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Structure factors of s=1/2 spin chains and magnetism at the edges of graphene ribbons Karimi, Hamed 2014

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Structure factors of S=1/2 spin chains and magnetism atthe edges of graphene ribbonsbyHamed KarimiB.Sc. Sharif University of Technology, 2007M.Sc. University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University Of British Columbia(Vancouver)September 2014c© Hamed Karimi, 2014AbstractIn this thesis we study two different one dimensional systems. The first projectis on the transverse dynamical structure factors of the XXZ spin chain and the sec-ond project is on magnetism of zigzag edges of graphene nano-ribbons.In chapter 2, we apply field theory methods, first developed to study x-ray edgesingularities, to interacting one-dimensional systems in order to include band cur-vature effects and study edge singularities at arbitrary momentum. We point outthat spin chains with uniform Dzyaloshinskii-Moriya interactions provide an op-portunity to test these theories since these interactions may be exactly eliminatedby a gauge transformation that shifts the momentum. However, this requires an ex-tension of these x-ray edge methods to the transverse spectral function of the XXZspin chain in a magnetic field.In chapter 3, by considering the Hubbard model in the weak coupling limit,U  t, for bearded as well as zigzag edges, we argue for existence of magneticedges. We first present an argument based on Lieb’s theorem. Then, projectingthe Hubbard interactions onto the flat edge band, we prove that the resulting one-dimensional model has a fully polarized ferromagnetic ground state. We also studyexcitons and the effects of second neighbor hopping as well as a potential energyterm acting on the edge only, proposing a simple and possibly exact phase diagramwith the magnetic moment varying smoothly to zero. Finally, we consider cor-rections of second order in U, arising from integrating out the gapless bulk Diracexcitations.iiPrefaceThis thesis is based almost entirely on notes written by myself during my PhDprogram. In addition some sections are based on notes written by my researchsupervisor Ian Affleck, and also publications authored by me and my supervisor.The concept and scope of the research was developed collectively by myself andmy supervisor. All the analytical and numerical calculations in this thesis wereconducted by me but heavily influenced by significant consultation with my super-visor.Chapter 2, is the study of transverse dynamical structure factors of XXZ spinchains. Most of this chapter is contained in the following paper : H. Karimi, I.Affleck, Physical Review B, 84, 174420 (2011).A version of chapter 3 has been published. H. Karimi, I. Affleck, PhysicalReview B, 86, 115446 (2012).This chapter contains more detailed version of calculation and longer introductorysections, which all written by me.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Anisotropic spin chain, XXZ model . . . . . . . . . . . . . . . . 21.2 Luttinger liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Beyond the Luttinger Liquid paradigm . . . . . . . . . . . . . . 51.4 Magnetism at the edges of graphene ribbons . . . . . . . . . . . . 71.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Spectral functions and Dzyaloshinskii-Moriya interactions in XXZspin chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Spectral function of the XXZ spin chain . . . . . . . . . . . . . . 152.2.1 Bosonization of interacting fermion model . . . . . . . . 182.2.2 Spectral function of XXZ spin chain using bosonization . 232.2.3 Effect of irrelevant band curvature operators . . . . . . . . 30iv2.3 X-Ray Edge method . . . . . . . . . . . . . . . . . . . . . . . . 352.4 Sub-dominant singularities . . . . . . . . . . . . . . . . . . . . . 472.5 Electron spin resonance with Uniform Dzyaloshinskii-Moriya In-teractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Toward rigorous proof of edge magnetism in graphene nano-ribbons 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Band structure of single layer graphene . . . . . . . . . . . . . . 643.3 Dirac fermions and low-energy effective hamiltonian . . . . . . . 683.4 Edge states of graphene . . . . . . . . . . . . . . . . . . . . . . . 713.4.1 Lieb’s theorem and edge magnetism . . . . . . . . . . . . 763.5 Projected Hubbard Model of the edge states . . . . . . . . . . . . 783.5.1 Ferromagnetic ground state . . . . . . . . . . . . . . . . . 813.6 Effects of the Excitations in presence of NNN-hopping and singlesite potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.7 Effective inter-edge and intra-edge interactions . . . . . . . . . . 893.8 Details of field theory calculations . . . . . . . . . . . . . . . . . 933.8.1 Integrating out bulk excitations . . . . . . . . . . . . . . . 993.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A Derivation of X-Ray edge Hamiltonian . . . . . . . . . . . . . . . . . 115B Positivity of spectral function . . . . . . . . . . . . . . . . . . . . . . 118vList of FiguresFigure 1.1 Energy of particle-hole excitations for fixed values of total mo-mentum q, but for different particle and hole momenta. Fig athe energy for non-linear dispersion. It is clear that the energydepends on the momentum of particle or hole. Fig b is the en-ergy for linearized dispersion; the energy is independent of themomentum of particle or hole . . . . . . . . . . . . . . . . . 6Figure 1.2 Honeycomb structure of graphene nanoribbon with an upperand lower zigzag edges. In this example, L = 5 and W = 6. . 8Figure 2.1 The band structure of non-interacting fermion model. To findlow energy field theory of interacting model we only keep ex-citation withing the cutoff ΛF . . . . . . . . . . . . . . . . . 19Figure 2.2 Singular points of zero temperature transverse spin spectralfunction of XXZ model, predicted by bosonization. The solidlines indicate diverging singularities and dashed line the van-ishing singularities. . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.3 Zero temperature transverse spectral function S+−(ω,q) pre-dicted by bosonization for fixed q. a) shows transverse spectralfunction for zero magnetic field and b) is for non-zero mag-netic field with q < H. . . . . . . . . . . . . . . . . . . . . . 28Figure 2.4 Various shapes of singularities of S+− and S−+ for differentrange of momentum q, with |q|,H 1, predicted by bosoniza-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29viFigure 2.5 The relevant Feynman graphs which contribute to correlationin first order of perturbation. . . . . . . . . . . . . . . . . . . 32Figure 2.6 In beyond Luttinger models, in addition to low energy excita-tions we also include the effect of a high-energy particle or adeep hole. In this case we have a hole at momentum k. . . . . 36Figure 2.7 The behaviour of S+−(ω,q) correlation for |q|< H. There aretwo different hole excitations which contribute to the spectralfunction; the lower energy hole produces a cusp-like singularity. 46Figure 2.8 Singular points of zero temperature transverse spectral func-tions of the XXZ model, predicted by X-ray edge method. Thesolid lines indicate diverging singularities and dashed line thevanishing singularities. The grey thick line indicates the rangeof momentum over which interactions should broaden the di-verging singularity. In Fig. [2.8a] and [2.8b] the lower dashedline is given by particle excitations and upper one is given byhole excitation. . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 2.9 The behaviour of S+−(ω,q) spectral function at finite temper-ature for the case νR = 1/2, νL = 0, corresponding to ∆ = 1,h = 0. We see that for small enough T the broadening is asym-metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.1 Honeycomb lattice and its Brillouin zone. Left: lattice struc-ture which is used. Right: the Brillouin zone and Dirac points 67Figure 3.2 A nanoribbon with an upper zigzag edge and lower beardededge and armchair edges on sides. In this example, L = 5 andW = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 3.3 Energy spectrum for graphene ribbon for different boundaryconditions. a) is energy spectrum for graphene ribbon withzigzag-zigzag boundary condition. b) is spectrum for ribbonwith zigzag-bearded (ZB) boundary condition. . . . . . . . . . 74Figure 3.4 Energy to add a spin down electron of momentum k for variousvalues of η ≡ ∆/U ≡ (t2−Ve)/U . . . . . . . . . . . . . . . . 86viiFigure 3.5 Lowest energy particle-hole state (circles) and bottom of theparticle-hole continuum (lines) for various values of η ≡∆/U ≡(t2−Ve)/U . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 3.6 Edge magnetization versus ∆≡ t2−Ve. . . . . . . . . . . . . 89Figure 3.7 Feynman diagrams inducing edge interactions from integratingout bulk states. . . . . . . . . . . . . . . . . . . . . . . . . . 91viiiGlossaryRG renormalization groupESR Electron Spin ResonanceLL Luttinger liquidDSF dynamical structure factorDMRG Density Matrix Renormalization GroupNNN next nearest neighbourDM Dzyaloshinskii-MoriyaARPES Angle-Resolved Photoemission SpectroscopySTM Scanning Tunneling MicroscopeZZ zigzag-zigzagZB zigzag-beardedixAcknowledgmentsFirst and foremost, I would like to express my deepest gratitude to my researchsupervisor Prof. Ian Affleck, for his excellent guidance, caring and patience, en-couraging and helping me to experience diverse field of Condensed Matter Physics.Thanks also for funding me for all these years.I would like to thank my supervisory committee, Prof. Mark Van Raamsdonk,Prof. Marcel Franz and Prof. Robert Kiefl for their criticism of my research,constructive feedback and guidance. Special thanks to Mark Van Raamsdonk forhelping me to develop my background in high energy physics during my M.Scprogram.To the University of British Columbia (UBC) for giving me the opportunity todiscover Vancouver, and funding me through the Graduate Entrance Fellowship,International Partial Tuition Scholarship, the PhD Tuition Fee Award and Univer-sity Four Year Fellowship.I will always be grateful to my parents for their everlasting encouragement andsupport and believing in me. Of my friends and colleagues, I could write at lengthhow each provided their own unique brand of fellowship and support.Finally, I would like to thank Elham, my best friend and beloved partner forher love and creating the best moments of life for me, whom I can not express thedebt of gratitude I owe her.xChapter 1IntroductionOne dimensional interacting quantum systems, due to strong quantum fluctua-tions in one dimension, have very interesting and unusual many body effects. Forexample because of strong quantum fluctuations, continuous symmetries can notbe broken spontaneously; as a result there is no long range order in the system.The order parameter is ill-defined and conventional phase transitions (Landau pic-ture), based on breakdown of continuous symmetries, are absent in such systems.Although there is no long range order in such systems, we could still define a formof ”quasi long-range” ordering for these systems and study different phases andphase transitions in one dimension.One dimensional quantum systems are not only relevant in theoretical realm.One dimensional spin chains, where in some three dimensional compounds dueto lattice structure, exchange interaction in one direction is higher than other di-rections [1, 2] are one example of extensively studied experimentally quantumsystems. Also edge states of some two dimensional systems such as Graphenewith zigzag edges [3] or Quantum Hall systems [4] and 2D Topological Insulators[5], are well localized near the edges and are effectively one dimensional. Studyof quantum system require understanding of both its ground state, to study phasetransitions, and correct description of excitations of the system to address dynam-ical properties, such as spectral functions which are measurable in photo emissionspectroscopy or dynamical structure factors which can be probed by inelastic neu-tron scattering experiments or Electron Spin Resonance (ESR).1In this thesis we study two different one dimensional quantum systems, in oneproject we study the dynamical structure factors of spin chains and in the other wefocus on ground state properties and try to prove theoretically and understand themagnetic properties of edge states of Graphene with zigzag edges.1.1 Anisotropic spin chain, XXZ modelThe anisotropic spin chain, XXZ model is one of the most studied one dimen-sional quantum systems. The Hamiltonian description of this model is given byH = J∑j[SxjSxj+1 +SyjSyj+1 +∆SzjSzj+1−hSzj], (1.1)where J is the exchange coupling, ∆ represents anisotropy in exchange interactionwhich could be present due to lattice structure, the last term is just the Zeemanterm due to presence of magnetic field which could be applied experimentally.For general values of ∆ the system only has U(1) symmetry and the well definedquantum number is total spin along the z-direction Sz = ∑ j Szj. The special case of∆ = 1 and h = 0 is called Heisenberg spin chain [6] which has SU(2) symmetryand has been studied extensively.The phase diagram of XXZ spin chain is well understood by using combina-tion of theoretical methods [7] and also using analytic and numeric solutions ofpowerful methods such as Bethe ansatz [8–10]. Although the spin can take anyinteger or half-integer values, we only focus for the case of S = 1/2 spin chain.In general the anisotropy in spin chain compounds is very weak and ∆ ≈ 1, butthere are many artificial systems whose effective Hamiltonian description could bemapped to XXZ Hamiltonian with arbitrary values of ∆, for example ultra coldbosonic atoms trapped in optical lattices [11] or Josephson junction arrays [12].By using Bethe ansatz methods the phase diagram of XXZ model as functionof ∆ and h is known exactly. The limiting case of ∆→ −∞ and ∆→ ∞ are thelimiting ferromagnetic and anti-ferromagnetic Ising chain, respectively. In thesecases the system is gapped and the excitations are the domain-walls.The range of ∆ which is more relevant is the case of −1 < ∆≤ 1, which is the2critical line. In this regime the system is gapless and also it does not have long-range order. The other two cases are both gapped. When ∆ > 1 we have gappedsystem and ground state is in Neel phase; when ∆ < −1 the ground state is in theferromagnetic phase.Although the ground state and also the spectrum of excitations can be derivedby using Bethe ansatz solutions, finding the dynamical quantities such as spin-spin structure factors is very difficult. The reason is calculating the correlationfunctions requires working with very large number of complicated wavefunctions,which number grows exponentially as 2N , where N is the number of sites. As aresult development of a field theoretical approach to study the spin chain systemsand in general, one dimensional quantum system is very useful to study the corre-lations and dynamical structure functions.1.2 Luttinger liquidBy using Jordan-Wigner transformation [2] the spin-1/2 XXZ model could bemapped to interacting one dimensional spinless fermion model. Thus in principleany theoretical methods relevant to study interacting fermion model could be usedto understand the spin chains.In two and three dimensions the Landau theory of fermi liquid is general methodto study the interacting models. In this picture the excitations of the interacting sys-tem are quasi-particles with some renormalized velocity and effective mass. Butin one dimension strong quantum fluctuations lead to breakdown of Fermi liquidtheory[13].The equivalent theoretical framework of Fermi liquid theory in one dimensionis Luttinger liquid (LL) and bosonization method [2, 7, 9, 14]. The Luttinger Liquidis the low energy fixed point of the one dimensional electron systems [15]. In thisapproach first the dispersion is linearized near the Fermi points ±kF . As we areonly interested in low energy description of the system we only keep excitationsnear the fermi energy. It also could be shown that higher order band curvatureterms of the dispersion are irrelevant in renormalization group (RG) sense. Then3by using bosonization method [16] we represent the fermionic fields in terms ofsome bosonic fields. Using bosonization representation the kinetic part of fermionmodels maps to kinetic part of the bosonic picture.But the important difference is that the interaction part of fermionic modelwhich usually has four fermion operator, maps to a quadratic term in bosonicrepresentation which means we could include the interaction effects exactly, notperturbatively . The resulting non-interacting bosonic Hamiltonian is called theLuttinger Hamiltonian [17] which only has two parameters in it. The first one isthe velocity of the bosonic excitations, and the second one is called “stiffness”.HLL =v2pi[K(∂xθ)2 +1K(∂xφ)2](1.2)By using the dictionary of bosonization and Jordan-Wigner transformation wecould represent the spin operators in terms of bosonic fields; when the correspond-ing bosonized Hamiltonian is non-interacting, the spin-spin correlations can befound [18]. And it turns out that spin-spin correlations decay with a power law, asfunction of distance, which confirms the non-existence of long range order and theexponent of the decay only depends on Luttinger parameter K.As a result it is crucial to find a relation between the parameters of Luttingermodel, v and K, in terms of the parameters of microscopic model, ∆, J and h. Forthe special case of h = 0 it is possible to find an exact analytical equation for K andv in terms of ∆ and J. But for general case of non-zero magnetic field it is possible,by using Bethe ansatz equations [9], to find numerically the Luttinger parameterfor any value of anisotropy.We should mention that the bosonization method is asymptotically exact aswe approach the Fermi point, or excitations with q→ 0. Although it is a pow-erful method to address many theoretical questions using it to find the dynamicalquantities at finite momentum q 6= 0 gives not very exact results [19].For example let’s look at dynamic structure factor, which is the imaginary partof retarded density-density correlation function. The Fourier transform of densityoperator at momentum q consists of particle-hole excitations with total momentumq, ρ(q) =∑p c†q+pcp. Then by using Lehmann representation it could be shown that4dynamical structure factor at momentum q and frequency ω , could be representedbyS(ω,q) =∑αδ (ω−E(α)p−h(q)), (1.3)where the summation is over all possible configuration of particle-hole excitationswith momentum q, and the energy of that excitation is given by E(α)p−h(q).Thus if we linearize the dispersion relation of fermions, with velocity vF , theenergies of all the possible particle-hole excitations are exactly the same, E(α)p−h(q)=vFq, Fig. 1.1. This means that the dynamical structure factor (DSF) is only deltafunction peak at zero temperature, and going to finite temperature leads to sym-metric Lorentzian broadening of it. But by using neutron scattering experimentsis was pointed out that the DSF has asymmetric double peak structure at finitetemperatures which is in contradiction to bosonization result[20].We could see that if we include the band curvature of the dispersion then theenergies of the particle-hole excitations not only depends on the total momentumbut also on the position of the particle and the hole. For example in Fig. 1.1 we seethat the energy of a particle-hole excitation for a hole near fermi point is smallerthan for the case where the particle is near the fermi point. As a result of this, theDSF is not a delta function peak and has structure even at zero temperature. This isan important observation, it tells us that the DSF is non-zero only at some energyinterval ωL(q) < ω < ωU(q), where ωL/U(q) are the threshold lower and upperfrequencies of particle-hole excitation for fixed q, and DSF has non-zero widthgiven by δω = ωU −ωL even at zero temperature. Although in general includingthe interaction terms will lead to many more p-h excitations and as a result tosome tails and smearing of the sharp feature of the DSF, but non-zero width andasymmetric shape of DSF remains intact.1.3 Beyond the Luttinger Liquid paradigmAs we mentioned in previous section, in order to fully understand the dynam-ical structure factors at finite momentum we need to include the effects of higherorder band curvature effects. Although these higher order terms are irrelevant inrenormalization group sense for asymptotic theory as q→ 0, they could be in-5EkF+q−xEFkF−xE p−hxωLωUq(a) Ep−h(q)EkF+q−xE FkF−xE p−hxωL ? ωUq(b) Ep−h(q)Figure 1.1: Energy of particle-hole excitations for fixed values of total mo-mentum q, but for different particle and hole momenta. Fig a the energyfor non-linear dispersion. It is clear that the energy depends on the mo-mentum of particle or hole. Fig b is the energy for linearized dispersion;the energy is independent of the momentum of particle or hole .6cluded perturbatively in the study of finite momentum DSFs. But it turns out thatthe study of these band curvature terms is not a trivial task and they cause someinfrared divergences in perturbation analysis, specially near the thresholds [21, 22].In bosonization approach the interactions were treated exactly while the ap-proximation was linearized dispersion. Using different approach, Pustilnik et al.