carried by each MF excitation is proportional to the square of the associatedmatrix element.In Figure 5.3, we plot the matrix elements of the electronic spin operator c1 j = 〈1|τz| j〉between the ground state and the excited states. We see that the mixing of the lower MFeigenstates is orders of magnitude smaller than the mixing of the upper eigenstates, withthe exception of the mixing of the ground state with the first excited state. This matrix ele-ment is included as a dashed line with the upper matrix elements. A rough argument can bemade explaining why matrix elements between the lower eigenstates are small. As the trans-verse magnetic field is increased, the electronic spin rolls over in the magnetic field, that is,〈~τ〉= 〈τz〉 → 〈~τ〉= 〈τz〉+ 〈τx〉, with the τx component of the ground state becoming increas-ingly large with the applied transverse field Bx. Application of the τz operator to the groundstate flips the x component of the electronic spin. When this is projected onto one of the lowerMF eigenstates there is little overlap, leading to a small matrix element, whereas when it is pro-jected onto one of the upper eigenstates, which have a τx component predominantly oppositethe applied field, there will be significant overlap.Plugging the expression for the unperturbed propagator back into the RPA Green’s function,we findG(k, iωn) =11g +Vk=−C2zz(χzz0 )2χzz2 [χzz0χzz2− (iωn)2− Vk(χzz0 )2χzz2], (5.13)and the lowest energy mode of the longitudinal electronic spectrum is given byωk =√χzz0χzz2√1−Vkχzz0 . (5.14)The spectrum softens to zero at a critical field defined implicitly by 1 = Vkχzz0 . In Figure 5.4,we plot the static (k = 0) longitudinal RPA susceptibility of LiHoF4, along with the minimumenergy gap in the excitation spectrum at zero wavevector, in the vicinity of the quantum critical980 1 2 3 4 5 6Bx (T)00.511.522.53Matrix Elements×10 -3                Longitudinal Electronic Spin Operator|c 1,3 ||c 1,4 ||c 1,5 ||c 1,6 ||c 1,7 ||c 1,8 |0 1 2 3 4 5 6Bx (T)00.20.40.60.81Matrix Elements   Longitudinal Electronic Spin Operator|c 1,2 ||c 1,9 ||c 1,10 ||c 1,11 ||c 1,12 ||c 1,13 ||c 1,14 ||c 1,15 ||c 1,16 |Figure 5.3: The plots above show the matrix elements of the electronic spin operator τz,with ci j = 〈i|τz| j〉, for the low temperature effective Hamiltonian for LiHoF4, givenin equation (5.1), in the MF basis. We plot the matrix elements as a function of thephysical transverse field Bx (in Tesla). Note the scale of the vertical axis in the plotto the left is three orders of magnitude smaller than that of the plot to the right.point. We see that the susceptibility diverges, and the gap softens all the way to zero, at thequantum critical point. Note that the susceptibility shown in Figure 5.4 is the susceptibility in along thin cylindrical sample consisting of a single domain. As discussed in Section 2.1.3, in asample consisting of multiple domains the susceptibility in the ferromagnetic phase is constant,and is dominated by the motion of the highly mobile domain walls.The excitation mode given in equation (5.14) is the lowest energy mode of the RPA spec-trum in the vicinity of the quantum critical point. The other modes of the spectrum may befound numerically by looking at the zeros of C2zzG−1 = 1g +Vk, where g is the unperturbedpropagator given in equation (5.9). At T = 0 the unperturbed propagator has poles located atthe energy difference between the ground state and each excited state of the MF Hamiltonian.The effect of the interaction is to shift the location of these poles. Explicitly, we findC−2zz G(k, iωn) =∑16j=2 |c1 j|22E j1∏s 6= j(E2s1− (iωn)2)Vk∑16j=2 |c1 j|22E j1∏s 6= j(E2s1− (iωn)2)−∏16j=2(E2j1− (iωn)2). (5.15)The poles of the Green’s function are given by the zeros of the polynomial in the denominatorof (5.15). Numerically, finding the poles can be quite time consuming because near a poleg ∼ αiE2i1−ω2, and, if αi = 2|c1i|2Ei1 is small, a very fine frequency scan must be made in orderto see the divergence. It is much faster to consider the denominator of equation (5.15) as acontinuous function of frequency (iωn→ z) and find the zeros of the polynomial. Points where|c1 j|2 vanish correspond to points where a pole in the RPA Green’s function carries no spectralweight. We see this by noting that if a factor of |c1 j|2 is zero then there is a common factor994.5 5 5.5 6Bx(T)0100200300400χzz (K-1 )      Susceptibility and Soft Mode of LiHoF400.050.10.150.20.25Soft Mode (K)Figure 5.4: The plot above shows the zero wavevector longitudinal susceptibility ofLiHoF4 in the random phase approximation, along with the lowest energy mode(at zero wavevector, k = 0) in the electronic (RPA) spectrum, as a function of theapplied transverse magnetic field Bx (in Tesla). We consider a long thin cylindricalsample. We see that the susceptibility diverges, and the lowest energy mode softensall the way to zero, at the quantum critical point.of E2j1− (iωn)2 in both the numerator and denominator that cancels reducing the degree of thepolynomial. Setting t = z2 = (iωn)2, we find the zeros of the polynomialQ(t) =Vk16∑j=2|c1 j|22E j1∏s 6= j(E2s1− t)−16∏j=2(E2j1− t). (5.16)In Figure 5.5, we plot the full RPA spectrum at zero wavevector (k = 0) as a function oftransverse magnetic field. At zero field, we see the upper and lower bands, corresponding toelectronic spin up and electronic spin down, respectively, split into clusters of eight by theinclusion of the nuclear degrees of freedom. These states are then mixed by the transversefield. In the absence of the nuclear spins, the middle mode that splits off from the group wouldsoften to zero in a quantum phase transition. We see that the mode is gapped by the nuclearspins; however, the lowest energy electronuclear mode softens to zero.1000 1 2 3 4 5 6Bx(T)012345678Energy (K)RPA Spectrum of LiHoF4Figure 5.5: The plot above shows the RPA spectrum (in Kelvin) of the low temperatureeffective Hamiltonian for LiHoF4 given in (5.1), at zero wavevector, as a functionof the transverse magnetic field Bx (in Tesla).We now consider the spectral weight of the RPA modes. We may rewrite the Green’sfunction asC−2zz Gk(z) =∑16j=2 |c1 j|22E j1∏s 6= j(E2s1− z2)∏15p=1[(z−E pk )(z+E pk )] , (5.17)where E pk denotes the pth RPA mode. Note that the Green’s function has the form C−2zz Gk(z) =P(z)Q(z) , where P(z) and Q(z) are polynomials. Performing a partial fraction decomposition (as-suming no degenerate modes) yieldsC−2zz Gk(z) =15∑p=1[P(E pk )Q′(E pk )1(z−E pk )+P(−E pk )Q′(−E pk )1(z+E pk )], (5.18)whereP(E pk ) = P(−E pk ) =16∑j=2|c1 j|22E j1∏s 6= j[E2s1− (E pk )2](5.19)Q′(E pk ) =−Q′(−E pk ) = 2E pk ∏s 6=p[(E pk )2− (Esk)2].101The spectral density is given by (at T = 0)ρk(ω) =−2Im[Gk(ω+ i0+)] = 2piN C2zz∑p|〈0|τzk |p〉|2[δ (ω−E pk )−δ (ω+E pk )], (5.20)where |p〉 denotes a many body RPA eigenstate of the Hamiltonian. Taking the imaginary partof the RPA Green’s function, we find thatρk(ω) = 2piC2zz15∑p=1P(E pk )Q′(E pk )[δ (ω−E pk )−δ (ω+E pk )]. (5.21)In Figures 5.6 and 5.7, we plot the intensities of the RPA modes of the longitudinal Green’sfunction. Figure 5.6 shows the soft mode E1, along with the upper set of modes E8 to E15.The modes not shown carry no spectral weight. We see that away from the phase transition,the spectral weight is carried by the upper electronuclear levels, which are plotted in Figure5.7. In the vicinity of the quantum critical point, the spectral weight is transferred to the lowestelectronuclear level. The spectral weight of this mode diverges at the quantum critical point,where the mode softens to zero.0 1 2 3 4 5 6Bx (T)00.511.52Intensity             Spectral Weight of the RPA Modes of LiHoF4E1E8E9E10E11E12E13E14E15Figure 5.6: The plot above shows the spectral weight of the RPA modes (E1 and E8 toE15) of LiHoF4, calculated from the longitudinal Green’s function (5.15), at zeromomentum~k = 0, as a function of the physical transverse magnetic field field Bx(in Tesla). We see the spectral weight of the lowest energy mode diverges at thequantum critical point. Above Bx ≈ 3T , most of the spectral weight is carried byE8, except near the phase transition. The modes not shown in the figure carry nospectral weight.1021 1.5 2 2.5 3Bx (T)00.050.10.150.20.250.30.350.4Intensity                     Spectral Weight of the Upper RPA Modes of LiHoF4E8E9E10E11E12E13E14E15Figure 5.7: The plot above shows the spectral weight of the upper RPA modes (E8 toE15) of LiHoF4, calculated from the longitudinal Green’s function (5.15), at zeromomentum~k = 0, as a function of the physical transverse field Bx between 1T and3.5T . Above Bx ≈ 3T , E8 is the dominant mode. The peaks in the intensities, inorder of decreasing amplitudes, correspond to E9, E10, ..., E15.In addition to considering the RPA modes as a function of transverse field, we may examinehow they vary in momentum space. We do so by tracing out the triangle [0,0,0]→ [pia ,0,0]→[pia ,0,pic ]→ [0,0,0], at the critical MF, ∆0c = 5.33T . At zero momentum, the spectral weight ofthe lowest mode diverges. As we vary the momentum, the weight carried by this mode decays,and it is the upper electronic mode that carries the spectral weight. Note that the eighth RPAmode, which carries the spectral weight away from k = 0, varies extremely rapidly as |~k| → 0.This is a result of the shape dependence of the dipole-wave sum near |~k|= 0.1030 0.5 1~k = [ piax, 0, 0]0246Energy (K)      RPA Spectrum of LiHoF4 kx0 0.5 1~k = [ pia, 0, picx]0246      RPA Spectrum of LiHoF4 kz00.51~k = [ piax, 0, picx]0246         RPA Spectrum of LiHoF4 kxkzFigure 5.8: The plot above shows the RPA spectrum of LiHoF4 in momentum space, atthe critical mean field ∆0c = 5.33T , calculated from the longitudinal Green’s func-tion given in equation (5.15). The central mode, separated from the rest, carries thespectral weight away from k = 0. Near k = 0, spectral weight is transferred to thelowest energy mode.0 0.5 1~k = [ piax, 0, 0]0123IntensitySpectral Weightmode 1mode 80 0.5 1~k = [ pia, 0, picx]0123Spectral Weightmode 1mode 800.51~k = [ piax, 0, picx]0123Spectral Weightmode 1mode 8Figure 5.9: The plot above shows the spectral weight of the RPA modes of LiHoF4 in mo-mentum space, calculated from the longitudinal Green’s function given in equation(5.15), at the critical mean field, ∆0c = 5.33T .SummaryIn this chapter, we examined some of the properties of the longitudinal Green’s function (rele-vant to neutron scattering experiments) for electronic spins in LiHoF4, using the random phaseapproximation (RPA). To begin, we solved the mean field (MF) Hamiltonian self consistently,and illustrated the MF energy levels, as well as the electronic and nuclear magnetizations.In Section 5.1, we considered the longitudinal electronic Green’s function in the low energylimit, as a function of the transverse magnetic field. We extracted the static (zero wave vector)longitudinal RPA susceptibility, and calculated the energy of the lowest energy excitation in thevicinity of the quantum critical point. We then considered the full RPA Green’s function, andextracted the full RPA spectrum, along with the spectral weight of each mode. In the absenceof the hyperfine interaction the dominant electronic mode softens to zero in a quantum phasetransition. We found that the hyperfine interaction gaps this mode, with spectral weight being104transferred down to the lowest electronuclear level, which softens to zero. A calculation ofthis low energy soft mode and its associated spectral weight does not exist in the literature. Inaddition, we examined the RPA modes in momentum space near the quantum critical point. Wefound that in momentum space the spectral weight carried by the lowest electronuclear leveldecays, and the spectral weight is transferred back to the upper gapped mode.105Chapter 6Field Theoretic Treatment of QuantumIsing SystemsIn this chapter, we develop a field theoretic treatment of quantum Ising systems with an arbi-trary single ion Hamiltonian. This formalism is suitable for a renormalization group treatmentof such systems. Furthermore, the formalism allows for diagrammatic techniques to be usedto incorporate the effects of fluctuations in quantum Ising systems perturbatively throughout asystem’s phase diagram, where the perturbation parameter is the inverse coordination number.This approximation is referred to as the high density approximation by Brout, who introducedit in a 1959 paper on random ferromagnetic systems [60, 61]. The formalism is well suited fordealing with systems such as LiHoF4 because we are able to easily include the nuclear degreesof freedom of the holmium ions, and, due to the dominant dipolar coupling between electronicspins, the effective coordination number is large.We obtain a field theory for a quantum Ising system by making use of the well knownHubbard-Stratonovich (HS) transformation. Mu¨hlschlegel and Zittartz (MZ), in 1963, werethe first to apply the HS transformation to an Ising system [95]. They considered a spin halfIsing system in a longitudinal applied field, and, after applying the the HS transformation, theyused a variational approach to analyze the resulting free energy. MZ were only concerned withthe static properties of the system. They did not consider the time dependence of the quantumspin operators. In this thesis, we consider the dynamic properties of a quantum Ising systemwith an arbitrary single ion Hamiltonian, and develop a diagrammatic method for calculat-ing corrections due to spatial and temporal fluctuations, rather than employing the variationalapproach of MZ. We will say more on this subject after a more general discussion of diagram-matic perturbation theory in the following paragraphs. In 1975, the HS transformation wasapplied by Young to the spin half transverse field Ising model to illustrate quantum effects inthe renormalization group approach to phase transitions [58]. We fully develop Young’s ap-106proach, generalizing to systems with an arbitrary single ion Hamiltonian, and thereby derive afield theory suitable for both renormalization group calculations, and for calculating the effectof fluctuations away from a system’s critical region. We note that Young’s work was carriedout independently of the more thorough work of Hertz on quantum phase transitions [31].Diagrammatic perturbation theory has proven to be an invaluable tool in the study of manybody systems. The book of Mattuck is a good introduction to the use of diagrams in Boseand Fermi systems [96]. Spin systems are complicated by the fact that the spin operators donot commute or anticommute, so the Wick reduction theorem does not apply in the usual way.Building on work on random ferromagnetic systems, Brout introduced a diagrammatic pertur-bation theory for ferromagnetic spin systems in 1960 [60, 61]. The theory was subsequentlydeveloped by Brout, Englert, Stinchcombe, and others, and is presented in two 1963 papers[97, 98]. The diagrammatic perturbation theory is analogous to that used in Bose and Fermisystems, with an important difference - each vertex corresponds to a spin cumulant, the ana-logue of a propagator in Bose and Fermi systems, and each line corresponds to an interactionbetween sites. Englert refers to the diagrams as the dual of Feynman diagrams [97]. Brout’sdiagrammatic perturbation theory is outlined in Appendix C.A theory similar to Brout’s, corresponding to a rearrangement of the terms in Brout’s the-ory, was introduced by Horwitz and Callen [99, 100]. Brout’s theory, treated in the randomphase approximation (RPA), suffers from unphysical behaviour in the ordered phase of thesystem. Brout accounts for this by enforcing a constraint on the total number of spins in thesystem [61], thereby introducing a chemical potential into the equations. The essential dif-ference between the Horwitz and Callen theory, and the theory of Brout, is that Horwitz andCallen renormalize the vertices of their theory (the spin cumulants) by summing over all treelevel diagrams, thereby working around the mean field (MF) ground state and avoiding anyunphysical behaviour. Horwitz and Callen do not introduce a chemical potential in their work.We note that a diagrammatic perturbation theory equivalent to that of Brout was developedindependently by Vaks, Larkin, and Pikin, and published in a 1968 paper [101].In this thesis, we avoid the unphysical behaviour in the theory of Brout by taking the inter-action to be V = −12 ∑i6= jVi j(Szi −〈Sz〉0)(Szj−〈Sz〉0), rather than V = −12 ∑i 6= jVi jSziSzj, wherethe subscript zero indicates the average is to be taken with respect to the MF Hamiltonian. Thisis equivalent to following the renormalization procedure of Horwitz and Callen. We apply theHS transformation to the interaction, and trace over all spin variables to obtain an effectivefield theory. This scalar field theory is then treated using the usual Feynman diagram methods.We find the results to be equivalent to those of the Brout theory, which has been applied to thetransverse field Ising model in a series of 1973 papers by Stinchcombe [26, 62, 63]. We findthe formalism presented here to be conceptually simpler than the formalism of Brout. Further-107more, the field theoretic formalism presented here allows for a renormalization group treatmentof a system’s critical region. In Brout’s theory, the behaviour of a system in its critical regionis plagued by unphysical divergences.Brout’s formalism has only been applied to systems with simple single ion Hamiltonians.This is because the spin cumulants, which correspond to the vertices of the diagrammatic the-ory, become increasingly difficult to calculate as the complexity of the single ion Hamiltonianis increased. We overcome this difficulty by working in a basis of MF eigenstates. By doing so,we are able to deal with systems having arbitrarily complicated single ion Hamiltonians, andsystems having additional degrees of freedom at each atomic site, such as the nuclear spins inLiHoF4. The MF basis is discussed in Appendix A. In order to calculate the spin cumulants inthe MF basis, we make use of a general reduction scheme of Yang and Wang [102]. The samereduction scheme is presented for spin operators in the appendix of the earlier paper by Vaks,Larkin and Pikin [101]; however, except for simple spin systems such as the Ising model in alongitudinal field, the reduction of spin operators is of little practical utility. The reduction ofspin operators working in the MF basis can be done quite generally, and we have carried outthe reduction for cumulants of up to four spin operators in Appendix E.The reduction of MF basis operators has been developed into a general formalism for theanalysis of magnetic systems with complicated single ion Hamiltonians by Wang and collab-orators, and is reviewed in a 1987 paper by Wentworth and Wang [103]. The result of theformalism is a high temperature series expansion that reduces to MF theory in the zero temper-ature limit. High temperature series expansions are of little use in the examination of quantumcritical behaviour. We note that the formalism of Wang et al. is complicated by the multiplesite indices appearing in the spin cumulants associated with each vertex in the diagrammaticversion of the theory [102, 103]. In Section 6.3.1, we show that it is only necessary to considercumulants of spins at a single site. Furthermore, in the formalism presented by Wentworth andWang, MF operators are grouped according to their dependence on imaginary time τ . The fieldtheoretic formalism presented in this thesis avoids this complication by decoupling the interac-tion between spins with a HS transformation, which leads to spin cumulants with no repeatedτ indices.Series expansion techniques for the zero temperature behaviour of spin half transverse fieldIsing systems are investigated in a 1971 paper by Pfeuty and Elliott [104]. The authors makeuse of perturbation theory to calculate the static properties of the model, with the perturbationparameter being the ratio of the strength of the exchange interaction to the strength of thetransverse field, J∆ , or its inverse. These techniques break down near the quantum criticalpoint, where the value of the perturbation parameter is