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Theory of nuclear magnetic relaxation in Haldane gap materials: an illustration of the use of (1+1)-dimensional… Sagi, Jacob S. 1995

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THEORY OF NUCLEAR MAGNETIC RELAXATION IN HALDANEGAP MATERIALS: AN ILLUSTRATION OF THE USE OF(1+1)-DIMENSIONAL FIELD THEORY TECHNIQUESByJacob. S. SagiB. Sc. (Physics) University of Toronto, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1995© Jacob. S. Sagi, 1995In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of PhysicsThe University of British Columbia6224 Agricultural RoadVancouver, B.C., CanadaV6T 1Z1Date:AbstractA comprehensive theory of nuclear magnetic relaxation in S = 1 Haldane gap materials is developed using nonlinear-a, boson and fermion models. We find that at temperatures much smaller than the lowest gap the dominant contribution to the relaxationrate comes from two magnon processes with T’ ‘- e_m/T, where Lm is the smallestgap corresponding to a polarization direction perpendicular to the field direction. Asthe gap closes, we find that the dominant contribution comes from one magnon processes, and the result depends on the symmetry of the Hamiltonian. Overall the modelsagree qualitatively, except near the critical regime, where the fermion model is shownto be the best description. We include a thorough discussion of the effects of interchaincouplings, nearest neighbour hyperfine interactions and crystal structure, and introducea new theory of impurities corresponding to broken chain ends weakly coupled to bulkmagnons. The work is then applied to recent measurements on NENP. We find overallfair agreement between available T’ data and our calculations. We finish by suggestingfurther experimental tests of our conclusions.IITable of ContentsAbstract iiTable of Contents iiiList of Figures vi1 Introduction and Background 11.1 Introduction 11.2 Spin-wave Theory 31.3 Non-Linear a (NLu) Model 71.4 Boson Model 151.5 Fermion Model 171.6 Nuclear Magnetic Relaxation Rate 202 Details of the Models 252.1 NLa Model: Temperature and Field Dependence of the Spectrum; ExactResults 252.1.1 Temperature Dependence of the Gap 252.1.2 Field Dependence of the Gap 292.1.3 Exact Results 342.2 The Free Boson and Fermion Models 372.2.1 Diagonalization 371112.2.2 Discussion: Comparison of Spectra and Spin Operator Matrix Elements 473 Model Predictions for Tj’ 533.1 T’ for h << h 583.1.1 Isotropic Symmetry 623.1.2 Axial Symmetry 633.1.3 Z2 x Z2 x Z2 Symmetry 663.2 Close to the Critical Field 673.3 Above the Critical Field 703.4 Summary 714 Material Properties and Possible Effects on Experiment 724.1 Hyperfine Tensor 724.2 Impurities 744.3 Interchain Couplings 844.4 Crystal Structure 865 NENP: Direct Comparison with Experiment 885.1 The Structure of NENP and Experimental Ramifications 885.2 Analysis of the Data 956 Suggested Experiments and Curious Predictions 1096.1 Experimentally Testable Conflicts Between Models 1096.2 Measuring Small Anisotropies 1136.3 ESRforNENP 1157 Concluding Remarks 119ivBibliography 120VList of Figures2.1 Free boson and free fermion dispersions with the gap parameters of NENP.Top graph: flIIb; bottom graph: flWc 322.2 Corrected free boson (t = z(H)) and free fermion dispersions with thegap parameters of NENP. Top graph: ilIb; bottom graph: ftIc 333.1 First non-vanishing contribution to relaxation due to the staggered partof the spin 543.2 Inter- vs Intrabranch transitions 563.3 The gap structure for 0(3) symmetry 623.4 The gap structure for U(1) symmetry. (a) /.S > zj; (b) LS <z . . 644.1 Impurity level diagram when D’ = 0 764.2 Impurity level diagrams for D’ 0 785.1 NENP 895.2 Local and crystallographic axes projected onto the ac-plane in NENP . 905.3 A projection of the NENP unit cell onto the ac-plane showing two chainsper unit cell 925.4 Boson F(h, T) for fields along the b and c chain directions 975.5 Fermion F(h, T) for fields along the b and c chain directions 985.6 Boson F(h, T) for fields along the b and a chain directions 995.7 Fermion F(h, T) for fields along the b and a chain directions 1005.8 Theoretical (lines) vs. experimental data (circles and squares) 102vi5.9 Theoretical (lines) vs. experimental data (circles and squares) 1035.10 Relaxation rate for field along the b-axis 1065.11 F(T)—the width of the impurity resonance 1076.1 Dispersions for the two chain conformations and sample resonant transitions for a uniform field placed 60° from the crystallographic c-axis. . . . 1166.2 Resonant field vs. field orientation in the ac-plane for .19 meV transitions. 