[23] studied dynamical structure factors by keeping the non-linearity of the disper-sion intact while doing perturbation in the interaction. Although perturbations leadto logarithmic divergences in all orders, using methods developed for X-ray edgesingularities in metals [24], these divergences could be managed in consistent way.The results were quite astonishing. Pustilnik et al. found that dynamical structurefactors have power law singularities near the threshold energies while the singu-larity exponents had momentum dependencies. The momentum dependencies ofexponents in this approach has no equivalent counterpart in linearized bosoniza-tion and Luttinger Liquid theory. Further study of that approach [25–30] lead todevelopment of nonlinear Luttinger liquid theory.Although this approach is perturbative in terms of interaction, the resultingfield-theoretical model could be used as a phenomenological framework to studythe DSF of a general one-dimensional system at finite momentum. Among all onedimensional interacting systems, the more attractive ones are those which are ex-actly solvable by Bethe ansatz. For this class of models it is possible to find theparameters of the phenomenological nonlinear Luttinger liquid model for any inter-action strength [22]. Combination of this new approach with Bethe ansatz for ex-actly solvable models leads to nontrivial tests of this approach, which could be doneusing numerical methods such as Density Matrix Renormalization Group (DMRG)[31].1.4 Magnetism at the edges of graphene ribbonsGraphene is a two dimensional allotrope of Carbon which has many interestingelectronic properties. In graphene the Carbon atoms are arranged in honeycomblattice Fig. 1.2. The sp2 hybridization of the atomic s orbital with two p orbitals7L ABFigure 1.2: Honeycomb structure of graphene nanoribbon with an upper andlower zigzag edges. In this example, L = 5 and W = 6.leads to trigonal planar structure, and to the formation of so called σ bond betweenthe neighboring carbon atoms, with distance equal to 1.42A˚. This σ band is re-sponsible for holding the carbon atoms in two dimensions and the robustness ofgraphene. The remaining p orbital is perpendicular to plane of this planar structureand the resulting covalent bond between p orbitals of the neighboring atoms leadto the formation of pi band, which is responsible for many low-energy electronicand transport properties of graphene.The first theoretical study of the band structure of graphene was done by P.R. Wallace in 1946 [32]; he showed the unusual semimetallic behavior of it. Themore systematic study of the low-energy excitations of graphene leads to its mostinteresting properties. Simple tight-binding study of the dispersion of graphene8shows that the dispersion vanishes linearly at two inequivalent points, these pointare so called Dirac points.Semenoff [33] showed that the low energy excitations near these two points aremassless chiral, Dirac fermions. Quite interestingly these excitations were prop-agating like massless fermions but with fermi velocity which is about 1/300 ofvelocity of light.In addition to interesting properties of excitations of bulk graphene, it turns outthat graphene with some specific boundary conditions has very interesting prop-erties too. Again by using simple tight-binding model it could be shown that asemi-infinite graphene sheet with zigzag boundary, or edge, supports zero energystates. More importantly it could be shown that these zero energy modes are lo-calized near the zigzag edge. As these edge modes have zero energy and have flatenergy band Fig. 3.3a then it’s quite natural to expect that even weak interactionscould have drastic effects on the ground state of these zero modes.Mean field theoretical study of these edge states for graphene with electron-electron interactions, Hubbard interaction [34, 35], revealed that the edge statesare spin polarized. In this work we first give an argument based on Lieb’s theo-rem that the turning on the Hubbard interaction could lead to the magnetization ofthe edge modes. Then by projecting out the bulk excitations we find an effectiveHamiltonian which describes the edge modes and then try to prove the magnetismof it’s ground state rigorously.1.5 OverviewIn this thesis we study two different one dimensional systems. The first one isthe study of transverse spin dynamical structure factors S+−(q,ω) of XXZ model.The goal is to study the shape of DSFs and to derive analytical expressions forsingularity exponents near the threshold frequencies. Pereira et al. [22, 29] exten-sively studied the DSF for longitudinal spin, Szz(q,ω), here we use same approachto study the transverse spin structure factors. Transverse structure factors are morecomplicated than the longitudinal spin structure factors. The reason is that in do-ing the Jordan-Wigner transformation the transverse spins are non-local in termsof fermion operators and even for non-interacting regime still there is no exact re-9sult for dynamical correlation functions. Transverse spin structure factors could bemeasured either by neutron scattering measurements in general, or using ElectronSpin resonance experiments for special class of spin chains with lower symmetrywhich supports uniform Dzyaloshinskii-Moriya (DM) [36, 37] interaction.In chapter 2 we first describe a possible experimental realization for the mea-surement of transverse spin structure factors by using ESR for special class of sam-ples to test the theoretical predictions. The reason for using ESR is that it is moreaccurate while neutron scattering measurements resolutions is not enough to seethe singularity features for small momentum. Then we use bosonization approachand treat the band curvature terms perturbatively, and will show that this approachleads to divergences near the threshold frequencies which means unreliability ofbosonization method for the study of singularity exponents.By using the beyond bosonization method we find the singularity exponentsand the shape of transverse structure factors for the XXZ model. Similar to thelongitudinal case we find momentum dependence in the singularity exponents. Oneimportant and nontrivial test of our result would be to look at the SU(2) symmetriccase, since in that case symmetry implies that both transverse and longitudinalstructure factors have to be same; we see that our result for singularity exponentsare in agreement with the results of longitudinal case [22].In chapter 3 we study a completely different one dimensional system: grapheneedge states. It is well known that the tight binding description of graphene withzigzag edges supports zero energy modes. By looking at the wave function ofthese modes it turns out that they are well localized at the zigzag edges. Fujita et al.[34] studied the graphene ribbons with zigzag edges with Hubbard interaction byusing mean field methods and quite astonishing they found that for any interactionstrength the edges are spin polarized.In that chapter by using the projection method [35] we study the effectiveHamiltonian of the edge modes, which we expect to become valid in the weaklyinteracting limit of the Hubbard model. We show rigorously that the ground stateof the projected Hamiltonian is ferromagnetic and as a result the edges are spin po-larized. Then we include the effect of unavoidable nearest neighbor hopping termsand study the stability of ferromagnetic state of the edge as function of next near-est neighbor hopping strength, and we show that the fully polarized state remains10intact for some range of next nearest neighbour (NNN) hopping term and havingpassed some critical value the polarization gradually decreases.Then in final section we take into account the effect of bulk excitations. Inte-grating out the bulk excitations leads to higher order interactions. The importantresult of doing so is that it leads to long range anti-ferromagnetic exchange inter-action between two edges of a graphene ribbon. This means that for a grapheneribbon with both edges in zigzag shape each edge is spin polarized and the totalspins of the edges are coupled anti-ferromagnetically.Another effect of integrating out the bulk excitation is to have higher order interac-tion correction to the projected Hamiltonian. By studying these correction we seethat they even make the magnetism of the edges more robust.11Chapter 2Spectral functions andDzyaloshinskii-Moriyainteractions in XXZ spin chains2.1 introductionOne dimensional (1D) interacting systems exhibit unusual correlation effectsdominated by strong quantum fluctuations. Fortunately, an array of powerful the-oretical methods exists to study this physics, which is finding many experimentalrealizations. One powerful method is based on bosonization [2, 7, 9, 14] leading tothe Luttinger liquid concept. Traditionally these methods are based on low energyeffective field theory and only apply to the low energy excitations occurring nearcertain wave-vectors (such as q = 0). In the case of fermion models they beginby linearizing the dispersion relation near the Fermi energy and ignoring irrelevantband curvature effects. However, in the last few years, these bosonization methodshave been significantly extended by using techniques first developed to study X-rayedge singularities [22, 23, 25, 27, 29].This has shown that band curvature effects, while formally irrelevant in therenormalization group sense [38], can nonetheless have important effects on line-shapes of spectral functions even at low energy. Perhaps even more importantly, by12combining these techniques with Bethe ansatz methods, it has become possible tomake exact predictions of critical exponents at arbitrary momentum [27, 29]. Thatis, spectral functions are predicted to have the form S(q,ω)→A[ω−ωL(q)]−η nearsingular energies ωL(q) for arbitrary q where both ωL(q), which is not small, andthe exponents η(q) are determined exactly using the Bethe ansatz. Recently, thisapproach has been extended to also obtain the amplitudes of correlation functions,A, using the Bethe ansatz [39]. This new approach has been applied to a numberof systems including fermions and bosons moving in the continuum [23, 25–27],the fermion spectral function for a tight binding model [22] and the longitudinalspectral function of the XXZ S=1/2 spin chain in a magnetic field [28, 29] withHamiltonian:H = JN∑i=1(Sxi Sxi+1 +Syi Syi+1 +∆Szi Szi+1)−h∑Szi (2.1)The longitudinal and transverse spectral function are given by :Szz(q,ω) = 1N ∑j∫ ∞−∞dte−iq j+iωt〈Szj(t)Sz0(0)〉Sss′(q,ω) = 1N ∑j∫ ∞−∞dte−iq j+iωt〈Ssj(t)Ss′0 (0)〉 (2.2)We set the lattice spacing a = 1, and (s,s′) = (+,−) and (−,+). For non-zero h,S+− and S−+ are different. In principle, at arbitrary q, neutron scattering experi-ments would normally measure a sum of Szz and the transverse spectral functions.The high neutron fluxes available at the Spallation Neutron Source may eventuallymake such experimental confirmation possible, here we explore another route.Quasi 1D magnetic compounds that do not have link parity symmetry (reflec-tion about the midpoint of a link) will generally have magnetic Hamiltonians con-taining anti-symmetric Dzyaloshinskii-Moriya [36, 37] interactions:δH =∑j~D j · (~S j×~S j+1). (2.3)The two standard cases are staggered ~D j = (−1) j~D and uniform interaction ~D j =13~D. Staggered DM interactions are invariant under site-parity (reflection about asite) but violate symmetry of translation by one site. On the other hand, uniformDM interactions have no parity symmetry whatsoever but respect full translationinvariance. We restrict our further discussion to the case where ~D j ∝ zˆ so thatthe DM vector is parallel to the easy (or hard) axis of the symmetric exchangeinteractions, a situation which is sometimes dictated by symmetry.With this configuration, it is possible to eliminate the DM interactions exactlyby a gauge transformation, yielding the standard XXZ model of Eq. (2.1) withmodified parameters. This approach was used to study staggered DM interactionsin Ref. [40, 41] and the theory was applied to a number of real materials. Herewe consider the case of uniform DM interactions, ~D j = zˆD and choose D > 0. Toshow this transformation, it is more convenient to write the interacting Hamiltonianin terms of S±, as the magnetic field is in the same direction as DM interaction,without loss of generality we could ignore the Zeeman term.H +δH == JN∑i=1(Sxi Sxi+1 +Syi Syi+1 +∆Szi Szi+1)+DN∑i=1(Sxi Syi+1−Syi Sxi+1)= J/2N∑i=1(S+i S−i+1 +S−i S+i+1 +2∆Szi Szi+1)+ i D/2N∑i=1(S+i S−i+1−S−i S+i+1)= 1/2N∑i=1((J + i D)S+i S−i+1 +(J− i D)S−i S+i+1 +2J∆Szi Szi+1)(2.4)In this case by defining a gauge transformation, local rotations around z-direction,as follow:S˜+j = e−iα jS+j S˜zj = Szj (2.5)whereα = tan−1(DJ). (2.6)the Hamiltonian Eq. (2.4) transforms to standard XXZ Hamiltonian:H =J∑i(S˜xi S˜xi+1 + S˜yi S˜yi+1 +∆e f f S˜zi S˜zi+1)(2.7)14where the new exchange coupling and anisotropy parameters are given byJ =√J2 +D2 ∆e f f = ∆cos(α). (2.8)The electron spin resonance (ESR) adsorption intensity, in standard Faradayconfiguration, is proportional to the transverse spectral function at q = 0, sincethe wave-vector of microwave photons is much less than the inverse lattice spac-ing. After the gauge transformation, the ESR intensity is therefore proportionalto S+− and S−+, for the Hamiltonian of Eq. (2.1) at q = α . [By using circularlypolarized microwave radiation both S+−(α,ω) and S−+(α,ω) could be measuredseparately]. Thus the edge singularities predicted by X-ray edge methods at a non-zero wave-vector α given by Eq. (2.6) are directly measured by ESR. ESR onspin chain compounds with uniform DM interactions therefore would provide apowerful probe of the new bosonization predictions. Quasi-1D spin-1/2 antiferro-magnetic insulators containing DM interactions with a uniform component includeCs2CuCl4 [42, 43] and KCuGaF6 [44]. This provides a strong motivation to ex-tend the X-ray edge methods to study edge singularities in the transverse spectralfunctions of the XXZ chain in a magnetic field.In the next section we review results on the transverse spectral function usingstandard bosonization and then show that band curvature effects (in the equivalentfermion model) render these results invalid close to edge singularities. In Sec. 2.3.we apply X-ray edge methods to the model obtaining new results on the leadingedge singularities. In Sec. 2.4 sub-dominant singularities are discussed. Section2.5 discusses ESR with uniform DM interactions, based partly on the results ofSec. 2.3. Sec. 2.6 contains conclusions and open questions.2.2 Spectral function of the XXZ spin chainThe XXZ S = 1/2 model of Eq. (2.1) is equivalent to an interacting spinlessfermion model by the Jordan-Wigner transformation:Szi = c†i ci−12S−j = (−1)j exp(ipi∑k< jc†kck)c j. (2.9)15The Hamiltonian (2.1) is transformed toH = −J2∑(c†i ci+1 +h.c.)−h∑c†i ci+ J∆∑(c†i ci−12)(c†i+1ci+1−12). (2.10)Note the factor of (−1) j in the second line of Eq. (2.9) is necessary for the firstterm in the fermionic Hamiltonian, Eq. (2.10), to have the standard minus sign. his the chemical potential of the fermionic model with h = 0 corresponding to half-filling. For the non-interacting case ∆ = 0, the Hamiltonian is just a free fermionmodel and by going to momentum spacecp =1√N∑c jeip jwhere p = 2pin/N for periodic boundary conditions, the energy spectrum of non-interacting fermion model is found to beH =∑(−J cos(p)−h)c†pcp. (2.11)For non-interacting case and using fermionic representation the longitudinal spec-tral function is given bySzz(ω,q) = 1N∫ ∞−∞dteiωt〈nq(t)nq(0)〉 (2.12)where nq = ∑ j eiq jn j = ∑p c†pcp+q. To find an analytical result for longitudinalspectral function, it is more convenient to use Lehmann representation of Eq.(2.12), as followSzz(ω,q) = 2piN ∑α|〈0|nq|α〉|2δ (ω−Eα −EΩ) (2.13)where |α〉 is an eigenstate with energy Eα and EΩ is the energy of the ground state.By using the fact that the system is non-interacting and doing some straightforward16algebra the final result for spectral function isSzz(ω,q) = θ(ω−ωL(q))θ(ωU(q)−ω)√(2J sin q2)2−ω2. (2.14)We see that Szz is non-zero only in finite frequency range for a fixed momentum q;the lower and upper thresholds for kF < pi/2 are given byωU(q) = 2J sin|q|2sin(kF +|q|2)ωL(q) = 2J sin|q|2sin(kF −|q|2) (2.15)and for kF = pi/2ωU(q) = 2J sin|q|2ωL(q) = J sin |q| (2.16)From Eq. (2.14), it is easily seen that the support of Szz, for ∆ = 0, is restricted tothe interval [ωL,ωU ]. The nature of these lower and upper thresholds is differentfor zero and non-zero magnetic field. For non-zero magnetic field and kF < pi/2the lower threshold is given by creating one deep hole with momentum q, whereasthe upper threshold is achieved by excitation of one high-energy electron with mo-mentum q. But for zero magnetic field, kF = pi/2, particle-hole symmetric case,the lower threshold is achieved either by creating one deep hole or one high energyelectron but higher threshold is given by symmetric particle hole excitation aroundFermi point with momentum of each q/2.In contrast to the longitudinal spectral function, even for ∆= 0 and having theentire spectrum at hand, finding the transverse spectral function is a very difficultproblem. The reason for this complication is the complicated form of the transversespin operators, which have the string operator in their fermionic form Eq. (2.9).The equal time transverse correlation function, in this case, is known exactly[45]but less is known about dynamical correlations. The non-local nature of stringoperator is such that we can not even evaluate simply the lower and upper threshold17for S−+ and S+−. By applying S+j to the vacuum, the exponential factor of it willallow creation of any number of particle hole excitationsS+i |0〉 = exp(ipi∑j<ic†jc j)c†i |0〉 = ∑n(ipi∑ j<i c†jc j)nn!c†i |0〉So it is obvious that potentially S−+(ω,q) could be non-zero for fixed q andany ω . However, in general, we expect an infinite number of progressively weakersingularities, extending down to zero energy, similar to the case of the longitudinalstructure function in a non-zero magnetic field [22]. After reviewing the standardbosonization method in next section and its results for spin structure factors insection 2.2.2, we look at the effects of irrelevant band curvature operators in section2. Bosonization of interacting fermion modelIn this part we first review the bosonization method and how to map a field the-ory from fermionic representation to bosonic form. The bosonization is powerfulmethod to study interacting one dimensional models. The essence of the approachis to study low-energy properties of a system; thus we only need to keep the lowenergy excitation. So let’s see how it works.Having done the Jordan-Wigner transformation, the fermionic model descrip-tion of the spin model is given byH = −J2∑(c†i ci+1 +h.c.)−h∑c†i ci+ J∆∑(c†i ci−12)(c†i+1ci+1−12) (2.17)As we discussed in section 2.2, for ∆ = 0, this is just free fermion model and theground state has all states filled up to momentum ±kF , with kF = cos−1(h/J).To find the low-energy description of Eq. (2.17) we only include the excitationsaround the two Fermi points Fig. 2.1, and linearize the energy dispersion near these182+ F2+ FEqEFk F−kFFigure 2.1: The band structure of non-interacting fermion model. To find lowenergy field theory of interacting model we only keep excitation withingthe cutoff ΛFpointsc j ≡ Ψ(x)≈ ψR(x)eikF x +ψL(x)e−ikF xε±kF+q = ±vF q+q22m+ · · · (2.18)where ψR/L are slowly varying fields, which represents excitations near±kF . vF =J sin kF is velocity of the excitations, m = (J cos kF)−1 is the effective mass atFermi level and · · · are higher order band curvature terms.19The continuum model representation of Hamiltonian 2.17 isH0 =∫dx : ψ†R(x) [vF(−i∂x)]ψR(x) : + : ψ†L(x) [−vF(−i∂x)]ψL(x) :(2.19)where we have dropped higher derivative terms, as they are irrelevant in the senseof renormalization group. Similarly we could write the interaction part of Hamil-tonian 2.17 in terms of continuum field as followHint = ∆J∫dx : Ψ†(x)Ψ(x) : : Ψ†(x+a)Ψ(x+a) := ∆J∫dx ρR(x)ρR(x+a)+ρL(x)ρL(x+a)+ ρR(x)ρL(x+a)+ρL(x)ρR(x+a)+[ei2kF aψ†R(x)ψL(x)ψ†L(x+a)ψR(x+a)+h.c.]