one. The diagrammatic techniquesused in this thesis go beyond the series expansion techniques of Pfeuty and Elliot [104] by108allowing for the calculation of the dynamic, rather than static, properties of a quantum Isingsystem. Furthermore, the field theory allows for a renormalization group treatment of thecritical behaviour.We have discussed the series techniques of Wang and collaborators, and of Pfeuty and El-liot, and have explained why, for the analysis of quantum critical behaviour, they are inferiorto the work carried out in this thesis. We now return to the work of Mu¨hlschlegel and Zittartz(MZ) [95], and elaborate on the differences between their work, related approaches, and thework carried out in this chapter. In the field theoretic treatment of MZ, the free energy of thespin half quantum Ising model was evaluated by assuming a quadratic form for the variationalfunction determining the energy of the fluctuating HS field [95]. This approach allows forthe effect of fluctuations to be incorporated through the variational parameters in the quadraticfunction; however, because the theory is Gaussian in nature, interactions between the fluctua-tions are ignored. In principle, it is an improvement over the standard Gaussian approximationbecause the parameters in the theory are renormalized. However, in practice, the variationalparameters turn out to be exactly what you would find by calculating the parameters from themicroscopic Hamiltonian. MZ also consider the leading order correction to the Gaussian re-sult; however, this is only possible due to the simple nature of the spin half Ising model in alongitudinal field that they base their work on.Mu¨hlschlegel and Zittartz make contact with the work of Brout by introducing a Lagrangemultiplier to enforce a constraint on the total number of spins in the system, µ[N−∑i(Szi )2] = 0.We note this constraint is only appropriate for spin half systems. Adding this constraint tothe Hamiltonian corresponds to shifting the interaction between spins, Vi j → Vi j+ 2µδi j, or,in momentum space, Vk → Vk+ 2µ . The renormalized interaction, which we will discuss inSection 6.4 of this thesis, is then assumed to have the form∑kTk(iωr = 0) =∑kVk+2µ1− (Vk+2µ)χzz0= 0, (6.1)where χzz0 is the longitudinal single ion susceptibility of the system. The assumption (6.1),which determines the Lagrange multiplier µ , is somewhat mysterious in the work of MZ.They simply state that it leads to the solution of Brout. We will have more to say about thenature of this assumption in the following paragraphs. The resulting expression for the freeenergy obtained by MZ is similar to the result to be given in equation (6.61) with the followingcaveats. First, in equation (6.61), the time dependence of the quantum operators will be takeninto consideration. In a quantum system, time and space are intrinsically linked, and the timedependence of operators cannot be ignored. To obtain the expression for the free energy givenin equation (24) of Section V of the MZ paper [95], the summation over Matsubara frequencies109in equation (6.61) must be dropped, and g(iωr) must be replaced with g(iωr = 0) = χzz0 (thelongitudinal component of the MF susceptibility). Second, the Lagrange multiplier shouldbe set to zero because, in this thesis, we work with the unconstrained theory. With µ = 0,the result is equivalent to the results of Horwitz and Callen [99]. MZ have compared theresults of Horwitz and Callen, and those of Brout, in the high temperature limit (βV0 1), andfound that differences due to the presence of the Lagrange multiplier µ appear at sixth order inperturbation theory. The results of the theory, with or without the constraint µ , would be thesame if the calculation was done exactly (to infinite order in perturbation theory). Constrainingthe total number of spins in the system by introducing µ will improve the results of the theoryobtained perturbatively; however, the work of MZ demonstrates that the improvement onlyoccurs at high orders of the perturbation parameter.We now return to assumption (6.1), and discuss its significance. Making use of the identity1N∑kχzzk =1N∑kχzz01− (Vk+2µ)χzz0= χzz0 , (6.2)where χzzk is the longitudinal susceptibility in the random phase approximation, and χzz0 is thesingle ion susceptibility, we may rewrite assumption (6.1) as1N∑kVk1− (Vk−α)χzz0= α, (6.3)with α = −2µ . As discussed in [105], equation (6.3) is one of the defining equations ofthe correlated effective field (CEF) approximation, which was introduced by Lines in 1972[106–108]. Note that our definition of α differs from that of Lines by a factor of V0, that is,α =V0αLines. Following [105, 107], we now proceed to outline the CEF approximation.To begin, we write the Ising Hamiltonian asH =−12∑iSzi[∑j 6=iVi j(〈Szj〉+Szj−〈Szj〉)]. (6.4)The fundamental assumption of the CEF approximation is that, because spins on differentsites are correlated, Szj−〈Szj〉= Ai j(Szi −〈Szi 〉). Making use of this approximation, we find theeffective field acting at each site to beHzi =∑j 6=iVi j(〈Szj〉+Ai j[Szi −〈Szi 〉]). (6.5)In a spin half system, where (Szi )2 = 1, we may drop the term involving Szi from equation (6.5)110because it simply contributes a constant shift to the ground state energy of the Hamiltonian.The effective field may be divided into its static and fluctuating parts by taking 〈Szi 〉= 〈Sz〉0+φi,where 〈Sz〉0 is determined from the static field HamiltonianH0 = Sz(V0−α)〈Sz〉0, (6.6)withα =∑jVi jAi j (6.7)being an unknown parameter representing the correlations with neighbouring spins, or, equiv-alently, a constraint on the total number of spins in the system. To leading order in the fluctua-tions, consideration of the RPA susceptibility leads to the condition1N∑kVk1− (Vk−α)χzz0= α. (6.8)Equations (6.6) and (6.8) may be used to solve self consistently for α and 〈Sz〉0. These equa-tions are identical to the RPA equations of the Ising model treated in standard MF theory aug-mented with a constraint on the total number of spins. This correspondence is noted by Linesin [107]. Lines also notes the equivalence of the CEF approximation and the Onsager reactionfield approach. As noted in [105], the CEF approximation may be viewed as a renormalizationof the single site susceptibility. That is, we define the renormalized single ion susceptibility tobeχzzr =χzz01+χzz0 V0α, (6.9)and the RPA susceptibility then follows from the usual equationχzzk =χzzr1−χzzr Vk . (6.10)Thus far, we have only considered static properties of spin systems. Spin dynamics withinthe CEF approximation are discussed in a 1975 paper of Lines [109]. Corrections to thedynamic susceptibility are obtained by replacing χzz0 with its dynamic counterpart, that is,χzz0 → χzz0 (ω) = −g(ω). The dynamic susceptibility in the correlated field approximation is111simply given by (in the Matsubara formalism)χzzk (iωn) =−Gzzk (iωn) =−g(iωn)1+g(iωn)(Vk−α) . (6.11)Correlations between spins lead to a static renormalization of the interaction between spins inthe RPA expression for the susceptibility.We have briefly outlined the CEF approximation, or, equivalently, an improvement of stan-dard MF/RPA theory via the introduction of a constraint on the number of spins in the system.We will now discuss its application to the dipolar ferromagnetic insulator LiHoF4, and the im-provements offered by the techniques developed in this chapter. The CEF approximation hasbeen used to study the static transverse susceptibility of LiHoF4, in the absence of a transversemagnetic field, by Page et al. in a 1984 paper [110]. Page et al. find that CEF theory offers asignificant improvement over standard MF theory in their calculation of the transverse suscep-tibility near the system’s critical temperature; however, the correlation parameter goes to zeroin the zero temperature limit, thus reducing the theory to the standard RPA. It would be inter-esting to see the CEF approximation applied to the LiHoF4 system in a transverse field at zerotemperature where the fluctuations are more significant. The main advantage of the high den-sity approximation is that it allows, in principle, for systematic improvements in the calculationby carrying the perturbation theory calculation to higher orders. Admittedly, this comes at thecost of a considerable amount of straightforward but tedious algebra. The CEF approximationallows for the effect of interactions between fluctuations to be accounted for to leading orderin a simpler manner, particularly at finite temperatures. Another advantage of the formalismdeveloped in this chapter is that it provides a rigorous method for obtaining an effective fieldtheory from the underlying Hamiltonian, thus allowing for a renormalization group treatmentof a system’s critical behaviour. We mention, for the sake of completeness, that Stinchcombe’swork on the high density approximation in spin half transverse field Ising systems has beenapplied to LiHoF4 in a paper of Page et al., separate from their CEF work [111]; however,in this paper, the approximations used in the calculations are very rough, meaning that thisapplication of the high density approximation should not be taken seriously.The CEF approximation does not take into consideration any time dependence of the cor-relation parameter α . An approximation, similar to the CEF theory, that makes use of the highdensity approximation of interest in this chapter, has been developed by Jensen [68, 112]. InJensen’s formalism, α becomes a time dependent parameter. In accord with the work of Jensen,our discussion will be given in terms of the imaginary time Green’s function Gzzk (iωn) discussedin Appendix D. Jensen introduces an effective medium coupling K(iωn) that renormalizes the112interaction between spins. The effective medium coupling is defined byK(iωn) =1N∑kVkGk(iωn)G(iωn)=1N∑kVk1+[Vk−K(iωn)]G(iωn), (6.12)whereG(iωn) =1N∑kGk(iωn) (6.13)is the single site Green’s function of the interacting theory. Equation (6.12) is a dynamicversion of equation (6.8) of the CEF approximation, with the single ion Green’s functionχzz0 = −g(0) in the MF approximation being replaced with the exact single site Green’s func-tion of the interacting theory given by equation (6.13). Note that G(iωn) is exact, and containsall corrections to the MF Green’s function g(iωn). Jensen proceeds by expanding the singlesight Green’s function in powers of the inverse coordination number (the high density approxi-mation), and solving the resulting equations self consistently. Jensen refers to the high densityapproximation without introducing the effective medium as the unconditional cumulant expan-sion. In both the unconditional cumulant expansion, and the effective medium theory of Jensen[68, 112], terms involving the fourth order cumulant contribute corrections of order 1z , termsinvolving the sixth order cumulant contribute corrections of order 1z2 , and so on. In Section 2of his 1994 paper on HoF3 [112], Jensen states: ”The unconditional cumulant expansion ac-counts correctly for the fourth-order cumulant term in the [single site Green’s function], but ananalysis of the sixth and higher-order terms shows that this procedure does not lead to a goodestimate of the higher order contributions in the single-site series.” We take this to mean that toorder 1z the considerably more complicated effective medium theory does not offer any advan-tage over the formalism developed in this chapter. For further discussion of Jensen’s effectivemedium theory, and its comparison with alternative formalisms, see Section 7.2 of the book byJensen and Mackintosh [65].The effective medium theory, to order 1z , has been applied to LiHoF4 by Jensen in the 2007paper of Rønnow et al. [67]. Rønnow et al. use a set of crystal field parameters, and choose avalue of exchange interaction that provide a good fit to the low temperature phase diagram. Weuse the same set of crystal field parameters throughout this thesis. Rønnow et al. proceed tocalculate the dominant mode in the low energy excitation spectrum at T = 0.31K and show that,after scaling the calculated energies by a factor of 1.15, it agrees with the spectrum measuredvia neutron scattering experiments. They note this crystal field mode is gapped by the presenceof the hyperfine interaction, with critical fluctuations deriving from a lower energy pole of theMF Hamiltonian, although this is not shown in their work.113In Chapter 5, we have provided an analysis of all the RPA modes of LiHoF4, and theirassociated spectral weights, which is not currently present in the literature. The formalismdeveloped in the current chapter is suitable for obtaining corrections to the RPA results, and, atorder 1z , it should be equivalent to the more complicated effective medium theory. In Chapter7, we use the formalism developed in the current chapter to calculate the zero temperaturemagnetization of LiHoF4, this being the simplest application of the theory. A calculation of thespectrum and the phase diagram, and its comparison to the effective medium theory, would beof interest.We begin the current chapter by using a heuristic model to illustrate the effect of exchangeanisotropy in a Heisenberg magnet in Section 6.1. We clearly show how the Goldstone modesgain mass when anisotropy breaks the rotational invariance of the system. Then, in Section6.2, we discuss first order phase transitions using a Landau energy function. This is relevantsince, as will be shown in Section 6.3, additional degrees of freedom beyond spin half may, inprinciple, lead to a cubic term in the energy function, resulting in a first order phase transition.Finally, in Section 6.3, we treat a quantum Ising system with an arbitrary single ion Hamilto-nian more rigorously using the HS transformation, deriving a diagrammatic perturbation theorysuitable for incorporating the effects of fluctuations.Field Theory: From Heisenberg to Ising SystemsConsider the classical Heisenberg model with uniaxial anisotropy in a magnetic field transverseto the easy axis of the systemH =− 12∑r,r′Jz(r− r′)Sz(r)Sz(r′)− 12∑r,r′J⊥(r− r′)(Sx(r)Sx(r′)+Sy(r)Sy(r′)) (6.14)−Bx∑rSx(r)+µ∑r(~S(r)2−1)2.We consider a continuous spin variable ~S, with the final term enforcing a constraint on the sizeof the spin, that is, it ensures that the maximum magnetization of the system is bounded. Taylorexpanding ~S(r′) around the point r, and an integration by parts, leads to a soft spin modelβH =∫d3~r[12rz(∇Sz)2+12rz0S2z +12r⊥((∇Sx)2+(∇Sy)2)+12r⊥0 (S2x+S2y)+14u0S4−∆Sx](6.15)where S4 = ((~S)2)2. Here the size of the spin, or the maximum magnetization, is unbounded,but there is an energy penalty for unphysically large values of the magnetization. Our goal114starting with equation (6.14) is to motivate the origin of the soft spin model. In what follows,we ignore the physical parameters in (6.14), and work with the parameters in the soft spinmodel instead. We may assume, with out loss of generality, that rz = 1. This corresponds to are-scaling of the spins (Sz→ 1√rzSz). This model is meant to illustrate the effects of anisotropicexchange in a Heisenberg ferromagnet. In particular, we will consider the limit J⊥(r− r′)→ 0,in which case we have an Ising system. It is a simple matter to include nuclear spins in such atheory; however, the resulting equations become quite cumbersome, and are not of interest tous here. The effect of the hyperfine interaction will be included when we treat quantum Isingsystems in Section 6.3.Minimizing the homogenous part of the Hamiltonian density with respect to the spins,βh=12rz0S2z +12r⊥0 (S2x+S2y)+14u0S4−∆Sx, (6.16)gives the MF magnetization, 〈~S〉MF = (mx,my,mz). We assume the system is anisotropic (rz0 6=r⊥0 ) and take u0 > 0. The two relevant parameter regimes are the paramagnetic regime, whererz0,r⊥0 > 0, and the ferromagnetic regime, where rz0,r⊥0 < 0. We ignore all other parameterregimes as they are not physically relevant to the systems of interest in this thesis. In theparamagnetic regime, mz = my = 0, and the transverse magnetization satisfies ∆ = mx(r⊥0 +u0m2x). In the ferromagnetic regime, my = 0, mx =∆r⊥0 −rz0, and the total magnetization is givenby m2 = −rz0u0.Paramagnetic RegimeWe expand our spins around the MF, ~S = ~m+~φ , where ~φ = (φx,φy,φz) are the fluctuations,and ~m = (mx,0,0) is the MF magnetization. This leads to an effective Hamiltonian for thefluctuations in the magnetization,βH = βHMF +∫d3~r[12(r⊥0 +3u0m2x)φ2x +12(r⊥0 +u0m2x)φ2y +12(rz0+u0m2x)φ2z (6.17)+r⊥2((∇φx)2+(∇φy)2)+12(∇φz)2+u0mxφx(φ2x +φ2y +φ2z )+u04(~φ2)2]where constant terms are included in HMF . Note that terms linear in the fluctuations vanish, asis to be expected as we’re looking at fluctuations around the MF (expanding about a minimum).The presence of the cubic term is a reflection of the fact that the transverse field breaks therotational symmetry in the x direction. The Hamiltonian remains symmetric under φz→−φzand φy → −φy, hence any non zero minimum in these fields will be degenerate. The MF115Hamiltonian βHMF(∆) =∫d3~r[ r⊥02 m2x +u04 m4x − ∆mx] doesn’t contain fluctuations; however,it is still a function of temperature and the transverse field, and is important for calculatingthermodynamic quantities.The effect of the transverse field is apparent in equation (6.17). We see that as the transversefield is increased, and hence the transverse magnetization, the energy cost of the fluctuationsincreases. Assuming we are in the Ising universality class, with |rz0|> |r⊥0 |, in the absence of thetransverse field, the system would go critical (in the Gaussian approximation) at rz0(T0c ) = 0.The transverse field suppresses the critical temperature. The system now goes critical when|rz0(T∆c )|= u0m2, with T∆c < T 0c .In Fourier space, the Gaussian component of the function is given byβH0 =121V ∑k[(r⊥0 +3u0m2x+ r⊥k2)φ2x +(r⊥0 +u0m2x+ r⊥k2)φ2y +(rz0+u0m2x+ k2)φ2z].