117viiChapter 1Introduction and Background1.1 IntroductionIn 1983, Haldane derived his famous result stating that integer spin one dimensionalHeisenberg antiferromagnets featured a gap in their low energy excitation spectrum [1].Since then, much effort has been devoted to further exploration of such systems, bothexperimentally and theoretically. The purpose of this work is to develop a theoreticalframework for the understanding of low energy experiments on one dimensional Haldanegap materials. In particular, we focus on the nuclear magnetic relaxation rate, T’,although the work has relevance to many other techniques. By studying this thesis, itis hoped that the reader can become familiar with the tools used to understand integerspin Heisenberg antiferromagnetic chains with anisotropies, and can apply these tools tothe analysis of real systems.There are, essentially, three important models that have so far been used to describethe system. In the later sections of this chapter we review the competing descriptionsof S = 1 antiferromagnetic spin-chains, paying some attention to their advantages andshortcomings. We start by outlining the traditional spin-wave theory’ used to modelantiferromagnetism in higher dimensions. After illustrating the deficiencies in this approach, we describe the Nonlinear a (NLa) model in some detail. This is followed byan analysis of a simplified yet closely related Landau-Ginsburg boson model. Last, wediscuss a free fermion model used recently to successfully treat the case of anisotropic1 [2] for a comprehensive discussion of this topic1Chapter 1. Introduction and Background 2spin-chains. We end the chapter with background on the nuclear magnetic relaxationrate, 1/T, for nuclear spins coupled to the spin-chain through hyperfine interactions.Chapter Two focuses on the details of the models, building the tools necessary for adetailed analysis. We discuss the temperature and magnetic field dependence of the NLumodel and its possible relevance to the spectrum of the boson model, as well as cite someexact results available in cases of high symmetry. We also diagonalize the free boson andfermion field theories, including on-site anisotropy effects. We derive matrix elements ofthe uniform part of the spin operator (fourier modes near wave vector zero) between oneparticle states of magnetic excitations. These are used to compare the different models.Chapter Three explicitly describes the calculation of NMR T1, considering varioussymmetries of the Hamiltonian. We identify the leading mechanisms for low temperaturerelaxation in the presence of a magnetic field. We discuss three regimes correspondingto different magnitudes of the applied external magnetic field, giving expressions for therate in each case. We discover that at temperatures much lower than the smallest gapthe uniform part of the spin operator contributes most to the relaxation rate; in theabsence of interactions, this corresponds to two magnon processes. The rate is foundto be T1’ e_m/T, where Lm is the smallest gap corresponding to a polarizationdirection perpendicular to the magnetic field. As the externally applied magnetic fieldapproaches a critical value, one of the gaps closes and we find the dominant process to beone magnon, corresponding to contributions from the staggered part of the spin operator(fourier modes near wavelength ir/a, where a is the lattice spacing). In this regime, weshow that the fermion model is the best description and that the expression for T1depends on the symmetry of the Hamiltonian.Chapter Four deals with intrinsic effects which must be taken into account whenanalyzing experimental data. We discuss nearest neighbour hyperfine interactions; weshow that these will contribute to order A/A, the ratio of the nearest neighbour couplingChapter 1. Introduction and Background 3to the local coupling. We also consider interchain couplings and show that they introducea natural infrared cutoff to the diverging density of states at the gap; for sufficiently longchains, they also densely fill the energy intervals between states along a finite chain.Finally, we introduce a new impurity theory to explain the effects of nearly free spin-ichain end degrees of freedom. We find that the states formed by such end spins in thegap, can give rise to non-trivial relaxation when coupled to the bulk excitations.Chapter Five applies the theory to recent experiments on the well studied material,Ni(C2H8)0(C104(NENP). We take a close look at the crystal structure of NENPand identify possible terms which may be present in the Hamiltonian. We also note thefact, hitherto neglected, that NENP possesses two inequivalent chains in each unit cell.The results of Chapters Three and Four are then used