+[e−i2kF (2x+a)ψ†R(x)ψL(x)ψ†R(x+a)ψL(x+a)+h.c.], (2.20)where ρR/L ≡ ψ†R/LψR/L are the density of right mover and left mover excitations.For general value of kF the last term of above equation, so called Umklapp scat-tering term, is oscillatory and in low-energy theory we can ignore it. But for thecase of half-filling, kF = pi/2, which corresponds to zero magnetic field regime itis non-oscillatory and we should include it in the study of the Hamiltonian.The bosonization of the Hamiltonian H0 +Hint is performed by expressing thefermionic operators ψR/L in terms of some bosonic fields as followψR(x)≈η√2piαe−i√2piφR(x) ψL(x)≈η√2piαei√2piφL(x) (2.21)where α = k−1F is a short distance cutoff, φR/L(x) are right and left component ofbosonic field and η is called Klein factor, which is introduced to ensure the correctanti-commutation relation for fermionic fields. The commutation relation for the20bosonic fields are[φR,φL] =i2[φR/L(x),φR/L(y)]= ±i2sign(x− y) (2.22)The fields φR/L could be expressed in terms of a bosonic field φ˜ and its dual fieldθ˜ as followφ˜ = φL−φR√2θ˜ = φL +φR√2(2.23)with commutation relation given by [φ˜(x),∂yθ˜(y)] = iδ (x−y). By using bosonizedrepresentation of fermionic operators it is easy to show that the right and left den-sities are proportional to derivatives of bosonic operators:ρR/L ≈∓1√2pi∂xφR/L (2.24)and total density is given byn(x) =Ψ†(x)Ψ(x)≈ σ + 1√pi∂xφ˜ +11piα cos(√4piφ˜ −2kFx)(2.25)where σ = kF/pi gives filling fraction, σ = 1/2 corresponds to half-filling. Simi-larly we could find the bosonized version of the kinetic part of Hamiltonian 2.19as followH0 =∫dx ivF(: ψ†R∂xψR−ψ†L∂xψL :)=vF2∫dx[(∂xφ˜R)2 +(∂xφ˜L)2](2.26)21and bosonized representation of terms in interaction part Eq. (2.20) areρR/L(x)ρR/L(x+a) =12pi (∂xφR/L)2ρR(x)ρL(x+a) = −12pi ∂xφR∂xφLψ†R(x)ψL(x)ψ†L(x+a)ψR(x+a) =−cos2kF2pi (∂xφR−∂xφL)2 +sin2kF3√2pi(∂xφR−∂xφL)3 + · · · (2.27)where · · · are higher order more irrelevant terms in the sense of renormalizationgroup, but the have important effect on spectral functions near the singular pointsas discussed in 2.2.3. By ignoring all the irrelevant and highly oscillatory terms thefinal form of low-energy bosonic Hamiltonian is given byH =vF2∫dx{(1+g42pivF)[∂xφR +∂xφL]2−g2pivF∂xφR∂xφL}(2.28)where g2 = g4 = 2J∆(1− 2cos(2kF)). The Hamiltonian 2.28 could be written inthe following form known as Luttinger Liquid HamiltonianHLL =v2∫dx[K(∂xθ˜)2+1K(∂xφ˜)2](2.29)where v is the renormalized velocity and K is the Luttinger parameter, and are givenbyv = vF√(1+g42pivF)2−(g22pivF)2≈ vF(1+2∆pi sinkF)K =√2pivF +g4−g22pivF +g4 +g2≈ 1−2∆pi sinkF (2.30)The expressions for v and K, in terms of microscopic model 2.17 are valid onlyfor ∆ 1. For zero magnetic field there is an exact expression for velocity and22Luttinger parameter, derived from Bethe ansatz calculations [9]v = Jpi2√1−∆2arccos∆, K = [2−2arccos(∆)/pi]−1. (2.31)For non-zero magnetic field, away from half-filling, there is no analytic expressionfor v and K, but they could be found by using numerical methods to solve Betheansatz equations.As we see the bosonization method is a powerful method to study one dimen-sional interacting systems, to study the new bosonic model we only need to knowtwo parameters, velocity v and the Luttinger parameter K. By evaluating thesetwo parameters either explicitly or by using numerical methods all the physicalproperties of the system could be expressed in terms of them.2.2.2 Spectral function of XXZ spin chain using bosonizationAs mentioned before, calculating the dynamical correlations for XXZ modelis very difficult, and is one of the most studied problems of one dimensional spinchains. By using field theory methods known as bosonization[2, 7, 9, 14] we couldget some information about low-energy effective description of these correlations.Here we review the results of bosonization for the transverse spectral function ofthe XXZ Hamiltonian Eq. (2.1).To use bosonization approach for finding the spectral functions of XXZ model,we need to write the spin operators in terms of the bosonic fields. From Jordan-Wigner transformation we have a relation between spin operators and fermion op-erators; by using the result of previous section we could find an expression for spinoperators in terms of bosonic fields.As we see from Eq. (2.9) the easy one is Szj:Szj ≈ m+ : ψ†LψL : + : ψ†RψR : +(ψ†LψR +h.c.)= m+√Kpi ∂xφ +Cz cos[√4piKφ +(2pim+pi)x](2.32)where : . . . : denotes normal ordering and m = 〈Szh〉 is the magnetization, which is23related to the Fermi wave-vector by the exact relation:2kF = 2pim+pi (2.33)For weak fields,m→ Kh/(piv). (2.34)Cz is a non-universal constant. To obtain the low energy representation of S+j fromEq. (2.9) we use:ψR/L ∝ e−i√pi/Kθ±i√piKφ (2.35)We also approximate the exponential of the Jordan-Wigner string operators using:ipi∑l< jc†l cl ≈ ipi∫ j−∞dy(kF +√K/pi∂yφ) = N+ ipikF j+ i√piKφ( j). (2.36)where N is just a constant. Following the standard bosonization approach [7]we have ignored the oscillating term in c†jc j in the exponential of the Jordan-Wigner string operator, but we will consider it in the next sub-section. Note thatexp[i∑ j<i c†jc j] is Hermitian, taking eigenvalues ±1. On the other hand, the ex-ponential of the continuum limit operator in Eq. (2.36) is not Hermitian. To dealwith this problem the standard approach[7] is to instead take the continuum limitof cos[pi∑l< j c†l cl]:cos[pi∑l< jc†l cl] ∝ eipikF jei√piKφ( j)+h.c. (2.37)Substituting these low energy limit formulas into the second of Eq. (2.9) gives:S−j ∝ (−1)je−i√pi/Kθ[eikF jei√piKφ + e−ikF je−i√piKφ]×[eikF jei√piKφ + e−ikF je−i√piKφ]= e−i√pi/Kθ[C(−1) j +C− cos(2pimx+√4piKφ)](2.38)where C and C− are two other non-universal constants. It can be seen from Eqs.(2.32) and (2.38) that the applied magnetic field induces a shift in momenta of24both transverse and longitudinal spin operators, but in different ways. For thelongitudinal operator, it shifts only the staggered part but for transverse spin it shiftsonly the uniform part. This shift is small for weak fields where the approximationof ignoring band curvature is valid.Let us focus on transverse structure function near q ≈ 0 for weak fields. Atzero temperature we have〈S+j (t)S−0 (0)〉 ∝e−iHx(vt + x− iε)2+2η(vt− x− iε)2η+eiHx(vt− x− iε)2+2η(vt + x− iε)2η(2.39)where x = j, H and η are given byH = 2pimη = (1−2K)28K(2.40)obeying η < 1/8 for |∆|< 1, and ε is a positive quantity of order the lattice spacing.By taking the Fourier transform, the spectral function S+− isS+−(q,ω) ∝ θ(ω− v|q+H|)(ω+ v(q+H))1+2η(ω− v(q+H))1−2η+ θ(ω− v|q−H|)(ω− v(q−H))1+2η(ω+ v(q−H))1−2η .Note that the first term has a diverging threshold at ω = v(q+H) for q+H > 0and a vanishing threshold at ω = −v(q+H) for q+H < 0. The second term isthe parity transform (q→ −q) of the first. Diverging and vanishing thresholdsare indicated by solid and dotted lines in Fig. [2.2a]. Note that, two divergingthresholds occur in S+−(q,ω) for |q|< H which cross each other at q = 0 but thatthere is only one diverging thresholds for |q| > H. S−+ is obtained from S+− bythe transformation H→−H. (Recall that we are assuming H > 0.)25(a) S+−(b) S−+Figure 2.2: Singular points of zero temperature transverse spin spectral func-tion of XXZ model, predicted by bosonization. The solid lines indicatediverging singularities and dashed line the vanishing singularities.26S−+(q,ω) ∝ θ(ω− v|q−H|)(ω+ v(q−H))1+2η(ω− v(q−H))1−2η+ θ(ω− v|q+H|)(ω− v(q+H))1+2η(ω+ v(q+H))1−2η (2.41)Its diverging and vanishing thresholds are shown in Fig. [2.2b]. Note that, for S−+,no diverging thresholds occur for |q| < H and a single diverging threshold occursfor |q| ≈ H and also near q≈ pi .For ESR applications we will be especially interested in the case ∆ slightly lessthan 1 and small H corresponding to K slightly greater than 1/2 and thus η  1.Then it is important to note that the η dependence of the constant factor in S+−and S−+ is ∝ sin2(2piη)Γ(−1− 2η)Γ(1− 2η) [46] which vanishes linearly withη ; here Γ is Euler’s Gamma function. To study the line shape at H = 0 and η→ 0,we take into account that this expression for S(q,ω) is only valid for a finite rangeof ω , v|q|< ω < Λ for an upper cut off Λ, of order J or less. We then use the factthatlimη→0∫ Λvq2η θ(ω− vq)(ω− vq)1−2η = limη→0Λ2η = 1. (2.42)Therefore we havelimη→02η θ(ω− vq)(ω− vq)1−2η = δ (ω− vq)and thus the term with a diverging threshold approachesS(q,ω) ∝ v|q|δ (ω− vq). (η → 0). (2.43)For a fixed momentum q and small η , S(q,ω) as function of ω is depicted inFig. [2.3]. Here we show only one term with a diverging threshold. It is zero forfrequencies such that ω < vq, and it has a local minima at point ω∗ given byω∗ =√1−ηη vq≈vq√η (2.44)So for η ≈ 0.1 we get ω∗ ≈ 3vq. We should also be careful about the cases ofvery small anisotropy; from Eq. (2.44) we see that for fixed momentum q, as27 0  1  2  3  4  5S+- (ω ,q)ω/vqω*ω4 η(a) h = 0 0  1  2  3  4  5S+- (ω ,q)ω/vqω1 ω2(b) h 6= 0Figure 2.3: Zero temperature transverse spectral function S+−(ω,q) pre-dicted by bosonization for fixed q. a) shows transverse spectral functionfor zero magnetic field and b) is for non-zero magnetic field with q < H.28anisotropy gets smaller and smaller ω∗ becomes larger and larger, so it seems thatwe are getting out of the region where bosonization is trustworthy. The results ofbosonization are reliable below some cutoff Λ; then the consistency relation ω∗ <Λ will gives us a restriction on momentum q such that we must have vq <√ηΛ.In fact, as shown above S+− and S−+ are a sum of two terms each with aseparate threshold for all q 6= 0. Depending on which spectral function we look atand the value of q, these thresholds can be both diverging, both vanishing or onediverging, one vanishing. The various shapes of S(q,ω) are sketched in Fig. [2.4].In the special case q = 0, there is a single term of diverging threshold type.(a) |q|. H for S+− (b) H . |q| for S+−(c) |q|. H for S−+ (d) H . |q| for S−+Figure 2.4: Various shapes of singularities of S+− and S−+ for different rangeof momentum q, with |q|,H 1, predicted by bosonization.29For the transverse Green’s function at q≈ pi we have〈S+(x, t)S−(0)〉 ≈ 〈S−(x, t)S+(0,0)〉∝eipix(vt− x− iε)1/4K(vt + x− iε)1/4K(2.45)where x = j. By taking the Fourier transform we obtain the transverse spectralfunction near q = pi:S+−(q,ω) ≈ S−+(q,ω) ∝ θ(ω− v|q−pi|)(ω2− v2(q−pi)2)1−1/4K(2.46)So we see that the singularity exponent for staggered part is 1−1/4K, which is dif-ferent than the exponent for uniform part. Now there is a single diverging thresholdat ω = v|q−pi| for either sign of q−pi for both S+− and S−+. It seems natural toassume that the diverging thresholds of S+− and S−+ starting at q = ±H can beinterpolated to the diverging thresholds terminating at q = pi . Such an assumptiongoes beyond the standard bosonization approach which is restricted to q near 0 andpi (and to small H) but we shall see in section 2.3, using X-ray edge methods, thatthis interpolation is correct.However we will also find that the exponents of the diverging thresholds thatare predicted by standard bosonization: 1−2η near |q| = H and 1−1/(4K) nearq = pi are both incorrect, as are the exponents of the vanishing thresholds.2.2.3 Effect of irrelevant band curvature operatorsIn previous sub-section we reviewed the prediction of bosonization for the sin-gularity exponent of transverse spectral functions. In this section by including theeffect of band curvature operators we show that the predictions of naive bosoniza-tion for singularity exponents are not reliable.In bosonization approach we treat the interactions exactly but we linearize thedispersion around Fermi points and neglect the effects of higher order band cur-vature terms. By power counting these terms are irrelevant in low energies andrenormalize to zero, but as discussed in [28], the effect of these operators is impor-tant near the singular thresholds for longitudinal spectral function. We will show30that the same argument works for transverse spectral function. In this section welook at the effect of these terms to lowest order for h 6= 0.As shown in [28, 38] by including the effect of band curvature corrections theHamiltonian becomesH =HLL +δH (2.47)where HLL is Luttinger Liquid Hamiltonian and δH is given byδH =√2pi6∫dx{η−[(∂xϕL)3− (∂xϕR)3]+ η+[(∂xϕL)2∂xϕR− (∂xϕR)2∂xϕL]} (2.48)Where to first order in ∆ we haveη− ≈1m(1+2∆pi sinkF)η+ ≈ −3∆pim sinkFwhere m = (J coskF)−1 is the effective mass of Fermi excitations. For weak inter-action we can neglect η+ and only include the effect of η− term in Eq. (2.48).Now let us evaluate the transverse spectral function using perturbation theory inδH . By ignoring terms proportional to η+ which mixes right and left operators,in general we are looking for the following kind of imaginary time correlationfunctionsGνν¯(x,τ) = GR(x,τ)GL(x,τ)GR(x,τ) = 〈ei√2piνϕR(x,τ)e−i√2piνϕR(0,0)〉GL(x,τ) = 〈ei√2piν¯ϕL(x,τ)e−i√2piν¯ϕL(0,0)〉 (2.49)Where ν and ν¯ could be written explicitly in terms of Luttinger parameter, K, butfor the following discussion we do not need their explicit form. The first order31correction from perturbation Eq. (2.48) modifies the correlation function toGνν¯(x,τ) = G(0)R (x,τ)G(0)L (x,τ)−iη−√2pi6∫dzdτ ′〈ei√2piνϕR(x,τ)(∂zϕR(z,τ ′))3e−i√2piνϕR(0,0)〉0G(0)L (x,τ)+iη−√2pi6∫dzdτ ′〈ei√2piν¯ϕL(x,τ)(∂zϕL(z,τ ′))3e−i√2piν¯ϕL(0,0)〉0G(0)R (x,τ)(2.50)To evaluate the correlation function Eq. (2.50), we focus on corrections to GR(x,τ);calculations for GL(x,τ) are exactly the same.So we haveGR(x,τ) = G(0)R (x,τ)−iη−√2pi6∫dzdτ ′〈ei√2piνϕR(x,τ)(∂zϕR(z,τ ′))3e−i√2piνϕR(0,0)〉0G(0)L (x,τ)(2.51)In diagrammatic way the non-zero contributions are depicted in Fig. [2.5] .(x, t) (0, 0) (x, t)(z, τ)(0, 0)+Figure 2.5: The relevant Feynman graphs which contribute to correlation infirst order of perturbation.32So the non-zero correction is given byδGR ∝∫dzdτ ′〈ei√2piνϕR(x,τ)(∂zϕR(z,τ ′))3e−i√2piνϕR(0,0)〉=∫dzdτ ′∫∑n,m(i√2piν)n(−i√2piν)mn!m!×〈ϕR(x,τ)n(∂zϕR(z,τ ′))3ϕR(0,0)m〉δGR =∫dzdτ ′∑m,n3n m(m−1)(i√2piν)n(−i√2piν)mn!m!×〈ϕR(x,τ)n−1ϕR(0,0)m−2〉〈ϕR(x,τ)∂zϕR(z,τ ′)〉〈∂zϕR(z,τ ′)ϕR(0,0)〉2(2.52)In going from second line to third line, we have used Wick’s theorem, and thefactor 3n m(m− 1) comes from the all possible number of contraction of fields.Thus we haveδGR ∝ −3i√(2piν)3G(0)R (x,τ)×∫dzdτ ′〈ϕR(x,τ)∂zϕR(z,τ ′)〉〈∂zϕR(z,τ ′)ϕR(0,0)〉2 (2.53)Now by using the fact that< ∂xϕR,L(x,τ)ϕR,L(0,0)>=12pi1vτ∓ ix (2.54)we can write Eq. (2.53) asδGR ∝ 3i√(2piν)3G(0)R (x,τ)∫dz dτ ′ 1v(τ ′− τ)− i(z− x)1(vτ ′− iz)2= −12pii√(2piν)3 1(vτ− ix)ν∫dτ ′ sgn(τ′)− sgn(τ ′− τ)(vτ− ix)2= −12pii√(2piν)3 t(vτ− ix)2+ν (2.55)33By replacing vτ → ((vτ− ix)+(vτ+ ix))/2 we can write δGR asδGR ∝−6pii√(2piν)3v(1(vτ−ix)1+ν +vτ+ix(vt−ix)2+ν)(2.56)With exactly the same calculations we will get results for left moving fields asδGL ∝−6pii√(2piν¯)3v(1(vτ+ix)1+ν¯ +vτ−ix(vτ+ix)2+ν¯)(2.57)Now by plugging all these results into Eq. (2.50), the final form of the correlationfunction to first order is given byGνν¯(x,τ) =1(vτ− ix)ν(vτ+ ix)ν¯ {1−4pi2η−√ν3v(1vτ− ix +vτ+ ix(vτ− ix)2)+4pi2η−√ν¯3v(1vτ+ ix +vτ− ix(vτ+ ix)2)} (2.58)Now by taking the Fourier transform of Eq. (2.58) and continuing to real frequen-cies, we haveGνν¯(ω,q) = (ω− vq)ν−1(ω+ vq)ν¯−1{1−4pi2η−√ν3v(ω− vq)(1+ω− vqω+ vq)+4pi2η−√ν¯3v(ω+ vq)(1+ω+ vqω− vq)} (2.59)Where η− ≈ 1/m. It is easily seen from the last term of Eq. (2.59) that, as ωapproaches vq, the perturbative corrections blow up like q2/m(ω − vq). This isexactly the reason that bosonization fails near the threshold. We also see that for|ω − vq|  q2/2m, perturbative corrections become small and we get the naivebosonization results, so we showed that irrelevant operators potentially will changethe singularity exponent of correlation functions near the threshold but their effectis negligible away from the threshold so they would not change the qualitativeshape of correlation function found by bosonization. In the following section byusing the X-Ray edge method we find the singularity exponents of transverse spec-tral functions.342.3 X-Ray Edge methodIn sub-section 2.2.2, by use of standard bosonization we found that the trans-verse spectral functions S+−(ω,q), for q near 0 has a diverging singularity withexponent 1− 2η . In sub-section 2.2.3 we argued that band curvature operatorswould change the result of bosonization for singularity exponents. In this sectionwe will explore this question by extending bosonization using X-ray edge methodsintroduced in [22, 23, 27, 29]. We find that by use of these methods, transversespectral functions have different critical exponents at singular energies than pre-dicted by standard bosonization. We also point out the existence of large numbersof sub-leading singularities with vanishing intensities, similar to the ones found bystandard bosonization.We use the notation and results of [22] and we skip the details of derivations;interested readers should see [22, 29] and also Appendix A for most detailed calcu-lations. In this approach we try to evaluate transverse spectral functions for fixedmomentum q by including the effect of a single high-energy particle or hole ex-citation. To do this we need to find relevant momenta of this excitation whichcontributes to the spectral functions at momentum q. In this section we find theeffective Hamiltonian for these excitations, and also those relevant momenta.Suppose that the momentum of this particle or hole excitation is k. (We willeventually use the notation kp for a particle and kh for a hole.) In X-Ray edgemethod we are interested in the high energy excitations near this momentum k andalso low-energy excitations around the fermi points Fig. 2.6. Thus we can writethe fermion operator in the following formc j ≈ ψReikF j +ψLe−ikF j +d eik j. (2.60)Where ψR,ψL and d vary slowly on lattice scale. Then by linearizing the dispersionaround the fermi points and bosonizing low energy fermions, and also linearizingthe dispersion around high-energy particle or hole excitation, we haveH = d† (ε− iu∂x)d +v2[(∂xϕL)2 +(∂xϕR)2]+1√2piK(κL∂xϕL−κR∂xϕR)d†d352+ F2+ FEqEF2+ k? F− ? FFigure 2.6: In beyond Luttinger models, in addition to low energy excitationswe also include the effect of a high-energy particle or a deep hole. Inthis case we have a hole at momentum k.Here the chiral fields ϕL and ϕR are the transformed ones defined by:φ = ϕL−ϕR√2θ = ϕL +ϕR√2.This Hamiltonian is described in [47–49] for Luttinger liquid coupled to an impu-rity. The parameters of above Hamiltonian are as follows; ε is the energy of thehigh energy particle or hole and, for ∆= 0, it is given by ε =−2J(cosk− coskF).u is the velocity of the heavy particle or hole and equals J sink at ∆= 0. v and K are36the boson velocity and Luttinger parameter, respectively. v may be regarded as theFermi velocity of the interacting fermion model. It is the only velocity appearingin the standard bosonization approach and plays the role of the “velocity of light”in the effective Lorentz invariant field theory. The velocity parameter u, describingthe high energy particle or hole is an important new parameter in the X-ray edgeapproach. Finally κR,L are the couplings between high energy fermion and bosonicfields and to first order in ∆ are given by κR,L = 2∆[1− cos(kF ∓ k)]. These cou-pling could be evaluated by Bethe ansatz calculations[9] for any ∆, H and k. Theyare important for finding the singularity exponents. Note that, in general, u, ε andκR,L all depend on k as well as ∆ and h.The Hamiltonian (2.61) looks complicated as it contains interactions betweenfermions and bosons. We can eliminate the interacting part of (2.61) by doing aunitary transformation given byU = exp[−i∫dx√2piK(γRϕR + γLϕL)d†d](2.61)Effect of unitary transformation on bosonic and fermionic fields is as follows∂xϕR,L = ∂xϕ¯R,L±γR,L√2piKd¯†(x)d¯(x)d = d¯ exp[−i√2piK(γRϕ¯R + γLϕ¯L)]ϕR,L = ϕ¯R,L±γR,L4√2piKN˜(x)where N˜(x) is defined by:N˜(x) =∫ ∞−∞sgn(x− y)d¯†(y)d¯(y)dy. (2.62)It is easy to see that unitary transformation leaves d†(x)d(x) invariant and we candecouple fermionic fields from the bosonic ones with appropriate choice of γR,L,given byγR,L =κR,Lv∓u(2.63)37Having done the unitary transformation the Hamiltonian will look likeH =v2[(∂xϕ¯L)2 +(∂xϕ¯R)2]+ d¯† (ε− iu∂x) d¯ + · · · (2.64)Where · · · means higher dimension irrelevant interactions that will be produced bydoing unitary transformation and which we ignore.We now consider the transverse Green’s function:S−+ = 〈S−j (t)S+0 (0)〉.By doing Jordan-Wigner transformation, we haveS−+j (t) = eipi j〈c j(t)cos[piN j(t)]cos[piN0(0)]c†0(0)〉where c j(t) is approximated as in Eq. (2.60) andN j(t) =∑l< jc†l (t)cl(t)To obtain the transverse spectral function, S−+(q,ω) at a wave-vector q far fromthe low energy regions, ±H, pi , the term that we are interested in isS−+j (t) = eipi jeikp j〈d( j, t)cos[piN j(t)]cos[piN0(0)]d†(0,0)〉. (2.65)We see that d must be chosen to be a particle operator and we have consequentlylabeled its momentum kp. Note that we have written the Jordan-Wigner stringoperator in manifestly Hermitian cos form, as in Sec. 2.2.2. Now we decompose38N j(t), into c-number, non-oscillatory and oscillatory partsN j(t) =kFpi j+ n˜( j, t)+m( j, t)n˜(x, t) =∫ x−ε∞dy : ψ†R(y, t)ψR(y, t) : + : ψ†L(y, t)ψL(y, t) : +d†(y, t)d(y, t)m(x, t) =∫ x−ε∞{ψ†R(y, t)d(y, t)ei(kp−kF )y +ψ†L(y, t)d(y, t)ei(kF+kp)y+ψ†L(y, t)ψR(y, t)e2ikF y +h.c.