(6.18)We identify the coefficients of the fields with the inverse of the connected two point correlationfunctions Sµν(k) ≈ 1β χµν(k), χ being the susceptibility (see Appendix D for a discussion ofthe susceptibilities and correlation functions). We findSxx(k) =1r⊥0 +3u0m2x+ r⊥k2(6.19)Syy(k) =1r⊥0 +u0m2x+ r⊥k2Szz(k) =1rz0+u0m2x+ k2.Alternatively, the transverse susceptibility,χxx(k = 0) =∂mx∂Bx=β(r⊥0 +3u0m2x), (6.20)is easily obtained by differentiating the equation for the magnetization, ∆ = βBx = mx(r⊥0 +u0m2x). Note that when the longitudinal susceptibility goes critical at rz0 =−u0m2 the transversesusceptibilities remain finite.116Ferromagnetic RegimeWe now expand our spin about the MF magnetization in the ferromagnetic regime,~S= ~m+m~φ ,with ~m= (mx,0,mz), and m= |~m|. We findβH = βHMF +∫d3~r[2u0mxmzm2φxφz+m22(u0m2x+3u0m2z + rz0)φ2z +m22(∇φ z)2 (6.21)+m22(u0m2z +3u0m2x+ r⊥0)φ2x +m22r⊥(∇φ x)2+m22(r⊥0 − rz0)φ2y +m22r⊥(∇φ y)2+u0mzm3φz(φ2x +φ2y )+u0mxm3φx(φ2z +φ2y )+u0mxm3φ3x +u0mzm3φ3z +u04m4(~φ2)2]where HMF contains constant (homogenous in space) terms. Again, there are no linear termsas we are looking at fluctuations around the ground state. The inclusion of a factor of m inour definition of the fluctuation means that the ~φ field represents only the direction of thefluctuation. We see that the symmetry of the system is broken in both the x and z directions;however, the Hamiltonian remains invariant under the transformation φy→−φy.In the Gaussian approximation, transforming to momentum space and dropping the con-stant MF component, the Hamiltonian is given byβH0 =m221V ∑k[(2u0m2z + k2)φ2z +(r⊥0 − rz0+2u0m2x+ r⊥k2)φ2x (6.22)+(r⊥0 − rz0+ r⊥k2)φ2y +4u0mxmzφxφz].In zero transverse field this reduces toβH0 =m221V ∑k[(2|rz0|+ k2)φ2z +(r⊥0 − rz0+ r⊥k2)φ2⊥](6.23)where φ2⊥ = φ2x +φ2y . The two point correlation functions are,Szz(k) =m−22|rz0|+ k2S⊥(k) =m−2r⊥0 − rz0+ r⊥k2. (6.24)In the absence of anisotropy, the transverse correlation functions would have poles at k = 0,corresponding to the Goldstone modes of the system. The effect of anisotropy is to give massto what were the Goldstone modes.117First Order Phase Transitions in an Ising SystemIn the Landau theory of phase transitions, the standard approach is to write down an energyfunction for the order parameter consistent with the symmetries of the system. In an Isingsystem, this is typically taken to mean that the cubic term in the theory should be neglected.However, a cubic term may be included by defining the function piecewiseβH =12 ∑k(r0+ k2)S2+g03 S3+ u04 S4 S≤ 012 ∑k(r0+ k2)S2− g03 S3+ u04 S4 S> 0. (6.25)A linear term is excluded because for r0 > 0 we want the function to have a minimum at S= 0.Equation (6.25) is a valid function, and is symmetric under the transformation S→−S despitethe cubic term. In Section 6.3.3 we will see that in a quantum system with more than two energylevels at each site, a cubic term may be present. In the formalism developed in Section 6.3,there will be either a plus or a minus sign associated with the cubic term that arises from a signambiguity in the Hubbard-Stratonovich transformation. If the cubic term is present, choosingto work with one sign or the other amounts to specifying in which direction the system willorder. In what follows, we will choose the minus sign.We consider a scalar field theory of the formβH =12∑k(r0+ k2)S2− g03 S3+u04S4. (6.26)Minimizing the homogenous (k = 0) part of the Hamiltonian yields the MF equation,m(r0−g0m+u0m2) = 0, (6.27)where m= 〈S〉MF .Consider r0 > 0. If g20 < 4u0r0, the potential has a single minimum at m = 0. As r0 isreduced, the potential develops an inflection point at m = g02u0 when g20 = 4u0r0. Reducing r0further causes a secondary minimum at m = g0+√g20−4u0r02u0to be lowered and shifted to theright (assuming g0 > 0). As r0 is reduced, the secondary minimum will become degeneratewith the minimum at m= 0, at which point the system undergoes a first order phase transition.When r0 = 0, m = 0 becomes an inflection point and m =g0u0becomes the unique global min-imum. The point where the two minima are degenerate occurs with m ∈ ( g02u0 ,g0u0); hence, thediscontinuity in the magnetization can be at most ∆m|max = g0u0 . When r0 < 0, m= 0 becomesa maximum, and there will be minima to either side at m= g0±√g20−4r0u02u0.118If we expand about the MF, S=m+φ , we may write the homogenous part of the Hamilto-nian asβH = βHMF +∫d3~r[12(r0+2g0m+3u0m2)φ2− 13(g0−3u0m)φ3+u04φ4]. (6.28)In the following section, we derive an expression analogous to the one above beginning witha microscopic model for a quantum Ising system. Unlike equation (6.28), the resulting theoryfully incorporates the spatial dependence and time dependence of the interactions between thefluctuations.Quantum Ising SystemsConsider a quantum spin system of the form H = H0 +H′, where H0 is the MF Hamiltonian,and may include any number of single ion terms such as a transverse field, single ion anisotropy,or hyperfine interactions. For example H0 could be the MF Hamiltonian of LiHoF4, H0 =∑iH iMF , withH iMF =−∆2Sxi −V0〈Sz〉0Szi + ~∆n ·~Ii+AzSzi Izi (6.29)+A⊥S+i I−i +A†⊥S−i I+i +A++S+i I+i +A†++S−i I−i .We take H′to be a longitudinal coupling between spin fluctuations,H′=−12∑i 6= jVi jS˜zi S˜zj, (6.30)where S˜z = Sz−〈Sz〉0. The subscript 0 indicates the average is to be taken with respect to theMF Hamiltonian, and 〈Sz〉0 is determined self consistently. In Fourier space, the interactionbecomesH′=−12∑kVkS˜zkS˜z−k =−12∑kVRk S˜zkS˜z−k, (6.31)where, in the final expression, we note that in, for example, an exchange coupled crystal thatlacks inversion symmetry, Vk = VRk + iVIk may have an imaginary component; however, sinceVk = V ∗−k the imaginary component vanishes upon summation. In what follows, we simplywrite Vk and it is to be understood we are talking about the real component. We assume theinteraction is ferromagnetic.In this section, we apply the Hubbard-Stratonovich transformation to H ′ in order to derive119an effective field theory for a quantum Ising system. We then proceed to use the field theory todevelop a diagrammatic perturbation theory for systematically including the effects of fluctua-tions in quantum Ising systems. We show that the diagrammatic form of the theory is equivalentto that of Brout [60], which was introduced in 1959, and applied to the spin half transverse fieldIsing model in a series of 1973 papers by Stinchcombe [26, 62, 63]. The field theoretic for-malism presented here generalizes Stinchcombe’s results to systems having an arbitrary singleion Hamiltonian, as well as offering a great deal of simplicity and clarity when compared toprevious work. Furthermore, the effective field theory is suitable for a renormalization grouptreatment of a system’s critical behaviour.Partition FunctionStatistical mechanics requires knowledge of the partition function,Z = Z0〈Tτ exp[− 1β∫ β0dτβH′(τ)]〉0. (6.32)We now apply the Hubbard-Stratonovich (HS) transformation,ex22 =∫ ∞−∞dy√2pie−y22 ±xy, (6.33)to the exponential, following the procedure in [58, 95]. Here we generalize Young’s work [58]to systems with more than two degrees of freedom, and we carefully work out coefficients of theresultant field theory to all orders. Note that there is a sign ambiguity in the HS transformation.As discussed in Section 6.2, the choice of sign may determine in what direction the field willorder in the resulting theory. In what follows, we choose the minus sign. We let xi = (BJ)i =∑ j 6=iBi jS˜zj be components of a vector, and note that12~x2 = 12 ∑i6= j βVi jS˜zi S˜zj, with (B2)i j = βVi j.We apply the HS transform to each component of the vector~x, at each imaginary time step τ .The partition function is thenZ = Z0∫DQ√2piexp(− 12β∫ β0dτ∑iQ2i (τ))×〈Tτ exp(− 1β∫ β0dτ∑i 6= jQi(τ)Bi jS˜zj)〉0.(6.34)What we have done here is replace the interactions between spins with interactions of asingle spin with a site dependent Gaussian random field. This is an analytic procedure forobtaining a functional integral representation of the theory. We will now proceed to integrateout the microscopic degrees of freedom from our Hamiltonian by completing the trace over thesingle ion Hamiltonian, thus obtaining an effective field theory from our microscopic quantum120model. Keep in mind that the single ion Hamiltonian may contain the hyperfine interaction. Byperforming this trace, the nuclear degrees of freedom are incorporated into the coefficients ofthe resulting field theory. We have arranged the factors of β to make our field dimensionless.The spatial Fourier transforms of the exchange interaction and the fields are given byBi j =1N∑kBke−ik(ri−r j) Bk =1N∑i jBi jeik(ri−r j) =∑jBi jeik(ri−r j) (6.35)Qi =1√N∑kQke−ikri Qk =1√N∑iQieikri.After Fourier transforming, we find the partition function to beZZ0=∫DQ√2piexp(− 12β∫ β0dτ∑k|Qk(τ)|2)×〈Tτ exp(− 1β∫ β0dτV̂ (τ))〉0, (6.36)whereV̂ (τ) =∑kQ−k(τ)BkS˜zk(τ), (6.37)where we integrate over the real and imaginary parts of the HS field separately. However,noting Qk = Q∗−k, we see that this double counts each degree of freedom. This may be dealtwith by restricting momentum space summations to half the Brillouin zone, or introducingfactors of one half.It is advantageous at this point to establish a relationship between the HS field, and theimaginary time connected longitudinal correlation function, Gk(τ− τ ′) =−〈Tτ S˜zk(τ)S˜z−k(τ ′)〉,as was done for the Hubbard model in [113]. We add a fictitious site and time dependentmagnetic field, H ′′ = ∑i hi(τ)S˜zi to our Hamiltonian so that our interaction becomesV̂ (τ) =∑k[h−k(τ)+Q−k(τ)Bk]S˜zk(τ). (6.38)A factor of β has been absorbed into the applied field in order to make it dimensionless. Now,by shifting the field, Q−k(τ) = Q′−k(τ)− h−k(τ)Bk , we may transfer all the dependence of thepartition function on the applied field to the Gaussian prefactor,ZZ0=∫DQ√2piexp(− 12∫ 10dτ∑k∣∣∣∣Qk(τ)− hk(τ)B−k∣∣∣∣2) (6.39)×〈Tτ exp(−∫ 10dτ∑kQ−k(τ)BkS˜zk(τ)〉0,where we take τ→ τβ in order to make the integrals dimensionless. The correlation function is121then given byGk(τ− τ ′) =− δ lnZδh−k(τ)δhk(τ ′)∣∣∣∣h=0. (6.40)By performing the derivatives we findGk(τ− τ ′) =− 1βVk[〈Qk(τ)Q−k(τ ′)〉Q−1], (6.41)where the subscript Q on the average is a reminder the average is now being taken using thepartition function of the HS field given in equation (6.36). In Fourier space, this becomesGk(iωn) = β∫ 10dτeiωnτGk(τ) =− 1Vk[〈|Qk(iωn)|2〉Q−1]. (6.42)This establishes a general result between the spin correlation function and the field. Typically,the HS field is governed by a much simpler equation of motion than the original spins, soworking with the field is advantageous. Also, note that〈S˜zk(τ)〉= 〈Szk(τ)〉−〈Szk(τ)〉0 =1√βVk〈Qk(τ)〉Q, (6.43)where, as before we take τ → τβ so that our dimensionless τ ∈ [0,1].We now return to the partition function given in equation (6.36), and perform a cumulantexpansion leading to,ZZ0=∫DQ√2piexp(− 12β∫ β0dτ∑k|Qk(τ)|2)(6.44)×Tτ[exp( ∞∑n=1(−1)nn!β nn∏i=1∫ β0dτi〈Mn(V̂ (τ))〉0)],where Mn is the nth cumulant of the interaction.Consider the second order cumulant〈M2(V (τ))〉0 = Tτ[∑k∑k′Q−k(τ1)Q−k′(τ2)BkBk′〈M2(S˜zk(τ1)S˜zk′(τ2))〉0]. (6.45)122Now,Tτ〈M2(S˜zk(τ1)S˜zk′(τ2))〉0= Tτ1N∑i je−ikrie−ik′r j〈M2(S˜zi (τ1)S˜zj(τ2))〉0(6.46)= Tτ〈M2(Sz(τ1)Sz(τ2))〉0δk,−k′.The average in the second expression is with respect to a single site Hamiltonian; hence, all theterms in the sum vanish except those with i = j. We are left with the expression on the right,in which the site index has been dropped as we are always dealing with averages of spins at asingle site. Note that we have used the fact that〈M2(S˜z(τ1)S˜z(τ2))〉0=〈M2(Sz(τ1)Sz(τ2))〉0At third order a quick calculation shows thatTτ〈M3(S˜zk1(τ1)S˜zk2(τ2)S˜zk3(τ3))〉0= Tτ1√N〈M3(Sz(τ1)Sz(τ2)Sz(τ3))〉0δk1+k2+k3,0. (6.47)We wish to establish the general resultTτ〈Mn(S˜zk1(τ1)S˜zk2(τ2) . . . S˜zkn(τn))〉0= Tτ1Nn−22〈Mn(Sz(τ1)Sz(τ2) . . .Sz(τn))〉0δ∑ni=1 ki,0.(6.48)We begin by writing the cumulant asTτ〈Mn(S˜zk1(τ1)S˜zk2(τ2) . . . S˜zkn(τn))〉0=TτNn2[∏n∑ine−ikrin]〈Mn(S˜zi1(τ1)S˜zi2(τ2) . . . S˜zin(τn))〉0(6.49)= Tτ1Nn−22〈Mn(S˜z(τ1)S˜z(τ2) . . . S˜z(τn))〉0δ∑ni=1 ki,0.The final line follows from the fact that spins belonging to different lattice sites are independentbecause the average is taken with respect to the single ion (MF) Hamiltonian; hence, cumulantsof spins belonging to different lattice sites are equal to zero, as discussed in Section C.1 ofAppendix C.We are left with the task of establishing Mn(∏ni=1 S˜z(τi)) =Mn(∏ni=1 Sz(τi)). This followsalmost directly from the definition of the cumulants; here we give a brief proof by induction.123We assume the result is true for Mn, and then considerMn+1(n+1∏i=1S˜zi)=〈n+1∏i=1S˜zi〉0− ∑n1+...+ni=n+1ni 6=1Mn1 . . .Mni, (6.50)where the lower order cumulants all satisify our desired result by the induction hypothesis, andwe exclude any terms containing M1(S˜z) as this term equals zero. The first term on the righthand side of the equation is the (n+1)st central moment of the spins, which may be expressedas the (n+1)st cumulant plus terms that cancel with the lower order cumulants on the far righthand side of the equation. This proves our result. The τ ordered averages of the spin cumulantsare dealt with in Appendix E.In frequency space we haveMn({ωri}) =n∏i=1[∫ β0dτieiωriτi]Mn({τi}) Mn({τi}) = 1β n ∑{ri}n∏i=1[e−iωriτi]Mn({ωri})(6.51)Q(iωr) =1β∫ β0eiωrτQ(τ) Q(τ) =∑re−iωrτQ(iωr),which givesn∏i=1∫ β0dτi〈Mn(V (τ))〉0 = ∑{ki}∑{ri}n∏i=1[Qki({iωri})B−ki]Mn({−iωri})δk1+...+kn,0 (6.52)We may now write the partition function asZZ0=∫DQ√2pie−H[Q] (6.53)where the Hamiltonian is understood to be dimensionless. The integration is over all complexfields Qk(iωr); however, Qk(iωri) =Q−k(−iωri)∗. Recall, this double counts all the degrees offreedom, and we must restrict our k summation, or introduce factors of one half. The Hamilto-nian is given byH[Q] =∞∑n=1[∑{ri}∑{ki}un({ki},{iωri})n!n∏i=1Qki(iωri)], (6.54)124withu22!=12[δiωr1 ,−iωr2 −1β(Vk1Vk2)12M2(−iωr1,−iωr2)]δk1,−k2, (6.55)where we have used Bk =√βVk. Note that M2, and the higher order spin cumulants, containadditional zero frequency contributions that vanish in the zero temperature limit, and in theparamagnetic phase of the system. The higher order coefficients are given byun =(−1)n+1Nn2−1[ n∏i=1(βV−ki)12]1β nMn({−iωri})δ∑ki,0. (6.56)Gaussian ApproximationWe now analyze our quantum Ising system in the Gaussian approximation. We begin by per-forming a momentum and frequency summation on u2, given in equation (6.55), to get∑r2,k2u2 = [1+Vkg(iωr)], (6.57)where we have used∑r2M2(−iωr1,−iωr2) =−βg(iωr1), (6.58)and the MF Green’s function,g(iωr) =− ∑n>mc2mnDmn2EnmE2nm− (iωr)2−β(∑mc2mmDm−[∑mcmmDm]2)δωr1 ,0, (6.59)is derived in Appendix E. The MF Green’s function contains poles at the differences betweeneach of the systems MF eigenstates, Enm = En−Em, as well as an additional zero frequencycontribution that vanishes in the paramagnetic phase, and in the limit T → 0. The cmn are theMF matrix elements of the Sz operator, and the population factors are Dmn = Dm−Dn, whereDm = Z−10 e−βEm .The partition function in the Gaussian approximation is given byZZ0=∫DQ√2piexp(− 12∑r,k[1+Vkg(iωr)]|Qk(iωr)|2)=∏r,k11+Vkg(iωr), (6.60)and the corresponding (Gibb’s) free energy, F =U−TS−HM, dF =−SdT −MdH, where H125is the externally applied field, isF = F0+1β ∑r,k12ln [1+Vkg(iωr)]. (6.61)The path integral double counts all the degrees of freedom, hence, there is an additional factorof 12 in the summations. The momentum space summations are over the entire Brillouin zone.Differentiating the free energy with respect to Vk we obtain∂F∂Vk=−〈τ˜kτ˜−k〉= 1β ∑rg(iωr)1+Vkg(iωr). (6.62)We identify the term after the summation with the imaginary time connected Green’s functionin the Gaussian approximation. Alternatively, taking the derivative before performing the pathintegral yields−〈τ˜kτ˜−k〉= 1β ∑rg(iωr)〈|Qk(iωr)|2〉Q, (6.63)where the average is now taken with respect to the partition functionZ0Q =∫DQ√2piexp(− 12∑r,kD−1k (iωr)|Qk(iωr)|2), (6.64)and the free field propagator is defined to beDk(iωr) =11+Vkg(iωr). (6.65)We find that〈|Qk(iωr)|2〉Q=∫D |Q|exp(−∑r,k>0D−1k (iωr)|Qk(iωr)|2)|Qk(iωr)|3∫D |Q|exp(−∑r,k>0D−1k (iωr)|Qk(iωr)|2)|Qk(iωr)|=Dk(iωr),(6.66)where the integral has been performed in polar coordinates, and the momentum summationshave been restricted to k > 0 to avoid factors of two. From this, we can relate the imaginarytime connected two point correlation function to the auxiliary field Q; we find, at the Gaussian126level,Gk(iωr) = g(iωr)〈|Qk(iωr)|2〉Q=g(iωr)1+Vkg(iωr). (6.67)This result can also be derived directly from equation (6.42).Higher order correlations between the HS fields may be generated fromZhQ =∫DQ√2piexp(− 12∑r,kD−1k (iωr)|Qk(iωr)|2+12∑r,k[h∗k(iωr)Qk(iωr)+hk(iωr)Q∗k(iωr)]).(6.68)Integrating, we find ZhQ = Z0QeW (h), whereW (h) =12∑k,rhk(iωr)Dk(iωr)h∗k(iωr). (6.69)The connected field correlation functions are then given by〈∏iQki(iωri)〉=δδhiW (h)∣∣∣∣h=0. (6.70)In the low temperature limit, the additional zero frequency contribution to g(iωr) vanishes,and we findDk(iωr)∣∣∣∣T=0= Dk(iωr) =∏n>1[E2n1− (iωr)2]∏n>1[E2n1− (iωr)2]−Vk∑n>1 |c1n|22En1∏p6=n,1[E2p1− (iωr)2](6.71)=∏n>1[E2n1− (iωr)2]∏p[(Epk )2− (iωr)2] ,where, in the final expression, we have factored the numerator into the RPA modes of thesystem. The spectral weight of each RPA mode follows fromAk(ω) =− 1pi Im[Dk(iωr→ ω+ i0+)] =∑pApk[δ (ω−E pk )−δ (ω+E pk )], (6.72)127whereApk =∏n>1[(E pk )2−E2n1]2E pk ∏s 6=p[(E pk )2− (Esk)2] . (6.73)The free energy is given bylimT→0F = F0− 12β ∑r,k∑p[ln [E2p1− (iωr)2]− ln [(E pk )2− (iωr)2]]. (6.74)Terms not included in the expression above decay exponentially with temperature. We mayperform the frequency summation by making use of the following trick−1β∂∂a∑rln(a2− z2r ) =−1β ∑r[1zr+a− 1zr−a]= nB(−a)−nB(a) =−coth(βa2), (6.75)where nB is the Bose-Einstein distribution function. Integrating the result with respect to ayields−1β ∑rln(a2− z2r ) =−2βln[sinh(βa2)]+C. (6.76)Subsequent analysis will show that the integration constant is zero. Applying this to the freeenergy we findlimT→0F = F0+1β ∑k ∑p[ln[sinh(βE pk2)]− ln[sinh(βEp12)]]. (6.77)Note that in the limitVk→ 0 we have F→F0 and E pk →Ep1. All the terms under the summationin the above equation vanish, giving the correct free energy. This justifies setting the integrationconstant discussed above to zero.The thermodynamics of the transverse field Ising model has been dealt with in some detailby Stinchcombe in [63]. Here we review some of the basics using the field theoretic formalism.The energy levels of the transverse field Ising model in the MF approximation are given byE± = ±E = ±12√∆2+H2, where ∆ is the applied transverse field, and the longitudinal fieldH = h+V0〈Sz〉0 includes an applied component h, and a contribution from the MF of the rest128of the crystal. The MF magnetization is defined implicity by〈Sz〉0 = tanh(βE) H2√∆2+H2(6.78)The lowest order term in the free energy per spin is thenf0 =F0N= Eg− 1β ln [2cosh(βE)], (6.79)where the ground state energy per spin is Eg = V02 〈Sz〉20. From this we find the MF magnetiza-tions to be−∂ f0∂h= 〈Sz〉0 −∂ f0∂∆ = 〈Sx〉0 = ∆H 〈Sz〉0. (6.80)In the paramagnetic phase of the system, the entropy in a fixed transverse field is given bys0 =−∂ f0∂T = ln [2cosh(βE)]−βE tanh(βE). (6.81)For the transverse field Ising model we have E21 = 2E, and the RPA spectrum is given byE2k = E221− 2|Vk|∆2E21. In the zero temperature limit we findF1∣∣T=0 =12∑k[Ek−E21]. (6.82)This term yields corrections due to quantum fluctuations around the MF ground state.The Interacting Field TheoryWe now systematically include corrections to the Gaussian results for a quantum Ising systemdue to the interactions between the fluctuating fields. We begin with a brief look at the cubicterm in the theory in Section 6.3.3. We find that this cubic term may be non-zero in the para-magnetic phase of a quantum Ising system with more than two degrees of freedom. If such isthe case, the system will undergo a first order phase transition. Unfortunately, in all systemsstudied by the author to date, the cubic term vanishes. After our look at the cubic term, weoutline how to derive corrections to thermodynamic qua due to all powers of the HS field, andcalculate the leading order corrections to the Green’s function in Section 6.3.3.129Cubic TermAt third order in our Hubbard-Stratonovich field we find the coefficient to beu(3) =1√Nβ32V12−k1V12−k2V12−k31β 3M3({ωri})δk1+k2+k3,0. (6.83)We restrict our attention to the low temperature limit of a system in its paramagnetic phase(above some quantum critical point), in which caselimT→0M3({ωri}) = 2β ∑p>n6=1Re[c1ncnpcp1] ∑P{ωri}Ep1En1− (iωr1)(iωr2)(E2p1− (iωr1)2)(E2n1− (iωr2)2)δωr1+ωr2+ωr3 ,0.(6.84)In a two level system, no such term exists because it involves matrix elements ci j betweenat least three MF energy levels. The leading order correction to the Gaussian results will bequartic in the Hubbard-Stratonovich field. With three or more levels, the cubic term may benon-zero in the paramagnetic phase of the system. This indicates the phase transition will befirst order. This is a surprising result - additional degrees of freedom, beyond spin one half,may lead to a first order phase transition in a quantum Ising system. This leads to the question:In what systems, if any, will this term be significant?First of all, we note that if the single ion Hamiltonian is diagonal in the Sz basis, there willbe no off-diagonal terms in the matrix elements of the Sz operator because ci6= j = 0. So, forexample, in a spin one Ising system with longitudinal single ion anisotropy (an (Szi )2 term)we don’t expect the physics discovered here to be relevant. Secondly, if the Sz operator onlycauses transitions between neighbouring single ion eigenstates, for example, a large spin in anapplied transverse magnetic field, the phase transition will not be first order. The cubic term ofthe effective field theory will be non-zero in the paramagnetic phase of the system when thereare non-zero matrix elements, ci j, such that c1ncnpcp1 is non-zero. It would be of interest todetermine if any such systems exist, or if they are forbidden by a general principle.130Systematic Corrections to the Gaussian ResultsThe calculation of spin correlation functions has been reduced to the calculation of correlationfunctions of an interacting field with generating functionalZhQ =∫DQ√2piexp(− 12∑r,kD−1k (iωr)|Qk(iωr)|2 (6.85)+12∑r,k[h∗k(iωr)Qk(iωr)+hk(iωr)Q∗k(iωr)]−V (Q)),where,V (Q) =1n!