to analyze experimental data. Wefind reasonable agreement for a magnetic field placed along the crystal c-axis of NENP,and an unexpected discrepancy for a magnetic field placed along the chain axis. Theimpurity theory is used to model low field data with qustionable results.The final chapter proposes further experimental tests of the theoretical predictions ofthis work. We suggest elastic neutron, electron spin resonance and further NMR studiesto verify our own.1.2 Spin-wave TheoryThe Heisenberg Hamiltonian describing the isotropic antiferromagnetic spin-chain isH=J• Ii+1 J>O (1.1)This arises naturally from the Hubbard model for insulators at ‘half’ filling [3]. Tounderstand where this might come from, we follow the case where there is a triplet ofpossible spin states per site. On each site there are a number of valence electrons (eightvalence electrons in the d shell of Ni+2, for example); the degenerate electronic levelsChapter 1. Introduction and Background 4are split in a way determined by Hund’s rules2 and the symmetry of the crystal fieldssurrounding the ion. In some special cases (as with Ni+2 in a field with octahedralsymmetry), a degenerate triplet of states lies lowest. The ensuing low energy physics canbe essentially described using effective spin 1 operators [4]. By ‘half’ filling, we mean thatthere is an effective S = 1 triplet of states for every site in the chain (ie. there are twosingly occupied orbitals on each site. Other orbitals are either empty or doubly occupied.Spins in singly occupied orbitals are aligned by Hund’s rules.) Antiferromagnetism comesfrom allowing a small amplitude for nearest neighbour hopping which is highly suppressedby coulomb repulsion from the electrons already occupying the site.In the quantum case, the spin operators have the commutation relations:[S,S] = j5 fabcc,.= s(s + 1) (1.2)where is the Kronecker Delta Function and is the completely antisymmetric LeviCivita symbol.It is easy to see that the classical Néel ground state with alternating spins is not thequantum ground state. To this end we write the Hamiltonian in terms of raising andlowering spin operators:S Sx±iSYH = j [s:s7+1 + + S1S1)] (1.3)The Néel ground state is composed of spins alternating in quantum numbers SZ betweensites.INéel>=s=+1,s=—1,s=+1,...,4=—1> (1.4)2Hd’s rules maximize the total electronic spin and the total angular momentum of the electrons inthe valence shell.Chapter 1. Introduction and Background 5This state is clearly not an eigenstate of the above Hamiltonian since upon acting onit, the StS1 terms in the Hamiltonian generate states with m = 0. To proceed inunderstanding the low energy properties one usually assumes that the ground state isapproximately Néel with quantum fluctuations. The picture is that of zero point motionabout the positions of the classical Néel spins. What we will shortly see is that theassumption of small fluctuations breaks down in one dimension.The conventional approach makes use of the Holstein-Primakov transformation. Onebegins by dividing the chain into two sublattices, “A” and “B”, with adjacent sites onseparate sublattices. On sublattice “A” one definesS=s—aa, S1=a/2s—aaj (1.5)On sublattice “B” we haveS1 = —s + S = b/2s — bb (1.6)a and b are usual bosonic operators with commutation relations:[a, at] = [b, bt] = 1, [a, a] = [b, b] = 0 (1.7)It can be checked that (1.5) and (1.6) preserve the correct spin commutation relationsand the constraint,= s(s + 1). The Néel ground state is one without bosons.So far no approximations have entered into the picture. However, to make progress, weassume that s is large. This is equivalent to a semi-classical approximation since fors —+ oo the commutator of the spins will have much smaller eigenvalues than the squareof the spin variables[Sa, Sb]=fabc5lc= 0(s) <<0(s2 (1.8)We expand the spin operators to giveS=aV’, Sj1=b/ (1.9)Chapter 1. Introduction and Background 6To leading order, the Hamiltonian reduces toH = J [_s + s(2aa2 + 2bb + ab + b_1a+ ba + ab_1)+ 0(1)] (1.10)Fourier transforming:Na3 = > ei27Nak (1.11)with a similar expression for b; N is the number of sites on each sublattice. Ignoring theconstant term, we rewrite (1.10) asH = 2Js> [aLak + l4bk + (1 +e2tha)(akb_k + b)aL)] (1.12)2a is the sublattice spacing and k = irn/Na for n e [—j, ]. We now make the Bogoliubov transformation,Ck = ukak— Vkb!kdk = Ukb_k — vka (1.13)where,Uk (1 + csc(ka))’12—ika/2Vk= (—1+csc(ka))1 2 (1.14)The d’s and c’s are spin wave operators corresponding to magnetic excitations (magnons)with SZ = ±1 respectively. This transformation preserves the commutation relations andthe Hamiltonian can now be written asH=2Jssin(ka) (cck+c4cik) (1.15)We