}∝(1i(kp− kF)ψ†R(x′, t)e−ikF x +1i(kF + kp)ψ†L(x′, t)eikF x)d(x′, t)eikpx+12ikFψ†L(x′, t)ψR(x′, t)e2ikF x +h.c.Where ε → 0+, x′ ≡ x− ε and in the third line we used the fact that both ψand d are slowly varying fields and most of the contribution of the integral comesfrom limiting point x− ε . At this point we will set the rapidly oscillating term,m(x, t), to zero. This will give the dominant divergent singularity in the transversespectral function. By Taylor expanding in powers of m(x, t) we obtain variousvanishing singularities as well as unimportant renormalizations of the amplitude ofthe divergent singularity, as we discuss in Sec IV. We may then decompose n˜(x, t)into its commuting high energy and Fermi surface part.n˜(x, t) ≡ n(x, t)+nd(x, t)n(x, t) ≡∫ x−ε∞: ψ†R(y, t)ψR(y, t) : + : ψ†L(y, t)ψL(y, t) :nd(x, t) ≡∫ x−ε∞d†(y, t)d(y, t)dy.Because all the d operators in n˜(x, t) are at points y < x, we have[n˜(x, t),d(x, t)] = 0and thus we may drop the nd terms leaving:S−+j (t) ∝ ei(kp+pi) j〈cos[kF j+pin( j, t)]d(x, t)d†(0,0)cos[pin(0,0)]〉39Following Eq. (2.36) this becomes:S−+j (t) ∝ ei(kp+pi) j〈cos[kF j+√piKφ( j, t)]d( j, t)d†(0,0)cos[√piKφ(0,0)]〉.Note that we have treated the Jordan-Wigner string operator in precisely thesame approximation as in the standard bosonization approach. We now make theunitary transformation of Eq. (2.62) so that the fermion and bosons are decoupled.Noting that N˜(x, t) annihilates the vacuum this leaves:S−+ ∝ 〈d¯(x, t)cos[kFx+√piKφ¯(x, t)]e−i√2piK [γRϕ¯R(x,t)+γLϕ¯L(x,t)]ei√2piK [γRϕ¯R(0,0)+γLϕ¯L(0,0)] cos[√piKφ¯(0,0)]d¯†(0,0)〉ei(kp+pi)xSeparating the fermionic and bosonic factors, this becomes:S−+j (t) = S(0)−+(x, t)〈d¯(x, t)d¯(0)†〉 (2.66)where x = j,S(0)−+(x, t) = ei(pi+kp−kF )xI−(x, t)+ ei(pi+kp+kF )xI+(x, t) (2.67)andI+(x, t) =〈e−i√2piν+R ϕ¯R(x)+i√2piν+L ϕ¯L(x)ei√2piν+R ϕ¯R(0)−i√2piν+L ϕ¯L(0)〉=(εε+ ivt− ix)ν+R ( εε+ ivt + ix)ν+LI−(x, t) =〈ei√2piν−R ϕ¯R(x)−i√2piν−L ϕ¯L(x)e−i√2piν−R ϕ¯R(0)+i√2piν−L ϕ¯L(0)〉=(εε+ ivt− ix)ν−R ( εε+ ivt + ix)ν−L(2.68)40Where ε is of order of the lattice spacing and ν±R,L are defined as followν±R =14(γRpi√K±√K)2ν±L =14(γLpi√K∓√K)2. (2.69)At zero magnetic field we have γR,L/pi = 1−K,[29] independent of momentum, sothe results will simplify toν±R =14K(1−K±K)2ν±L =14K(1−K∓K)2 . (2.70)Having done the unitary transformation, the d¯ fields act as free particle so we have〈d¯(x, t)d¯†(0)〉 ≈ e−iε(kp)t∫ Λ−Λdk2pi eik(x−ut) ≈ e−iε(kp)tδ (x−ut). (2.71)We can now turn to the question of how the momentum of the high energyparticle, kp, should be chosen to study S−+(q,ω) near the threshold for arbitraryq. From Eq. (2.67) we see that a high energy particle of momentum kp givesterms in the transverse Green’s function oscillating at wave-vectors pi + kp− kFand pi+ kp + kF . Thus we see that there may actually be two choices for kp whichwill give a contribution to the transverse Green’s function oscillating at a specifiedwave-vector q:k±p ≡ q+pi∓ kF . (2.72)In general, both must be considered in calculating the singularity behaviour ofG−+(q,ω). However, the kp’s are restricted by the requirement that they are al-lowed particle momenta,kF < kp < 2pi− kF , (mod 2pi). (2.73)(Recall that kF = pi/2+ pim > pi/2.) Thus we see that the high energy particleof momentum k+p contributes to S−+(q,ω) for q in the range [2kF −pi,pi] and the41particle of momentum k−p contributes for q in the range [−pi,pi−2kF ]. For a givenq there is at most one possible high energy particle momentum contributing toS−+(q,ω). Having identified the appropriate high energy particle momentum wemay now complete the calculation by Fourier transforming Eq. (2.66) using Eqs.(2.68)-(2.71). The result is:S−+(q,ω) ∝∫dxdtei[ω−ε(k+p )]tδ (x−ut)(vt− x− iε)ν+R (vt + x− iε)ν+L, (2kF −pi < q < pi)∝∫dxdtei[ω−ε(k−p )]tδ (x−ut)(vt− x− iε)ν−R (vt + x− iε)ν−L, (−pi < q < pi−2kF).(2.74)The x-integrals may be done trivially using the δ -functions.S−+(q,ω) ∝∫dtei[ω−ε(k+p )]t[(v−u)t− iε]ν+R [(v+u)t− iε]ν+L, (2kF −pi < q < pi)∝∫dtei[ω−ε(k−p )]t[(v−u)t− iε]ν−R [(v+u)t− iε)ν−L, (−pi < q < pi−2kF).(2.75)The t-integrals can now be done by contour methods. Note that, if v > u, they areonly non-zero for ω > ε(k+p ) and ω > ε(k−p ) respectively, corresponding to a lowerthreshold. Using the free particle cosine dispersion relation v> u is always satisfiedfor particle excitations. While this dispersion relation is known to be exact, apartfrom an overall factor, including interactions for h = 0 it is in general modified.We might expect that v > u remains true for particles, at least for small enough h.However, see below. Assuming this, we obtain:S−+(q,ω) ∝ θ [ω−ωL(q)][ω−ωL(q)]µ(q)(2.76)where the singular energies are given by:ωL(q) = ε(k+p ), (2kF −pi < q < pi)= ε(k−p ), (−pi < q < pi−2kF). (2.77)42The critical exponents are given byµ(q) = 1−ν+R (k+p )−ν+L (k+p ), (2kF −pi < q < pi)= 1−ν−R (k−p )−ν−L (k−p ), (−pi < q < pi−2kF) (2.78)with ν±R,L(k) given by Eqs. (2.69) and (2.70). Note that the same phase shift param-eters γR/L determine both longitudinal and transverse spectral functions; howeverwe need to know them at the momentum of the high energy particle or hole whichis not the same for longitudinal and transverse spectral functions, for given q. Ingeneral, the phase shift parameters depend on momentum as well as field, becom-ing momentum independent at h = 0. It follows from parity that κL(k) = κR(−k).Since, for q > 0, k−p (−q) = −k+p (q), (mod 2pi) ν−L (−q) = ν+R (q) and ν−R (−q) =ν+L (q) and hence µ(−q) = µ(q).It is interesting to compare both the singular energies and exponents to thosepredicted by standard bosonization as q approaches the zero energy points q ≈±H = ±(2kF − pi) and q ≈ pi . Near these zero energy points we may linearizethe ε(k) giving: ωL(q) ≈ v|q∓H| and v|q− pi|, precisely the singular energiespredicted by standard bosonization. This X-ray edge calculation also confirms theconjecture made in Sec. 2.2.2 that the diverging singular energies at the differentlow energy momenta are smoothly connected. On the other hand, the exponentsappear to disagree with the standard bosonization results for all ∆ and h, a similarobservation to the one in Ref. [25]. This can be seen, for example. by consideringthe limit ∆→ 0. In this case γ±L,R → 0, K → 1 so ν±L,R → 1/4 and µ → 1/2, forall q and h. On the other hand the standard bosonization result from Eq. (2.40)and (2.41) is η → 1/8 and hence µ → 3/4. We can also compare the zero fieldpredictions for general ∆. From Eq. (2.70)µ = 2−1/(2K)−K, (h = 0,∀q). (2.79)On the other hand, standard bosonization predicts µ = 1− 2η = 2− 1/(4K)−Kfor q ≈ 0 and µ = 1− 1/(4K) for q ≈ pi . We expect that standard bosonizationfails to predict critical exponents correctly for the transverse spectral function, as43discussed in sub-section 2.2.3. Given this situation, it is useful to check the SU(2)symmetric case, h = 0, ∆ = 1. In this case X-ray edge methods predict, from Eq.(2.79) ν−R = ν+L = 0, ν+R = ν−L = 1/2, µ = 1/2 for the transverse spectral functionindependent of q. The same exponents were found earlier[29] for the longitudinalspectral function [50] (In this case, they agree with standard bosonization nearq = pi but not near q = 0.)By doing similar calculations we could find S+−, which is different than S−+for h 6= 0. Following the same procedure, Eq. (2.66) is replaced by:S+− ∝ 〈d¯†(x, t)cos[kFx+√piKφ¯(x, t)]ei√2piK [γRϕ¯R(x,t)+γLϕ¯L(x,t)]e−i√2piK [γRϕ¯R(0,0)+γLϕ¯L(0,0)] cos√piKφ¯(0,0)]d¯(0,0)〉ei(−kh+pi)xwhere d now annihilates a particle in a filled state below the Fermi energy withmomentum kh, i.e. creates a hole. This can again be factorized as:S+−j (t) = S(0)+−(x, t)〈d¯†(x, t)d¯(0)〉 (2.80)whereS(0)+−(x, t) = ei(pi−kh+kF )xI−(x, t)+ ei(pi−kh−kF )xI+(x, t) (2.81)and I∓(x, t) are the same functions defined in Eq. (2.68), except that kh must lie ina different range, |kh|< kF . Thus defining:k∓h ≡ pi−q∓ kF (2.82)we see that the first term in Eq. (2.81) is non-zero for q in the range [pi − 2kF ,pi]while the second is non-zero for q in the range [−pi,2kF − pi]. These are widerranges than occur in S−+. (Recall that we assume H ≥ 0 and hence kF ≥ pi/2.) Inparticular, both terms can contribute for |q| < H. Again, as q approaches the zeroenergy points, ±H and pi the singular energies approach those predicted by stan-dard bosonization. Again as anticipated in sub-section 2.2, the singular energies atthese zero energy points can be smoothly connected.Another interesting feature is the shape of the singularity. For S−+ the singu-larity was one-sided, vanishing for ω < ωL. This was a consequence of the fact44that the velocity of the high energy particle always obeys u < v assuming this fea-ture of the non-interacting dispersion relation is unchanged by interactions. On theother hand for holes, again using the non-interacting dispersion relation, u < v isonly obeyed if |kh| < pi − kF ; there is a range of hole momentum near kF wherethe high energy hole has a higher velocity than the Fermi velocity. |k+h | < pi − kFcorresponding to 0 < q < 2pi−2kF = pi−H. Thus, in this region the singularity isone-sided, ∝ θ(ω−ωL). On the other hand for −H < q < 0 and pi−H < q < pi ,where u > v the integral in Eq. (2.75) is also non-zero and gives the same criticalexponent with a different amplitude for ω < ωL. In this case we see that ωL(q)is not a lower threshold. There is also spectral weight below this frequency. Thequalitative shape of S+−(q,ω) for 0 < q < H is sketched in Fig. [2.7] .We emphasize that a singularity is one-sided for u < v and two-sided for u > vwhere v is the Fermi velocity and u is the velocity of the high energy particle orhole. In general, these velocities depend on ∆ and also h, being strongly renormal-ized by interactions. In [28], Fig. [15], it was illustrated that for ∆ = 1 and smallnon-zero field one of the cubic term in the bosonized Hamiltonian, due to bandcurvature effects, has a coupling constant η− < 0. This may indicate a reversal ofthe sign of the effective mass due to (strong) interaction effects, implying a reversalof the sign of u−v as the energy of the high energy particle or hole approaches theFermi energy, corresponding to |q| → |2kF−pi| (or q→ pi). Thus in this parameterrange, the one-sided and two-sided nature of the 3 singularities in S+− and S−+,discussed above, would be reversed.Given that the singular energy, ωL(q) is sitting inside a region of non-zero spec-tral weight, for certain ranges of q, we might ask whether it is reasonable to expectsingular behaviour at this energy or whether the infinite peak might be broadenedand made finite due to some sort of decay process for this high energy excita-tion. This important question also arises for the longitudinal structure function andfor the fermion spectral function. It has been suggested [22, 26] that integrabilitymight prevent this excitation from decaying, for some range of momentum, eventhough it is kinematically allowed, leaving the singularity intact. This is true be-cause three-body scattering processes are required for it to decay and these areexpected not to occur in this integrable model. This seemed to be consistent withDensity Matrix Renormalization Group results for the fermionic spectral function45Figure 2.7: The behaviour of S+−(ω,q) correlation for |q| < H. There aretwo different hole excitations which contribute to the spectral function;the lower energy hole produces a cusp-like singularity.[22].In fact, we should also consider processes in which the heavy particle or holedecays by producing 3 other high energy quasi-particles, a 2-body process which isexpected to be present even in this integrable model. Using the −cosk dispersionrelation, this is kinematically allowed for high energy holes[51] in the region u < v,corresponding to 0< |q|< pi−H (the upper branch for 0< |q|<H) but not allowedfor high energy particles for any q [22].Putting these observations together, we expect the sharp one-sided singularityof S−+(q,ω) to be present for H < |q|< pi and the 2-sided singularity of S+−(q,ω)for 0< |q|<H and pi−H < |q|< pi to be present. The one-sided singularity of S+−for H < |q| < pi −H should be broadened by higher order interaction effects[22]46not taken into account in this treatment. See Fig. [2.8]. Again we emphasizethat the precise region of q over which singularities are broadened depends on thedispersion relation, which is modified by interactions; here we have just stated itusing the −cosk dispersion relation, valid at small ∆.So far, we have set the rapidly oscillating operator m(x, t), defined in Eq. (2.66)to zero. The effects of including it are discussed in the following section. It basi-cally leads to additional terms in the transverse spectral function which have singu-larities at different energies, including lower ones. However, these singularities areof vanishing type, with exponent µ < 0, dashed lines of Fig. [2.8]. The relativelysimple approach we have taken here is just sufficient to give the diverging singularterms.The situation is considerably simpler at zero field, h = 0. In this case, the freedispersion relation is known to be exact, apart from an overall change of ampli-tude, 2t → v. Thus the condition u < v is always satisfied so S−+ = S+− has onlyone single-sided singularity at ωL = vsinq with v given in Eq. (2.31) and criticalexponent given by Eq. (2.79). In this case, no decay processes are kinematicallyallowed and no additional singularities occur, since the single hole or particle hasthe lowest possible energy for given wave-vector.2.4 Sub-dominant singularitiesIn Sec. 2.4 and 2.3 we ignored rapidly oscillating terms, m(x) of Eq. (2.65) inthe Jordan-Wigner string operator in calculating the transverse structure functionand also the effects of Umkalpp term which oscillates as ei4kF x. We consider theeffect of including these terms here. Let us begin with the term:m(x) =12ikFψ†L(x′, t)ψR(x′, t)e2ikF x +h.c. (2.83)Actually, this term represents a correction to standard bosonization, even withoutusing X-ray edge methods, so we consider its effects there. To make things assimple as possible we also consider zero field, kF = pi/2. Then, after bosonizing47(a) S+−(b) S−+Figure 2.8: Singular points of zero temperature transverse spectral functionsof the XXZ model, predicted by X-ray edge method. The solid linesindicate diverging singularities and dashed line the vanishing singular-ities. The grey thick line indicates the range of momentum over whichinteractions should broaden the diverging singularity. In Fig. [2.8a] and[2.8b] the lower dashed line is given by particle excitations and upperone is given by hole excitation.48m, the standard bosonized expression for S− in Eq. (2.38) is modified to:S−j ∝ e−i√pi/Kθ(x)[C(−1) j +C− cos(√4piKφ(x))]×exp[i(−1) jC′ sin(√4piKφ(x))]. (2.84)for a non-universal constant C′. We now Taylor expand the exponential and usedouble angle formulas. We see that the staggered and uniform parts of S−j have aseries in increasingly irrelevant operators:S−s = e−i√pi/Kθ(x) ∑n∈Za2nei2n√4piKφS−u = e−i√pi/Kθ(x) ∑n∈Za2n+1ei(2n+1)√4piKφ . (2.85)The effect of including the eim factor is simply to renormalize the coefficient of theleading operator in S−u and S−s together with producing the irrelevant corrections.Now consider non-zero field. Eq. (2.84) get replaced by:S−j ∝ e−i√pi/Kθ(x)[C(−1) j−C− cos((2kF −pi) j+√4piKφ(x))]×exp[iC′e2ikF j+i√4piKφ(x)+h.c.]. (2.86)We again get a series of irrelevant operators but now all at different wave-vectors:S−j = e−i√pi/Kθ(x) ∑n∈Zanei(2nkF+pi) j+in√4piKφ . (2.87)It is interesting to note that this expansion contains precisely the same terms as theone derived by Haldane[15] for a boson annihilation operator in a Luttinger liquid.We also see why the replacement of the exponential Jordan-Wigner string operatorby a cosine form, its Hermitian part, is not really necessary. Keeping the completeexpansion in Eq. (2.87), we get the same set of operators either way.Now consider the effect of the term in Eq. (2.83) in the X-ray edge approach.After the unitary transformation of Eq. (2.62) the term in S−j linear in the d¯ opera-49tor, with momentum k, is:S−j ∝ d¯ e−i√2piK (γRϕ¯R+γLϕ¯L) ∑n∈Za′nei[(2n+1)kF+pi+k] j+i(2n+1)√piKφ¯ . (2.88)This expansion is similar to the one derived by Haldane[15] for a fermion annihi-lation operator. The momentum q at which the nth term contributes to the spectralfunction is:q = (2n+1)kF +pi+ k (2.89)where the momentum k must correspond to that of a high energy hole, |k|< kF incalculating S+− or to that of a high energy particle, kF < |k| < pi , in calculatingS−+. Note that by ignoring m(x) in Sec. 2.2.2 we only considered the n = 0 andn =−1 terms in the sum of Eq. (2.88). The nth term in the expansion of S−j in Eq.(2.88) leads to a singular term in the transverse spectral function,S(q,ω) ∝ 1|ω−ω(q)|µ (2.90)at the energy ω(q) given by the energy of the corresponding particle or hole:ωn(q) =±ε[q− (2n+1)kF −pi]. (2.91)We see that, for general kF , these thresholds can occur at arbitrarily low energy.(For rational kF there is a finite number of them and there a lowest one at a non-zero energy.) The corresponding critical exponent is given by:µ(n) = 1−ν(n)R −ν(n)L (2.92)where ν(n)L,R are the left and right scaling dimension of the nth operator in Eq. (2.88).These obey:ν(n)R +ν(n)L =14[(γRpi√K+(2n+1)√K)2+(γLpi√K− (2n+1)√K)2]=12[(2n+1)√K +γR− γL2pi√K]2+12(γL + γR2pi√K)2. (2.93)50Thus the two largest values of µ occur for n = 0 and −1 for |γR− γL|/(2piK)< 1.At zero field, γL = γR so this condition is satisfied. Also at small ∆, γL,R are O(∆) sothe condition is again satisfied. It should remain satisfied for a large range of fieldand ∆ quite possibly including the entire Luttinger liquid regime, but without deter-mining the γL,R explicitly we can’t determine this range. It certainly includes weakfields h J relevant to most ESR experiments. For zero field we have explicitly:ν(n)R +ν(n)L =K2(2n+1)2 +(1−K)22K(2.94)so we see that the exponents for sub-dominant singularities (n 6= 0, −1) obey µ <−3/2 for all K in the Luttinger liquid regime K > 1/2. At ∆= 0,ν(n)R +ν(n)L =12(2n+1)2 (2.95)and µ <−7/2 for all sub-dominant singularities. We expect that µ < 0 for all sub-dominant singularities a wide range of field and ∆ including the weak field regime.So the approximation of dropping m(x) made in sub-section 2.2.2 appears quitegenerally valid.However, we must also consider the other terms in m ∝ ψ†Rd, ψ†Ld. Thesegive contributions to the transverse spectral functions proportional to matrix ele-ments in the d¯ space containing more operators. Consider, for example, the caseof a high energy particle. Then, to first order in these operators we either obtaind¯(−ε)d¯†(0)|0 >= 0 or else a matrix element:〈0|d¯(x, t)d¯(x− ε, t)d¯†(−ε)d¯†(0)|0〉. (2.96)In fact, this is also zero as ε → 0 as follows from Wick’s theorem and translationinvariance:〈0|d¯(x, t)d¯(x− ε, t)d¯†(−ε)d¯†(0)|0〉= (2.97){〈0|d¯(x, t)d¯†(0,0)|0〉2−〈0|d¯(x− ε, t)d¯†(0,0)|0〉2} ε→0−−→ 0.Now consider the higher order expansion in the terms in m proportional to d¯ and51d¯†. Even orders in the expansion give[d¯(−ε)d¯†(−ε)]nd¯†(0)|0〉 ∝ d¯†(0)|0〉. (2.98)The factor in the Green’s function involving Fermi surface excitations has S−dressed by n ψL,R operators and n ψ†L,R operators, all at the same point. The(ψ†LψR)n and (ψ†RψL)n terms just give contributions the same as Eq. (2.88), mod-ifying the a′n coefficients. Other products give higher dimension operators using:ψ†LψL ∝ A+B∂xφR +C∂xφL (2.99)et cetera. Odd terms in the expansion in the terms in m, which are proportional to d¯and d¯†, give zero as ε→ 0, since they are proportional to the same matrix element,Eq. (2.96).Till now we have considered the effect of particle excitation on S−+. In gen-eral high-energy hole excitations, also could contribute to S−+ as well as particleexcitations to S+−. Now we will find the effect of hole excitations on S−+. Holeexcitations only give us higher order corrections to S−+, which have sub-dominantsingularities, but to complete our discussion we find the vanishing singularity expo-nent of hole excitation to S−+; similar argument holds for particle excitation effectson S+−. In addition to the Eq. (2.65), there is another term which contributes toS−+ of the following formS−+j (t) = eipi jeik f j〈ψR( j, t)cos[piN j(t)]cos[piN0(0)]ψ†R(0,0)〉. (2.100)by similar argument as discussed in section 2.3 we could do the canonical andunitary transformations to decouple bosonic fields from high-energy excitations.Now the zero order term gives us the naive bosonization results; the interestingcontribution comes from the first order expansion of m ∝ ψ†Rdh, ψ†Ldh, where dh ishole creation operator. The first order correction of expansion has the following52formS−+ ∝ 〈ψ¯R(x, t)ψ¯R(x− ε, t)d¯†h(x− ε, t)cos[kFx+√piKφ¯(x, t)]×ei√2piK [γRϕ¯R(x,t)+γLϕ¯L(x,t)]e−i√2piK [γRϕ¯R(0,0)+γLϕ¯L(0,0)]×cos[√piKφ¯(0,0)]d¯h(ε,0)ψ¯†R(ε,0)ψ¯†R(0,0)〉ei(2kF−kh+pi)xSimilar to Eq. (2.67) we could decompose S−+ to two terms as followS−+(x, t) = ei(pi−kh+kF )xI−h (x, t)+ ei(pi−kh+3kF )xI+h (x, t) (2.101)where the expression for I±h is similar to Eq. (2.68) with exponents given byν¯±R =14(γRpi√K+2√K+(2±1)√K)2ν¯±L =14(γLpi√K+2√K− (2±1)√K)2(2.102)In general, the higher order correction will include more powers of ψ†RψL and gen-eral expression would beν¯(n)±R =14(γRpi√K+2√K+(2n±1)√K)2ν¯(n)±L =14(γLpi√K+2√K− (2n±1)√K)2(2.103)with momentum given byk(n) = pi− kh +(2n+1)kF (2.104)and the energy of the excitation for given momentum q isω¯n(q) =±ε[−q+(2n+1)kF +pi]. (2.105)ωn and ω¯n for n = 1, are depicted by dashed lines in Fig(2.8), which represents53vanishing singularities of spectral function. The sum of exponents isν¯(n)R + ν¯(n)L =12[(2n+1)√K +γR− γL2pi√K]2+12(γL + γR2pi√K+2√K)2(2.106)We see that our results Eq. (2.93) and (2.106) are the same as Eq. (17) of Ref [27],which is the singularity exponent for boson creation operator, upon identifyingδ±→ γR/L/√K. The actual values of these phase shift parameters are in generaldifferent in the two models however, being determined by Galilean invariance inthe Bose gas model. This correspondence might have been anticipated since aboson creation operator is related to the corresponding fermion one by a Jordan-Wigner string operator[15] just as is the S+j operator. Furthermore, the XXZ modelis equivalent to a lattice boson model with an infinite on-site repulsion which re-stricts the occupancy to 0 or 1.Now we look at the effect of Umklapp scattering term at zero magnetic field.The Umklapp scattering term is in the following formHU =−g[e−4ikF xψ†R(x)ψ†R(x)ψL(x)ψL(x)+h.c](2.107)At non-zero magnetic field we could ignore this term as it is highly oscillatory, dueto ei4kF x prefactor. At zero field we have 4kF = 2pi; thus this term does not oscillateand we need a more careful treatment. In the bosonized form of the XXZ model, itcan be seen that the Umklapp term is irrelevant for 0 < ∆< 1, and is marginal for∆ = 1, thus in this regime we could look at the effect of this term perturbatively.We focus on the effect of this term on S−+. In general the higher order Umklappterm could be written as followU2m¯+m =m+m¯∏i=1(ψ†R(zi))2(ψL(zi))2m¯∏j=1(ψ†L(z′j))2(ψR(z′j))2 (2.108)Where m and m¯ are arbitrary integers, and this term is actually 2m¯+m order inthe Umklapp perturbation. These operators gives us zero corrections unless wekeep higher powers of the m(x) term, Eq. [2.83], at least to power 2m. Thus the54most general non-zero term of both Umklapp and oscillatory term m(x) is in thefollowing formPn,m,m¯ =U2m¯+m(ψ†L(x)ψR(x))m(ψ†L(0)ψR(0))m (2.109)Now by bosonizing the above expression and plugging it into the definition of S−+we haveS−+ ∝∫〈d¯(x)cos(kF +√piKϕ(x))e−i√2piK (γRϕR(x)+γLϕL(x))ei√4piK(m−n)ϕ(x)e−i2√4piK∑m+m¯ ϕ(z j)ei2√4piK∑m¯ ϕ(yi) cos(√piKϕ(0))ei√2piK (γRϕR(0)+γLϕL(0))ei√4piK(n+m)ϕ(0)d¯†(0)〉m¯∏d2yim+m¯∏ d2z jThe effect of this term on the singularity exponents of S−+ could be evaluated bypower counting and the result is given by the following expressionνR =(γR2pi√K−n√K±√K2)2+K(m2 +2m+4m¯)− (m+2m¯)νL =(γL2pi√K+n√K∓√K2)2+K(m2 +2m+4m¯)− (m+2m¯)(2.110)The−(m+2m¯) term in above equations comes from the integration variables z j,yi.The overall exponent is given by the summation of these two exponents, thus wehaveνR +νL =12(γR + γL2pi√K)2+12(γL− γR2pi√K+(2n∓1)√K)2+ 2Km2 +2(2m¯+m)(2K−1) (2.111)As we are considering the effect of Umklapp term at zero field, 4kF = 2pi; theseare singularity exponents of the spectral function at threshold frequencies given byEq. [2.91]. Compared to Eq. [2.93], we see that for fixed n higher order Umklappterms give larger and larger exponents, and are more irrelevant. Therefore, at each55threshold energy, we get a set of singularities with progressively weaker exponents.In this case, these corrections seem truly unimportant. That is, we don’t get anynew singular energies, just sub-dominant corrections to the singularities at the en-ergies we already have.One important point is that if we take n,m = 0 but arbitrary m¯, at half filling andfor ∆ = 1 we have 2K− 1 = 0; thus higher order non-chiral Umklapp correctionsdo not change the singularity exponents, based on power counting. But we shouldbe careful at that limit, because our result was based on power counting; in generalupon evaluating the integrals more carefully, there could be some logarithmic cor-rections to the correlations functions which could change the behaviour of spectralfunctions near singular frequencies.Till now we only considered the effect of Umklapp terms only at zero magneticfield. At finite magnetic field the Umklapp interaction has the following formHU = gU cos4(√piKϕ(x)+(kF −pi/2a)x) (2.112)Where a is the lattice spacing. In general this term is oscillatory and could bedropped out at low energies. But for weak enough magnetic field, the wavelengthof the oscillation is very long, therefore in that limit this term should be treatedcarefully. We claim that at low enough temperatures and weak magnetic field theUmklapp term affects neither the threshold frequency nor the singularity expo-nents, but it will change the overall behaviour of the spectral function; the reasonis as follow. If we include the effect of Umklapp term perturbativley, it could beeasily shown that such higher order terms can not change the oscillation wave-vector of the spectral functions. It only modifies the non-oscillatory part of thespectral function without changing the oscillatory part. Therefore, if the oscilla-tions wave-vector remains intact the threshold frequency does so.The singularity exponents does not change because, to find the singularity ex-ponents of spectral functions, we need to study the behaviour of spectral functionsat frequencies, ω , around the threshold frequencies, ωL. In principle the probe56frequency could be chosen as close as possible to the singular frequency such thatthe Umklapp term effects would be irrelevant at those energy difference scales,|ω−ωL|  |(kF −2pi/a)v|. Thus the Umklapp term would not change the singu-lar exponent at ωL; we expect a cross over regime where the effect of Umklapp willbe important at energies near to |ω−ωL| ≈ (kF −2pi/a)v|.2.5 Electron spin resonance with UniformDzyaloshinskii-Moriya InteractionsElectron spin resonance provides a sensitive probe of spin dynamics. A mi-crowave field is weakly Zeeman coupled to the q = 0 components of the spin op-erators. In the standard (Faraday) configuration, the microwave field is polarizedperpendicular to a static magnetic field. For simplicity we restrict ourselves to therelatively simple situation of Eq. (2.1), with DM vector and magnetic field in thez-direction. Then, as discussed in Sec. 2.1, a uniform DM interaction added tothe XXZ model of Eq. (2.1) simply shifts the parameters J and ∆ and the momen-tum, q in the transverse spectral function. Therefore the ESR adsorption intensityis proportional to the transverse spectral function at q = α = arctan(D/J). Lowtemperature ESR measurements on quasi-1D antiferromagnets with uniform DMinteractions could therefore probe the edge singularities predicted by X-ray edgemethods that disagree with standard bosonization results due to the effects of bandcurvature. Such ESR results would be especially useful if they were done with cir-cularly polarized microwave radiation since then S−+ and S+− could be measuredseparately.A further major challenge for such ESR experiments would be that the theo-retical predictions give Sss′(α,ω,h) for fixed h as a function of ω . However, inan ESR experiment, ω is normally fixed at the resonant frequency of a microwavecavity and h is varied. ω can only be varied by using a sequence of microwavecavities with different resonant frequencies. Alternatively, theoretical line-shapescould be produced for fixed ω and varying h but these would be complicated sincethe critical exponent α varies with h. For simplicity, we just discuss the line shapeversus frequency at fixed h here.We begin by discussing the T = 0 limit. The ESR adsorption intensity can be57simply read off from the results of Sec. 2.2.2, 2.3. We first consider the case of zerostatic field with the microwave field in the xy plane. Then the adsorption intensityhas a lower threshold near which:I(ω) ∝ θ(ω− vsinα)(ω− vsinα)µ (2.113)with µ = 2−1/(2K)−K. Here v and K are determined in terms of ∆e f f =∆cos(α)by Eq. (2.31). For small α we expect the results of standard bosonization to applyat somewhat higher energies, ω− vα  α3J. In this region we obtain:I(ω) ∝ (ω+ vα)1+2η(ω− vα)1−2η (2.114)with η = (1− 2K)2/(8K). Note that 1− 2η = 2− 1/(4K)−K 6= µ , a differentexponent than occurs at the threshold. As discussed in sub-section 2.2.2, I(ω) inEq. (2.114) is non-monotonic, eventually passing through a minimum and startingto increase again as ω increases. However, since the formula is only valid in thelow energy regime, ω  J, whether or not this minimum occurs in the frequencyregion where the formula is valid depends on α and ∆e f f .If ∆e f f ≥ cosα , the value resulting from ∆ = 1, then the minimum predictedby standard bosonization is not in the region where the approximation is valid.In this case the intensity is monotone decreasing up to high frequencies where ourtechniques break down. ∆e f f is typically close to 1. In fact, with some assumptionsabout the higher energy levels of the magnetic ion, it is exactly one [52]. In thiscase, the transverse spectral function becomes the same as the longitudinal onediscussed extensively in [28, 29, 31, 53]. The edge exponent has the value µ = 1/2,first obtained from the 2-spinon approximation [53] in this case and the standardbosonization prediction of Eq. (2.114), with η = 0, reduces to a δ -function whichfails to capture many features of the actual spectral function for non-zero α . Inparticular there is a narrow peak of width ∝ α3 followed by a slowly decaying tailat higher ω .At non-zero field, h, we may again use the results of Sec. 2.2.2, 2.3, whichare less complete in this case. The X-ray edge results of Sec. 2.3 imply a quantumphase transition as the magnetic field is increased, occurring when the field-induced58magnetization, m(h), obeys α = H ≡ 2pim ≈ Kh/(pi/v) as can be seen from Fig.[2.8]. S−+ has a threshold singularity at a frequency ωL≈ v(α−H) for H <α 1.A threshold singularity was also predicted in III for S+− in this field range, at ahigher frequency, of approximately v(α+H). However, as discussed at the end ofthat sub-section, we expect this to be broadened. On the other hand, for H > α ,we expect S+− to have a sharp 2-sided singularity at a frequency of approximatelyv(H−α). The other threshold singularity predicted for S+− at v(H +α) in 2.2.2is likely to be broadened. (The presence of two peaks at these energies was firstpredicted in [54] and was observed experimentally in [43].) The precise energiesand critical exponents for these singularities could be predicted by numerical Betheansatz calculations but analytic expressions are not available. Assuming α , H 1,we expect the spectral functions to cross over to the form predicted by standardbosonization at energies somewhat higher than the singularities, (H −α)2aJ v|H−α|  J:S−+ ∝[ω+ v(H−α)]1+2η[ω+ v(α−H)]1−2η (H < α)S+− ∝[ω+ v(α−H)]1+2η[ω+ v(H−α)]1−2η (H > α). (2.115)As discussed above and sketched in Fig. [2.2], these functions are non-monotonicbut the minimum only occurs in the energy region ω J, where the approximationholds, for a certain parameter range of α , ∆ and h. In addition to these dominantsingularities, as discussed in 2.4, we expect many additional weaker vanishing sin-gularities extending down to low energies.Finite Temperature BroadeningAt finite temperature the sharp peaks are broadened. This can be calculatedusing standard bosonization for (H −α)2aJ  T  J. At finite T the spectralfunction of Eq. (2.114) becomes:I ∝ Im{sin(2piη)(2piT )ηB(1+η− i(ω+ vα)/(4piT ),−1−2η)×B(η− i(ω− vα)/(4piT ),1−2η)}. (2.116)59where B(x,y) = Γ(x)Γ(y)/Γ(x+y) is the Euler beta function. For weak anisotropy,small α and 1−∆e f f , and hence η  1, a Lorentzian line-shape occurs:I ∝1(ω− vα)2 +(4piTη)2 (2.117)with a similar broadening at finite H. The width of the peak is 4piTη ≈ 2T (1−∆e f f )/pi for ∆e f f close to 1. This is essentially the same result derived in [40] forthe ESR width due to exchange anisotropy parallel to the magnetic field.In section 2.3 we found the singularity exponent and behaviour of the trans-verse spectral function near the thresholds at zero temperature using X-ray edgemethods. Now we look at the finite temperature effects on the spectral functions,and we find that finite temperature results in a non-Lorentzian broadening of thetransverse spectral functions near the thresholds for |ω− ε(q)|,T  q2/2m. Non-zero temperature has two effects on the spectral function Eq. (2.80). First, theGreen’s function for excitations near the fermi surface, S(0) is modified to the usualfinite T form by a conformal transformation. Secondly, the Green’s function forthe d¯ operators would be modified to finite T form, with step functions θ(ε) re-placed by Fermi functions nF(ε). This would allow a hole contribution to S−+ anda particle contribution to S+−. However, at low T  ε(kp) these would be neg-ligible. So the only important effect may be in S(0). Thus in this regime and for(2kF −pi < q < pi) we haveS−+(q,ω) ∝∫dxdtei[ω−ε(k+p )]tδ (x−ut)(2piT )ν+R +ν+L(sin(2piT (ε+ i(t− x/v))))ν+R (sin(2piT (ε+ i(t + x/v))))ν+L,By doing the integration over x we haveS−+(q,ω) ∝∫ ∞−∞dtei[ω−ε(k+p )]t(2piT )ν+R +ν+L(sin(2piT (ε+ i(1−u/v)t)))ν+R (sin(2piT (ε+ i(1+u/v)t)))ν+L(2.118)By evaluating the above integral we can get the finite T behaviour of the spectral60S-+(ω ,q)ω-ε(q)T=0.1T=0.2T=0.3T=0.4T=0.5Figure 2.9: The behaviour of S+−(ω,q) spectral function at finite tempera-ture for the case νR = 1/2, νL = 0, corresponding to ∆ = 1, h = 0. Wesee that for small enough T the broadening is asymmetric.function. From the above equation it can readily be seen that the spectral functionis a pure real number; in the Appendix B we will prove that it is positive too. Letus look at the behaviour of above spectral function for 2kF −pi < q < pi , at h = 0and ∆e f f ≈ 1, small anisotropy η  1 so that νR = 1/2−η and νL = 0. In thisregime we haveS−+ ∝1T 1−νRRe[e−ipiνR/2B(−iω− ε(k+p )4piT (u− v) +νR2,1−νR)](2.119)In Fig. [2.9] we have depicted, the spectral function for different values of tem-perature. As temperature gets higher the broadening increases; as can be seen, forsmall T the broadening is asymmetric and so is non-Lorentzian. Furthermore, thewidth is O(T ), not suppressed by a factor of 1−∆e f f as predicted by standardbosonization. Since the effects predicted by the new theoretical methods occur at61very low energy scales, a highly one-dimensional spin compound would probablybe needed to observe them, in order that three dimensional exchange processeswould be negligible. Furthermore, materials like KCuGaF6 with both uniform andstaggered intra-chain DM interactions may not be suitable since the staggered DMinteractions tend to have a larger effect than the uniform ones[40]. Thus identifyingthe right material to test these predictions remains an open challenge. An alterna-tive approach might be to study metallic quantum wires with spin-orbit couplings.It was shown in [54] that analogous phenomena occur in that system.2.6 ConclusionsBy applying X-ray edge techniques, we have obtained results on the transversespectral function of the XXZ spin chain in a magnetic field. We illustrated whystandard bosonization techniques fail near threshold energies, even when these oc-cur at low energy. In the zero field case we have exactly determined a critical ex-ponent governing the lower edge singularity, for all |∆| < 1 and all wave-vectors.For the finite field case, we have shown how this exponent can be determined fromparameters which can be obtained from solving Bethe ansatz equations and whichalso determine the behaviour of the longitudinal structure function, fermion spec-tral function and the finite size spectrum. We have argued that, for general magne-tization, a large number of increasingly weaker singularities occur in the spectralfunction, extending all the way down to zero energy. We derived results for the fi-nite temperature spectral function using X-ray edge methods, obtaining strikinglydifferent behaviour than that given by standard bosonization at 0 < 1−∆e f f  1.The line-shape is non-Lorentzian and the line width is O(T ), unsuppressed by1−∆e f f . We pointed out that electron spin resonance measurements on spin chaincompounds with uniform Dzyaloshinskii-Moriya interactions would provide a wayof experimentally confirming, for the first time, the new bosonization results beingobtained on spin chains, using X-ray edge techniques.In the case of staggered DM interactions, the most interesting ESR signal oc-curs when the magnetic field is transverse to the DM vector. This may also be thecase for uniform DM interactions, but we leave this for future work.62Chapter 3Toward rigorous proof of edgemagnetism in graphenenano-ribbons3.1 IntroductionGraphene is a two dimensional allotrope of Carbon which has many interestingelectronic properties. In graphene the Carbon atoms are arranged in honeycomblattice. The sp2 hybridization of the atomic s orbital with two p orbitals lead totrigonal planar structure, leads to the formation of so called σ bond between theneighboring carbon atoms, with distance equal to 1.42A˚. This σ band is responsiblefor holding the carbon atoms in two dimensions and the robustness of graphene.The remaining p orbital is perpendicular to plane of this planar structure and theresulting covalent bond between p orbitals of the neighboring atoms lead to theformation of pi band, which is responsible for many low-energy electronic andtransport properties of graphene.The first theoretical study of the band structure of graphene was done by P.R. Wallace in 1946 [32]; he showed the unusual semimetallic behavior of it. Themore systematic study of the low-energy excitations of graphene leads to its mostinteresting properties. Semenoff [33] showed that the low energy excitations are63massless chiral, Dirac fermions. Quite interestingly these excitations were prop-agating like massless fermions but with fermi velocity which is about 1/300 ofvelocity of light.In addition to interesting properties of excitations of bulk graphene, it turns outthat graphene with some specific boundary conditions has very interesting proper-ties too. It could be shown that a semi-infinite graphene sheet with zigzag boundaryor edge, supports localized zero energy states at the boundary. Mean field theoret-ical study of these edge states for graphene with electron-electron interactions,Hubbard interaction [34, 35], revealed that the edge states are spin polarized. Inthis work we rigorously study this problem and show that the ground state of theeffective edge Hamiltonian is spin polarized.In the following section we look at the band structure of the bulk grapheneand its low energy excitations, in section 3.3 we derive the low-energy effectiveHamiltonian which describes the excitations and effect of interactions on low en-ergy behavior of it. In section 3.4 we look at the edge states of graphene and willgive a hand-waving argument based on Lieb’s theorem about the magnetism ofedge states.In section 3.5 we derive effective Hamiltonian description of edge states andprove the ferromagnetism of its ground state. Then we take into account the effectof next-nearest-neighbor hopping terms and stability of ferromagnetic ground statein 3.6. Finally in section 3.7 by taking into account the effect of bulk excitationsin proper field theoretical method, we find the effective higher order corrections toedge Hamiltonian by integrating out these excitations.3.2 Band structure of single layer grapheneGraphene is two-dimensional layer of carbon atoms arranged in honeycomblattice. Honeycomb lattice is a bipartite lattice of two triangular sub-lattices, whichin this work we distinguish as sub-lattice A and B. The basis vectors of triangularBravais Lattice are given by~a1 =a2(3,√3)~a2 =a2(3,−√3)(3.1)64where a≈ 1.42A˚ is the distance between two neighbor carbon atoms. Graphene ismade of two copies of triangular lattice generated with ~a1,~a2, which are displacedby ~δ1. Each carbon atom has three nearest-neighbours, where their positions aregiven by~δ1 =a2(1,√3)~δ2 =a2(1,−√3)~δ3 =−a(1,0) (3.2)The tight-binding Hamiltonian of pi band of graphene is derived from the over-lapping of the wavefunction of pz orbital of each Carbon atom with its neighboringatoms, which gives the hopping parameter of this orbital. In general there couldbe hopping between any two carbon atom but as the distance between the atomsincreases the hopping amplitude decreases and the only important terms are thenearest-neighbor hopping t ≈ 2.8eV and the next-nearest-neighbor hopping t ′ ≈0.1eV .The tight-binding Hamiltonian for graphene is given byH =−t ∑〈i, j〉,σ(a†i,σb j,σ +h.c.)− t ′ ∑〈〈i, j〉〉,σ(a†i,σa j,σ +b†i,σb j,σ +h.c.)(3.3)where ai and b j are electron annihilation operators on sub-lattice A and B, respec-tively, and 〈· · · 〉 means summation only over nearest neighbors and 〈〈· · · 〉〉 repre-sents summation over next-nearest-neighbors. Location of the NNN neighbors ofeach Carbon is given by ±a1, ±a2 and ±(a2−a1).The spectrum of Eq. (3.3) can be easily derived by using Fourier representationof annihilation operatorsa~r =1√NA∑~kei~k.~ra~k b~r =1√NB∑~lei~l.~ra~l (3.4)In this representation we haveH =−t∑k,σ(φ(~k) a†~k b~k +h.c.)−2t ′∑k,σψ(~k)(a†~ka~k +b†~kb~k)(3.5)65whereφ(~k) = ∑~δ jei~k.~δi = 2cos(√32kya)exp(ikxa2)+ exp(−ikxa)ψ(~k) = cos(~k.~a1)+ cos(~k.~a2)+ cos(~k.(~a2−~a1))(3.6)The diagonalization of the Hamiltonian Eq. (3.5), gives us the spectrum of theexcitationsE±(~k) =±t|φ(~k)|−2t ′ψ(~k) (3.7)By looking at Fig. 3.1 , there are two special points in the Brillouin Zone (BZ),where two energy bands touch each other, ~K and ~K′.~K =2pi3a(1,1√3)~K′ =2pi3a(1,−1√3)(3.8)As we see the band structure is gapless, and excitations near these points have lineardispersion. These point are named Dirac points which leads to many interestingproperties of graphene.From Eq. (3.7) it is clear that when t ′= 0, the energy spectrum is symmetric aroundzero energy and we have particle-hole symmetry. This could also be seen in thelattice representation of Hamiltonian 3.3, particle-hole transformation for bi-partitelattice is given byai,σ −→ a†i,σ bi,σ −→−b†i,σ (3.9)We see that the first term in Eq. (3.3) is invariant under this transformationwhile the second term is not. It is also instructive to look at the dispersion near theDirac points; to do so, we need to expand the dispersion 3.7 near ~K or ~K′. Letting~k = ~K +~q, for small momentum q |~K|E±(q)≈±vF |q|+O[(q/K)2], (3.10)where vF ≈ 106m/s is the Fermi velocity. This is exactly similar to the dispersionof relativistic massless particles, with the only difference that speed of excitations66L +1+2+3a2a1KK 'b1b2k xk yFigure 3.1: Honeycomb lattice and its Brillouin zone. Left: lattice structurewhich is used. Right: the Brillouin zone and Dirac pointsis two orders of magnitude smaller than the velocity of light. This very importantand interesting property of graphene, enables the study of properties of relativisticparticles in graphene.In addition to nearest neighbor hopping term, there is also small amount ofnext-nearest-neighbor hopping term. But the only effect of including the next-nearest-neighbor term is that it shifts the energy of Dirac points and changes higherorder terms of the dispersion of the excitations:E±(q) ' 3t′± vF |q|−3q2a24(3t ′±t2sin(3θq))θq = arctan(qxqy)(3.11)67As we see, to leading order the dispersion has rotational symmetry around q = 0,while higher order terms break that symmetry and in general the energy dependsboth on magnitude and the direction of momentum.3.3 Dirac fermions and low-energy effective hamiltonianWe saw in the previous section that the graphene band structure is gapless andthe excitations near the Dirac points have linear dispersion. Following [33, 55]in this section we explicitly show that these excitations could be described by aHamiltonian which has the same form as the Hamiltonian describing relativisticfermions, so called Dirac Hamiltonian.As the only effect of t ′ is to shift the Dirac points energy without changingthe leading order low-energy dispersion, we could safely assume that t ′ = 0. Tofind the effective Hamiltonian for excitations near the Dirac points, we write ourfermion operators as followan,σ ' e−i~K.