∞∑n=3[∑{ri}∑{ki}u(n)({ki},{iωri})n∏i=1Qki(iωri)], (6.86)contains all the higher interactions between the fields. In what follows, we will consider thethird and fourth order terms dropping all higher order interactions. For notational convenience,we define g ≡ u(3)({ki},{iωri}) and u ≡ u(4)({ki},{iωri}), suppressing all momentum andfrequency dependence. Note that these functions are symmetric under permutations of thefrequencies and momenta. Our partition functional is thenZQ =∫DQ√2piexp(− 12∑r,kD−1k (iωr)|Qk(iωr)|2 (6.87)− 13![∑{ri}∑{ki}g3∏i=1Qki(iωri)]− 14![∑{ri}∑{ki}u4∏i=1Qki(iωri)]).We begin by calculating the leading order correction to the magnetization〈Szk(τ)〉= 〈Szk(τ)〉0+1β ∑re−iωrτ1√βVk〈Qk(iωr)〉, (6.88)with 〈Qk(iωr)〉=δδh∗klnZhQ∣∣∣∣h=0. (6.89)We treat the interaction perturbatively. The leading order correction involves one power of g,131and is given by〈Szk(τ)〉1 =−1√βVk1β ∑re−iωrτ13! ∑{ri}∑{ki}g({ri}{ki})〈Qk(iωr)Qk1(iωr1)Qk2(iωr2)Qk3(iωr3)〉0.(6.90)Note that because g is symmetric in the momenta and frequencies, the coefficient for the variousWick contractions of the field will be the same. Recall that the path integral double counts eachdegree of freedom; therefore, we must restrict the momentum summations to half the Brillouinzone, or introduce a factor of one half. In the following, we introduce an additional factorof one half and leave the momentum space summations unrestricted. As we are consideringunrestricted momentum summations, we must consider, for example, contractions of k2 =−k3and k3 = −k2 separately. Hence, all combinatoric factors cancel. Contracting the fields, wefind〈Szk(τ)〉1 =−12√N∑re−iωrτDk(iωr)∑r′,k′Tk′(iωr′)1β 3M3(iωr, iωr′,−iωr′), (6.91)where we define the renormalized interaction, or T matrix, to beTk(iωr) =VkDk(iωr) =Vk1+g(iωr)Vk. (6.92)The role of the HS field is to renormalize the bare interaction between spins. Integrating overthe imaginary time we find〈Szk〉1 =∫ β0dτ〈Szk(τ)〉1 =−Dk(0)√N ∑r′,k′βTk′(iωr′)1β 3M3(0, iωr′,−iωr′). (6.93)This is the leading order correction to the magnetization of the system.We now turn to 〈|Qk(iωr)|2〉, and calculate the leading order corrections to the spin correla-tion function. Expanding the interaction, the leading order corrections will come from a singlepower of u, and from two powers of g. We write〈|Qk(iωr)|2〉= 〈|Qk(iωr)|2〉0+ 〈|Qk(iωr)|2〉u+ 〈|Qk(iωr)|2〉g2 + · · · (6.94)Wick contracting the auxiliary fields, we find the leading order correction due to the quartic132coupling to be〈|Qk(iωr)|2〉u = 12NβVkD2k (iωr)∑k′,r′βTk′(iωr′)1β 4M4(iωr,−iωr, iωr′,−iωr′), (6.95)where, as with the magnetization calculation, all combinatoric factors cancel. We have in-troduced an additional factor of one half to compensate for double counting the degrees offreedom in the path integral. Equation (6.95) is conveniently written as〈|Qk(iωr)|2〉u =− gu(iωr)Vk[1+g0(iωr)Vk]2, (6.96)withgu(iωr) =− β2N ∑k′,r′βTk′(iωr′)1β 4M4(iωr,−iωr, iωr′,−iωr′). (6.97)We find the field correlation function to approximately be given by〈|Qk(iωr)|2〉0+ 〈|Qk(iωr)|2〉u = 11+g0Vk[1− guVk1+g0Vk]≈ 11+(g0+gu)Vk(6.98)We now consider the contribution from two powers of g. There are two different diagramscorresponding to this contribution, 〈|Qk(iωr)|2〉g2 = 〈|Qk(iωr)|2〉L+ 〈|Qk(iωr)|2〉B, which weconsider separately. The first, 〈|Qk(iωr)|2〉L, comes from a loop diagram where the externalfields are each coupled to a separate vertex. The second, 〈|Qk(iωr)|2〉B, comes from a ”balloon”diagram where the external fields are coupled to the same vertex. As before, all combinatoricfactors cancel. The loop diagram is given by〈|Qk(iωr)|2〉L =− gL(k, iωr)Vk[1+g0(iωr)Vk]2, (6.99)withgL(k, iωr) =− β2N∑k1∑r1,r2β 2Tk1(iωr1)Tk−k1(iωr2) (6.100)× 1β 6M3(iωr, iωr1, iωr2)M3(−iωr,−iωr1,−iωr2).In the low temperature limit, where we need only consider overall momentum conservation,133δiωr+iωr1+iωr2 ,0, equation (6.100) reduces togL(k, iωr) =− β2N ∑k′,r′β 2Tk′(iωr′)Tk−k′(−iωr− iωr′) (6.101)× 1β 6M3(iωr, iωr′,−iωr− iωr′)M3(−iωr,−iωr′, iωr+ iωr′).As with gu, we have introduced an additional factor of one half to compensate for doublecounting the degrees of freedom in the path integral.The balloon diagram is given by〈|Qk(iωr)|2〉B =− gB(iωr)Vk[1+g0(iωr)Vk]2, (6.102)wheregB(iωr) =− β2N∑k′ ∑r1,r2β 2T0(iωr1)Tk′(iωr2)1β 6M3(iωr,−iωr, iωr1)M3(−iωr1, iωr2,−iωr2).(6.103)In the low temperature limit, where we need only consider overall momentum conservation,we havegB(iωr) = β 2V01β 3M3(iωr,−iωr,0)〈Sz〉1, (6.104)where〈Sz〉1 = 1√N〈Szk=0〉1 =−R02N ∑r′,k′βTk′(iωr′)1β 3M3(0, iωr′,−iωr′), (6.105)and R0 = D0(0) is the ratio of MF to RPA energy levels at zero wavevector as discussed inAppendix D.Combining these corrections, we find〈|Qk(iωr)|2〉 ≈ 11+(g0+gu+gL+gB)Vk , (6.106)which, using equation (6.42), leads toGk(iωr)≈[g0+gu+gL+gB1+(g0+gu+gL+gB)Vk]. (6.107)134The diagrams corresponding to the corrections gu, gL, and gB are given in Figure 6.1.Expressing the correlation function in the form given in equation (6.107) allows for easy com-parison to the work of Stinchcombe [26, 62, 63]. We find the function G ≈ g0+gu+gL+gBto be equivalent to equation (2.7) of [62]. The approach presented here is simpler than Stinch-combe’s approach, and it clearly illustrates how the fluctuations screen the bare interactionbetween spins. We have also generalized Stinchcombe’s work to systems with an arbitrarysingle ion Hamiltonian.Figure 6.1: The figure above shows the one loop diagrams that contribute to the leadingorder correction (order 1z , z being the coordination number) to the connected twopoint longitudinal correlation function of a quantum Ising system given in equation(6.107). The left most diagram is the balloon contribution gB, the center diagram isthe loop contribution gL, and the rightmost diagram corresponds to gu.We close this section with a statement of the momentum space Feynman rules for calculat-ing corrections to the Gaussian results for the Hubbard Stratonovich field to all orders.1. We associate an external leg of our diagram with momentum k and frequency iωr witheach of the Hubbard-Stratonovich fields in our correlation function. Each external legcarries a factor of√βVkDk(iωr).2. We consider vertices of all orders. A vertex of order n has a factor of order 1β nMn associ-ated with it. We join the vertices to each other, and to all the external legs, in all possibleways that leave the diagram connected. The order of each diagram is determined by thenumber of free momentum summations it contains as discussed in Appendix C.3. Each internal line has a momentum k and frequency iωr associated with it and con-tributes a factor of βVkDk(iωr). Energy and momentum conservation at each vertex isdetermined by the associated spin cumulant.4. Sum over each internal frequency, and for each internal momenta, add a summation 1N ∑k.135The Renormalized InteractionAs noted following equation (6.92), the role of the Hubbard-Stratonovich field is to renormalizethe bare interaction between spins. As the renormalized interaction (The T matrix),Tk(iωr) =VkDk(iωr) =Vk1+g(iωr)Vk(6.108)plays a central role in calculating corrections to the RPA results, we present it here in somedetail. A similar analysis is carried out by Stinchcombe in [62] for the case of a spin halfsystem. The expressions below generalize Stinchcombe’s result to systems with an arbitrarysingle ion Hamiltonian.We begin by writing the renormalized interaction asTk(iωr) =VkDk(iωr)+T 0k δiωr,0. (6.109)In terms of the MF energy levels of our system, and MF matrix elements of the longitudinalspin operator, we findDk(iωr) =∏n>m(E2nm− (iωr)2)∏n>m(E2nm− (iωr)2)−Vk∑p>q c2qpDqp2Epq∏ n>mn,m6=p,q(E2nm− (iωr)2)(6.110)=∏n>m(E2nm− (iωr)2)∏p((Epk )2− (iωr)2)andT 0k =Vk∏p(Epk )2∏n>mE2nm−βVk[∑m c2mmDm−(∑m cmmDm)2] − Vk∏n>mE2nm∏p(E pk )2 = (6.111)=βV 2k[∑m c2mmDm−(∑m cmmDm)2]∏n>mE4nm∏p(Epk )2[∏p(Epk )2−βVk[∑m c2mmDm−(∑m cmmDm)2]∏n>mE2nm]where in the final expression for Dk(iωr) we have written the denominator in terms of the RPAmodes E pk . Note that T0k vanishes in the limit T → 0 and in the paramagnetic phase of thesystem.We see that Dk(iωr) has zeros at the MF energy levels of our system. This fact proves usefulwhen performing frequency summations in the fluctuation analysis because when Dk(iωr) ismultiplied by a function with poles at the MF energy levels, we may ignore those poles.136SummaryWe introduced this chapter with a general discussion of the problem of including the effects offluctuations in quantum Ising systems, and have emphasized the simplicity and utility of thefield theoretic approach developed here. After the general discussion, in Section 6.1, we intro-duced a classical heuristic model of an anisotropic Heisenberg spin system. We showed that afield transverse to the easy axis in this model leads to a reduction in the critical temperature ofthe system, and derived expressions showing how the energy cost of fluctuations is affected byanisotropy. Finally, we showed that the effect of anisotropy is to give mass to what were theGoldstone modes. In Section 6.2, we briefly discussed how a cubic term in an effective freeenergy function for a system leads to a first order phase transition. This may be relevant toquantum Ising systems with more than a spin half degree of freedom.In Section 6.3, we developed a field theoretic formalism for treating quantum Ising systemswith an arbitrary single ion Hamiltonian. We then proceeded to analyze the resulting theoryin the Gaussian approximation in Section 6.3.2. In section 6.3.3, we discussed systematiccorrections to the Gaussian result, beginning with a look at the cubic term. We found that, inprinciple, the cubic term may be present in the paramagnetic phase of a system with more thana spin half degree of freedom at each site. If such a term exists, this would lead to a first orderphase transition. We proceeded to use the field theoretic formalism to calculate the leadingorder corrections to the magnetization, and the longitudinal Green’s function, then we statedthe Feynman rules for systematically obtaining higher order corrections.137Chapter 7Corrections to Mean Field MagnetizationIn this chapter, we calculate the leading order correction to the mean field (MF) magnetizationof quantum Ising systems. Recall, the magnetization is given by 〈Sz〉 ≈ 〈Sz〉0 + 〈Sz〉1, where〈Sz〉0 is the MF magnetization, and the leading order correction is given by equation (6.105)〈Sz〉1 =− R02N∑r,kβTk(iωr)1β 3M3(0, iωr,−iωr). (7.1)This result is the 1z term, z being the coordination number, in the high density approximationdeveloped in Chapter 6. The diagram corresponding to equation (7.1) is shown in Figure 7.1.We may compare this result to equation (2.10) of Stinchcombe [62]. The result of Stinchcombeis missing the prefactor R0 because Stinchcombe does not screen the interaction correspondingto the zero momentum and frequency line in Fig. 1d of his paper (the vertical line in Figure 7.1of this thesis). In Section 7.2, we will deal explicitly with the spin half transverse field Isingmodel, and demonstrate the prefactor is necessary in order to obtain the correct leading ordercorrection to the magnetization.We restrict our attention to the zero temperature limit, in which case the renormalizedinteraction (the T matrix) is given byTk(iωr)∣∣∣∣T=0=Vk∏n>1(E2n1− (iωr)2)∏p((Epk )2− (iωr)2) , (7.2)and the prefactor R0 is the zero wave vector and zero frequency component of the free fieldpropagator Dk(iωr) given byR0 =11+g(0)Vk=0∣∣∣∣T=0=∏n>1E2n1∏p(Epk=0)2 , (7.3)138Figure 7.1: The diagram above corresponds to the leading order correction to the magne-tization of a quantum Ising system in the high density approximation (an expansionin the inverse coordination number).The third order spin cumulant, derived in Appendix E, is given byM3(0, iωr,−iωr)∣∣∣∣T=0=∑n>1(c11− cnn)|c1n|2A01(0, iωr,−iωr)+ (7.4)+ ∑n>1p>nRe[c1ncnpcp1]A02(0, iωr,−iωr),whereA01(0, iωr,−iωr) =−2β3E2n1− (iωr)2[E2n1− (iωr)2]2(7.5)A02(0, iωr,−iωr) = 4β[En1Ep1(E2n1− (iωr)2)+Ep1En1(E2p1− (iωr)2)+Ep1En1+(iωr)2(E2n1− (iωr)2)(E2p1− (iωr)2)].Note that A01 and A02 are functions of the summation indices, n and p, as well. We suppress theseindices for the sake of compactness. Combining terms, we may write the zero temperature139correction to the magnetization as〈Sz〉1 = R02N∑kVk[∑n>1(c11− cnn)|c1n|2An3+ ∑p>n6=1Re[c1ncnpcp1]Anp4], (7.6)whereAn3 =−2β ∑r3E2n1− (iωr)2E2n1− (iωr)2∏m6=n,1(E2m1− (iωr)2)∏p((Epk )2− (iωr)2) (7.7)Anp4 =4β ∑r1∏l(E lk)2− (iωr)2[En1Ep1∏m6=n,1(E2m1− (iωr)2)+Ep1En1∏m 6=p,1(E2m1− (iωr)2)+(Ep1En1+(iωr)2) ∏m 6=n,p,1(E2m1− (iωr)2)].We see that all the poles at the MF energy levels vanish from Anp4 due to the zeros of thescreened interaction; however, we still must deal with MF poles in An3. Furthermore, we musttreat points at which the MF energy levels and the RPA energy levels are degenerate carefullybecause the order of the associated poles in An3 and Anp4 will change. There is also the possibilitythat a pair of the RPA modes are degenerate. In such cases, we simply shift one of the RPAmodes by a small amount to avoid the degeneracy.Our next task is to perform the frequency summations. We do so in the usual way, viz., wewriteA j =− 1β ∑rf j(z= iωr) =∑prpnB(zp) (7.8)where rp is the residue of f j at the pth pole zp, and nB(zp) is the Bose-Einstein distributionfunctionnB(z) =1eβ z−1 . (7.9)Note that f j(z) = f j(−z); hence, Res[ f j(z0)] =−Res[ f j(−z0)]. This simplifies our task some-what because it means we only need to consider positive poles. The Anp4 function has simplepoles at each of the RPA energy levels. Making use of the fact nB(z)−nB(−z) = coth(β z2 ), and140using Res[ f (z);c] = limz→c(z− c) f (z) to find the residue of f at c, we find thatAnp4 =∑l2E lk∏q6=l(Eqk )2− (E lk)2[En1Ep1∏m6=n,1(E2m1− (E lk)2)+Ep1En1∏m6=p,1(E2m1− (E lk)2) (7.10)+(Ep1En1− (E lk)2) ∏m6=n,p,1(E2m1− (E lk)2)]coth(βE lk2).In the zero temperature limit, coth(βElk2 ) is simply equal to one. Analysis of the An3 functionrequires more care than analysis of the Anp4 function because we must consider the possibilitythat the RPA modes and the MF energy levels are degenerate. If there are no degeneracies,there are simple poles at each RPA mode and the nth MF energy level, and the result of thefrequency summation isAn3|En 6=E pk =−2[En1∏m 6=n,1(E2m1−E2n1)∏p((Epk )2−E2n1)coth(βEn12) (7.11)+∑p3E2n1− (E pk )2E2n1− (E pk )2∏m 6=n,1E2m1− (E pk )22E pk ∏q6=p(Eqk )2− (E pk )2coth(βE pk2)].In order to deal with the case where the nth MF energy level is degenerate with one of the RPAmodes, En1 = Eqk , we begin by rewriting An3 asAn3 =−1β ∑rP(z= iωr)(Eqk − iωr)2Q(z= iωr), (7.12)whereP(z) = 2[3(Eqk )2− z2] ∏m 6=n,1[E2m1− z2] (7.13)Q(z) = [Eqk + z]2∏p6=q[(E pk )2− z2].We now have a second order pole at the Eqk , and simple poles at all the other RPA modes. Theresidue at the second order pole is given byRes|Eqk = limz→EqkddzP(z)Q(z)=P′(Eqk )Q(Eqk )−P(Eqk )Q′(Eqk )Q(Eqk )2 , (7.14)141whereP′(Eqk ) =−4Eqk ∏m 6=n,1[E2m1− (Eqk )2]−8(Eqk )3 ∑m6=n,1∏r 6=m,n,1[E2r1− (Eqk )2] (7.15)Q′(Eqk ) = 4Eqk ∏p6=q[(E pk )2− (Eqk )2]−8(Eqk )3 ∑p6=q∏r 6=p,q[(Erk)2− (Eqk )2].This gives the following result for the frequency summationAn3|En=Eqk =P′(Eqk )Q(Eqk )−P(Eqk )Q′(Eqk )Q(Eqk )2 coth(βEqk2) (7.16)− ∑p6=q3(Eqk )2− (E pk )2[(Eqk )2− (E pk )2]2∏m 6=n,1(E2m1− (E pk )2)E pk ∏r 6=p,q((Erk)2− (E pk )2)coth(βE pk2)To summarize, we have found the leading order correction to the magnetization at zerotemperature to be〈Sz〉1 = R02N∑kVk[∑n>1(c11− cnn)|c1n|2An3+ ∑p>n6=1Re[c1ncnpcp1]Anp4], (7.17)with An3 given by equation (7.11) if there are no degeneracies between the RPA modes of thesystem and the MF eigenstates, and An3 given by equation (7.16) if such degeneracies do exist.In the paramagnetic phase of the system, the contribution from An3 vanishes, and we are leftwith only the contribution from Anp4 , given in equation (7.10). In the event that two of the RPAmodes of the system are degenerate, we simply shift one of the modes by a small amount,rather than deal with a higher order pole in the associated frequency summation.Landau TheoryRecall from Section 6.3.3 that the partition function governing the Hubbard-Stratonovich fieldcorresponding to fluctuations in a quantum Ising system is given by (to fourth order in the field)ZQ =∫DQ√2piexp(− 12∑r,kD−1k (iωr)|Qk(iωr)|2 (7.18)− 13![∑{ri}∑{ki}g3∏i=1Qki(iωri)]− 14![∑{ri}∑{ki}u4∏i=1Qki(iωri)]),142where, in the low temperature limit,Dk(iωr)∣∣∣∣T→0= Dk(iωr) =∏n>1[E2n1− (iωr)2]∏p[(Epk )2− (iωr)2] , (7.19)and the higher order coefficients are functions of the fluctuating Matsubara frequencies andmomenta, g = g({ki},{iωri}) and u = u({ki},{iωri}). In a renormalization group treatmentof the system, the frequency and momentum dependence of g and u is irrelevant, and we maysimply writeZQ =∫ [√βV0N2piDQ]e−βV0NL[Q], (7.20)with the Landau energy function beingL[Q] =12∑r,kD−1k (iωr)|Qk(iωr)|2 (7.21)− g˜03[∑{ri}∑{ki}3∏i=1Qki(iωri)]+u˜04[∑{ri}∑{ki}4∏i=1Qki(iωri)],The fields have been rescaled (Q→√βV0NQ). This eliminates any explicit dependence onβ or N from the resulting Landau energy function, although, the function still has implicittemperature dependence because the prefactors are functions of the temperature dependentspin cumulants. The prefactors in the Landau theory, g˜0 and u˜0, are obtained by setting thefrequency and momentum dependence of g and u to zero, and rescaling them as followsg˜0 =g({ki = 0},{iωri = 0})2√βV0N(7.22)u˜0 =u({ki = 0},{iωri = 0})6βV0N.In Landau MF theory, we assume the system is uniformly magnetized, and ignore the frequencyand momentum dependence of the quadratic term, takingD−1k (iωr)→ r˜0 = R−10 =∏p(Epk=0)2∏n>1E2n1. (7.23)143Using the results of E we find, in the low temperature limit,g˜0∣∣∣∣T→0= 3V 20[∑n>1(c11− cnn)|c1n|2 1E2n1− ∑p>n6=1Re[2c1ncnpcp1]1Ep1En1](7.24)u˜0∣∣∣∣T→0=−4V 30[∑n>1|c1n|2E3n1[(c11− cnn)2−|c1n|2]+ ∑n6=p>1c11c1ncnpcp11Enp(1E2n1− 1E2p1)+ ∑m 6=p>1Re[cmmcm1c1pcpm]1E2m1Ep1− ∑p>n>1|c1n|2|c1p|2En1Ep1(1Ep1+1En1)−∑n>1∑p>1p6=n∑q>1q6=n,pc1ncnpcpqcq11En1EnpEq1].It follows from equation (6.88) that the magnetization in terms of the rescaled field is〈Sz〉= 〈Sz〉0+ 〈Q〉Q, (7.25)with the subscript 0 indicating an average with respect to the MF Hamiltonian, and the sub-script Q indicating an average with respect the Landau free energy function given in equation(7.21), which ignores the frequency and momentum dependence of the fluctuations, and theirinteractions. We use the tilde on the coefficients of the effective field theory because they rep-resent the energy cost of the fluctuation field Q. As shown in equation 6.28, these parametersare related to the parameters of the field theory for 〈Sz〉 byr0 = r˜0+2g˜0〈Sz〉0+9u˜0〈Sz〉20g0 = g˜0+3u˜0〈Sz〉0u0 = u˜0. (7.26)The Landau energy function given in equation (7.21), which is derived from a microscopicmodel, is amenable to treatment via the renormalization group. The methods used here aremore rigorous then obtaining an effective field theory based solely on symmetry considerations.The Landau coefficients, r˜0, g˜0 and u˜0, may be used to estimate the size of the region inwhich fluctuations of the order parameter will play a significant role in a quantum Ising system.In order for the Gaussian approximation to be valid, we require r˜0 g˜0, u˜0. In Figure 7.2, weplot the Landau coefficients for the spin half transverse field Ising model (TFIM) as a functionof the transverse magnetic field. The left hand plot shows the coefficients in the absence ofa longitudinal magnetic field, and the right hand plot shows the coefficients in the presenceof a longitudinal field having the same strength as the nearest neighbour exchange interaction144between spins. The TFIM is discussed in Section 7.2. We find that, in the absence of a longitu-dinal field, the point at which r˜0 and g˜0 cross corresponds to the point at which corrections dueto fluctuations begin to have significant impact on the longitudinal magnetization (the orderparameter) of the system, shown in Figure 7.4. When a longitudinal field is applied, we seethat r˜0 > g˜0, u˜0 for any value of the transverse field, hence the system is stabilized against orderparameter fluctuations, in line with our expectations.0 2 4 6 8 10∆/Jnn-0.8-0.6-0.4-0.200.20.40.60.81Landau Exponents            Landau Exponents of the TFIMr˜02/3 g˜01/2 u˜00 2 4 6 8 10∆/Jnn-0.4-0.200.20.40.60.81Landau