see that this low energy description implies the spins are in some coherent stateof a’s and b’s built on the Néel state, but there are long wavelength Goldstone modesChapter 1. Introduction and Background 7with dispersion relation = 2JsIka which allow each of the sublattice magnetizationvectors to make long wavelength rotations. To ensure consistency, we now look at theexpectation value of the magnetization (say, on the “A” sublattice), hoping to see that weget the semi-classical result < S >= s — 0(1). We invert the Bogoliubov transformationto getak =ukck+vkdk (1.16)and compute<S>=s—<aa>=s—jZ<aak>= s—af---IvkI = s— af —(csc(ka) —1) (1.17)The last line follows from (1.16) and the fact that the true ground state has zero spinwave occupation number. The problem is now apparent: low wavelength modes cause< s — SZ > to diverge logarithmically. The semi classical picture of a Néel-like groundstate is completely off. This is special to 1 dimension and actually arises as a generalconsequence of Coleman’s theorem [5]; it states that in (1 + 1) dimensions, infrareddivergences associated with Goldstone bosons will always wash out the classical value ofthe order parameter rendering spontaneous symmetry breaking of continuous symmetriesimpossible. One is therefore forced to look elsewhere in order to describe the low energyphysics of the Heisenberg model.1.3 Non-Linear a (NLa) ModelThe most consistent continuum model derivable from Eqn. (1.1) is the Nonlinear a model.In addition to being a continuum model (valid only in the long wave length limit), it isalso based on a large s approximation. One introduces two fields:— corresponding toChapter 1. Introduction and Background 8the Néel order parameter, and r the uniform magnetization. The spin variable, S isdefined in terms of these fields via(x) = (_1)xs(x) + Y(x) (1.18)The conventional derivation defines these variables on the lattice and a continuum limitis taken in the semi-classical large s approximation to arrive at the NLu model Hamiltonian [6]. There are several problems with this approach. First, parity is broken inintermediate steps and is eventually restored in the continuum limit. Second, and moreimportantly, the crucial topological term which is found in the continuum Hamiltonianis derived without clear notions of how 1/s corrections may be made. A much moreelegant approach will be reviewed here. We will make use of path integral formalismbased on independent derivations by Haldane [7] and Fradkin and Stone [8]. These weremotivated by similar questions about topological terms in 2-D quantum models thoughtuseful in attempting to describe high temperature superconductors.One begins by defining a coherent basis of states for the S = s representation ofSU(2) [9]:n > + s> (1.19)where + s > is the eigenstate of 5Z with eigenvalue s, ñ = cos 6 and (xñ) is a unitvector perpendicular to both i and . We see that I > is the state with spin s in theñ direction (ie. i >= sñ >.) This basis is over-complete and not orthogonal. Wenow make use of two identities:<nun2 >= eis 1,2)(1 + fli n2) (1.20)1= 2s± fd3ñ.— 1)In>< (1.21)3We note that the derivation in reference [8] is somewhat in error. We correct their mistakes using asimilar derivation due to Ian Affleck (unpublished).Chapter 1. Introduction and Background 9(n1,n2) is the area enclosed by the geodesic triangle on the sphere with vertices at, n2 and the north pole. There is an ambiguity of 4ir in this definition, but this makeslittle difference when exponentiated since e48 = 1 for s integer or half-integer. The firstidentity is most easily proved by using the , ..., > representation of SU(2) while thesecond follows from the first. We can now use these states to write the partition functionor path integral of the system:= rf [N_i 2s± 1d3()] <(Tm)Ie_T(Tm_i)> (1.22)1=1 m=2A1IT = /3, Tm— Tm_i = T, T1 = TNWith zXT —+ 0, we expand the exponential to order O(z.T) to get<ñ(Tm)Ie_THI1i(Tm_i) >< (Tm)ñ(Tm_i) >— <fl(Tm)IHfl(Tm) > LT= ei (ñ(Tm),(Tml))(1 + (Tm) (Tm_i))8— <(Tm)IHI(Tm) > T (1.23)In the limit N —* cc, the path integral can be written,Tre’ cc f [Dñ(T)] e_S (1.24)S = [<(Tm)IHI(Tm) > T — (1((Tm)(Tm_i) +ln(1 +(Tm) . (Tm_l)))]m=2The last term can be written to second order in LT asSZT ST——i— f d-(i-) Ofl() = —i--. f dT (ôñ(T))2 (1.25)This vanishes in taking the limit tT —÷ 0. The sum over the phases is just the areaenclosed by the vector ñ(T) as it traces its periodic path on the surface of the sphere.Parametrizing ‘i as= (sinOcosq, sinOsinq5, cosO) (1.26)Chapter 1. Introduction and Background 10we can write—is f dA== _isf dq(1—cosO) = —is fdt (1—cosO) (1.27)The Hamiltonian, Eqn. (1.1), is a sum over a chain of