~RnψA,n,σ + e−i~K′.~Rnψ ′A,n,σbn,σ ' e−i~K.~RnψB,n,σ + e−i~K′.~Rnψ ′B,n,σ (3.12)where ~K, ~K′ are the two Dirac points, ψA/B,n,σ and ψ ′A/B,n,σ represent excitationsnear ~K, ~K′ respectively, and are slowly varying operators in position space. FromEq. (3.3) we need to evaluate terms like a†nbn′ where n,n′ are two neighboring sites∑~δia†nbn+~δi =∑~δie−i~K.~δiψ†A(~Rn)ψB(~Rn +~δi)+ e−i~K′.~δiψ ′†A (~Rn)ψ′B(~Rn +~δi) (3.13)where we have ignore terms proportional to ei(~K−~K′).~Rn , as they are varying veryfast and can be neglected in study of low-energy Hamiltonian. By expanding theslowly varying field ψB(~Rn +~δi) around ~Rn we haveψB(~Rn +~δi)≈ ψB(~Rn)+~δi.∇ψB(~Rn)+O[∇2] (3.14)68where higher order terms are irrelevant in RG sense. By using above equation andEq. (3.13) we get∑~δia†nbn+~δi = ∑~δie−i~K.~δiψ†A(~Rn)ψB(~Rn)+ e−i~K′.~δiψ ′†A (~Rn)ψ′B(~Rn)+ ∑~δie−i~K.~δiψ†A(~Rn) ~δi.∇ψB(~Rn)+ e−i~K′.δiψ ′†A (~Rn) δi.∇ψ′B(~Rn)By using the following identities we could simplify previous equation∑~δiei~K.~δi = ∑~δiei~K′.~δi = 0∑~δiei~K.~δi~δi = −3a2ei4pi3 (1, i)∑~δiei~K′.~δi~δi = −3a2ei4pi3 (1,−i) (3.15)we get∑~δia†nbn+δi =−3aei4pi32(ψ†A(~Rn)(∂x + i∂y)ψB(~Rn)+ψ′†A (~Rn)(∂x− i∂y)ψ′B(~Rn))(3.16)The low-energy Hamiltonian will beH '−3taei4pi32×∫dxdy(ψ†A(r)(∂x + i∂y)ψB(r)+ψ′†A (r)(∂x− i∂y)ψ′B(r))+h.c.(3.17)By defining vF = 3ta/2 andΨ1(x)≡(e−i2pi3 ψAei2pi3 ψB)Ψ2(x)≡(e−i2pi3 ψ ′Aei2pi3 ψ ′B)(3.18)69we getH =−ivF∫dx dy[Ψ†1(~r)σ .∇Ψ1(~r)+Ψ†2(~r)σ¯ .∇Ψ2(~r)](3.19)where σ = (σx,σy), σ¯ = (σx,−σy), are Pauli matrices. This Hamiltonian is DiracHamiltonian for massless two dimensional Dirac fermions. We have two copies ofDirac Hamiltonian for each ~K, ~K′ which are known as two “valleys”. This represen-tation of theory is more convenient to study the effect of interactions perturbatively.For this work we are interested to study the effect of Hubbard interaction onedges of graphene. Let’s first look at the effect of this interaction on bulk graphene.The hubbard interaction, which is screened on-site Coulomb repulsion, is given byHU =U∑ic†↑,ic↑,ic†↓,ic↓,i (3.20)where U is so called Hubbard interaction strength. One important point to mentionis that, in graphene density of states vanishes linearly and thus the Coulomb inter-action is not fully screened. But effectively longer range Hubbard interactions areweaker and as a first step we could ignore them. Also as we will show shortly evenon-site Hubbard repulsion is irrelevant in renormalization group sense and withmore involved argument it could be shown that long range Coulomb repulsion ismarginally irrelevant [56] in renormalization group (RG) sense.In low-energy picture the Hubbard interaction term isHU =U∫dxdy Ψ†↑(~r)ΠA/BΨ↑(~r)Ψ†↓(~r)ΠA/BΨ↓(~r) (3.21)where ΠA/B is projection matrix to sub-lattice A/B. To study the RG flow of thisinteraction for weak interactions U  t, we need to find the scaling dimension ofthe fields Ψ. To find the dimension of Ψ we look at the kinetic term in Lagrangianpicture, and the term will beL ∝∫dx dy dt Ψ†σ .∇Ψ (3.22)70from this equation we see that dimension of fermionic field is D[Ψ] = L−1, whereL means dimension of length, as length and energy are inversely related, D[Ψ] = E.The interaction term in Lagangian picture isLU =∫dxdydt Ψ†↑(~r)ΠA/BΨ↑(~r)Ψ†↓(~r)ΠA/BΨ↓(~r) (3.23)This term has scaling dimension of length, D[LU ] = L−1 ≡ E. As the dimensionof this term is proportional E, in renormalization flow picture this means that thisterm re-normalizes to zero as we integrate out high-energy terms, and terms likethis are called irrelevant in study of low-energy properties of the system. If thescaling dimension of some terms became E−n, those terms are called relevant, inRG sense, and term with no scaling dimension are called marginal.As a result the Hubbard interaction term is irrelevant for weak interactions; thismeans for small U/t the low energy picture is semimetal with fermionic excitationsgoverned by Eq. (3.19). Many numerical studies of phase diagram of tight-bindinghoneycomb lattice with Hubbard interaction confirm this picture for finite range ofinteraction strength U <Uc, Uc ≈ 3.8t, where for U >Uc the ground state is Anti-Ferromagnetic Mott Insulator [57] . In this work we are interested to study theeffect of weak Hubbard interaction on magnetic properties of edges of graphene.3.4 Edge states of grapheneIn previous section we studied the band structure of bulk, infinite size, graphene.We showed that low-energy excitations of graphene are massless Dirac like fermionsand also weak Hubbard interaction is irrelevant and does not change the qualitativepicture of low-energy physics. One of the many fascinating properties predictedfor graphene [55] is that a non-interacting nanoribbon with zigzag edges has bandsof states with energy exponentially small in the ribbon width, localized at the edges[34, 58–62]. Unzipping of carbon nanotubes has recently provided a technique forproducing nano-ribbons with clean edges and Scanning Tunnelling Microscopy(STM) on such ribbons [63] has shown evidence for interacting edge states.71WL A BFigure 3.2: A nanoribbon with an upper zigzag edge and lower bearded edgeand armchair edges on sides. In this example, L = 5 and W = 6.Among the infinite possibilities for edges of a graphene sample, there are threepossible edge configurations which support translational invariance over short dis-tance, so called Zigzag, Armchair and Bearded edges, Fig 3.2. Among these three,the zigzag and bearded edges lead to zero-energy bands.To show the existence of zero-mode states, we look at a semi-infinite graphenesheet with zigzag edge, by taking limit of L,W → ∞ in Fig 3.2. In this geometrythe tight-binding Hamiltonian is given byH = −t∞∑m=−∞∞∑n=0a†σ (m,n)bσ (m,n)+a†σ (m,n)bσ (m−1,n)+ a†σ (m,n)bσ (m,n−1)+h.c (3.24)72again a and b are electron annihilation operators for A sites and B sites, and m, nare position indices along x-direction and y-direction, respectively . By using trans-lational invariance along the x-direction and Fourier transforming the operators inthat direction we haveH = −t∑n,ka†σ (k,n)bσ (k,n)(1+ eika′)+a†σ (k,n)bσ (k,n−1)+h.c(3.25)where a′=√3a. By using the fact that k is a conserved quantum number, in generalany eigenstate of above Hamiltonian could be written as follow|E,k,σ〉=∞∑n=0(αEn a†σ (k,n)+βEn b†σ (k,n))|0〉 (3.26)To find the coefficients αn and βn of zero-energy mode we should solve H|0,k,σ〉=0. It is easy to show that the following state is the solution to this equation|0,k,σ > = ∑nαk,na†σ (k,n)|0 >αk,n = Nk(−2coska′2)neikna′/2 (3.27)where Nk is the renormalization factor for wave function, and is given byN 2k =1∑∞n=0 |2coska′2 |2n(3.28)This equation tells us that for semi-infinite zigzag sheet, normalizable zero-energystates exist if and only if |2coska′2| < 1, which restricts the values of k to k ∈[ 2pi3a′ ,4pi3a′ ]. We should notice the fact that the weight of the wavefunction of edgestates is non-zero only on A-sub-lattices.This result is only valid for semi-infinite graphene sheet with zigzag edge; ifwe have a finite ribbon with zigzag edges on both sides the result is different.Then we have two edges and if we try to solve eigenvalue problem for Eq. (3.26)we see that it is coupled equation and there is no exact result for αE and βE forE = 0. Actually numerical solution of those equation does not give a band of zero-73-4-2024-4-2024Figure 3.3: Energy spectrum for graphene ribbon for different boundary con-ditions. a) is energy spectrum for graphene ribbon with zigzag-zigzagboundary condition. b) is spectrum for ribbon with ZB boundary condi-tion.modes in contrast to semi-infinite sheet [64, 65]. The intuitive picture is that theoverlapping of the wave-functions of each edge lifts the degeneracy of zero-modesand leads to exponentially small energies for the same range of momemtum. Theband structure for finite ribbon with Zigzag-Zigzag (ZZ) edges are depicted in Fig.3.3a.74The other possible and interesting boundary condition is the case that we havea finite ribbon with zigzag edge on, let’s say, upper edge and bearded edge onlower edge similar to Fig. 3.2. For this geometry as we see the number of atomsin A-sublattice and B-sublattice are different, NA − NB ∝ L. The tight-bindingHamiltonian with only nearest-neighbor terms is a matrix which transforms an NA-dimensional space on NB-dimensional space; from linear algebra we know that thisHamiltonian should have |NA−NB| zero eigenvalues. This can be shown explicitlyand the exact form of the wave function is exactly the same as Eq. (3.27)|0,k,σ〉 = ∑nαk,na†σ (k,n)|0〉αk,n = Nk(−2coska′2)neikna′/2 (3.29)where Nk is the renormalization factor for wave function, and is given byN 2k =1−|2cos ka′2 |2Ny1−|2cos ka′2 |2. (3.30)As Ny is finite, which is the number of unit cells along the width of the ribbon, thereis no normalization restriction on k. One interesting and important point to notice,is that for k∈ [ 2pi3a′ ,4pi3a′ ] we have |2coska′2 |< 1 and from second line of Eq. (3.29) wesee that for this range of k the weight of wave-function decreases exponentially asn increases, which means the wave function is localized around n = 0, or the zigzagedge. But for k ∈ [− 2pi3a′ ,2pi3a′ ] we get |2coska′2 |> 1 and this leads to wave-functionswhich are localized around n = Ny, or Bearded edge. The numerical result of bandstructure of this configuration is depicted in Fig. 3.3b.If we impose periodic boundary condition along the x-direction and assumingthere are L unit cells along that direction, then we have L zero-energy states withmomenta given byk =2pi jLa′j = 0,1 · · · ,L−1 (3.31)To summarize the results, we showed that the tight-binding model of graphenenano-ribbons with ZB edge has exact zero energy modes, one-third of these modes75are localized around the zigzag edge and two-third are localized around the Beardededge.3.4.1 Lieb’s theorem and edge magnetismIn condensed matter physics there aren’t many exact statements about the prop-erties of the ground states of many-body systems. Lieb’s theorem is one of thosefew which convinces us about the magnetism of edges states of graphene with Hub-bard interaction. Lieb’s theorem is an exact statement about the magnetic proper-ties of the ground state of the Hubbard model at half-filling:Lieb’s Theorem: Consider a tight-binding repulsive Hubbard model U > 0 onbipartite lattice and hopping term only between A and B sites and half-filled band.Then the ground state is unique, up to spin degeneracy, and has spin S = |NA−NB|/2.We saw that for graphene ribbon with ZB boundary, numbers of atoms are dif-ferent in two sub-lattices and NA−NB = L. Then Lieb’s theorem tells us that if weturn on on-site Hubbard repulsion, no matter how small U is, the ground state ofthis system has to have spin S = L/2.We showed in section 3.3 that weak Hubbard interactions are irrelevant in low-energy effective picture of bulk graphene and the ground state of bulk grapheneremains paramagnetic semi-metal. So if the bulk is not magnetized then wheredoes this spin L/2 live?The only physical explanation of the Lieb’s theorem result seems to be that thefully polarized edge state multiplet persists as the (unique) ground state as we turnon U , also the spin of the ground state predicted from Lieb’s theorem is in agree-ment with the number of edge modes. This picture can be further substantiatedby calculating the weak interaction between the upper and lower edges, of orderU2/(tW 2). This interaction is found to be ferromagnetic, as we show below. Thusthere must be a spin ≈ L/6 on the zigzag edge and ≈ L/3 on the bearded edge,with these two spins coupled ferromagnetically.76Now consider replacing the bearded edge by a second zigzag edge. Lieb’s the-orem [66] now implies a zero spin ground state since we have equal numbers of Aand B sites. Now, at U = 0, we have approximately L/3 edge states on both lowerand upper edges. These mix to form two bands, with 2pi/3 ≤ k ≤ 4pi/3, with en-ergies exponentially small in W and symmetric around E = 0. Ignoring inter-edgeinteractions, we expect spin≈ L/6 on both upper and lower edges. In this case, theintra-edge interaction of order U2/(tW 2) is antiferromagnetic, implying a zero spinground state consistent with the result from Lieb’s theorem. A further consistencycheck can be obtained by going smoothly between zigzag and bearded lower edgesby turning on the hopping term, t ′ on the hairs. Lieb’s theorem implies spin L/2for all t, t ′ and U > 0. For t ′ = 0 we have a ZZ ribbon together with L decoupledsites sitting below the lower edge. The ZZ ribbon has spin 0 but we can obtain astate with total spin L/2 by polarizing the electron spins at the decoupled sites. Al-though the zigzag ribbon has total spin 0, for large W we expect that the upper andlower edges have spin ≈ L/6 with antiferromagnetic inter-edge coupling. Turningon t ′ produces an effectively antiferromagnetic coupling between the spin ≈ L/6on the lower zigzag edge and L/2 on the nearly decoupled sites. This gives a mo-ment ≈ L/3 which is now ferromagnetically coupled to the upper edge, giving atotal spin of L/2 as required by Lieb’s theorem.Assuming that the ground state remains an unpolarized Dirac liquid up to U =Uc, the magnetism of the edges seems to follow from Lieb’s theorem for large W .If the transition at Uc is into a bulk antiferromagnetic state (with, for example, spinup on A sites and spin down on B sites) then the edge magnetism should persist,since it is of this type, and may be regarded as a sort of precursor of the bulkantiferromagnetic order. The simplest possibility is that the system goes from Diracliquid into Mott-Hubbard insulator at Uc but numerical evidence [67] has beenpresented for a spin liquid phase at intermediate U , of unknown edge magneticproperties.We note that the above arguments also apply to carbon nanotubes [59, 68, 69].Indeed, since we have been considering periodic boundary conditions in the x-direction, we have actually been discussing tubes, of circumference L and lengthW . The magnetic moments exist on the upper and lower caps (i.e. rings) of the nan-77otubes with ferromagnetic or antiferromagnetic inter-ring coupling for a beardedor zigzag lower ring respectively. (The half-filled bulk of the nanotube might be ina 1D version of a Mott-Hubbard insulating state but this only serves to weaken theeffects of bulk states on edge states.)This is an intuitive argument and need to be examined carefully, in the follow-ing section we will find an effective Hamiltonian which describes the edge modesin presence of Hubbard interaction and will prove that the ground state of thatHamiltonian is ferromagnetic state.3.5 Projected Hubbard Model of the edge statesIn this section we find the effective Hamiltonian which describes the dynamicsof edge states. As we showed the exact form of edge states is only known for eithersemi-infinite graphene sheet with zigzag edge or finite ribbon with zigzag-beardededges. In this section we focus on the case of semi-infinite sheet with zigzag edge,similar result holds for ZB case.To find the effective low-energy model describing the edge states, we keep allthe edge states and only bulk states near the Dirac points, within cut-off given byΛ. To write the fermion operator in term of low-energy excitations, edge states andnear the Dirac point excitations, we need to consider that for semi-infinite sheet theedges are only on the sub-lattice A. Thus we haveaσ (k,y) = αk,yeσ (k)+Θ(k)φA,σ (k,y)+Θ′(k)φ ′A,σ (k,y)bσ (k,y) =Θ(k)φB,σ (k,y)+Θ′(k)φ ′B,σ (k,y) (3.32)Where a(k,y) and b(k,y) are original electron operator on each sub-lattice, whichare Fourier transformed along the x direction. Θ(k) = θ(Λ− |k− 2pi/3a|) andΘ′(k) = θ(Λ− |k− 4pi/3a|) are sharp cut-off for the excitations near the Diracpoints and φA/B,σ (k,y) are slowly varying operators as function of y, also αk,y isgiven by Eq. (3.27)αk,n = (−2coska2)neikna/2αk,0 (3.33)78The equation 3.32 could be written in a compact form asΨσ (k,y) = αk,yeσ (k)(10)+Θ(k)Ψ1,σ (k,y)+Θ′(k)Ψ2,σ (k,y) (3.34)By Fourier transforming the operators along the x direction the Hubbard interactionEq. (3.20) becomesHU =UNx∑a†↑(m,k+q)a↑(m,k)a†↓(m,k′−q)a↓(m,k′)+a↔ b (3.35)Now by plugging in the Eq. (3.38) into the Hubbard Hamiltonian Eq. 3.35, we willhave different kinds of interacting terms as follow: a) 4− edge interaction, whichdescribes the first order induced interaction on the edges. b) 2−edge 2−bulk term,which describes the interaction between two edge operator with two bulk operator,c) 3− edge 1−bulk term and finally d) 1− edge 3−bulk term. We describe eachterm in detail as follow:4-edge interaction:By only keeping the edge operators, the first order effective interaction of edgeexcitations is given byHU =UNx∑k,k′,qΓ(k,k′,q)e†↑(k+q)e↑(k)e†↓(k′−q)e↓(k′)Γ(k,k′,q) = ∑mα∗k+q,mαk,mα∗k′−q,mαk′,m (3.36)The summation over m can be done exactly and the result isΓ(k,k′,q) =|(1− (2cos k2)2)(1− (2cos k+q2 )2)(1− (2cos k′−q2 )2)(1− (2cos k′2 )2)|121−16cos k2 cosk+q2 cosk′−q2 cosk′2(3.37)where all the momenta, k,k′,k+q,k′−q are restricted to the interval [2pi/3,4pi/3],as there is no edge mode outside this interval.792-edge, 2-bulk interaction:For 2− edge 2−bulk interactions we have 6 different terms which we need tostudy. To simplify the terms we write the electron operators on A sites as followaσ (k,y) = αk,yeσ (k)+ΦA,σ (k,y)ΦA,σ (k,y) =Θ(k)φA,σ (k,y)+Θ′(k)φ ′A,σ (k,y) (3.38)by using this notation we have all six term as followH1 =UNx∑λ (k+q,k,y)e†↑(k+q)e↑(k)Φ†A,↓(k′−q,y)ΦA,↓(k′,y)+ ↑↔↓H2 =−UNx∑λ (k+q,k′,y)e†↑(k+q)e↓(k′)Φ†A,↓(k′−q,y)ΦA,↑(k,y)+ ↑↔↓H3 =−UNx∑ λ˜ (k+q,k′−q,y)e†↑(k+q)e†↓(k′−q)ΦA,↑(k,y)ΦA,↓(k′,y)+h.c(3.39)Whereλ (k+q,k,y) = α∗(k+q,y)α(k,y)λ˜ (k+q,k,y) = α∗(k+q,y)α∗(k,y) (3.40)To first order, mean-field level, we could replace the bulk operators in Eq. (3.39)by their expectation values, where at half-filling 〈a†σ (k+q,y)aσ ′k,y〉= δσ ,σ ′ δq,0/2and by using Eq. (3.38) we have〈Φ†A,σ (k′−q,y)ΦA,σ ′(k′,y)〉 =12δσ ,σ ′δq,0(1−|αk,y|2)〈ΦA,σ (k,y)ΦA,σ ′(k′,y)〉 = 0 (3.41)From first line and knowing the fact that αk,y exponentially goes to zero as functionof y, we see that the bulk states remain at half filling. By using this result and also80Eq. (3.36) the final form of the projected edge Hamiltonian isH = −U2Nx∑k,k′,qΓ(k,k′,0)e†σ (k)eσ (k)+UNx∑k,k′,qΓ(k,k′,q)e†↑(k+q)e↑(k)e†↓(k′−q)e↓(k′) (3.42)In section 3.7 we come back and consider in more details the effect of fluctuationsof bulk excitations. They lead to higher order interaction terms between and on theedges, and also interesting results about the correlation between the two edges forfinite ribbon. Let’s first study the ground state properties of Hamiltonian Ferromagnetic ground stateAs we see the Hamiltonian 3.42 has very complicated form but we will seethat it could be written in a special form, such that we could rigorously prove theferromagnetism of its ground state at half-filling.Hamiltonian 3.42 could be written in more symmetric form, which is manifestlySU(2) symmetric and particle-hole symmetric.H =12 ∑k,k′,qΓ(k,k′,q)[∑σe†σ (k+q)eσ (k)−δq,0]×[∑σ ′e†σ ′(k′−q)eσ ′(k′)−δq,0]+E0E0 = −12∑k,k′Γ(k,k′,0) (3.43)Here eα(k) annihilates an electron in an edge state with momentum k and spin α =↑or ↓. The interaction function is given by Eq. (3.37), and the sum over k, k′ and qis restricted to the band in which 2pi/3 < k, k′, k+q, k′−q < 4pi/3 and Γ(k,k′,q)is strictly positive [70]. Periodic boundary conditions in the x-direction imply k =2pin/L so the number of wave-vectors N ≈ L/3. Note that we are considering thecase of half-filling in the entire lattice and that the edge Hamiltonian is therefore81invariant under the particle-hole symmetry transformation:eα(k)↔ e†α(k). (3.44)This is highly unusual since normally a particle-hole symmetry transformation re-lates a particle and hole at different wave-vectors. Here with an exactly flat band,the particle and hole operators occur at the same wave-vectors. It is important tonote that Γ(k,k′,q) arose from summing the wave-function of the edge states oversites at arbitrary distance n from the zigzag edge and can be written:Γ(l,k,q) =∞∑n=0gn(k) gn(l) gn(l +q) gn(k−q) (3.45)wheregn(k)≡ θ(1−|2cosk2|)√1−(2cosk2)2(2cosk2)n. (3.46)Thus, dropping the constant E0, we may write:H =12∑n,qO†n(q)On(q) (3.47)withOn(q)≡∑kgn(k)gn(k+q)[∑σe†σ (k+q)eσ (k)−δq,0](3.48)It follows that all eigenstates of H are non-negative. It is easy to show that fullyspin-polarized state is a zero energy eigenstate and therefore a ground state. Tocheck this consider, for example, the representative fully polarized state where allelectron spins are in the up direction. Then clearly On(q) annihilates this state forall non-zero q since the spin up terms in On(q) try to produce a spin up electron inan