Exponents                        Landau Exponents of the TFIM in a Longitudinal Fieldr˜02/3 g˜01/2 u˜0Figure 7.2: In this figure, we plot the Landau exponents, r˜0, g˜0, u˜0 given in equations 7.23and 7.24, of the effective field theory for the transverse field Ising model at zerotemperature, as a function of the transverse field ∆. The figure on the left shows thesystem in the absence of a longitudinal field, whereas the figure on the right showsthe exponents in the presence of a longitudinal field of h = Jnn, where Jnn is thenearest neighbour exchange interaction between the spins.In Figure 7.3, we plot the Landau coefficients of the spin half spin half model (SHSH)in the left hand plot, and for LiHoF4 in the right hand plot. The SHSH model is given inequation (7.37). We take ε = 0, so that there is no transverse field acting directly on thenuclear spins, and consider a weak isotropic hyperfine interaction, Az = A⊥ = 0.01Jnn, whereJnn is the strength of the exchange interaction. In this limit, one might expect the Landaucoefficients to reduce to those of the TFIM; however, comparing the u˜0 function of the SHSHmodel with that obtained from the TFIM, we find this is not the case. The discrepancy betweenthe two plots persists even when the hyperfine interaction is further reduced. The reason forthis is that, at the MF level, when the hyperfine interaction is taken to zero in the SHSH modelthe Hamiltonian consists of two disjoint copies of the TFIM, with each copy correspondingto a different nuclear spin state. The interaction term in the Hamiltonian couples these twosubspaces. That is, a fluctuation in one nuclear subspace may have an effect on the othernuclear subspace. These additional degrees of freedom are not present in the TFIM. In theplot of the Landau coefficients of LiHoF4 on the right hand side of Figure 7.3, it appears that145the fluctuations will only play a role in a narrow region around the quantum critical point. Aglance at the plot of the longitudinal magnetization of LiHoF4 shown in Figure 7.6 indicatesthis is not the case. The problem is that the Landau MF field theory fails to take into accountthe frustrated long range nature of the dipolar interaction, which causes the LiHoF4 system tobe significantly more susceptible to the effect of fluctuations than is indicated by the LandauMF theory. The condition r˜0  g˜0, u˜0 is necessary for the effect of fluctuations to be small;however, as the LiHoF4 system demonstrates, it is not sufficient.0 2 4 6 8∆/Jnn-0.500.511.52Landau ExponentsLandau Exponents of the SHSH modelr˜02/3 g˜01/2 u˜05 5.1 5.2 5.3 5.4 5.5 5.6Bx (T)00.050.10.15Landau ExponentsLandau Exponents of LiHoF4r˜02/3 g˜01/2 u˜0Figure 7.3: In this figure, we plot the zero temperature Landau exponents, r˜0, g˜0, u˜0 givenin equations 7.23 and 7.24, of the effective field theory for the spin half spin halfmodel (left) and LiHoF4 (right), as a function of the transverse magnetic field. Forthe spin half spin half model, given in equation (7.37), we assume a nearest neigh-bour exchange interaction Jnn between spins, and we assume there is no effectivefield acting directly on the nuclear spins (ε = 0). We assume a weak isotropic hyper-fine interaction Az = A⊥ = 0.01Jnn. The LiHoF4 Hamiltonian is given in equation(7.39).146Spin Half Transverse Field Ising ModelWe now apply the theory developed for including the effects of fluctuations in quantum Isingsystems to find the leading order zero temperature correction to the magnetization of the spinhalf transverse field Ising model. This may be compared to the result of Stinchcombe given inequation (2.43) of [63]. We note that Stinchcombe’s result is obtained by differentiating thefree energy given in equation (6.82), rather than evaluating equation (7.1), so the followingcalculation is an independent consistency check of the theory.For a spin half system, the leading order zero temperature correction to the magnetizationis given by〈Sz〉1 = R02N∑kVk(c11− c22)|c12|2A3, (7.27)whereR0 =E221E2k=0, (7.28)and, assuming the RPA mode and mean field energy level are not degenerate,A3 =−2E21E2k −E221− 3E221−E2kE221−E2k1Ek. (7.29)The MF matrix elements, energy levels, and RPA mode are derived in Appendix B.The correction to the MF magnetization vanishes in the paramagnetic phase of the sys-tem, so we will focus on the ferromagnetic phase. Following Stinchcombe [63], we make thefollowing definitionsx=2∆V0γ(k) =VkV0, (7.30)where ∆ is the transverse field acting on the system. In terms of these parameters, in theferromagnetic phase of the system, we findE21 =∆xEk =∆x√1− x2γ(k). (7.31)Plugging these values into A3 yieldsA3 =− 2x∆γ(k)[1+ 12x2γ(k)√1− x2γ(k) −1](7.32)147For the prefactor and matrix elements we findR0(c11− c22)|c12|2 = x241√1− x2 (7.33)Putting everything together yields〈Sz〉1 =−12N√1−x2 ∑k[1+ 12 x2γ(k)√1−x2γ(k) −1]∆< ∆c0 ∆≥ ∆c(7.34)which is in agreement with equation (2.43) of Stinchcombe [63]. We see that the prefactor R0,which is missing from equation (2.10) of Stinchcombe [62], is required to obtain the correctresult. We see that the approximation breaks down as x→ 1, or, equivalently, as ∆→ ∆c. Thisis to be expected because in the critical region fluctuations contribute at all orders, and cannotbe neglected. The Landau function discussed in Section 7.1 is suitable for a renormalizationgroup treatment of a system’s critical behaviour.We now specialize to the case of a simple cubic crystal, with lattice spacing a, and a nearestneighbour exchange interaction. In this case, we haveVk = 2J[cos(kxa)+cos(kya)+cos(kza)],and γk = Vk6J . The MF magnetization in the ferromagnetic phase of the system is〈Sz〉0 = 12√1− x2, (7.35)with x = ∆3J , and the leading order correction to the longitudinal magnetization in the ferro-magnetic phase is〈Sz〉1 =− 14N〈Sz〉0∑k[1+ 12x2γ(k)√1− x2γ(k) −1]. (7.36)In Figure 7.4, we plot the MF magnetization of the transverse field Ising model, alongwith the leading order correction calculated from equation (7.36). We see that the correctionbecomes unreliable, and, in fact, diverges near the quantum critical point, where fluctuationsbecome important. The leading order correction to the critical value of the transverse fieldfollows from the point at which the corrected magnetization reaches zero. As discussed byStinchcombe in [63], the correction to the transverse field is consistent with the result obtainedby considering the transverse magnetization, or the longitudinal static susceptibility, neitherof which suffer from the divergence seen in the corrected magnetization. We find for thetransverse Ising model the critical transverse field is about 95% of its MF value.1480 1 2 3 4 5∆J00.10.20.30.40.5〈 Sz 〉                  Longitudinal Magnetization of the TFIMFigure 7.4: In this figure, we plot the mean field longitudinal magnetization of the trans-verse field Ising model (dashed line), along with the leading order correction in thehigh density approximation calculated from equation (7.36), as a function of thetransverse field ∆. We consider a simple cubic crystal with exchange interactionstrength J. We see the theory breaks down in the vicinity of the critical transversefield where fluctuations become more important. The point at which the correctedmagnetization reaches zero gives the leading order correction to the critical trans-verse field. We find ∆c ≈ 2.84J.Spin Half Spin Half ModelIn this section, we calculate the leading order correction to the MF magnetization of the spinhalf spin half modelH =−12∑i 6= jVi jSziSzj−∆∑i(Sxi − εIxi)+Az∑iIzi Szi +A⊥2 ∑i(I+i S−i + I−i S+i ). (7.37)We include a transverse field ∆n = ε∆ acting directly on the nuclear spins because in systemssuch as LiHoF4, the effective transverse field acting on the nuclear spins can be a significantfraction of the effective field splitting the electronic levels. We take our interaction to beVi j = JDDzzi j − Jnnδi j, (7.38)149with Dzzi j being the longitudinal dipolar interaction, and Jnn being a nearest neighbour exchangeinteraction. We assume a long thin cylindrical sample, or needle like domain, in a material witha simple cubic Bravais lattice. In this case, as discussed in Chapter 3, the dipolar interaction isgiven by (in momentum space)a3Dk =4pi3 : k = 04pi3 +4pi3(1−3 k2zk2): k 6= 0with a being the unit cell length. At k= 0, we have the Lorentz local field, and no contributionfrom the demagnetization field, which is zero in a long cylinder. Away from k = 0, we use theresult for a spherical sample, which is valid provided kR >> 1, with R being the system size.In momentum space, our total interaction will be Vk = JDDzzk − 2Jnn[cos(kxa) + cos(kya) +cos(kza)]. In this toy model, we allow both JD and Jnn to be tunable parameters. This allowsus to investigate how the competition between dipolar and exchange energies (ferromagneticfor Jnn < 0, or antiferromagnetic for Jnn > 0) affects the magnitude of the corrections to MFtheory. We may also reduce the overall strength of the interaction, Vk → pVk, with p < 1, inaccordance with what is expected if some of the magnetic ions in the crystal are doped withnon-magnetic impurities.We take JD = 0 and ε = 0, and we set Az = A⊥. This corresponds to an exchange coupledtransverse field Ising model with an isotropic hyperfine interaction. The calculation of theleading order correction to the magnetization, given in equation (7.17), involves summations ofcomplicated functions over the MF energy levels and the Brillouin zone. We do the summationsnumerically. In order to test that the numerical summations are correct, we consider the SHSHmodel with Az = A⊥ = 0.01Jnn. With this small value of the isotropic hyperfine interaction,we expect that the results will be similar to the result for the transverse field Ising model withno hyperfine interaction, which is illustrated in Figure 7.4, and indeed, this is the case. Theprogram used to calculate the summations in the SHSH model is easily adapted to systemswith Hamiltonians containing more MF eigenstates, such as LiHoF4. In Figure 7.5, we plot theMF magnetization and corrected magnetization of the SHSH model with Az = A⊥ = 0.01Jnn.We leave further investigation of the effects of fluctuations in the SHSH model as the subjectof future work.1500 0.5 1 1.5 2 2.5 3 3.5∆Jnn00.10.20.30.40.5〈 Sz 〉                       Longitudinal Magnetization of the SHSH ModelAz/Jnn=A⊥/Jnn=0.01Figure 7.5: In this figure, we plot the MF magnetization of the spin half spin half modelgiven in equation (7.37), along with its leading order correction in the high densityapproximation calculated from equation (7.17). We set the dipolar interaction tozero, and assume no transverse field acting directly on the nuclear spins. We assumean isotropic hyperfine interaction, with Az = A⊥ = 0.01Jnn.LiHoF4In this section, we obtain the leading order correction to the longitudinal magnetization ofLiHoF4, using the effective low temperature HamiltonianH =− ∆2∑iτxi −12JDC2zz∑i6= jDzzi jτzi τzj +12JnnC2zz ∑<i j>τzi τzj+ (7.39)+∑i~∆n ·~Ii+Az∑iτzi Izi +A⊥∑iτ+i I−i +A†⊥∑iτ−i I+i ++A++∑iτ+i I+i +A†++∑iτ−i I−i .All the parameters in the model are given in Chapter 2. The longitudinal magnetization is givenby Jz ≈Czz〈τz〉0+Czz〈τz〉1, where 〈τz〉0 is the MF expectation value of the Pauli operator τz,and 〈τz〉1 is given by equation (7.17). The interaction in equation (7.17) is given by Vk =C2zz[JDDzzk −Jnnγk], and all the relevant matrix elements, MF energy levels, and RPA modes arecalculated in Chapter 5.151Bx 4.0T 4.9T 5.4T 6.0T〈Jz〉0 3.2293 1.7784 2.8 ∗10−7 3.0 ∗10−8N=20〈Jz〉0+ 〈Jz〉1 1.5102 -2.8062 -1.0 ∗10−5 -8.2 ∗10−8N=30〈Jz〉0+ 〈Jz〉1 1.4951 -2.8502 -1.0 ∗10−5 -8.3 ∗10−8N=40〈Jz〉0+ 〈Jz〉1 1.4873 -2.873 -1.0 ∗10−5 -8.4 ∗10−8N=50〈Jz〉0+ 〈Jz〉1 1.4825 -2.8869 -1.0 ∗10−5 -8.4 ∗10−8Table 7.1: In this table, we show corrections to the mean field magnetization of LiHoF4calculated using the high density approximation. Each column corresponds to adifferent value of the applied transverse magnetic field Bx, and each row correspondsto a finer division of the Brillouin zone (a larger value of N in equation (7.40)).This calculation is complicated by the Brillouin zone summations. We perform the summa-tions by brute force, summing over finer and finer divisions of the Brillouin zone until suitableconvergence is achieved. We carve up the positive quadrant of the Brillouin zone as follows,kx,ky ∈[0,kx,kyN,pia]kz ∈[0,kzN,pic], (7.40)where a = 5.175A˚, and c = 10.75A˚, are the transverse, and longitudinal, lattice spacings ofLiHoF4. All other Brillouin zone points may be included via symmetry considerations. We il-lustrate the convergence by looking at four values of transverse magnetic field, 4.0T,4.9T,5.4Tand 6.0T . These represent points in the ferromagnetic phase of the system, at the experimentalvalue of the critical transverse field, at the MF value of the critical transverse field, and in theparamagnetic phase of the sytem, respectively. In Table 7.1, we list the corrected magnetiza-tion for N = 20,30,40, and 50. The point to take from the table is that, even for our mostcoarse division of the Brillouin zone (N=20), the sums are converging to within two significantfigures, with the least reliable correction being near the experimental critical point where thecorrection diverges.In Figure 7.6, we plot the longitudinal MF magnetization of LiHoF4, along with the leadingorder correction in the high density approximation calculated from equation (7.17). Due tothe frustrated long range nature of the dipolar interactions in LiHoF4, we expect the effectof fluctuations to be significant, and indeed, we find this to be the case. We also expect,due to the long range nature of the interaction, that the leading order correction in a highdensity approximation will be accurate, except in the vicinity of the critical point where the152formalism breaks down. The experimental value of the critical transverse field is Bcx = 4.9T.The leading order correction to the magnetization gives a critical value of the transverse field ofabout Bcx = 4.4T, significantly underestimating the experimental value. It is not clear whetherthis discrepancy is due to the choice of crystal field parameters, and the magnitude of theexchange interaction, or whether it is a shortcoming of the approximation. The reduced criticaltemperature in this approximation is about 81% of its MF value.0 1 2 3 4 5 6Bx (T)012345〈 Jz 〉Longitudinal Magnetization of LiHoF4Figure 7.6: The figure above shows the MF longitudinal magnetization of LiHoF4(dashed line), along with the leading order correction calculated from equation(7.17), as a function of the transverse field Bx. The point at which the correctedmagnetization reaches zero gives the leading order correction to the critical trans-verse field. We find Bcx ≈ 4.4T. The experimental value of the critical transversefield is Bcx = 4.9T; hence, with our choice of crystal field parameters, and near-est neighbour exchange interaction, the leading order correction underestimates thecritical transverse field by about as much as it is overestimated by MF theory.153SummaryIn this chapter, we have the high density approximation developed in Chapter 6 to find theleading order correction to the magnetization of several quantum Ising systems. We began bydiscussing the form of the correction to the magnetization for a general quantum Ising system,and then, in Section 7.1, we discussed how to derive a Landau free energy function from ourmore general formalism. The coefficients of the Landau theory may be used to estimate whenfluctuations will have a significant impact on a quantum Ising system.In Section 7.2, we applied the theory to the spin half transverse field Ising model (TFIM),making contact with the results derived by Stinchcombe in [63]. In Section 7.37, we appliedthe formalism to the spin half spin half (SHSH) model. The calculation of the corrections tothe MF magnetization involves Brillouin zone summations over complicated functions. Byconsidering the SHSH model with a small hyperfine interaction, we reproduced the resultsobtained for the TFIM, which verifies the program used to calculate the numerical summationsis correct. Section 7.4 is concerned with the application of the theory to the magnetic insulatorLiHoF4. We found the corrections to MF theory to be quite large, which is to be expected in adipolar coupled system.154Chapter 8Conclusions and Future WorkThe work in this thesis can be divided into two main parts: (1) the properties of the magneticinsulating crystal LiHoF4, and (2) the development of a formalism for incorporating the effectof fluctuations in general quantum Ising systems, of which LiHoF4 is a particular example.Regarding (1), we have introduced a new effective low temperature Hamiltonian for LiHoF4that fully incorporates the nuclear degrees of freedom in Chapter 2. We find that the dominantmixing of the hyperfine states is due to an effective transverse field acting directly on thenuclear spins. This field is an order of magnitude larger than the effective transverse componentof the hyperfine interaction. The origin of this effective field is a shift in the 4 f electroncloud of each Ho3+ ion due to an applied transverse magnetic field. Chapter 3 contains ananalysis of the dipole-dipole interaction that is the dominant coupling between the electronicdegrees of freedom in LiHoF4. The Fourier analysis of the long range dipolar interaction,taking into consideration the underlying lattice of the LiHoF4 crystal, is an important aspect ofthe theoretical investigation of the material.In Chapter 4, we present the spin half spin half (SHSH) modelH =−12∑i 6= jVi jSziSzj−∆∑iSxi +Az∑iSzi Izi +A⊥2 ∑i(S+i I−i +S−i I+i ), (8.1)a toy model meant to illustrate the effects of an anisotropic hyperfine interaction in a transversefield Ising system. We find that Az > A⊥ leads to an enhancement of the single ion susceptibil-ity, an increase in the critical transverse field, as well as an enhancement of an applied longi-tudinal field. The enhancement of a longitudinal field has not been previously noted. With theaddition of a transverse field acting directly on the nuclear spins, equation (8.1) serves as a toymodel for LiHoF4. We go on to analyze the zero temperature spectrum of the SHSH model inthe random phase approximation (RPA). We find that what would have been the electronic softmode is gapped by the hyperfine interaction, with spectral weight being transferred to a lower155energy electronuclear mode that fully softens at the quantum critical point. In Chapter 5, weperform an RPA analysis of the LiHoF4 system, observing behaviour of the low energy modessimilar to that of the toy model.Regarding (2), we have made use of the well known Hubbard-Stratonovich transformationto derive an effective field theory for quantum Ising systems with an arbitrary single ion Hamil-tonian in Chapter 6. This formalism is used to derive a diagrammatic perturbation theory forincorporating the effects of fluctuations in quantum Ising systems beyond the RPA. We findthe formalism to be equivalent to the high density approximation of Brout [60, 61], which hasbeen applied to the spin half transverse field Ising model by Stinchcombe [26, 62, 63]; how-ever, the field theoretic derivation offers significant simplicity and clarity when compared toprevious approaches. Basically, in this thesis, we have taken a ground up approach. We haveused the Hubbard-Stratonovich transformation to obtain an effective field theory suitable forrenormalization group analysis of a system’s critical behaviour, and we have worked in a basisof mean field eigenstates (Hubbard operators) in order to rigorously obtain the theory. We findthat a regular nuclear spin bath, or any other regular modification to the single ion Hamilto-nian, will not fundamentally alter the nature of the quantum phase transition in a transversefield Ising system. The strength of the field theoretic formalism developed in this thesis is itsversatility. It is easily applied to real quantum Ising ferromagnets such as LiHoF4, and theformalism may be generalized to study antiferromagnetic materials and spin glass, as well asoffering the possibility of studying quantum Ising systems in time dependent potentials. Wehave demonstrated the validity of the formalism by applying it to the calculation of correctionsto the zero temperature magnetization of LiHoF4 in Chapter 7.Future WorkThe work presented in this thesis, in particular the field theoretic formalism presented in Chap-ter 6, has many interesting applications beyond the scope of what is presented in this thesis.Furthermore, there remain many fascinating possibilities for further research on LiHoF4. Weconclude this thesis with a discussion of this future work. To begin, we discuss some easy ap-plications of the random phase approximation (RPA) to LiHoF4 that we have left undeveloped.We proceed to consider a fluctuation analysis of LiHoF4 that goes beyond the magnetizationcorrections presented in Chapter 7, and some interesting modifications of the Hamiltonian forfuture deliberation. We move on from LiHoF4 by discussing some possible improvements ofthe field theoretic formalism presented in Chapter 6, and outlining a few of its most significantapplications.Our primary focus in this thesis has been on the behaviour of quantum Ising systems in156the zero temperature limit. The RPA results for the spin half spin half (SHSH) model, andfor LiHoF4, presented in Chapters 4 and 5 respectively, are easily generalized to finite tem-peratures. The SHSH model will yield analytic expressions for the temperature dependenceof experimentally relevant quantities, such as the paramagnetic susceptibility. Furthermore,our analysis of the SHSH model and LiHoF4 has been focused on the dynamic longitudinalelectronic susceptibility, which is relevant to, for example, neutron scattering experiments. Acalculation of the transverse electronic susceptibility, and the specific heat, is straightforward,and would be of experimental interest, as would a calculation of the nuclear susceptibilities.The dynamic nuclear susceptibilities, or equivalently, the nuclear correlation functions, are ofparticular interest as they are relevant to magnetic resonance experiments that may be used toprobe the low energy properties of the LiHoF4 system. See the paper of Schechter and Stampfor further details [38].Our fluctuation analysis of quantum Ising systems in Chapter 7 is limited to the zero tem-perature magnetization, this being the simplest application of the field theoretic formalism. Wewould like to see these calculations generalized to finite temperatures, and we would like tosee the field theoretic formalism applied to the calculation of dynamic correlation functions,which yield the energies and lifetimes of excited states. The groundwork for such calculationshas been laid out in Chapter 6 and Appendix E. In particular, it would be interesting to use thefield theoretic formalism to see if the longitudinal hyperfine interaction in LiHoF4 stabilizes thesystem against the disordering effects of thermal fluctuations, or a transverse magnetic field.In low transverse fields, this may account for the discrepancy between the experimental phasediagram and the phase diagram that has been obtained from Monte Carlo simulations [69].Using the effective low temperature Hamiltonian for LiHoF4, derived in Chapter 2, it wouldalso be of interest to compare results obtained from the field theoretic formalism of Chapter6 to results obtained using the effective medium theory of Jensen, results obtained using thecorrelated effective field approximation of Lines, and results obtained via Monte Carlo simula-tions. The formalisms of Jensen and Lines are discussed in the introduction to Chapter 6. Therole of fluctuations when there is competition between dipolar and exchange interactions, suchas in LiHoF4, is another interesting avenue for future work. The SHSH model, as presented inSection 7.3, is amenable to such an investigation.Regarding the LiHoF4 Hamiltonian derived in Chapter 2, we would like to see it analyzedwith modifications to account for doping of the system with non-magnetic yttrium ions, andthe inclusion of an oscillator bath environment. A preliminary theoretical investigation of theeffect of doping has been carried out by Schechter and Stamp [37, 38], and of the effects ofan oscillator bath environment by Banerjee and Dattagupta [94]; however, none of this worktakes full account of the complexity of the low temperature LiHoF4 Hamiltonian. In the work of157Banerjee and Dattagupta, only a longitudinal hyperfine interaction is considered. Schechter andStamp consider transverse hyperfine interactions; however, the effective transverse field actingdirectly on the nuclear spins upon application of a physical transverse field is not apparent intheir work. Furthermore, we would like to see the model used to calculate the dynamics of thesystem with time dependent parameters, thus going beyond the thermodynamics that has beenthe primary focus of this thesis. In particular, we would like to see the model used to calculateLandau-Zener transitions, and the effects of an AC magnetic field.We have discussed some possibilities for future research on LiHoF4 relevant to the workdone in this thesis. We now consider the field theoretic formalism of Chapter 6, and some ofits potential applications apart from the LiHoF4 system. The primary drawback of the fieldtheoretic formalism is the algebraic complexity of the resulting equations, and the need to per-form Brillouin zone summations over complicated functions. The algebraic difficulties stemfrom the calculation of the spin cumulants that make up the coefficients of the field theory.Simplifications, or approximations, of the spin cumulants would be of considerable use. Inorder to utilize the field theoretic formalism, it would be useful to optimize the time required toperform the Brillouin zone summations necessary for acquiring results. In this thesis, the Bril-louin zone summations have been performed by brute force. Consideration of the symmetry ofthe underlying crystal lattice will lead to improvements over the brute force approach.A simple application of the field theoretic formalism, to a system with more complexitythan the spin half transverse field Ising model, is to the Blume-Capel modelH =−D∑i(1− (Szi )2)+ ~H∑i~Si+ J∑〈i j〉SziSzj, (8.2)where we take ~S to be spin one. In a spin one system, the spin cumulants, and hence thecoefficients of the effective field theory, will be more complicated than in the spin half case;however, they will be much simpler than in a system with four or more single ion energy levels.The parameter D may be used to control the number of Sz = 0 states occuring in the system.With D=−∞ the model reduces to the spin half case, and with D= 0 we have a standard spinone Ising system. This model was introduced in 1966 by Blume in an attempt to model theantiferromagnetic insulator UO2 [114], and more generally in a series of papers by Capel toaccount for the behaviour of triplet (S = 1) spin systems [115–117]. A variant of the modelwas introduced by Blume, Emery, and Griffiths in an attempt to model He3−He4 mixtures[118]. As discussed in the book of Cardy, the Blume-Capel model may also be used to studya spin half system with mobile vacancies corresponding to the Sz = 0 states [119]. This modelis of interest in the study of phase transitions because the order of the phase transition dependson the parameter D. We would like to see the field theoretic formalism of Chapter 6 applied to158the interesting and versatile Blume-Capel model.In the development of the field theoretic formalism of Chapter 6, we have limited ourselvesto the consideration of ferromagnetic interactions between spins. We may equally well considerantiferromagnetic interactions; however, the field in the resulting theory is no longer a validorder parameter. Instead, if the system has a bipartite lattice, we must consider the staggeredmagnetization which corresponds to making the field transformation Q→ (−1)‖~r‖Q, where‖~r‖ is even or odd depending on which subsytem the vector~r points to. For example, in theantiferromagnetic Ising system DyPO4, discussed by Wright et al. in reference [120], ‖~r‖is the minimum number of lattice steps between the origin and the lattice point at~r. With asuitable modification of ‖~r‖ it is possible to study layered antiferromagnetic materials such asFeCl2, the properties of which are discussed in a chapter of the book by Barbara et al. [71].This material is of particular interest because the Fe ions may be replaced with non-magneticMg ions in FexMg1−xCl2. This leads to random field effects in the presence of a longitudinalmagnetic field, as discussed by, for example, Fishman and Aharony, and Cardy [121, 122].These random field effects are reminiscent of the random field effects in LiHoxY1−xF4, inwhich dilution leads to a random field via the off diagonal components of the dipolar interaction[37].The quenched nature of the disorder in dilute quantum Ising systems, such as FexMg1−xCl2and LiHoxY1−xF4, may be accounted for using the formalism of Chapter 6 by making use ofAnderson’s replica trick [43]. As discussed by Stephen and Aharony [123], by considering a setof n replicas of the system, and first averaging over the disorder, then expanding and resummingthe resulting equation, the partition function may be written Z(n) = Z0〈exp(−βH )〉, withH =∑i j∑lKl(i j) fl(i j). (8.3)The coefficients Kl(i j) are functions of the interaction and the concentration of dopents, andf1(i j) =n∑α=1SziαSzjα f2(i j) = ∑α<βSziαSzjαSziβSzjβ (8.4)fl(i j) = ∑α1<α2···αlSziα1Szjα1Sziα2Szjα2 · · ·SziαlSzjαl ,where the summations are over the n replicas of the system. A Hubbard-Stratonovich trans-formation may now be applied to each of the l terms in equation (8.3), which yields a set ofcompeting order parameters for the system. The work in this thesis provides a way to carry outthe procedure for real magnetic systems such as LiHoxY1−xF4.In the work presented in this thesis, we have not considered Hamiltonians with time de-159pendent parameters. A time dependent longitudinal magnetic field is easily included in thefield theoretic formalism Chapter of 6, with the assumption being that the system remains inthermodynamic equilibrium so that at each moment in time Z(t) = 〈exp(−βH(t))〉 yields thedistribution of the system. Note that in the Matsubara formalism we haveZ = Z0〈Tτ exp[∫ β0dτV (τ)]〉0. (8.5)This encodes corrections to the free energy, F = − 1β lnZ, due to both spatial and temporalfluctuations. With the time dependent part of the Hamiltonian included in V (τ), our basis ofeigenstates is time independent. However, the formalism contains memory effects through thetime dependence of V (τ). A preliminary investigations shows that the time dependence yieldsa shifted free energy with a complex part corresponding to the decay rate of the collective spinwave excitations. We see this as an interesting and fruitful avenue for future research.We have discussed further research to be done on the magnetic insulator LiHoF4, as well assome of the future applications of the field theoretic formalism developed in this thesis, whichis suitable for treating general quantum Ising systems. This includes applications to disorderedsystems, and systems with time dependent parameters. We think these topics will be amongthe most important physics of the 21st century, and would welcome the opportunity to pursuethis research further.160Bibliography[1] Bertrand Russell. The History of Western Philosophy. Simon and Schuster, 1st edition,1945. → pages 1[2] Stephen Blundell. Magnetism: A Very Short Introduction. Oxford University Press, 1stedition, 2012. → pages 1[3] W Heisenberg. Mehrko¨rperproblem und Resonanz in der Quantenmechanik.Zeitschrift fu¨r Physik, 38:411–426, 1926. → pages 2[4] PAM Dirac. On the Theory of Quantum Mechanics. 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Springer-Verlag, 1stedition, 1994. → pages 186169Appendix AMean Field Basis OperatorsIn this appendix, we explore some of the properties of the mean field (MF) operators, oftenreferred to as standard basis operators, or Hubbard operators, in the literature. These operatorscan be traced back to the 1965 work of Hubbard, in which operators acting between a discreteset of energy levels at each atomic site were used to analyze correlated electron systems [124].In 1972, Haley and Erdo¨s reintroduced these operators, and presented a formalism that wassubsequently developed by Yang and Wang in 1974 [102, 125]. Here, we give some of thebasic properties of the MF operators, and present Yang and Wang’s general reduction schemefor time ordered products of the MF operators. Further discussion of the MF basis operatorformalism is provided in the introduction to Chapter 6. We only consider products of MF basisoperators belonging to a single site because, as is shown in Section 6.3.1, considering MF basisoperators belonging to different sites is unnecessary in the formalism developed in this thesis.We use the reduction scheme to calculate general expressions for the correlation function ofproducts of up to four MF operators. These are necessary for deriving the spin cumulants ofup to four spin operators in Appendix E.Consider a Hamiltonian of the formH0 =∑i∑nEnLinn, (A.1)where we define the MF operators to beLmn = |m〉〈n|, (A.2)where |m〉 and |n〉 are MF eigenstates of the system. The superscript i is a site index, and thesummation in n is over all MF eigenstates of the system. Note that any single site operator may170be expressed in this basis; for example, we may writeSz =∑mncmnLmn. (A.3)In the Heisenberg picture, the MF operators will be time (or imaginary time τ) dependentLmn(τ) = eH0τLmne−H0τ . (A.4)Following reference [102], we note the following identitiesLmn(τ) = e(Em−En)τLmn e−βH0Lmn(τ) = e−β (Em−En)Lmn(τ)e−βH0 (A.5)and[Lpq(τ ′),Lmn(τ)] = e(Em−En)(τ−τ′)[Lpq(τ ′),Lmn(τ ′)]. (A.6)We now define the imaginary time ordered correlation function of the MF operators to be−〈TτLnm(τ ′)Lmn(τ)〉= DmnK0mn(τ ′− τ). (A.7)Noting that 〈Lnm(τ ′)Lmn(τ)〉 = 〈Lmn(τ)Lnm(τ ′)+ [Lnm(τ ′),Lmn(τ)]〉, and using the identities(A.5) and (A.6), we findK0mn(τ′− τ) ={e−(Em−En)(τ ′−τ)[1+n(Em−En)], τ ′ > τe−(Em−En)(τ ′−τ)n(Em−En), τ ′ < τ (A.8)where n(E) = (eβE−1)−1 is the Bose distribution function. We will refer to K0mn(τ ′−τ) as theMF Green’s function. The coefficient Dmn is a population factor, with Dmn = Dm−Dn, whereDm = 〈Lmm〉0 = e−βEmZ0, (A.9)and the MF partition function isZ0 = Tr[e−βH0] = Tr[e−β ∑nEnLnn] =∑ne−βEn. (A.10)Note that all the above is valid provided we are not dealing with degenerate energy levels. If171En = Em, we find−〈TτLnm(τ ′)Lmn(τ)〉=−Dn. (A.11)Accounting for degenerate levels, we may write−〈TτLnm(τ ′)Lmn(τ)〉= DmnK0mn(τ ′− τ)−Dmδmn. (A.12)Note that Dmn = 0 when the energy levels are degenerate. In this thesis we are primarilyconcerned with systems with a discrete set of non-degenerate single ion energy levels; hence,the degenerate case is ignored in subsequent discussion.In frequency space, the MF Green’s function K0mn is given byK0mn(iωr) =12∫ β−βdτK0mn(τ)eiωrτ =∫ β0dτK0mn(τ)eiωrτ =1Em−En− iωr (A.13)whereωr = 2rpiβ are the usual Bose-Matsubara frequencies. Note that K0mn(iωr) =−K0nm(−iωr).The inverse Fourier transform of the MF Green’s function is given byK0mn(τ) =1β ∑re−iωrτEm−En− iωr (A.14)Furthermore, we define the MF propagator for transitions between energy levels m and n to beKmn(τ) = [K0mn(τ)+K0mn(−τ)] Kmn(iωr) =2EmnE2nm− (iωr)2. (A.15)This function will be frequently encountered in our analysis of magnetic systems.Finally, for completeness, we consider the case Kmn(τ = τ ′). We find K0mn(0) = 12 +n(Em−En) and−〈TτLnm(τ)Lmn(τ)〉= Dmn2 +Dmnn(Em−En) =−12(Dm+Dn). (A.16)The expressions above are sufficient for calculating properties of spin systems within therandom phase approximation (RPA). In order to include the effects of fluctuations beyond theRPA, it is necessary to consider higher order correlation functions of the MF operators. Webegin by stating Yang and Wang’s general reduction theorem for time ordered averages of the172MF operators [102]. Let O(τ) = Lmn(τ) be an arbitrary MF operator. Then〈TτO(τ1) · · ·Lmn(τ) · · ·O(τi)〉0 (A.17)= K0mn(τ1− τ)〈Tτ [O(τ1),Lmn(τ1)] · · ·O(τi)〉0+K0mn(τ2− τ)〈TτO(τ1)[O(τ2),Lmn(τ2)] · · ·O(τi)〉0· · ·+K0mn(τi− τ)〈TτO(τ1) · · · [O(τi),Lmn(τi)]〉0.In the above expression, we have reduced an average over a set of i+ 1 MF operators to asum of averages over i MF operators. This process can be repeated until we are left with onlyaverages of diagonal operators Lmm(τ), which simply contribute population factors. Note thatthe reduction is not unique because we may start the process with whichever MF operator wechoose, leading to different, but equivalent, algebraic expressions for the relevant correlationfunction. Subsequent calculations in the zero temperature limit are often simplified by startingthe reduction with Lmn such that m is minimal, which is the convention we will adopt hereunless otherwise noted. We now use the reduction scheme to calculate correlation functions ofthree and four MF operators.At third order we must consider the following terms:〈TτLmm(τ1)Lmm(τ2)Lmm(τ3)〉0 〈TτLmm(τ1)Lmn(τ2)Lnm(τ3)〉0〈TτLmn(τ1)Lnp(τ2)Lpm(τ3)〉0.Of course, under the time ordering operator, any permutation of the operators is also allowed.The first term simply gives a factor of Dm. In the following, we set τi− τ j = τi j for brevity.The second term yields (assume n> m)〈TτLmm(τ1)Lmn(τ2)Lnm(τ3)〉0 =−DmnK0mn(τ12)K0mn(τ31)−DmK0mn(τ32), (A.18)and, for the third term, we find〈TτLmn(τ1)Lnp(τ2)Lpm(τ3)〉0 (A.19)=−DpnK0mn(τ31)K0pn(τ23)+DpmK0mn(τ21)K0pm(τ23).173At fourth order we must consider:〈TτLmm(τ1)Lmm(τ2)Lmm(τ3)Lmm(τ4)〉0 (A.20)〈TτLmm(τ1)Lnn(τ2)Lmn(τ3)Lnm(τ4)〉0 〈TτLmm(τ1)Lmm(τ2)Lmn(τ3)Lnm(τ4)〉0〈TτLmm(τ1)Lmn(τ2)Lnp(τ3)Lpm(τ4)〉0 〈TτLmn(τ1)Lnm(τ2)Lmn(τ3)Lnm(τ4)〉0〈TτLmn(τ1)Lnm(τ2)Lmp(τ3)Lpm(τ4)〉0 〈TτLmn(τ1)Lnp(τ2)Lpq(τ3)Lqm(τ4)〉0.Under the time ordering operator all permutations of these operators are also allowed. The firstterm simply gives a factor of Dm. The second and third terms are given by〈TτLmm(τ1)Lmm(τ2)Lmn(τ3)Lnm(τ4)〉0 (A.21)= K0mn(τ13)〈TτLmn(τ1)Lmm(τ2)Lnm(τ4)〉0+K0mn(τ23)〈TτLmm(τ1)Lmn(τ2)Lnm(τ4)〉0−DmK0mn(τ43)〈TτLmm(τ1)Lnn(τ2)Lmn(τ3)Lnm(τ4)〉0= K0mn(τ13)〈TτLmn(τ1)Lnn(τ2)Lnm(τ4)〉0−K0mn(τ23)〈TτLmm(τ1)Lmn(τ2)Lnm(τ4)〉0,where the contractions of three operators may be read from equations (A.18) and (A.19). Theremaining fourth order contractions are given as follows〈TτLmm(τ1)Lmn(τ2)Lnp(τ3)Lpm(τ4)〉0 (A.22)= K0mn(τ12)〈TτLmn(τ1)Lnp(τ3)Lpm(τ4)〉0−K0mn(τ32)〈TτLmm(τ1)Lmp(τ3)Lpm(τ4)〉0〈TτLmn(τ1)Lnm(τ2)Lmn(τ3)Lnm(τ4)〉0= K0mn(τ23)〈TτLmn(τ1)[Lnn(τ2)−Lmm(τ2)]Lnm(τ4)〉0+K0mn(τ43)〈TτLmn(τ1)Lnm(τ2)[Lnn(τ4)−Lmm(τ4)]〉0〈TτLmn(τ1)Lnm(τ2)Lmp(τ3)Lpm(τ4)〉0=−K0mn(τ21)〈TτLmm(τ2)Lmp(τ3)Lpm(τ4)〉0+K0mn(τ41)〈TτLnm(τ2)Lmp(τ3)Lpn(τ4)〉0〈TτLmn(τ1)Lnp(τ2)Lpq(τ3)Lqm(τ4)〉0=−K0mn(τ21)〈TτLmp(τ2)Lpq(τ3)Lqm(τ4)〉0+K0mn(τ41)〈TτLnp(τ2)Lpq(τ3)Lqn(τ4)〉0.By starting the reduction with Lnp in the first equation in expression (A.22), expressions for174spin correlations and cumulants simplify in the low temperature limit. Starting with Lnp wefind〈TτLmm(τ1)Lmn(τ2)Lnp(τ3)Lpm(τ4)〉0 (A.23)= K0np(τ23)〈TτLmm(τ1)Lmp(τ2)Lpm(τ4)〉0−K0np(τ43)〈TτLmm(τ1)Lmn(τ2)Lnm(τ4)〉0.The expressions derived here for contractions of three and four MF operators will be usedin Appendix E to derive spin cumulants. It is the spin cumulants that play a central role in theanalysis of fluctuations in magnetic systems, as was shown in Chapter 6 of this thesis.175Appendix BTransverse Ising ModelHere we consider the spin half transverse Ising model in the absence of any coupling to nuclearspinsH =−12∑i6= jVi jSziSzj−∆∑iSxi −h∑iSzi , (B.1)where ~S= 12~τ are spin half operators, and~τ are the Pauli matrices. We analyze the model usingthe mean field (MF) operators presented in Appendix A. The results here are well known;however, we present them to illustrate the use of the MF operators, and so that the resultingequations are available for reference in this thesis.The Hamiltonian may be divided into the MF part and the fluctuating part, H =HMF +H ′. The MF part of the Hamiltonian is given byHMF =−∆∑iSxi −H∑iSzi , (B.2)where H = h+V0〈Sz〉. If we consider a simple cubic crystal with nearest neighbour interac-tions, the interaction strength is V0 = ∑ jVi j = 6J. The fluctuations are given byH ′ =−12∑i 6= jVi jS˜zi S˜zj, (B.3)where S˜zi = Szi −〈Sz〉0, where the subscript zero denotes the average is to be taken with respectto the MF Hamiltonian. Note that a constant contribution to the ground state energy has beendropped.176Mean Field Operator ApproachWe work in a basis consisting of eigenstates of the MF HamiltonianHMFi =(−H2 −∆2−∆2 H2). (B.4)The eigenvalues of this matrix are E± =±E =±√∆24 +H24 and the associated eigenvalues are|Ψ1〉= α1(H2 +∆2 +E−H2 + ∆2 +E)|Ψ2〉= α2(−H2 + ∆2 +E−H2 − ∆2 −E), (B.5)where 1 corresponds to the E− state and 2 corresponds to the E+ state. If we define the MFoperators to be Li j = |Ψi〉〈Ψ j|, then the z component of the electronic spin operator is given bySz = c11[L11−L22]+ c12[L12+L21], (B.6)wherec11 =H4Ec12 =∆4E. (B.7)MagnetizationThe ground state MF magnetization is given by〈Ψ1|Sz|Ψ1〉= 〈Sz〉0 = h+V0〈Sz〉02√∆2+(h+V0〈Sz〉0)2. (B.8)Inspection of this equation immediately reveals that in order to have 〈Sz〉0 = 0, we requireh= 0. With h= 0, we see that 〈Sz〉0 = 0 is a possible solution. Another possible solution is〈Sz〉0 = 12√1−(2∆V0)2(B.9)provided that ∆< V02 . This solution corresponds to the ferromagnetic phase of the system.177Susceptibility and Correlation FunctionsThe longitudinal component of the static susceptibility is given by χzz = ∂ 〈Sz〉0∂h . Differentiat-ing the ground state MF magnetization with respect to h, we find the zero temperature staticsusceptibility to beχzz =χzz0 (H)1−V0χzz0 (H), (B.10)where the single ion susceptibility isχzz0 (H) =∆22(∆2+H2)32≈ 12∆[1− 3H22∆2+O(H4∆4)]. (B.11)As will be shown, the static susceptibility may also be obtained from the imaginary time con-nected two point correlation function.Using the Matsubara formalism, the cumulant part of the two point correlation function, orconnected Green’s function, for the system is given byG(k,τ) =−〈Tτ S˜zk(τ)S˜z−k(0)〉=−〈Tτ S˜zk(τ)exp(− ∫ β0 dτV (τ))S˜z−k(0)〉0〈Tτ exp(− ∫ β0 dτV (τ))〉0(B.12)where in the final expression the averages are taken with respect to HMF . We define theunperturbed propagator to beg(τ) =−〈Tτ S˜z(τ)S˜z(0)〉0(B.13)=−c212〈L12L21+L21L12〉0− c211〈L11L11+L22L22〉0+ 〈Sz〉20,where the τ dependence of the operators has been suppressed. Note that the cross terms be-tween the Li j and Lii operators, as well as the terms between the Lii and L j j operators withi 6= j, are identically zero. Contracting the operators and transforming to frequency space wefind−g(iωn) = c212D122E21E221− (iωr)2+βc211(D1+D2−D212)δωr,0, (B.14)with E21 = 2E =√∆2+H2. Note that at T = 0, we have −g(0) = χzz0 , where χzz0 is the static178single ion susceptibility given in (B.11). We find that when 〈Sz〉 6= 0, at finite temperatures, thespectrum develops a purely elastic zero frequency component.In the random phase approximation, the Green’s function is given byGRPA(k, iωn) =g(iωn)1+g(iωn)Vk. (B.15)We see that in the zero temperature limit the static susceptibility given in equation (B.10)follows from the RPA Green’s functionχzz =−GRPA(0,0)∣∣∣∣T=0=−g(0)1+g(0)V0∣∣∣∣T=0. (B.16)For low energies, in the limit T → 0, we find−g(iωn)≈ A+B(iωn)2, (B.17)withA=2c212D12E21B=2c212D12E321. (B.18)The Green’s function is given byG(k, iωn) =11g +Vk=−A2B[AB − (iωn)2− VkA2B ], (B.19)and the spectrum is given byωk =√AB− VkA2B= E21√1−Vkχzz0 (H). (B.20)In the absence of a longitudinal field, the spectrum softens to zero at a critical transverse field∆c defined implicitly by 1 = V0χzz0 |H=0. To leading order in H∆ , the gap in the spectrum at ∆cdue to a longitudinal field will beω0 ≈ H√3V04∆c=√32H. (B.21)These expressions prove useful for comparison with the results of the spin half spin half modeldealt with in Chapter 4.179Appendix CA Diagramatic Expansion for SpinSystemsWe begin this appendix with a general discussion of spin cumulants and their associated gen-erating functions. Spin cumulants play a central role in the treatment of interacting systemsof spins using diagrammatic perturbation theory. The development of a diagrammatic methodfor treating quantum Ising systems is a primary accomplishment of this thesis; hence, spincumulants are discussed for reference in Section C.1.A diagrammatic expansion for spin systems was introduced by Brout in 1959 [60, 61].The history of this formalism, its application by Stinchcombe to the spin half transverse fieldIsing model [26], and related approaches, are discussed in Chapter 6, where we develop a fieldtheoretic formalism for quantum Ising systems, and a corresponding set of rules to performdiagrammatic perturbation theory, and show that the resultant theory is equivalent to that ofBrout. We present an overview of Brout’s theory for easy reference in Section C.2.Cumulants and the Generating FunctionFollowing a 1963 paper of Englert [97], we consider the expressionln〈etx〉=∞∑n=1tnn!Mn(x). (C.1)The function Z = 〈etx〉 is the generating function for x since its derivatives ∂ n∂ tn |t=0Z = 〈xn〉 gen-erate the moments of x, and Mn(x) are the associated cumulants. A moment can be expressed180in terms of cumulants by considering all possible subdivisions of the moment in question,Mn(x) = 〈xn〉− ∑n1+n2+...+nk=nMn1Mn2 . . .Mnk . (C.2)The first four cumulants of x are given in terms of the moments byM1(x) = 〈x〉 (C.3)M2(x) = 〈x2〉−〈x〉2M3(x) = 〈x3〉−3〈x2〉〈x〉+2〈x〉3M4(x) = 〈x4〉−4〈x3〉〈x〉−3〈x2〉2+12〈x2〉〈x〉2−6〈x〉4.For a spin system, we generalize to the joint cumulant generating functionlnZ = ln〈Tτ exp(∑ihµi (τ j)Sµi (τ j))〉0, (C.4)where we have given each imaginary time τ a label j. This is to indicate we are dealing with adiscrete set of imaginary times; summation over the set is implicit. That ishµi (τ j)Sµi (τ j) =n∑j=1hµi (τ j)Sµi (τ j). (C.5)The superscript µ = (x,y,z) or (+,−,z), and again, summation over the repeated index isimplicit. Joint cumulants for spin systems are discussed in a 1963 paper of Stinchcombe et al.[98]. The cumulants are given byn∏j=1∂∂hµ(τ j)∣∣∣∣h=0lnZ =Mn(Tτn∏j=1Sµ(τ j)). (C.6)Note that in the definition of the cumulant, we have left off the site index of the spin. Thisis because cumulants of spins belonging to different sites are equal to zero due to their statis-tical independence (averages are with respect to the single ion Hamiltonian H0). Indeed, thestatistical independence of spins at different sites leads toln〈Tτ exp[∑ihµi (τ j)Sµi (τ j)]〉0=∑iln〈Tτ exp[hµi (τ j)Sµi (τ j)]〉0, (C.7)because spins on different sites commute. We see that any mixed partial derivatives of ourcumulant generating function involving spins at different sites yields zero. Hence, any joint181cumulant of spins at different sites is equal to zero.We may generalize our discussion of spin cumulants to a continous formulation by defin-ing the generating functional to be Z =〈Tτ exp(∫ β0 dτ f (τ))〉0. The cumulant generating func-tional is then given bylnZ =∑n1n!∫ β0dτ1 . . .∫ β0dτnMn(Tτ f (τ1) . . . f (τn)). (C.8)This is known as Kubo’s generalized cumulant expansion, and is reviewed in a clear 1962 paperof Kubo [126]. If we take f (τ) = hµ(τ)Sµ(τ) then the moment of a time ordered product ofspins and the associated cumulants are obtained by taking functional derivatives with respectto hµ(τ), rather than taking ordinary derivatives as in equation (C.6).The Diagramatic ExpansionAverages of spin operators are of primary importance in magnetic systems. The two pointcorrelation function yields the excitation spectrum of the system and is intimately related withthe magnetic susceptibility, as will be discussed in Appendix D. Brout, building on his workon random ferromagnets, has introduced a diagrammatic perturbation theory for calculatingaverages of spin operators where the perturbation parameter is 1z , z being the effective numberof neighbours felt by each spin [60, 61]. This is known as the high density approximation. Webriefly outline Brout’s formalism here.Taking Qn(τ1, . . . ,τn) = Tτ∏i Sµini (τi), with ni being a site index, and µi ∈ (+,−,z), we’reinterested in calculating averages of the form〈Q〉=〈Tτ exp(− ∫ β0 dτV ′(τ))Q〉0〈Tτ exp(− ∫ β0 dτV ′(τ))〉0, (C.9)where, suppressing the imaginary time dependence, V ′ =−12 ∑i 6= jV µνi j Sµi Sνj . The prime on theinteraction is meant to indicate i = j is excluded from the summation. We now introduce afactor of ξ into the average, V ′(τ)→ ξV ′(τ) and Q→ ξQ, to keep track of the order of termsin a series expansion, and we bring the denominator of the above expression into the numerator182as followsξ 〈Q〉=∑∞n=0(−1)nn! ξn+1 ∫ β0 . . .∫ β0 dτ1 . . .dτn〈TτV ′(τ1) . . .V ′(τn)Q〉0∑∞n=0(−1)nn! ξ n∫ β0 . . .∫ β0 dτ1 . . .dτn〈TτV ′(τ1) . . .V ′(τn)〉0(C.10)=∞∑n=1(−ξ )n−1(n−1)!∫ β0. . .∫ β0dτ1 . . .dτn−1Mn(V (τ1) . . .V (τn−1)Q),where V =−12 ∑i jV µνi j Sµi Sνj now includes terms where i= j. In the first line, the spatial sumsin the interaction V ′ exclude terms with i= j. In order to obtain the cumulants in our diagram-matic theory, we add and subtract the missing terms, which leads to the unrestricted sum in theinteraction V in the final line. Now that ξ has served its purpose, we set it to one. Note thateach power of the interaction V carries a factor of −1 which cancels with the overall factorof (−1)n−1; hence, for a ferromagnet, the overall sign of each diagram is positive. Mn in theabove expression is to be taken as the cumulants of the interaction V and Qn, i.e.,M1(Q) = 〈Q〉0 M2(Vτ1Q) = 〈Vτ1Q〉0−〈Vτ1〉0〈Q〉0, (C.11)and so on. Recall that the interaction V involves a double sum over crystal sites, and, whenthe average is taken with respect to H0, spins at different sites are statistically independent.By expanding the above expressions in terms of cumulants at single sites, we are led to thefollowing graphical representation of the series in equation (C.10) [98]:1. There are primary and secondary vertices. The primary vertices are labelled by the spinsin Q, and the secondary vertices are labelled by spins from the interaction. Draw a circlearound each primary vertex.2. Each bond carries a factor of i, j,τi, and makes a contribution of 12Vµνi j from the inter-action. The bonds representing an interaction between the z components of two spinsare given by a wavy line. The bonds representing an S+ or S− interaction are given bystraight lines with an arrow.3. Join the bonds to the primary vertices and each other in all possible ways. Neglect anydiagrams in which the bonds are completely independent of the primary vertices. Eachvertex represents a cumulant of the spin operators. The cumulant is determined by thenumber of bonds flowing into it; each wavy line is a factor of Sz, each straight lineflowing into the vertex is a factor of S−, and each straight line flowing out of a vertex is afactor of S+. The spin from the primary vertex should be included in the cumulant from183the primary vertex.4. Each graph carries a symmetry factor. We get a factor of (n−1)! from permutations ofthe interactionVτ , that cancels with the factor of 1(n−1)! in the perturbation expansion. Wealso get a factor of 2nl , where nl is the number of longitudinal bonds, from swapping thespins in each longitudinal interaction. Finally, since not all permutations lead to distinctdiagrams, we must divide by a factor of g, the symmetry factor for each diagram.5. Last of all, we integrate over all imaginary times, and sum over all spatial indices, fromthe secondary vertices. The sum over the spatial indices is unrestricted.Fourier transforming each spin into frequency space, and each interaction into momentumspace, leads to the same set of rules, only now we label each bond with momentum q andMatsubara frequency iωn. We conserve momentum and frequency at each vertex and sum overmomenta and frequencies from each secondary vertex. Each momentum summation carries afactor of 1N , where N is the total number of sites in the sample. Note that in each diagram, allof the momentum dependence comes from the interaction, which is associated with the lines inthe diagram. The vertices, which are associated with the spin cumulants, carry no momentumdependence. The spatial index on the spins is no longer relevant, as we are now dealing withcumulants of spins at a single site.We have calculated the moment of a time ordered product of n spins 〈Qn〉. In our diagram-matic rules for 〈Qn〉, we allow the diagrams to have disjoint parts provided that each disjointpart contains at least one primary vertex, viz., we must sum over every possible subdivisionof the primary vertices. Recall that a moment can be expressed in terms of cumulants byconsidering all possible subdivisions of the moment in question. That is,〈Qn〉=Mn+ ∑n1+n2+...+nk=nMn1Mn2 . . .Mnk , (C.12)where the second term represents the sum over the product of all possible cumulants of lowerorder. From this it follows that the nth order cumulant Mn is given by all totally connecteddiagrams, where each primary vertex is connected to every other primary vertex.In Chapter 6 of this thesis, we derive an effective field theory for quantum Ising systems.By performing a perturbation expansion of the field theory in the standard way, we are able toderive a set of diagrammatic rules equivalent to those listed above. The field theory is concep-tually much simpler than the approach of Brout, and offers the advantage of a renormalizationgroup treatment of a system’s critical behaviour.Recall that the perturbative parameter in our treatment of spin systems is 1z , z being theeffective number of neighbouring spins. In momentum space, the order of each diagram is184determined by the number of free momentum summations it contains. Indeed, following ref-erence [127], if r0 is the effective range of the interaction, and a is the density of spins, thenz∼ r30a, and the range in reciprocal space will be q0 ∼ 1r0 ∼ (az )13 . In one dimension, the frac-tion of terms in the first Brillouin zone for which q < q0 isn0n =q02pia . Similarly, the fractionof terms N0N for which q< q0 in a momentum summation over the first Brillouin zone in threedimensions isN0N=43piq30(2pi)3a∼ 1z. (C.13)Now, if g(q) is some function with range q0 we find1N∑qg(q)∼ N0Ng¯= g¯1z, (C.14)where g¯ is the average of g(q) with q≤ q0.185Appendix DSusceptibilities and Correlation FunctionsConsider a Hamiltonian for a spin system of the form H = H0 +V , where H0 = ∑iHµ0i is asum of single ion Hamiltonians, and V =−12 ∑i 6= jV µνi j Sµi Sνj is an interaction between spins atdifferent sites. The indices are µ,ν = x,y,z, or µ,ν = z,+,−, depending on what proves moreconvenient, and we sum over repeated indices. In the Matsubara formalism, the cumulant partof the imaginary time spin spin correlation functions, or Green’s functions, are given byGµνi j (τ1− τ2) =−〈Tτ S˜µi (τ1)S˜νj (τ2)〉=−〈Tτ S˜µi (τ1)exp(− ∫ β0 dτV (τ))S˜νj (τ2)〉0〈Tτ exp(− ∫ β0 dτV (τ))〉0, (D.1)where S˜µi = Sµi −〈Sµi 〉. The averages in the final expression are taken with respect to H0. Asis conventional, we define the Green’s function with an overall minus sign. In the absenceof the minus sign in our definition of G, we use the symbol χ = −G because, upon analyticcontinuation to real times, χ will yield the dynamic susceptibility of the spin system. The polesof the Green’s functions yield the resonance peaks seen in, for example, neutron scatteringexperiments. A goal of condensed matter physics, and a primary accomplishment of this thesis,is to develop ways of evaluating expressions like equation (D.1). In the following, we willexplore some of its properties, and define other relevant, related, functions. The discussion inthe introduction to this appendix can be found in many textbooks. We reproduce the materialhere to establish notation and conventions, primarily following references [5, 128].186Transforming to Fourier space we haveSµi (τ) =1√N∑ke−ikriSµk (τ) Sµk (τ) =1√N∑ieikriSµi (τ) (D.2)Sµk (τ) =1β ∑neiωnτSµk (iωn) Sµk (iωn) =∫ β0dτe−iωnτSµk (τ)for the spins, andGµνi j (τ) =1N∑ke−ik(ri−r j)Gµνk (τ) Gµνk (τ) =1N∑i jeik(ri−r j)Gµνi j (τ) (D.3)Gµνk (τ1− τ2) =1β ∑ne−iωn(τ1−τ2)Gµνk (iωn) Gµνk (iωn) =∫ β0dτeiωnτGµνk (τ)for the correlation function. The interaction, in Fourier space, is given byV µνi j =1N∑qV µνq e−iq(ri−r j) V µνq =1N∑i jV µνi j eiq(ri−r j), (D.4)and ∫ β0dτV (τ) =−12∫ β0dτ∑i6= jV µνi j S˜µi (τ)S˜νj (τ) (D.5)=− 12β ∑q ∑nV µνq S˜µq (iωn)S˜ν−q(−iωn).As previously mentioned, the dynamic susceptibility (response function) of the system fol-lows from the imaginary time correlation function. The relation isχµνk (ω) =−Gµνk (iωn→ ω+ i0+), (D.6)where, in real space,χµνi j (t− t ′) =−Gµνi j (t− t ′) = iθ(t− t ′)〈[S˜µi (t), S˜νj (t ′)]〉. (D.7)The spectral density is given byρµνk (ω) = 2Im[χµνk (ω)](D.8)=2piNZ ∑m,nδ (ω+En−Em)〈n|Sµk |m〉〈m|Sν−k|n〉(e−βEn− e−βEm),187or, in terms of the Green’s functions,ρµνk (ω) = i[Gµνk (iωn→ ω+ i0+)−Gµνk (iωn→ ω− i0+)]. (D.9)Note that the spectral density is often written in terms of Aµνk (ω) =12piρµνk (ω) in whichcaseχµνk (ω) =−∫ ∞−∞dω ′Aµνk (ω′)ω−ω ′+ i0+ . (D.10)The real and imaginary parts of of the dynamic susceptibility are related by a Kramers-KronigtransformRe[χµνk (ω)] =P∫ ∞−∞dΩpiIm[χµνk (Ω)]Ω−ω . (D.11)The connected real time correlation function is given bySµνi j (t− t ′) = 〈S˜µi (t)S˜νj (t ′)〉. (D.12)This function is related to the spectral density by the fluctuation dissipation theoremSµνk (ω) =ρµνk (ω)1− e−βω . (D.13)Finally, we consider the static susceptibility, χµν(k) = Re[χµνk (ω = 0)]. In the high tem-perature limit (ω T ), we find it to be related to the equal time correlation function as follows,βSµνk (t = 0)≈∫ dωpiIm[χµνk (ω)]ω= χµν(k), (D.14)where the Kramers-Kronig relation is used to obtain the final expression. The classical expres-sion for the static susceptibility may also be obtained by differentiating the free energyχµνi j =−∂ 2F∂hµi ∂hνj, (D.15)where hµi is a field conjugate to spin Sµi . We differentiate between static and dynamic ex-pressions for the susceptibility by the absence of a time (or frequency) argument in the staticcase.188Ising Model: Structure of the Green’s FunctionIn this section, we specialize to the case of the Ising model, that is, we only consider a longi-tudinal interaction between spins on different sites. The formalism discussed in Appendix C,was applied to the Ising model in a transverse field in a series of 1973 papers by Stinchcombe[26, 62, 63]. Here, we review the structure of the connected longitudinal correlation function,as presented in reference [26].Recall from Appendix C that the connected two point longitudinal correlation function,or, equivalently, the second order cumulant, may be expressed as a sum of totally connecteddiagrams. If we define an irreducible diagram as a diagram that cannot be seperated into twoparts by severing a single bond, then the correlation function may be written asGµνk (iωn) = Gµνk (iωn)−G µzk (iωn)VkGzνk (iωn) (D.16)= G µνk (iωn)−G µzk (iωn)VkGzνk (iωn)1+VkG zzk (iωn),where G µν is the sum over all irreducible diagrams. This is an exact result; all the complexityof the problem is contained in the calculation of G µν . Note that µ and ν may refer electronicspin components, or they may equally well refer to nuclear spin components. The z indexrefers strictly to an electronic spin, and comes from the longitudinal interspin interaction. Ourdefinition of G differs from that of Stinchcombe by a minus sign. What Stinchcombe refersto as G, we refer to as χ . Furthermore, our G µν is equivalent to Stinchcombe’s −M µν , andStinchcombe uses a different Fourier transform convention than what is used in this thesis. Thisaccounts for the factors of β appearing in Stinchcombe’s work that aren’t present here. Theconventions adopted in this thesis are meant to eliminate numerical factors from subsequentcalculations.The longitudinal component of our Green’s function reduces toGzzk (iωn) =G zzk (iωn)1+VkG zzk (iωn). (D.17)To lowest order we have G zzk (iωn)→ g(iωn) = −〈Sz(iωn)Sz(−iωn)〉0, where the average istaken with respect to the single ion Hamiltonian. Making this substitution yields the longitudi-nal Green’s function in the random phase approximation (RPA),Gzzk (iωn)∣∣∣∣RPA=g(iωn)1+Vkg(iωn). (D.18)In order to analyze the collective excitations of a spin system beyond the RPA, the sum189over irreducible diagrams may be separated into its real and imaginary parts G zzk (ω + i0+) =Rezzk (ω)+ iImzzk (ω). This yieldsGzzk (ω+ i0+) =Rezzk (ω)+ iImzzk (ω)1+VkRezzk (ω)+ iVkImzzk (ω). (D.19)We define the collective mode Ek to be a solution of1+VkRezzk (Ek) = 0. (D.20)The lifetime Γpk associated with the pth collective mode E pk follows from the pth (complex)pole of G(ω), zp = E pk − iΓpk . It follows thatΓpk =VkImzzk (Epk ). (D.21)Alternatively, we may work with a self energy by defining G zzk = g[1+Σ]−1 which yieldsGzzk (iωn) =g(iωn)1+Vkg(iωn)+Σk(iωn). (D.22)RPA Green’s FunctionHere we examine the structure of the longitudinal RPA Green’s function of an Ising systemGk(iωn)≡ g(iωn)1+Vkg(iωn) , (D.23)making use of the mean field (MF) operators discussed in Appendix A. Writing the Green’sfunction as G−1k (iωn)= g−1(iωn)+Vk we see that the RPA Green’s function essentially consistsof a momentum dependent shift in the MF Green’s function due to the interaction Vk. The MFGreen’s function (to be derived in Appendix E) is given byg(iωr) =− ∑n>mc2mnDmn2EnmE2nm− (iωr)2−gelδωr,0. (D.24)This function contains poles at the differences between each of the systems MF eigenstates,Enm = En−Em, as well as an additional elastic contribution,gel = β(∑mc2mmDm−[∑mcmmDm]2), (D.25)190that vanishes in the paramagnetic phase, and in the limit T → 0. The cmn are the mean fieldmatrix elements of the Sz operator, and the population factors are Dmn = Dm −Dn, whereDm = Z−10 e−βEm .Making use of the identitylimε→0+1x± iε =P(1x)∓ ipiδ (x), (D.26)we take the real and imaginary parts of g(ω+ iε). We findRe[g(ω)] =− ∑n>mc2mnDmnP[2EnmE2nm−ω2]−gelδω,0, (D.27)whereP denotes the principle value, andIm[g(ω)] =−pi ∑n>m∣∣∣∣〈m|Sz|n〉0∣∣∣∣2Dmn(δ (Enm−ω)−δ (Enm+ω)). (D.28)Note that at zero temperature, the real part of g corresponds to the Van Vleck contribution tothe susceptibility, that is, it is a contribution arising solely from the splitting of the energy levelsof the system’s eigenstates.To understand the elastic contribution to the spectrum, it is advantageous to writeIm[g(ω)] =−pi∑nm∣∣∣∣〈m|Sz|n〉0∣∣∣∣2Dmnδ (Enm−ω), (D.29)where the summation over n and m is now unrestricted. If ω 6= 0, all terms with n= m vanish,as Dmnδ (ω) = 0; however, at ω = 0 the n = m terms lead to the elastic contribution to thespectrum, as we will show below. The Kramers-Kronig relation tells usRe[g(ω)] =P∫ ∞−∞dΩpiIm[g(Ω)]Ω−ω . (D.30)The integration is straightforward except near ω = 0 where we havegel = limε→0∫ ε−εdΩpiIm[g(Ω)]Ω. (D.31)For further discussion of the elastic contribution to the spectrum see the book Rare EarthMagnetism, by Jensen and Mackintosh [65].191We define the free field propagator to beDk(iωr) =11+g(iωr)Vk. (D.32)If the Ising model is treated using the Hubbard-Stratonovich transformation, the expressionabove will be the propagator for the free auxiliary field, hence the name. Nomenclature aside,we divide the free propagator into two parts, Dk(iωr) = Dk(iωr)+D0kδiωr.0, whereDk(iωr) =∏n>m[E2nm− (iωr)2]∏p[(Epk )2− (iωr)2] (D.33)=∏n>m[E2nm− (iωr)2]∏n>m[E2nm− (iωr)2]−Vk∑n>m c2mnDmn2Enm∏p,q6=m,n[E2pq− (iωr)2],andD0k =gelVk∏n>mE4nm∏p(Epk )2[∏p(Epk )2−gelVk∏n>mE2nm] . (D.34)It is convenient to factor the denominator of Dk(iωr) because, in the zero temperature limit,this yields the RPA modes of the system. At finite temperatures, there are additional polescorresponding to excitations between excited states. It is a matter of convenience to includethe iωr = 0 term in Dk(iωr), and then subtract it from D0k . Note that D0k ∝ gel vanishes as thetemperature goes to zero, and in the paramagnetic phase of the system. The ratio of the productof the mean field energy levels to the product of RPA energy levels occurs quite frequently,hence we will designate itRk =∏n>mE2nm∏p(Epk )2 . (D.35)In terms of Rk, we findD0k =gelVkR2k1−gelVkRk . (D.36)We now consider the inelastic part of the RPA Green’s function G = gD, where g is givenby equation (D.24) with gel = 0 and D is given by equation (D.33). D has zeros at all the MFenergy levels, while g has simple poles at all these levels. Hence, the poles of g contributenothing to the pole structure of the longitudinal RPA Green’s function. This is a rather trivialobservation; however, it is worth noting that any function F(z), with simple poles at the MF192energy levels, will not alter the pole structure of F(z)D(z). This fact proves useful whenexamining corrections to the RPA result.We now decompose the RPA Green’s function into a sum over its constituent modes, inorder to obtain the spectral density of each mode. The RPA Green’s function is given byGk(iωr) = G˜k(iωr)−Gelk δωr,0, (D.37)whereG˜k(iωr) =− ∑n>mc2mnDmn2Enm∏t>s 6=nm[E2ts− (iωr)2]∏n>m[E2nm− (iωr)2]−Vk∑n>m c2mnDmn2Enm∏p,q6=m,n[E2pq− (iωr)2], (D.38)and the elastic contribution isGelk = gelRk[1+2VkRk∑n>m c2mnDmnE−1nm1−gelVkRk]. (D.39)The denominator of G˜k(iωr) has been written out in full, rather than being expressed in termsof the RPA modes. This is to illustrate the fact that if one of the MF matrix elements cmnvanishes, there will be a common factor of E2nm− (iωr)2 in the numerator and denominator ofG˜k(iωr). Hence, when a MF matrix element is zero, there will be a pole at the correspondingMF energy level in the RPA spectrum; however, this pole will carry no spectral weight. Theinelastic part of the RPA Green’s function has the form G˜k(z) =P(z)Q(z) , where P(z) and Q(z) arepolynomials in z, and deg[P(z)]< deg[Q(z)]. Assuming no degenerate modes, we may performa partial fraction decomposition to obtainG˜k(z 6= 0) =∑pP(E pk )Q′(E pk )[1(z−E pk )− 1(z+E pk )], (D.40)whereP(E pk ) = P(−E pk ) = ∑n>mc2mnDmn2Enm ∏t>s 6=nm[E2ts− (E pk )2] (D.41)Q′(E pk ) =−Q′(−E pk ) = 2E pk ∏s 6=p[(E pk )2− (Esk)2].The spectral density isAk(ω) =− 1pi Im[Gk(iωr→ ω+ iδ )] =∑pP(E pk )Q′(E pk )[δ (ω−E pk )−δ (ω+E pk )], (D.42)193and the dynamic structure factor is given bySk(ω) =2pi1− e−βω ∑pP(E pk )Q′(E pk )[δ (ω−E pk )−δ (ω+E pk )]. (D.43)Integrating over frequencies, we find the equal time correlation function to be given bySk(t = 0) =∫ ∞−∞dω2piSk(ω) =∑pP(E pk )Q′(E pk )sinh(βE pk )cosh(βE pk )−1+1βGelk , (D.44)where, as with the mean field Green’s function, we have included the elastic contribution stem-ming from the zero frequency pole. We may compare this expression to its classical counterpart(ω << T ). We findχk = βSck(t = 0) =∑p2E pkP(E pk )Q′(E pk )+Gelk . (D.45)As a consistency check of the formalism presented here, consider the imaginary time dy-namic correlation functionχk(iωr) =−Gk(iωr) =−∑p2E pkP(E pk )Q′(E pk )[1(iωr)2− (E pk )2]+Gelk δωr,0. (D.46)Summing over Matsubara frequencies should yield the equal time correlation function,Sk(τ = 0) =1β ∑rχk(iωr)T→0=∫ ∞−∞dω2pi ∑p2E pkP(E pk )Q′(E pk )[1ω2+(E pk )2]∣∣∣∣T=0. (D.47)By performing the frequency summation, or the integral in the zero temperature limit, equation(D.44) may be recovered, as expected.As a final note, we point out how the analytic structure of the Green’s function changeswhen the effect of fluctuations beyond the Gaussian approximation (RPA) are included byrenormalizing the system at its critical point. In the Gaussian approximation, the Green’sfunction has the formGk(iωn)∣∣∣∣RPA∝1(iωn)2−E2k. (D.48)194Under renormalization, this changes toGk(iωn) ∝[(iωn)2−E2k]−1+η2. (D.49)This shows that at the critical point, rather than having the usual quasiparticle pole in theGreen’s function, there is a branch cut corresponding to a continuum of excitations. Thisimplies the excitations at the quantum critical point are overdamped, and the system will showrelaxational dynamics [28–30].195Appendix ESpin CumulantsSpin cumulants play a central role in the analysis of interacting spin systems. In this appendix,we calculate cumulants of up to four spin operators for a system with an arbitrary single ionHamiltonian. Cumulants of this order are necessary for calculating corrections of order 1z inthe high density approximation developed in Chapter 6. We calculate the cumulants using theformalism discussed in Appendix A. Our focus will be on cumulants of the Sz operator; how-ever, as any operator may be expanded in terms of mean field (MF) eigenstates, the results arequite general. Essentially, with a little modification, the results of this appendix are completelygeneral expressions for the cumulants, or correlation functions, of operators acting on a systemwith a discrete set of energy levels.The calculation of the cumulants is straightforward, but rather tedious, and the resultingequations are quite cumbersome. We present them here for reference, and note a significantsimplification that occurs in the zero temperature limit. In writing out the cumulants we willmake extensive use of the function (and products of these functions)K0nm(iωr) =1En−Em− iωr , (E.1)the properties of which are discussed in Appendix A.We begin by writing Sz in the MF basisSz =∑mcmmLmm+ ∑m6=ncmnLmn. (E.2)The lowest order cumulant is given byM1(Sz) = 〈Sz〉0 =∑mcmmDm. (E.3)196Two Spin CumulantDefining Si ≡ Sz(τi) and Limn ≡ Lmn(τi), we find at second orderM2(TτS1S2) =∑m,ncmmcnn〈TτL1mmL2nn〉0+ ∑P{1,2}∑n>m|cmn|2〈TτL1mnL2nm〉0−[∑mcmmDm]2(E.4)where P{1,2, . . .n} denotes the set of all permutations. Contracting the MF operators we find〈TτL1mmL2nn〉0 = Dmδmn ∑P{1,2}〈TτL1mnL2nm〉0 =−DnmKnm(τ12) (E.5)where we’ve set τi j ≡ τi− τ j for brevity, and defined the MF propagator to beKnm(τ) = K0nm(τ)+K0nm(−τ). (E.6)We now transform our cumulant to frequency spaceM2(ωr1,ωr2) =F{M2(TτS1S2)}=2∏i=1∫ β0dτieiωriτiM2(TτS1S2). (E.7)Note that2∏i=1∫ β0dτieiωriτiKnm(τ12) = β∑λsKnm(λs)δωr1 ,λsδωr2 ,−λs = βKnm(ωr1)δωr1+ωr2 ,0, (E.8)whereKnm(ωr1) =2EnmE2mn− (iωr1)2. (E.9)Our final result for the second order cumulant isM2(ωr1,ωr2) =β ∑n>m|cmn|2Dnm 2EmnE2mn− (iωr1)2δωr1+ωr2 ,0 (E.10)+β 2(∑mc2mmDm−[∑mcmmDm]2) 2∏i=1δωri ,0.197Performing a frequency summation, we obtain the MF correlation function of the Sz operatorg(ωr1)≡−1β ∑r2M2(ωr1,ωr2) (E.11)=− ∑n>m|cmn|2Dmn 2EnmE2nm− (iωr1)2−β(∑mc2mmDm−[∑mcmmDm]2)δωr1 ,0,which can be written in terms of Knm(iωr) as−g(ωr1) = ∑n>m|cmn|2DmnKnm(iωr)+β(∑mc2mmDm−[∑mcmmDm]2)δωr1 ,0. (E.12)In the zero temperature limit, this expression reduces to−g(ωr1) =∑n>1 |c1n|22En1∏m 6=nE2m1− (iωr1)2∏nE2n1− (iωr1)2. (E.13)Three Spin CumulantAt third order we haveM3(TτS1S2S3) =〈TτS1S2S3〉0−M2(TτS1S2)〈S3〉0−M2(TτS1S3)〈S2〉0 (E.14)−M2(TτS2S3)〈S1〉0−〈S1〉0〈S2〉0〈S3〉0Expanding in the mean field basis, we find〈TτS1S2S3〉0 =∑mc3mm〈TτL1mmL2mmL3mm〉0 (E.15)+ ∑P{1,2,3}[∑m6=ncmm|cmn|2〈TτL1mmL2mnL3nm〉0+ ∑m6=n6=pcmncnpcpm〈TτL1mnL2npL3pm〉0].In the case of a two level system, the final term in this expression is zero, and the third ordercumulant vanishes altogether above the MF critical temperature. In principle, in a system withmore than two levels, this cumulant may be non-zero, even in the paramagnetic phase of thesystem. In the summation in the third term, we sum over all m, and for each m we sum overall n, p>m such that n 6= p. Using the formalism in Appendix A to contract the MF operators,198we find∑P{1,2,3}〈TτL1mmL2mnL3nm〉0 =−Dmn ∑P{1,2,3}Kmnmn (τ12;τ31) (E.16)−Dm[Kmn(τ21)+Kmn(τ31)+Kmn(τ32)]∑P{1,2,3}〈TτL1mnL2npL3pm〉0 =−Dpn ∑P{1,2,3}Kpnmn(τ31;τ23)+Dpm ∑P{1,2,3}Kpmmn (τ21;τ23),where we’ve definedKmn(τ)≡ K0mn(τ)+K0mn(−τ) Kpqmn(τ1;τ2)≡ K0mn(τ1)K0pq(τ2). (E.17)We now transform our cumulant to frequency spaceM3(ωr1,ωr2,ωr3) =F{M3(TτS1S2S3)}=3∏i=1∫ β0dτieiωriτiM3(TτS1S2S3) (E.18)=F{〈TτS1S2S3)〉0}−βM2(ω1,ω2)〈S3〉0δωr3 ,0−βM2(ω1,ω3)〈S3〉0δωr2 ,0−βM2(ω2,ω3)〈S3〉0δωr1 ,0−β3〈S〉303∏i=1δωi,0.For reference, we note thatF{Kmn(τ12)}= β 2∑sKmn(λs)δωr1 ,λsδωr2 ,−λsδωr3 ,0 = β2Kmn(ωr1)δωr1+ωr2 ,0δωr3 ,0 (E.19)andF{Kpqmn(τ12;τ31)}= β∑stK0mn(λs)K0pq(λt)δωr1 ,λs−λtδωr2 ,−λsδωr3 ,λt (E.20)= βKpqmn(−ωr2;ωr3)δωr1+ωr2+ωr3 ,0,where, in the last line, we defineK0mn(ωr)K0pq(ωs)≡ Kpqmn(ωr;ωs). (E.21)Making use of these expressions we find our Fourier transformed spin averages to be given byF{〈TτL1mmL2mmL3mm〉0}= β 3Dm3∏i=1δωri ,0, (E.22)199andA˜1 =F{∑P{1,2,3}〈TτL1mmL2mnL3nm〉0}= βDmn ∑P{ωri}Kmnnm (ωr1;ωr2)δωr1+ωr2+ωr3 ,0 (E.23)−β 2Dm[Kmn(ωr2)δωr2+ωr3 ,0δωr1 ,0+Kmn(ωr1)δωr1+ωr3 ,0δωr2 ,0+Kmn(ωr1)δωr1+ωr2 ,0δωr3 ,0]A2 =F{∑P{1,2,3}〈TτL1mnL2npL3pm〉0}= βDpm ∑P{ωri}Knmmp(ωr1;ωr2)δωr1+ωr2+ωr3 ,0+βDpn ∑P{ωri}Knmpn (ωr1;ωr2)δωr1+ωr2+ωr3 ,0.Note that we have made use of the fact K0mn(−ωr) =−K0nm(ωr).Some algebra will show that, in the low temperature limit, terms in 〈TτS1S2S3〉0 cancel withthe lower order cumulants, significantly reducing the complexity of the expression. We are leftwithlimT→0M3(ωr1,ωr2,ωr3) = ∑m6=ncmm|cmn|2A1(ωr1,ωr2,ωr3)+ ∑m 6=n6=pcmncnpcpmA2(ωr1,ωr2,ωr3)(E.24)withA1(ωr1,ωr2,ωr3) = βDmn ∑P{ωri}Kmnnm (ωr1;ωr2)δωr1+ωr2+ωr3 ,0 (E.25)A2(ωr1,ωr2,ωr3) = βDpm ∑P{ωri}Knmmp(ωr1;ωr2)δωr1+ωr2+ωr3 ,0+βDpn ∑P{ωri}Knmpn (ωr1;ωr2)δωr1+ωr2+ωr3 ,0.This may be further reduced by considering the population factors in the Ais. We findlimT→0M3(ωr1,ωr2,ωr3) =∑n>1(c11− cnn)|c1n|2A01(ωr1,ωr2,ωr3) (E.26)+ ∑n>1p>nRe[c1ncnpcp1]A02(ωr1,ωr2,ωr3),200whereA01(ωr1,ωr2,ωr3) = β ∑P{ωri}K1nn1 (ωr1;ωr2)δωr1+ωr2+ωr3 ,0 (E.27)A02(ωr1,ωr2,ωr3) = 2β ∑P{ωri}Ep1En1− (iωr1)(iωr2)(E2p1− (iωr1)2)(E2n1− (iωr2)2)δωr1+ωr2+ωr3 ,0.In the terms involving transitions between three MF eigenstates, we have used the fact thatcnp = c∗pn for any np to combine terms in the sum.Four Spin CumulantWe now calculate the fourth order cumulant. We start our reduction of the MF operators withLmn such that m is minimal, unless otherwise noted. We findM4(TτS1S2S3S4) = 〈TτS1S2S3S4〉0−〈S1〉0〈S2〉0〈S3〉0〈S4〉0 (E.28)−M2(TτS1S2)M2(S3S4)−M2(TτS1S3)M2(TτS2S4)−M2(TτS1S4)M2(TτS2S3)−M3(TτS1S2S3)〈S4〉0−M3(TτS1S2S4)〈S3〉0−M3(TτS1S3S4)〈S2〉0−M3(TτS2S3S4)〈S1〉0−M2(TτS1S2)〉0〈S3〉0〈S4〉0−M2(TτS1S3)〉0〈S2〉0〈S4〉0−M2(TτS1S4)〉0〈S2〉0〈S3〉0−M2(TτS2S3)〉0〈S1〉0〈S4〉0−M2(TτS2S4)〉0〈S1〉0〈S3〉0−M2(TτS3S4)〉0〈S1〉0〈S2〉0Expanding in terms of the MF operators, we find the fourth order spin correlation function tobe given by〈TτS1S2S3S4〉0 =∑mc4mm〈TτL1mmL2mmL3mmL4mm〉0 (E.29)+ ∑m 6=nc2mm|cmn|2B˜1+ ∑n>mcmmcnn|cmn|2B˜2+ ∑n>m|cmn|4B˜3+ ∑m 6=n6=pcmmcmncnpcpmB˜4+ ∑p>n>m|cmn|2|cmp|2B˜5+∑m∑n>m∑p>mp6=n∑q>mq6=n,pcmncnpcpqcqmB6,201whereB˜1 = ∑P{2,3,4}〈TτL1mmL2mmL3mnL4nm〉0+ ∑P{1,4}〈TτL2mmL3mmL1mnL4nm〉0 (E.30)+ ∑P{1,3}〈TτL2mmL4mmL1mnL3nm〉0+ ∑P{1,2}〈TτL3mmL4mmL1mnL2nm〉0B˜2 = ∑P{1,2,3,4}〈TτL1mmL2nnL3mnL4nm〉0B˜3 = 〈TτL3mnL1nmL4mnL2nm〉0+ 〈TτL1mnL3nmL2mnL4nm〉0+ ∑P{1,2}∑P{3,4}〈TτL1mnL2nmL3mnL4nm〉0B˜4 = ∑P{1,2,3,4}〈TτL1mmL2mnL3npL4pm〉0B˜5 = ∑P{1,2,3,4}〈TτL1mnL2nmL3mpL4pm〉0B6 = ∑P{1,2,3,4}〈TτL1mnL2npL3pqL4qm〉0The terms with repeated MF operators must be dealt with carefully, so as to not over countthe possible unique time ordered averages of the operators. We place a tilde over the first fivefunctions to differentiate them from the expressions obtained in the low temperature limit.Contracting the MF operators leads to rather lengthy expressions which are listed here forreference. We begin by making the following definitionsrsKpqmn(τab;τcd;τe f ) = K0mn(τab)K0pq(τcd)K0rs(τe f ), (E.31)andK˜pqmn(τab;τcd) = Kpqmn(τab;τcd)+Kpqmn(τba;τdc) (E.32)rsK˜pqmn(τab;τcd;τe f ) = rsKpqmn(τab;τcd;τe f )+ rsKpqmn(τba;τdc;τ f e).202We findB˜1 =−Dmn[mnK˜mnmn (τ12;τ23;τ34)+ mnK˜mnmn (τ12;τ31;τ43)+ mnK˜mnmn (τ12;τ34;τ41) (E.33)+ mnK˜mnmn (τ12;τ24;τ31)+ mnK˜mnmn (τ12;τ23;τ41)+ mnK˜mnmn (τ12;τ24;τ43)+ mnK˜mnmn (τ13;τ34;τ42)+ mnK˜mnmn (τ13;τ32;τ24)+ mnK˜mnmn (τ13;τ32;τ41)+ mnK˜mnmn (τ13;τ41;τ24)+ mnK˜mnmn (τ14;τ43;τ32)+ mnK˜mnmn (τ14;τ42;τ23)]−Dm[K˜mnmn (τ12;τ31)+ K˜mnmn (τ12;τ24)+ K˜mnmn (τ12;τ23)+ K˜mnmn (τ12;τ41)+ K˜mnmn (τ13;τ32)+ K˜mnmn (τ13;τ41)+ K˜mnmn (τ13;τ34)+ K˜mnmn (τ14;τ42)+ K˜mnmn (τ14;τ43)++K˜mnmn (τ23;τ34)+ K˜mnmn (τ23;τ42)+ K˜mnmn (τ24;τ43)]−Dm[Kmn(τ12)+Kmn(τ13)+Kmn(τ14)+Kmn(τ23)+Kmn(τ24)+Kmn(τ34)],andB˜2 = ∑P{1,2,3,4}[Dmn mnKmnmn (τ23;τ12;τ41)+Dmn mnKmnmn (τ13;τ21;τ42) (E.34)+DmKmnmn (τ23;τ42)+DnKmnmn (τ13;τ41)],203andB˜3 = Dmn[K0mn(τ23)(Knmnm (τ24;τ12)+Kmnmn (τ21;τ42))(E.35)+K0mn(τ32)(Knmnm (τ34;τ13)+Kmnmn (τ31;τ43))+K0mn(τ43)(K˜nmnm (τ42;τ14)+ K˜mnmn (τ41;τ24))+K0mn(τ34)(K˜nmnm (τ32;τ13)+ K˜mnmn (τ31;τ23))+K0mn(τ13)(Knmnm (τ14;τ21)+Kmnmn (τ12;τ41))+K0mn(τ42)(Knmnm (τ43;τ14)+Kmnmn (τ41;τ34))++K0mn(τ24)(K˜nmnm (τ23;τ12)+ K˜mnmn (τ21;τ32))+K0mn(τ14)(K˜nmnm (τ13;τ21)+ K˜mnmn (τ12;τ31))]−Dn[K˜nmmn (τ43;τ12)+ K˜nmmn (τ34;τ12)+K0mn(τ24)Knm(τ13)+K0mn(τ14)Knm(τ23)+Kmn(τ23)K0nm(τ14)+Knmmn (τ13;τ24)+Knmmn (τ42;τ13)]+Dm[K˜mnmn (τ43;τ12)+ K˜mnmn (τ34;τ12)+K0mn(τ24)Kmn(τ31)+K0mn(τ14)Kmn(τ32)+Kmn(τ32)K0mn(τ41)+Kmn(τ13)K0mn(τ42)].The B˜1 and B˜3 terms are more complicated then the rest, as the sum over the permutations ofthe imaginary time indices is restricted. For the remaining terms, as in B˜2, we may consider thesum of all possible permutations of the imaginary time index because each of the MF operatorsappearing in the averages is unique. For B4, we begin our reduction with Lnp, as the resultingequation is simpler in the low temperature limit. We findB˜4 = ∑P{1,2,3,4}[Dmn mnKmnnp (τ43;τ12;τ41)−Dmp mpKmpnp (τ23;τ12;τ41) (E.36)+Dm(Kmnnp (τ43;τ42)−Kmpnp (τ23;τ42))].204The final two functions are given byB˜5 = ∑P{1,2,3,4}[Dmp mpKmpmn (τ21;τ23;τ42)−Dmn nmKmpmn (τ41;τ43;τ42) (E.37)−Dnp npKmpmn (τ41;τ23;τ42)+DmKmpmn (τ21;τ43)],B6 = ∑P{1,2,3,4}[Dmq qmKnpmn(τ21;τ32;τ34)+Dqp(qpKmpmn (τ21;τ42;τ34)− qpKnpmn(τ41;τ42;τ34))+Dqn qnKnpmn(τ41;τ32;τ34)].In Fourier space, the fourth order cumulant becomesM4({ωri}) =4∏i=1∫ β0dτieiωriτiM4(TτS1S2S3S4) (E.38)=F{〈TτS1S2S3S4〉0}− β6 ∑P{ωri}M3(ωr1,ωr2,ωr3)〈S4〉0δωr4 ,0− β24 ∑P{ωri}M2(ωr1,ωr2)〈S〉20δωr1+ωr2δωr3δωr4− 18 ∑P{ωri}M2(ωr1,ωr2)M2(ωr3 ,ωr4)−β 4〈S〉404∏i=1δωri ,0We findB˜1 =βDmn ∑P{ωri}nmKmnmn (ωr1;ωr1 +ωr2;ωr3)δ∑ωri ,0 (E.39)+β 2Dm ∑P{ωri}Knmmn (ωr1;ωr2)δωr1+ωr2+ωr3 ,0δωr4 ,0− β3Dm4 ∑P{ωri}Kmn(ωr1)δωr1+ωr2 ,0δωr3 ,0δωr4 ,0,B˜2 =− ∑P{ωri}[2βDmn mnKmnnm (ωr1 ;ωr2 +ωr3;ωr2)δωr1+ωr2+ωr3+ωr4 ,0 (E.40)+β 2(Dm+Dn)Kmnnm (ωr2 ;ωr3)δωr1 ,0δωr2+ωr3+ωr4 ,0],205B˜3 =βDmn ∑P{ωri}mnKnmnm (ωr1;ωr2;ωr3)δ∑ωri ,0 (E.41)− β2Dn2 ∑P{ωri}Kmnnm (ωr1;ωr3)δωr1+ωr2 ,0δωr3+ωr4 ,0+β 2Dm2 ∑P{ωri}Kmnmn (ωr1;ωr3)δωr1+ωr2 ,0δωr3+ωr4 ,0,B˜4 = ∑P{ωri}[β(Dmn mnKmnnp (ωr1;ωr2;ωr2 +ωr3)+Dmp pmKmpnp (ωr1;ωr1 +ωr2;ωr3))δ∑ωri ,0(E.42)−β 2DmKpmmn (ωr1;ωr2)δωr1+ωr2+ωr3 ,0δωr4 ,0]B˜5 = ∑P{ωri}[β(Dmp mpKpmnm (ωr1;ωr2;ωr3)+Dmn mnKpmnm (ωr1;ωr2;ωr3)−Dnp npKpmnm (ωr1 ;ωr2 ;ωr1 +ωr3))δ∑ωri ,0+β 2DmKmpmn (ωr1;ωr3)δωr1+ωr2 ,0δωr3+ωr4 ,0]B6 = ∑P{ωri}β[Dmq mqKnpnm(ωr1;ωr2 +ωr3;ωr2)+Dqp(qpKmpnm (ωr1;ωr2 +ωr3;ωr2)− qpKpnnm(ωr1;ωr2;ωr3))+Dqn qnKpnnm(ωr1;ωr2;ωr2 +ωr3)]δ∑ωri ,0In the final term of B˜4, we have permuted the Matsubara frequencies, and made use of the factthat under the delta function ωr3 =−ωr1−ωr2 , to obtain∑P{ωri}(Kmnnp (ωr1;ωr2)+Kpmnp (ωr1;ωr2))δωr1+ωr2+ωr3 ,0 =− ∑P{ωri}Kpmmn (ωr1;ωr2)δωr1+ωr2+ωr3 ,0.(E.43)This facilitates taking the low temperature limit, which we now consider. Some tedious algebrawill show that, as was the case with the previous cumulants, all the terms in M4 involving lower206order cumulants cancel with terms in 〈S1S2S3S4〉0. We are left withlimT→0M4({ωri})≈ ∑m 6=nc2mm|cmn|2B1+ ∑n>mcmmcnn|cmn|2B2 (E.44)+ ∑n>m|cmn|4B3+ ∑m6=n6=pcmmcmncnpcpmB4+ ∑p>n>m|cmn|2|cmp|2B5+∑m∑n>m∑p>mp 6=n∑q>mq6=n,pcmncnpcpqcqmB6,whereB1 =βDmn ∑P{ωri}nmKmnmn (ωr1;ωr1 +ωr2;ωr3)δ∑ωri ,0 (E.45)B2 =−2βDmn ∑P{ωri}mnKmnnm (ωr1 ;ωr2 +ωr3;ωr2)δ∑ωri ,0B3 =βDmn ∑P{ωri}mnKnmnm (ωr1;ωr2;ωr3)δ∑ωri ,0B4 = ∑P{ωri}β(Dmn mnKmnnp (ωr1;ωr2;ωr2 +ωr3)+Dmp pmKmpnp (ωr1;ωr1 +ωr2 ;ωr3))δ∑ωri ,0B5 =β ∑P{ωri}[Dmp mpKpmnm (ωr1 ;ωr2 ;ωr3)δ∑ωri ,0+Dmn mnKpmnm (ωr1;ωr2;ωr3)δ∑ωri ,0−Dnp npKpmnm (ωr1 ;ωr2 ;ωr1 +ωr3)δ∑ωri ,0]B6 =β ∑P{ωri}[Dmq mqKnpnm(ωr1;ωr2 +ωr3;ωr2)δ∑ωri ,0+Dqp(qpKmpnm (ωr1;ωr2 +ωr3;ωr2)− qpKpnnm(ωr1;ωr2;ωr3))δ∑ωri ,0+Dqn qnKpnnm(ωr1;ωr2;ωr2 +ωr3)δ∑ωri ,0]207Making use of the population factors, this further reduces tolimT→0M4({ωri})≈ ∑n>1c211|c1n|2B01a+∑n>1c2nn|c1n|2B01b+∑n>1c11cnn|c1n|2B02+∑n>1|c1n|4B03(E.46)+ ∑n6=p>1c11c1ncnpcp1B04a+ ∑m 6=p>1cmmcm1c1pcpmB04b+ ∑m 6=n>1cmmcmncn1c1mB04c+ ∑p>n>1|c1n|2|c1p|2B05({ωri})+∑n>1∑p>1p6=n∑q>1q 6=n,pc1ncnpcpqcq1B06,whereB01a =β ∑P{ωri}n1K1n1n (ωr1;ωr1 +ωr2;ωr3)δ∑ωri ,0 (E.47)B01b =−β ∑P{ωri}1nKn1n1 (ωr1;ωr1 +ωr2;ωr3)δ∑ωri ,0B02 =−2β ∑P{ωri}1nK1nn1 (ωr1 ;ωr2 +ωr3;ωr2)δ∑ωri ,0B03 =β ∑P{ωri}1nKn1n1 (ωr1;ωr2;ωr3)δ∑ωri ,0B04a =β ∑P{ωri}[1nK1nnp(ωr1;ωr2;ωr2 +ωr3)+ p1K1pnp (ωr1;ωr1 +ωr2;ωr3)]δ∑ωri ,0B04b =−β ∑P{ωri}m1Km11p (ωr1;ωr2;ωr2 +ωr3)δ∑ωri ,0B04c =−β ∑P{ωri}1mKm1n1 (ωr1;ωr1 +ωr2;ωr3)δ∑ωri ,0B05 =β ∑P{ωri}[1pKp1n1 (ωr1;ωr2;ωr3)+ 1nKp1n1 (ωr1;ωr2;ωr3)]δ∑ωri ,0B06 =β ∑P{ωri}1qKnpn1 (ωr1;ωr2 +ωr3;ωr2)δ∑ωri ,0.208

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