occupied state with wave-vector k+q while the spin down terms try to annihilatea spin down electron in a vacant state of wave-vector k. On(0) also annihilates thisstate since the occupancy of each single particle state is precisely 1.It is also possible, though more difficult, to argue that the fully polarized multi-plet, of spin S = L/6, are the unique groundstates of the projected 1D Hamiltonian.To prove that fully polarized states are the unique ground states of H we need to82prove that the only states annihilated by On(q)†On(q) for all n and q are fully polar-ized (that is, have maximal total spin). For convenience, in this paragraph, we takeall momenta to be in the region of [−pi/3,pi/3], which can be obtained by shiftingall of them by pi . Suppose |ψ〉 is such that for any n,q , On(q)|ψ〉 = 0. Then wehave∑kgn(k)gn(k+q)[e†σ (k+q)eσ (k)−δq,0]|ψ〉= 0(3.49)and by knowing that gn is even function and gn(k)gn(k+q) = gn(−k)gn(−k−q),we rewrite above equation as∑k>0gn(k)gn(k+q)[e†σ (k+q)eσ (k)+ e†σ (−k)eσ (−k−q)−2δq,0]|ψ〉= 0(3.50)(Repeated spin indices are summed in this section.) For fixed q and using thedefinition of gn we have, for any n that∑k>0(4sin(k2)sin(k+q2))n|ψ(q)k 〉= 0 (3.51)|ψ(q)k 〉 ≡√1− (2sink/2)2√1− (2sin(k+q)/2)2×[e†σ (k+q)eσ (k)+ e†σ (−k)eσ (−k−q)−2δq,0]|ψ〉 (3.52)Since n runs from 0 to ∞, the number of independent momenta is L/3 and all the(4sin( k2)sin(k+q2 ))are different, the determinant of the Vandermonde matrix isnon-zero so Eq. (3.51) is satisfied if and only if for any k,q we have[e†σ (k+q)eσ (k)+ e†σ (−k)eσ (−k−q)−2δq,0]|ψ〉= 0 (3.53)First, using Eq. (3.53) for q = 0, we get n(k)+n(−k) = 2, thus the only possibleterms have n(k) = n(−k) = 1 or n(k) = 0 and n(−k) = 2 or vice versa. In general83|ψ〉 could be written as a linear combination of Fock states ∏c†σ (k)|0〉. We firstwill show that in the expansion of |ψ〉 in terms of such states there is no Fockstate which for any momentum k we have a vacancy or double occupancy in thatmomentum state. In other words, in the expansion of |ψ〉, with condition 3.53and n(k)+n(−k) = 2, only Fock states with singly occupied momentum states areallowed.Suppose that there is a state which has the property n(k)+n(−k) = 2 for any k andhas double occupancy at momentum l (vacancy at momentum −l); call this stateφ :|φ〉= | · · · , 0︸︷︷︸−l, · · · , ↓↑︸︷︷︸l, · · · 〉 (3.54)and we suppose 〈φ |ψ〉 6= 0. We impose the condition 3.53 on |ψ〉 for q =−2l andk = l. Thus we should have e†σ (−l)eσ (l)|ψ〉= 0. Let us first look at the action ofe†σ (−l)eσ (l) on φ ,e†σ (−l)eσ (l)|φ〉= e†σ (−l)eσ (l)| · · · , 0︸︷︷︸−l, · · · , ↓↑︸︷︷︸l, · · · 〉= | · · · , ↑︸︷︷︸−l, · · · , ↓︸︷︷︸l, · · · 〉− | · · · , ↓︸︷︷︸−l, · · · , ↑︸︷︷︸l, · · · 〉 (3.55)Now, in order to satisfy the condition e†σ (−l)eσ (l)|ψ〉 = 0, we should have someother Fock states in the expansion of |ψ〉 such that the action of e†σ (−l)eσ (l) onthem could cancel the terms created in the second line of Eq. (3.55). e†σ (−l)eσ (l)only acts on states with momentum l and−l, as a result does not change the spinconfigurations of the other (singly occupied) states. There are only three possibleFock states which have the same configurations of the singly occupied states are :|1〉 = | · · · , ↓↑︸︷︷︸−l, · · · , 0︸︷︷︸l, · · · 〉|2〉 = | · · · , ↑︸︷︷︸−l, · · · , ↓︸︷︷︸l, · · · 〉,|3〉 = | · · · , ↓︸︷︷︸−l, · · · , ↑︸︷︷︸l, · · · 〉 (3.56)84if we operate e†σ (−l)eσ (l) on these states, we gete†σ (−l)eσ (l)|1〉 = 0e†σ (−l)eσ (l)|2〉 = | · · · , ↓↑︸︷︷︸−l, · · · , 0︸︷︷︸l, · · · 〉e†σ (−l)eσ (l)|3〉 = −|· · · , ↓↑︸︷︷︸−l, · · · , 0︸︷︷︸l, · · · 〉 (3.57)None of these states are able to cancel the terms created on (3.55). Then theassumption is wrong and we have to have 〈φ |ψ〉 = 0. Having proven this, weshow that in the expansion of |ψ〉 in terms of singly occupied Fock states, onlysymmetric combinations like the following are acceptable| · · · , ↑︸︷︷︸k1, · · · , ↓︸︷︷︸k2, · · · 〉+ | · · · , ↓︸︷︷︸k1, · · · , ↑︸︷︷︸k2, · · · 〉 (3.58)Suppose that, the Fock expansion of |ψ〉 has a term like| · · · , ↑︸︷︷︸k1, · · · , ↓︸︷︷︸k2, · · · 〉Now by chosing k = k1 and q = k2− k1 we should have(e†σ (k2)eσ (k1)+ e†σ (−k1)eσ (−k2)|ψ〉= 0We also havee†σ (k2)eσ (k1)| · · · , ↑︸︷︷︸k1, · · · , ↓︸︷︷︸k2, · · · 〉 = | · · · , 0︸︷︷︸k1, · · · , ↓↑︸︷︷︸k2, · · · 〉e†σ (k2)eσ (k1)| · · · , ↓︸︷︷︸k1, · · · , ↑︸︷︷︸k2, · · · 〉 = −|· · · , 0︸︷︷︸k1, · · · , ↓↑︸︷︷︸k2, · · · 〉(3.59)Thus we see that the symmetric combinations leads to zero, while the anti-symmetricones give us a non-zero result. Thus |ψ〉must be fully symmetric under exchangingall spins and is therefore of maximal spin.852.5 3.0 3.5 4.0 k- HkLUΗ=0 Η=0.05 Η=0.10 Η=0.109 Η=0.115 Η=0.125 Η=0.175Figure 3.4: Energy to add a spin down electron of momentum k for variousvalues of η ≡ ∆/U ≡ (t2−Ve)/U .3.6 Effects of the Excitations in presence ofNNN-hopping and single site potentialIn this section we investigate the effects of NNN hopping and single site poten-tial on the stability of the ferromagnetic ground state. One interesting quantity thatfollows from the Hamiltonian is the energy to add a spin down electron of momen-tum k, which is the same as the energy to remove a spin up electron of momentumk:εk =U2L∑k′Γ(k,k′,0). (3.60)This quantity is plotted (at L→ ∞) in Fig. 3.4. It vanishes linearly at the Diracpoints, k = 2pi/3, 4pi/3. In principle, εk could be measured in Angle-ResolvedPhotoemission Spectroscopy (ARPES) experiments. The corresponding electronaddition or removal energy is given by εk. The corresponding density of states∝ 1/|dεk/dk| could be measured by Scanning Tunneling Microscope (STM). Witha spin-polarized STM tip and an edge fully polarized in the z-direction, it wouldonly be possible to tunnel in a spin-down electron or tunnel out a spin-up electron.We have calculated numerically the lowest energy particle-hole state of total860.5 1.0 1.5 2.0 q-Η=0 Η=0.05 Η=0.10 Η=0.109Figure 3.5: Lowest energy particle-hole state (circles) and bottom of theparticle-hole continuum (lines) for various values of η ≡ ∆/U ≡ (t2−Ve)/U .momentum q, for L up to 602 (N = 200). This is plotted in Fig. 3.5 along withthe bottom of the particle-hole continuum. We see that a strongly bound excitonexists for most values of q, as might be expected in this strongly interacting system.However, the binding energy vanishing at q = ±2pi/3. This vanishing can be un-derstood from the fact that Γ(k,k′,q) vanishes when k or k′ is at a band edge 2pi/3or 4pi/3 so the zero energy particle and hole become non-interacting at wave-vector2pi/3 and −2pi/3 or vice versa.While Lieb’s theorem continues to imply a fully polarized ground state at suf-ficiently small U for any hopping terms between opposite sub-lattices (A to B),adding a small [O(U)] second neighbor hopping term, t2, may destroy the fully po-larized state. Likewise, a single site potential, Ve, acting at the edge of the ribbononly, could destroy the fully polarized state. Temporarily ignoring interactions,the zigzag edge states survive at finite t2 and V but develop a non-zero dispersiongiven, to first order in ∆ ≡ t2−Ve , by [71]: ε2(k)− εF = ∆(2cosk + 1). break-ing the particle-hole symmetry. Here we are assuming, for simplicity, that thebulk chemical potential is at the energy of the bulk Dirac points, which becomes87εF = 3t2. (Shifting εF away from the Dirac points, the Hubbard interactions havea larger effect in the bulk rendering the edge model approach more questionable.)Including a small U , the energy to add a spin down electron or remove a spin upelectron at momentum k now becomesEp/h(k) = εk±∆(2cosk+1). (3.61)respectively, where εk is given in Eq. (3.60). Ep(k) is plotted in Fig. 3.4 forseveral values of ∆. We see that for |∆| < ∆c ≈ 0.109U , the energy to add anelectron or hole remains positive, so the edge states remain undoped. A localminimum at k = pi develops in Ep(k) for ∆> .087U , and Ep(k) becomes negativein the vicinity of k = pi for ∆ > ∆c. The lowest energy of a particle-hole state,and the bottom of the particle-hole continuum for various values of ∆ are shownin Fig. 3.5. We see that the exciton becomes unbound except for wave-vectorsnear zero, as |∆| increases. For |∆| > ∆c the edge states become doped, adsorbingelectrons or holes from the bulk. The simplest assumption for |∆| > ∆c is that aFermi sea of spin-down electrons or spin up holes forms near k = pi , for ∆ > ∆cor ∆ < −∆c respectively. This assumption is reasonable since there appear to beno bound excitons for ∆ > ∆c. We also calculated for ∆ near ∆c and L ≤ 74, thelowest energy state with M = N/2− 2, finding no states below the 2-particle, 2-hole continuum, consistent with this assumption. These are exact eigenstates of theHamiltonian of Eq. (3.43), (3.61) with the added holes or particles non-interacting.The corresponding exact result for magnetization versus ∆ is plotted in Fig. 3.6,given this assumption. The non-interacting nature is a simple consequence of thefact that the on-site Hubbard model only gives interaction between electrons ofopposite spin. On the other hand, we cannot rule out the possibility that the groundstate for ∆> ∆c contains a finite density of spin-up holes as well as the spin-downelectrons (and similarly for ∆ < −∆c). In that case, Hubbard interactions havea non-trivial effect. In any event, adding a nearest neighbour Coulomb repulsionterm to the bulk Hamiltonian has no effect on the projected edge Hamiltonian, sincesuch a term acts between A and B sites whereas the zigzag edge states live entirelyon one sublattice. On the other hand, a second neighbour Coulomb repulsion,U2, produces interactions between electrons with parallel spins in the projected880.05 0.10 0.15 DU0.050.100.15MLFigure 3.6: Edge magnetization versus ∆≡ t2−Ve.edge Hamiltonian. While this doesn’t change our conclusions qualitatively in theundoped phase, it will produce interaction effects in the ground state for the dopedcase even if contains only particles or only holes. However, we might expect U2U , in which case these effects could be quite small. Thus in general we expect aone or two component Luttinger liquid for |∆| > ∆c. On the other hand the edgephase occurring for |∆| < ∆c is definitely not a Luttinger liquid. Instead, it mightbe described as a fully spin-polarized semi-metal since all levels are filled withspin-up electrons and there is a non-zero electron and hole addition energy for allwave-vectors accept the band-edges, 2pi/3 and 4pi/3.3.7 Effective inter-edge and intra-edge interactionsThere are also important effects of O(U2/t) which arise from the interactionsbetween bulk and edge states. We can consider integrating out the bulk states toobtain a low energy effective action for the edge states. Due to the gapless natureof the bulk Dirac spectrum, this produces long range retarded interactions amongthe edge excitations. Decay processes of edge into bulk electrons are forbiddenby energy-momentum conservation but the Feynman diagrams of Fig. 3.7 inducequartic interaction terms. For large W and low energies we may calculate these89interactions keeping only the low energy bulk states near the Dirac points, usingthe corresponding Dirac propagators. Note that we ignore interaction effects inthe bulk, as discussed above. This is rather similar to an RKKY interaction. Theinteraction involving the dynamical spin operators [72] ~SU/L(ω,q) on the upperand lower edge, respectively is:Sinter =∫dqdω(2pi)2~SU(ω,q) ·~SL(−ω,−q)Jinter(ω,q,W ) (3.62)whereJinter(ω,q,W ) = 2 U2∫dω ′dk(2pi)2 G(ω′,k,0,W )G(ω−ω ′,q− k,0,W ). (3.63)Here G(ω,k,0,W ) is the bulk free electron Green’s function with momentum kin the x-direction at y = 0 and y = Ly with appropriate zigzag or bearded bound-ary conditions and projected onto the sublattices corresponding to the upper andlower edge (A−A for zigzag-bearded or A−B for zigzag-zigzag (ZZ)). Using thelinearized, Dirac dispersion relation, which is valid at small 1/W , ω/t and q,GZB(ω,kx,y = 0,y′ =W ) =2iv2FW ∑n(−1)nk2nωε2(kx,kn)[ω2 + ε2(kx,kn)](3.64)where ε(~k) = vF |~k| is the Dirac dispersion relation. GZZ is given by the sameexpression with ω replaced by iε(kx,kn) in the numerator inside the sum. The sumover n can be taken up to an arbitrary ultra-violet cut-off whose value doesn’t affectthe behavior at small 1/W , ω/t and q. kn = pin/W for the ZB case. Although thewave-vectors of edge modes are phase-shifted from these values in the ZZ case,this can be ignored at leading order in 1/W , allowing us to again use kn = pin/W .It is straightforward to evaluate Jinter(ω,q,W ) numerically with the two typesof edges. The characteristic scales for the ω and q dependence of Jinter are setby t/W and 1/W , respectively. Since the energy scale of the inter-edge interactionin Eq. (3.64) is U2/(tW 2), it should be permissible to ignore the retardation, andevaluate Sinter at ω = q = 0 to calculate the properties of low energy states. This90Figure 3.7: Feynman diagrams inducing edge interactions from integratingout bulk states.91gives:JZB/ZZ(W ) = ∓cU2t1W 2(3.65)where the positive constant c is given by the convergent sums and integral:c≡√3pi ×∞∑n,m=1(−1)n+m×∫ ∞−∞dκ n2m2(κ2 +m2)(κ2 +n2)1√κ2 +m2 +√κ2 +n2≈ 0.20(3.66)(A similar result was obtained in [73, 74] for the ZZ case.) We see that the groundstate for the zigzag-bearded ribbon has spin L/2 while that for the zigzag-zigzagcase has spin 0, as shown above rigorously using Lieb’s theorem. The remarkablefact that the change in sign of this tiny coupling drastically changes the spin of theground state provides evidence for the polarized nature of the edge spins. Thereis also a large manifold of low energy states, which are simply the eigenstates ofJZZ~ST ·~SB with ST = SB = L/6 (in the ZZ case).Another important effect of O(U2/t) is the intra-edge interaction, independentof W . For a zigzag edge by integrating out the low-energy bulk excitations, thespin part is:Sintra =∫dqdω(2pi)2~S(ω,q) ·~S(ω,q)Jintra(ω,q) (3.67)withJintra(ω,q) =2 U2∫k,ω ′/v f <Λdω ′dk(2pi)2 G(ω′,k,y = y′ = 0)G(ω−ω ′,q− k,y = y′ = 0)(3.68)Now the free bulk Green’s function, with zigzag edge boundary conditions, maybe evaluated for a semi-infinite system, giving, at small kx (measured from a Dirac92point) and small ω:G(ω,kx,y = y′ = 0)≈ 2iv2F∫dky2pik2y(vFk)2ωω2 +(vFk)2(3.69)By using this green function, the Jintra of Eq. (3.68) is ultraviolet divergence andthe integral should be cut off at some point Λ. Although the resulting Jintra(ω,q)is cut off dependent, by ignoring the weak retardation, the corrections to the en-ergy of the excitons is proportional to Jintra(0,q)− Jintra(0,0), which is cut offindependent. For small ω and q these O(U2) intra-edge interactions become moresingular than the O(U) terms by logarithmic factors of q2 lnq,εex(q)≈ 0.36 Uq2−√3(4−pi)(2pi)2 (U2/t)q2 lnq2 (3.70)As mentioned above, Eq. (3.68) only includes the effect of low energy bulkexcitations; it is still possible that the high energy bulk excitations could wipeout this singularity of the exciton dispersion relation. By using the exact formof the bulk wavefunctions one can determine the exact intra-edge interaction ofO(U2). This has a more complicated form than Eq. (3.68). Nonetheless, it can beshown that the only part of this interaction which contributes to this lnq singularcorrection to the exciton dispersion relation is the contribution from low energybulk excitations of the form of Eq. (3.68).We leave a more detailed study of these effects of O(U2) and higher for thefuture. A reasonable approach might be to ignore the bulk interactions, since theyare irrelevant, but analyze the bulk-edge Hubbard interactions using the renormal-ization group. This corresponds to a novel type of boundary critical phenomena inwhich the bulk is a massless (2+1) dimensional Dirac liquid and the edge is a one-dimensional spin-polarized semi-metal. The arguments based on Lieb’s theoremimply that the edge magnetic moment remains stable against weak interactions.3.8 Details of field theory calculationsIn previous section we gave the result of integrating out bulk excitations and theeffective induced higher order inter− and intra− edge interactions. In this section93we give the more detailed procedure of finding those higher-order interactions. Wewill consider two cases of graphene ribbon with zigzag-bearded and zigzag-zigzagedges.As we discussed in section 3.3 the effective Hamiltonian describing bulk excita-tions is Dirac HamiltonianH =−ivF∫dx dy[Ψ†1(~r)σ .∇Ψ1(~r)+Ψ†2(~r)σ¯ .∇Ψ2(~r)](3.71)For infinite graphene sheet, we could simply work in momentum space and fermionpropagators are simply given by〈Ψ1(~q,ω)Ψ†1(~q,ω)〉=1~q.σ + iω 〈Ψ2(~q,ω)Ψ†2(~q,ω)〉=1~q.σ¯ + iω (3.72)But for graphene ribbons, we do not have translational invariance along thewidth of the ribbon. What we have is a field theory with boundaries, and bound-ary conditions will change the form of single particle Green functions, which areneeded to evaluate Feynman diagrams of Fig. 3.7. Let’s find the effect of boundaryconditions on low-energy field operators and also the green functions.To find the impact of boundary conditions on low-energy field operators repre-sentation we follow the same approach as [75] . We start by Hamiltonian 3.71, andby using translational invariance along the x−direction we haveΨ1(r) = eikxx(φA(y)φB(y)), Ψ2(r) = eikxx(φ ′A(y)φ ′B(y))(3.73)The eigenfunction equation around K is given by(0 kx−∂ykx−∂y 0)(φA(y)φB(y))= ε˜(φA(y)φB(y))(3.74)By solving the above equations we will find thatφB = Cezy +De−zyε˜φA = (kx−∂y)φB (3.75)94where the eigenvalue equation is ε˜2 = k2x−z2, and z could be found by imposing thecorrect boundary conditions(bc). The boundary conditions for ZB and ZZ ribbonsareφB(0) = φ ′B(0) = φB(Ly) = φ ′B(Ly) = 0φB(0) = φ ′B(0) = φA(Ly) = φ ′A(Ly) = 0 (3.76)where Ly is the width of the ribbon, first line is bc for ZB ribbon and the secondline is for ZZ ribbon. The reason for this choice of boundary condition is morevivid if we look at Fig. 3.2. From that figure we see that the first row is zigzagedge which starts with atoms only on A sub-lattice. We could imagine that somefictitious atoms are connected to these atoms from some fictitious previous row, ifit exists, but the weight of wave-function is zero on those atoms. Similar argumentholds for other edge.For both ZB and ZZ edges, which zigzag edge starts at first row and is only onA sub-lattice we have, φB(0) = 0; this impose the condition that C+D = 0 for bothcases. By using this condition and also Eq. (3.75) we getφB = sin(z y)ε˜φA = (kx sin(z y)− zcos(z y)) (3.77)For ZB ribbon, boundary condition for lower edge is φB(Ly)= 0 resulting in sin(z Ly)=0 which restricts the values of z to zn = npiLy . This is very interesting result, whichconfirms the existence of zero mode edge states. By using ε˜2 = k2x − z2, for anyvalue of n if we choose kx = zn then we get zero energy state.For ZZ ribbon the boundary condition is φA(Ly) = 0, gives us the following tran-scendental equation to solve for z:tan(zLy) =zkx(3.78)As we see, in contrast to ZB ribbon where zn was independent of kx, for ZZ ribbonfirst of all there is no simple expression for zn, and also the values of zn depends onkx; for each kx by solving Eq. (3.78) we get a set of solutions for zn. This implies95zn should be a function of kx, but its dependence on kx is very weak and in generalwe could write zn(kx) = [npi + δn(kx)]/Ly, where δn(kx) ∈ [0,pi/2] and is slowlyvarying function of kx.Having found the effect of boundary conditions on the low-energy wave-functionsand taking number of the unit cells along the width of the ribbon is Ny, which leadsto Ly = 3aNy/2, we haveΦB(kx,y) =1√Ny∑kysin(kyy)ckΦA(kx,y) =v f√Ny∑ky1Ek[kx sin(kyy)− ky cos(kyy)]ck (3.79)where Ek =±v f√k2x + k2y , and we changed the notation from z to ky, and values ofky depends on the boundary conditions.We will see shortly that, to integrate out bulk excitations, we need these two sin-gle particle Green functionsT 〈ΦA(kx,τ,y= 0)Φ†A(kx,0,y=Ly)〉 andT 〈ΦA(kx,τ,y=0)Φ†B(kx,0,y = Ly)〉, for ZB ribbon and ZZ ribbon , respectively, so let’s evaluatethem here.For the case of zigzag-bearded we needGZB ≡ T 〈ΦA(kx,τ,y = 0)Φ†A(kx,0,y = Ly)〉=v2fNy∑kyk2y cos(kyLy)E2kT 〈ck(τ)c†k(0)〉 (3.80)96In imaginary time representation we have ck(τ) = e−Ekτck, thus we getGZB(kx,τ,y = 0,y = Ly)=v2fNy∑kyk2y cos(kyLy)[1E2kΘ(τ)e−Ekτ〈ckc†k〉−1E2kΘ(−τ)e−Ekτ〈c†kck〉]=v2fNy∑kyk2y cos(kyLy)E2k[Θ(τ)Θ(Ek)e−Ekτ −Θ(−τ)Θ(−Ek)e−Ekτ]=v2fNy∑kyk2y cos(kyLy)ε2k[Θ(τ)e−εkτ −Θ(−τ)eεkτ](3.81)Where εk = v f√k2x + k2y . Now by taking the Fourier transform, we needGZB(kx,ω,y = 0,y = Ly) =−v2fNy∑kyk2y cos(kyLy)ε2k[1iω− εk+1iω+ εk]=2iv2fNy∑kyk2y cos(kyLy)ε2kωω2 + ε2k(3.82)For the case of zigzag-zigzag ribbon we haveGZZ ≡ T 〈ΦA(kx,τ,y = 0)Φ†B(kx,0,y = Ly)〉=−v fNy∑kyky sin(kyLy)EkT 〈ck(τ)c†k(0)〉 (3.83)In imaginary time representation we have ck(τ) = e−Ekτck thus we getGZZ(kx,τ,y = 0,y = Ly)=−v fNy∑kyky sin(kyLy)[1EkΘ(τ)e−Ekτ〈ckc†k〉−1EkΘ(−τ)e−Ekτ〈c†kck〉]=−v fNy∑kyky sin(kyLy)[1EkΘ(τ)Θ(Ek)e−Ekτ −1EkΘ(−τ)Θ(−Ek)e−Ekτ]=−v fNy∑kyky sin(kyLy)εk[Θ(τ)e−εkτ +Θ(−τ)eεkτ](3.84)97by doing the Fourier transformation we getGZZ(kx,ω,y = 0,y = Ly) =v fNy∑kyky sin(kyLy)εk[1iω− εk−1iω+ εk]=−2v fNy∑kyky sin(kyLy)εkεkω2 + ε2k(3.85)The boundary conditions for ZB , ΦB(y = 0) = ΦB(y = Ly) = 0, restricts thevalues of ky to ky = npiLy where n is an integer between 0, · · · ,Ny, thus for ZB casewe have coskyLy = (−1)n. But for ZZ case the boundary conditions, ΦB(y = 0) =ΦA(y = Ly) = 0, gives us tankyLy = ky/kx.For the case of ZZ case we could write k(n)y Ly = npi + δ (n,kx) where nowδ (n,kx) ∈ [0,pi/2], and is given by tanδ (n,kx) = k(n)y /kx . By this representationwe havesink(n)y Ly = sin(npi+δ (n,kx)) = (−1)n sinδ (n,kx)= (−1)ntanδ (n,kx)√1+ tan2 δ (n,kx)= (−1)nk(n)y√k2x +(k(n)y )2(3.86)Thus the green functions areGZB(kx,ω,y = 0,y = Ly) =2iv2fNy∑nk2n(−1)nε2(kn,kx)ωω2 + ε2(kn,kx)GZZ(kx,ω,y = 0,y = Ly) =−2v2fNy∑n(k(n)y )2(−1)nε2(k(n)y ,kx)ε(k(n)y ,kx)ω2 + ε2(k(n)y ,kx)(3.87)Where kn = npiLy and k(n)y = npiLy +δ (n,kx)Ly. By knowing that δ (n,kx) ∈ [0,pi/2], toleading order in Ly, we could approximate k(n)y = npiLy and green functions formula98simplifies toGZB(kx,ω,y = 0,y = Ly) =2v2f iNy∑nk2n(−1)nε2(kn,kx)ωω2 + ε2(kn,kx)GZZ(kx,ω,y = 0,y = Ly) =−2v2fNy∑nk2n(−1)nε2(kn,kx)ε(kn,kx)ω2 + ε2(kn,kx)(3.88)3.8.1 Integrating out bulk excitationsIn this section we focus on the case of graphene ribbon with ZB boundarycondition; similar calculation with slightly minor changes applies for ZZ ribbon.From section 3.5 we saw that by representing electron operators in terms of edge-modes and bulk excitations, to first order in U we get the following edge-bulkinteractionsH1 =UNx∑λ (k+q,k,y)e†↑(k+q)e↑(k)Φ†A,↓(k′−q,y)ΦA,↓(k′,y)+ ↑↔↓H2 =−UNx∑λ (k+q,k′,y)e†↑(k+q)e↓(k′)Φ†A,↓(k′−q,y)ΦA,↑(k,y)+ ↑↔↓H3 =−UNx∑ λ˜ (k+q,k′−q,y)e†↑(k+q)e†↓(k′−q)ΦA,↑(k,y)ΦA,↓(k′,y)+ h.c. (3.89)Whereλ (k+q,k,y) = α∗(k+q,y)α(k,y)λ˜ (k+q,k,y) = α∗(k+q,y)α∗(k,y) (3.90)where α(k,y) is exponentially localized at the zigzag edge for k∈ I1≡ [2pi/3,4pi/3],and localized at bearded edge for k ∈ I2 ≡ [−2pi/3,4pi/3]. ΦA,σ (k,y) is represent-ing low-energy bulk excitations, as a result of this, values of k should be near thetwo Dirac points ±2pi/3.As we mentioned before ΦA/B,σ (k,y) is slowly varying function of y. Thus inequation (3.89) we could assume that ΦA/B,σ (k,y) is more or less independent of y99and keeps its values at either zigzag edge, y = 0, or the bearded edge y = Ly. Thenwe need to evaluate sum of λ (k+q,k,y) and λ˜ (k+q,k,y) over y. Let us defineλ (k+q,k) = ∑yα∗(k+q,y)α(k,y)λ˜ (k+q,k) = ∑yα∗(k+q,y)α∗(k,y) (3.91)It is easy to checked that for large N both λ (k+q,k) and λ˜ (k+q,k) are exponen-tially small if k,k + q belong to different edges. Now let us look at the effect ofmomentum restrictions from ΦA. From Eq. 3.89 we could see that forH1 andH2,q is either near 0 or ±2pi/3a. For q≈ 0 we have λ (k+q,k) = 1 up to first order inq, thusH(0)1 =UNx∑k∈I1e†↑(k+q)e↑(k)Φ†A,↓(k′−q,y≈ 0)ΦA,↓(k′,y≈ 0)+ ↑↔↓+UNx∑k∈I2e†↑(k+q)e↑(k)Φ†A,↓(k′−q,y≈ Ly)ΦA,↓(k′,y≈ Ly)+ ↑↔↓H(0)2 =−UNx∑k∈I1e†↑(k+q)e↓(k)Φ†A,↓(k′−q,y≈ 0)ΦA,↑(k′,y≈ 0)+ ↑↔↓+−UNx∑k∈I2e†↑(k+q)e↓(k)Φ†A,↓(k′−q,y≈ Ly)ΦA,↑(k′,y≈ Ly)+ ↑↔↓(3.92)For q≈ 0, it could be shown that it leads to terms which are less relevant and couldbe ignored. Now that we have the expression for edge-bulk interaction Eq. 3.92and also the bulk propagators Eq. (3.88), we could integrate out the bulk excita-tions.Inter-edge interaction :To find the inter-edge effective interactions we only need to consider the loopintegrals which connect zero-mode excitations of different edges. An example of100these kind of terms to second order in H (0)1 is given byδH =(−iUNx)2∑e†↑(k+q,τ)e↑(k,τ)e†↑(l +q′,0)e↑(l,0)×〈Φ†A,↓(k′−q,y≈ 0,τ)ΦA,↓(k′,y≈ 0,τ)×Φ†A,↓(l′−q′,y≈ Ly,0)ΦA,↓(l′,y≈ Ly,0)〉 (3.93)Where operators are at time τ and 0. This is not the only possible term, to find thefull inter-edge interaction we need to find four more terms, which could be foundwith the same approach.The correlation function in the second line of Eq. (3.93) could be written asG =1Nx〈Φ†A,↓(k′−q,y≈ 0,τ)ΦA,↓(k′,y≈ 0,τ)∗Φ†A,↓(l′−q′,y≈ Ly,0)ΦA,↓(l′,y≈ Ly,0)〉=1Nx〈Φ†A,↓(k′−q,y≈ 0,τ)ΦA,↓(l′,y≈ Ly,0)〉×〈ΦA,↓(k′,y≈ 0,τ)Φ†A,↓(l′−q′,y≈ Ly,0)〉=−δk′−q,l′δk′,l′−q′NxGZB(k′−q,−Ly,τ)GZB(k′,Ly,−τ) (3.94)The Kronecker delta function leads to q =−q′ and the minus sign in the fifth linecomes from the anti-commuting nature of fermionic operators, which lead to theminus sign for fermion loops. The Green function GZB(k,Ly,τ) was derived inprevious sub-section and is given by Eq. (3.88). For the rest of this section wedrop the Ly dependence to simplify the notation.By taking the time Fourier transform of above equation we see that we need toevaluate the following integralf (ω,q) =∫dq′ dω ′(2pi)2 GZB(ω′,q′)GZB(ω ′−ω,q′−q) (3.95)And if we do the same calculation for the rest of the terms and carefully add themtogether we find that the effective intra-edge interaction induced by integrating out101bulk excitation has the formδHz−b =−U2aNxv f∑ω,qf (ω,q)[2SU(ω,q).SL(−ω,−q)+12NU(ω,q)NL(−ω,−q)](3.96)where SU/L and NU/L are the total spin and total number of electrons for up-per(zigzag) and lower(bearded) edges, respectively. Before going forward to eval-uate the function f (ω,q) there are few important points that we need to consider.First, we see that the induced interaction has ”retardation”, meaning that itcouples operators at different times and its Fourier transform contain a functionof f (ω,q). This retardation was expected from the beginning because the bulkexcitations were gapless. Second, in addition to spin-spin exchange interactionbetween the edges, as we see in Eq. (3.96) there is also a long range Coulomb-likeinteraction proportional to NU ∗NL, which is also retarded.In general retardation and long-range interactions complicate the study of groundstate properties, but in this case we could ignore the retardation and ignore the fre-quency dependency of these interactions. The reason for such assumption is that,the characteristic scale for ω and q is t/Ly and 1/Ly (where t is the hopping ampli-tude). But if we calculate f (ω,q), we see that the strength of induced interactionsis proportional to U2t∗L2y. In our perturbative study ,we assumed that U/t  1; thismeans that for ribbons with large width, Ly a, the strength of induced interactionis weakened by factors of U/t and (a/Ly)2. This justifies the assumption that wecould take ω = 0 and q = 0, and only evaluate f (0,0).fZB(0,0) =−4v4fN2y∑n,m(−1)n+m∫dl dω(2pi)2k2nk2mε2nε2mω2(ω2 + ε2n )(ω2 + ε2m)=−2v4fN2y∑n,m(−1)n+m∫dl(2pi)k2nk2mε2nε2m1(εn + εm)(3.97)Where εn = v f√k2n + l2 and kn = npi/Lx, going from the first line to second line issimple contour integration. This integral could be done analytically but the sum-102mation is only possible numerically, and the result was discussed in section 3.7 ,Eq. (3.65).For ZZ ribbon we only should replace the GZZ from Eq. (3.88) into Eq. (3.95),from which we could find fZZ(0,0) which has exactly the same result as Eq. (3.97)but with important different sign. This difference in the sign of induced exchangeinteraction leads to Ferromagnetism and Anti-Ferromagnetism of two edges for ZBand ZZ, respectively.Integrating out bulk excitations, in addition to giving inter-edge interactionswhich gives makes the total spin of ground consistent with Lieb’s theorem, alsogives us induced intra-edge interactions, which we briefly discuss here.intra-edge :Similar to the inter-edge case, we look at second order perturbation of Eq. (3.92),and only keep loop diagrams which connects edge modes which are localized onsame edge, e.g zigzag edge at y = 0 this gives terms like :δH =(−iUNx)2∑e†↑(k+q,τ)e↑(k,τ)e†↑(l +q′,0)e↑(l,0)× 〈Φ†A,↓(k′−q,y≈ 0,τ)ΦA,↓(k′,y≈ 0,τ)∗Φ†A,↓(l′−q′,y≈ 0,0)ΦA,↓(l′,y≈ 0,0)〉 (3.98)By using similar calculations, of section 3.8, we find the needed green functionGintra(ω,kx) =2v2f iNy∑nk2nε2(kn,kx)ωω2 + ε2(kn,kx)(3.99)Because we are only interested to find effective interaction terms on single edge, wecould simply work with semi-infinite graphene sheet and replace the sum with theintegral in above equation. The rest of calculations is exactly similar to inter−edgecase we find similar interaction as Eq. (3.96), with the following function forf (ω,q)f (ω,q) =∫dq′ dω ′(2pi)2 Gintra(ω′,q′)Gintra(ω ′−ω,q′−q) (3.100)103The effect of this term on excitation’s spectrum was discussed in section 3.7. Andthey make the ferromagnetic state more robust while changing the dispersion ofthe excitations.3.9 ConclusionIn this chapter by using the projection method [35] we study the effectiveHamiltonian of the edge modes of graphene. We expect this method to becomevalid in the weakly interacting limit of the Hubbard model. We show rigorouslythat the ground state of the projected Hamiltonian is ferromagnetic and as a resultthe edges are spin polarized. Then we include the effect of nearest neighbor hop-ping terms and study the stability of ferromagnetic state of the edge as functionof next nearest neighbor hopping strength, and we show that the fully polarizedstate remains intact for some range of NNN hopping term and having passed somecritical value the polarization gradually decreases.Then in final section we take into account the effect of bulk excitations. Inte-grating out the bulk excitations leads to higher order interactions. The importantresult of doing so is that it leads to long range anti-ferromagnetic exchange inter-action between two edges of a graphene ribbon. This means that for a grapheneribbon with both edges in zigzag shape each edge is spin polarized and the totalspins of the edges are coupled anti-ferromagnetically.Another effect of integrating out the bulk excitation is to have higher orderinteraction correction to the projected Hamiltonian. By studying these correctionwe see that they even make the magnetism of the edges more robust.104Chapter 4ConclusionIn this thesis we study two different one-dimensional interacting systems. Inchapter 2 we try to understand the transverse dynamical structure factors of XXZspin chain which could teach us about the excitations of the system. In chapter 3we look at another interesting one dimensional system, which is the edge states ofgraphene with zigzag boundaries, and the problem of the fully polarized magneticground state of this system.In chapter 1 we showed that using standard bosonization method and LuttingerLiquid theory is no reliable way to study the nontrivial line shape of dynamicalstructure factors. We showed in section 2.2.3 that for the study of DSFs the bandcurvature terms, which are irrelevant in standard method, give rise to divergentterms, in higher orders of perturbative approach, near some threshold frequenciesand make the standard method unreliable. If someone persists to use standardmethod to study the line shapes the only way could be to sum up all these divergentterms, which appear at all orders of perturbation. Keeping track of and summingall these terms if not impossible, is very difficult and cumbersome approach.The more practical approach is beyond bosonization methods, X-ray edge method.In this approach 2.3 the band dispersion is kept intact, while the perturbation ismade in terms of interaction strength. In this approach the parameters of the neweffective Hamiltonian could be derived perturbatively in terms of the interactionstrength, but for exactly solvable models like XXZ spin chain, by using Bethe105ansatz solutions the parameters could be derived for any interaction strength. Butthe resulting theory could work as a phenomenological ground for the study ofmore diverse systems.By using this method we obtained results on the transverse spectral function ofXXZ spin chain in a magnetic field. In the zero field case we have exactly deter-mined a critical exponent governing the lower edge singularity, for all |∆|< 1 andall wave-vector. For the finite field case, we have shown how this exponent canbe determined from parameters which can be obtained from solving Bethe ansatzequations and which also determine the behavior of the longitudinal structure func-tion, fermion spectral function and the finite size spectrum. We have argued that,for general magnetization, a large number of increasingly weaker singularities oc-cur in the spectral function, extending all the way down to zero energy. We derivedresults for the finite temperature spectral function using X-ray edge methods, ob-taining strikingly different behaviour than that given by standard bosonization at0 < 1−∆e f f  1. The line-shape is non-Lorentzian and the line width is O(T ), un-suppressed by 1−∆e f f . We pointed out that electron spin resonance measurementson spin chain compounds with uniform Dzyaloshinskii-Moriya interactions wouldprovide a way of experimentally confirming, for the first time, the new bosoniza-tion results being obtained on spin chains, using X-ray edge techniques.In the case of staggered DM interactions, the most interesting ESR signal oc-curs when the magnetic field is transverse to the DM vector. This may also be thecase for uniform DM interactions, but we leave this for future work.In chapter 3 we look at different 1D system. The tight binding honeycomblattice with zigzag boundaries has been known to have states localized near theedges and so effectively are one dimensional, and interestingly they have zero-energy flat bands. Thus they are highly susceptible to the effect of interactions. Ahand-waving argument based on Lieb’s theorem gives some intuition that includingHubbard interactions could make the edges magnetized. In section 3.5 we foundthe effective projected Hamiltonian of the edge states, this Hamiltonian is non-localand has very complicated form, in that section we proved that the ground state ofthe effective edge Hamiltonian at half filling is fully polarized ferromagnetic state,which confirms the Lieb’s theorem prediction.106The study of the particle addition or removal energy εk could be probed ex-perimentally. In principle εk could be measured by using ARPES experiments.The density of states of the edge modes could be measured by using STM exper-iment, and as the ground state is predicted to be fully polarized state (suppose inz-direction), it would be only possible to tunnel in a spin-down electron or tunnelout a spin-up electron.Then we studied the effects of unavoidable NNN hopping terms and showedthat those terms would not change the ground state up to some critical value ∆c andhaving passed that value the magnetization decreases monotonically to zero. Forlonger range Coulomb repulsion, we see that the next neighbor repulsion could notchange our results but second and higher order terms in principle could affect ourprediction, however we expect U2U0 and their effects are quite small.As the excitations of bulk graphene are gapless, simply projected Hamiltonianis not enough. In section 3.7 by integrating out the bulk excitations we foundinduced interactions between the edges of graphene ribbons and also some higherorder interactions on a single edge too.The inter-edge interaction depends on the boundary conditions for the ribbon.For a ribbon with two zigzag edges the resulting interaction is spin exchange in-teraction with antiferromagnetic coupling, while for a ribbon with one zigzag edgeand other bearded edge the interaction term is ferromagnetic exchange. Again bothresults are in complete agreement with Lieb’s theorem prediction. The intra-edgemakes the ferromagnetic ground state more robust but it contributes some singularcorrections of like q2 lnq to the energy of the excitons. We leave a more detailedstudy of these effects of O(U2) and higher for the future. A reasonable approachmight be to ignore the bulk interactions, since they are irrelevant, but analyze thebulk-edge Hubbard interactions using the renormalization group. This correspondsto a novel type of boundary critical phenomena in which the bulk is a massless(2+1) dimensional Dirac liquid and the edge is a one-dimensional spin-polarizedsemi-metal. The arguments based on Lieb’s theorem imply that the edge magneticmoment remains stable against weak interactions.107Bibliography[1] P. Fazekas, Lecture Notes on Electron Correlation and Magnetism (WorldScientific, Singapore, 1999). → pages 1[2] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press,England, 2003). → pages 1, 3, 12, 23[3] A. H. Castro Neto, F. Guinea, and N. M. R. Peres. 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B, 79:235433, Jun 2009. → pages 92[75] H. A. Fertig and L. Brey. Luttinger liquid at the edge of undoped graphene ina strong magnetic field. Phys. Rev. Lett., 97:116805, Sep 2006. → pages 94114Appendix ADerivation of X-Ray edgeHamiltonianThe bosonization method which discussed in 2.2, is a powerful method to findthe correlation and spectral functions of particular model at low-energies in thelimit of q→ 0. But as we discussed in 2.2.3 it has its limitation to find the singular-ity exponents of spectral functions near their singular points. The band curvatureterms which were irrelevant by power counting, became divergent near those sin-gular points. Thus the result of bosonization for the singularity exponents of SF forfinite momentum q, are no longer trustworthy and we need look for new models orextension of existing models to deal with it.During the derivation of effective model in 2.2, we defined a sub-band cutoffΛF  J near the two Fermi points and only kept the excitations within those sub-bands. When we are dealing with correlations at finite momentum q, it is naturalto think that particle or hole excitations with momentum close to q should havesignificant effect on the correlations and in the naive bosonization method as weare looking at the limit ΛF → 0, we implicitly ignore the effect of these importantexcitations.The natural extension results of 2.2 would be to define three sub-bands forfermionic excitations, two sub-bands for excitations near the Fermi points ΛF andthird sub-band Λd for high-energy particle or a deep hole, see Fig 2.6. By including115these sub-bands the fermion annihilation operator isc j ≡ Ψ(x)≈ ψR(x)eikF x +ψL(x)e−ikF x +d eikx (A.1)Where ψR/L are excitations near the Fermi points and d is either a high-energyparticle or a deep hole with momentum, k = kF +q for particle and k = kF −q forhole. We focus for particle excitation for the rest of this section, similar resultsholds for a hole excitations.The procedure is similar to what we did in 2.2, first we derive the low-energyeffective Hamiltonian in terms of ψR,L and d, then by using bosonization we bosonizethe ψR,L while keeping the high-energy excitations in Fermionic form.By using equations Eq (3.71), the kinetic part of effective Hamiltonian isH0 ≈vF2[(∂xφR)2 +(∂xφL)2]+d†(−ε− i u∂x−∂ 2x2m)d (A.2)where vF = J sinkF is the fermi velocity, ε = J cos(kF + q) is the energy of high-energy particle with velocity u = J sin(kF +q) and effective mass m = [Jcos(kF +q)]−1. Using Eq (??) the density operator could be written asn(x) = Ψ†(x)Ψ(x)≈ ρR +ρL +(ei2kF xψ†LψR +h.c.)+d†d+[ei(kF−k)d†ψR + e−i(kF+k)d†ψL+h.c.](A.3)By writing the density operator in bosonized form, the interaction term becomesHint =g44pi[(∂xφR)2 +(∂xφL)2]−g22pi ∂xφR∂xφL+2∆√2pid†d [1− cos(kF − k)]∂xφR+2∆√2pid†d [1− cos(kF + k)]∂xφL (A.4)where g2 = g4 = 2∆[1− cos(2kF)]. Now by using Eq. (2.23) and writing φR,L in116terms of bosonic field φ˜ and its dual field θ˜ we haveH =v2[K(∂xθ˜)2+1K(∂xφ˜)2]+d†(−ε− iu∂x−∂ 2x2m)d+12√pi((κ˜L− κ˜R)∂xθ˜ +(κ˜L + κ˜R)∂xφ˜)d†d (A.5)where κ˜R,L = 2∆ [1− cos(kF ∓ k)]. By doing a canonical transformation that rescalesthe field φ˜ →√Kφ and θ˜ → θ/√K we haveH =v2[(∂xφR)2 +(∂xφL)2]+d†(−ε− iu∂x−∂ 2x2m)d+1√2piK(κL∂xφL +κR∂xφR)d†d (A.6)withκR,L =(1+K2)κ˜R,L−(1−K2)κ˜L,R (A.7)This describes the Luttinger Liquid coupled to an impurity and have been discussedin [47–49].117Appendix BPositivity of spectral functionIn this appendix we will prove that spectral function Eq. (2.118) is positive.We haveS−+(q,ω) ∝∫ ∞−∞dtei[ω−ε(kp)]t(2piT )νR+νL(sin(2piT (ε+ i(1−u/v)t)))νR(sin(2piT (ε+ i(1+u/v)t)))νL=∫ ∞0dtei[ω−ε(kp)]t(2piT )νR+νL(sin(2piT (ε+ i(1−u/v)t)))νR(sin(2piT (ε+ i(1+u/v)t)))νL+∫ 0−∞dtei[ω−ε(kp)]t(2piT )νR+νL(sin(2piT (ε+ i(1−u/v)t)))νR(sin(2piT (ε+ i(1+u/v)t)))νL(B.1)By defining the branch cut on (−i∞,−iε)⋃(iε, i∞) and doing the change of vari-able t→−t in second integral, we haveS−+(q,ω) ∝ Re{exp(−ipi2(νL + sgn(1−u/v)νR))∫ ∞0dt(2piT )νR+νL−1 ei[ω−ε(kp)]t/(2piT )|sinh((1−u/v)t)|νR |sinh((1+u/v)t)|νL} (B.2)We will prove shortly that above integral is positive in general, but let us first lookat the special cases of either νR = 0 or νL = 0, which is relevant for weak anisotropyat zero magnetic field, so u < v.118Suppose that νL = 0 thus we haveS−+(q,ω) ∝ (2piT )νR−1Re[exp(−ipi2νR)∫ ∞0dtei[ω−ε(kp)]t/(2piT )|sinh((1−u/v)t)|νR]∝ (2piT )νR−1Re(exp(−ipi2νR)B[−iω− ε(kp)4piT (1−u/v) +νR/2,1−νR])Where B[x,y] is Euler’s beta function. Let’s define W ≡ ω−ε(kp)4piT (1−u/v) , and by usingthe definition of beta function in terms of gamma function we haveS−+(q,ω) ∝ (2piT )νR−1Re(exp(−ipi2νR) Γ[−iW +νR/2]Γ[1−νR]Γ[1− (iW +νR/2)])(B.3)Then by using the identity that Γ[1− z]Γ[z] = pi/sin(piz) we haveS−+(q,ω) ∝ (2piT )νR−1Γ[−iW +νR/2]Γ[iW +νR/2]Γ[1−νR]×Re(exp(−ipi2νR)sinpi(iW +νR/2))(B.4)by writing the sine function in exponential form, we finally getS−+(q,ω) ∝ (2piT )νR−1 |Γ[iW +νR/2]|2Γ[1−νR]epiW sin(piνR) (B.5)so we see the spectral weight is positive, for all values of νR. This expression alsoholds for the case of u > v.Actually what we have shown is that the Fourier transform of 1/sin(2piT (ε+ i(1−u/v)t))ν is given by a real positive function, in the following formF.T[1sin(2piT (ε+ i(1−u/v)t))ν]=(2piT )ν−1∣∣∣∣Γ[ν/2+ i ω4piT (1−u/v)]∣∣∣∣2Γ[1−ν ]epiω/(4piT (1−u/v)) sin(piν)119Where F.T stands for Fourier Transform. Now by using above equation and takingthe convolution of Eq. (2.119), we haveS−+(q,ω) ∝ (2piT )νR+νL−2Γ[1−νR] sin(piνR)Γ[1−νL] sin(piνL)∫ ∞−∞dωR dωL δ (ω− ε(kp)−ωR−ωL)epiωR/(4piT (1−u/v))epiωL/(4piT (1+u/v))×∣∣∣∣Γ[νR/2+ iωR4piT (1−u/v)]×Γ[νL/2+ iωL4piT (1+u/v)]∣∣∣∣2(B.6)We see that all the functions in above equation are positive; thus whole the integralis positive.120


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