spins. We must thereforeextend the path integral to all sites. This is done by indexing each of the coherent stateswith a position label, x, and making the substitution(Tm) >+ ® fi(Tm,3J)> (1.28)Note that<ñ(T,x)I (x)n(r,x) >= sñ(’r,x) (1.29)It is useful to write I(r, x) in terms of a staggered and a uniform part which are slowlyvarying in the limit of large s:i(r,x) (—1)(r,x) + tr,x)/s (1.30)To leading order in 1/s and derivatives of the slowly varying fields, this produces theconstraints(r, x) . (r, x) = 1 (r, x). t(T, x) = 0 (1.31)Setting zx = 1 (the fact that the fields vary slowly over this interval is justified aposteriori), we find that the leading contribution of the Hamiltonian to the action in thecontinuum limit isffdxdr (()24r/2) (1.32)We add up the phase terms by combining them in pairs:—isA = f dx (A ((T, x + 1) + t(r,x + 1)/sj + A [_(r,x) + 1(T, x)/s]) (1.33)Chapter 1. Introduction and Background 11Because A is an oriented area with respect to its argument, changing the sign of theargument also changes the sign of the area. Eqn. (1.33) can be written,-isA =-f f dx (A [(r, x) + ó(T, x)j - A [(r, x)}) (1.34)where to leading order, ö(r, x) = O(r, x) + 2 f/s. This then gives,—isA =—- f dx ((r, x) x S(T, x))-f f dx dr (r, x). (6(T, x) x 8(r, x))= f dx dr ((r x) (ô(T, x) x ôT(T, x)) - 2 ((r, x) x 8(T, x))) (1.35)If we compactify so that —* constant for 1x2 + -r2 —÷ oc, and maintain the constraint= 1 (valid to 1/s2), one can recognize the integralQ=_-fdtdxc.(84x8) (1.36)as measuring the winding number of the sphere onto the sphere. The integrand is theJacobian for the change of variables from compactified coordinates on the plane to thoseon i-space (also a sphere). Q is an integer corresponding to one of the countably manytopologically inequivalent ways there are to smoothly map the sphere onto the sphere;thus the phase term can be written as —2irisQ. For s an integer, a sum over all possibletopological configurations will not affect the path integral. For s half-integer, however,we can expect a drastic difference, since the path integral will be the difference betweenpartition functions with even and odd Q’s. It is important to stress that this is a purelyquantum mechanical result which has no analogue in the 2-D finite temperature classicalHeisenberg model (there is a well known equivalence between (d, 1)-dimensional quantumfield theory and d + 1-dimensional finite temperature classical statistical mechanics [10]).Chapter 1. Introduction and Background 12A detailed discussion of how a half-integer s will affect the physics will be omitted here;the reader is instead referred to [6] and references therein.We can solve the equations of motion for f—. i /—* —*\ (1.37)gvNot surprisingly, ris the generator of rotations. After integrating out the rfields, thefinal action is,S= —2irisQ+ ffdxdr (8)2+ !ff’drdx (Ovrc5)2 (1.38)Where we now define,v = 2Js g = -, (1.39)The action can be writtenS = 2irisQ +- f dx dr 8D’ (140)It is clear how 1/s corrections entered into the calculation of the topological term. Moreover, we did not break parity to derive (1.40).We are interested in integer s (in fact, s = 1). To this end we may ignore thetopological term in the action, as discussed, and consider the nonlinear u-model:Js2 2£ = = 1 (1.41)We now motivate the idea that, contrary to spin-wave theory, this model features a gapin its low energy spectrum. We first do this in the spirit of reference [10]. We can dealwith the constraint by incorporating it into the path integral as a Lagrange multiplier:-. _ij-fd2x (o+A(;_1))Z cc J DebVAe (1.42)Chapter 1. Introduction and Background 13The constraint is now enforced by the equation of motion for ). The fields can beintegrated out in the usual way to givecc (1.43)where N is the number of components of . As N —* cc the path integral is concentratednear the smallest value of the argument of the exponential. Minimizing this argumentwith respect to ). we solve for the saddle point, ):Js2 Nv 1—=—<xI (1.44)2 2using the standard rules for functional differentiation. The RHS of the above is simplythe Green’s function for a boson field with mass v\/;1 1 •1 rd2k 1<xl < - > Ix >= r (1.45)i (2K)kPk+ ,1 A2= —log-=4K Awhere A is an ultraviolet cutoff, d2k = dk dw/v, and kJLk = k2 +2/v. Solving for thesquare of the mass, A:A = A2e_46 (1.46)Another way to see the presence of a mass gap is to integrate out ultraviolet modesand apply the renormalization group. We start with the Lagrangian Eqn. (1.41) andparametrize the fluctuations in terms of slow and fast modes. One then integrates out thefast fields. This calculation is logarithmically infrared divergent. One then renormalizesby subtracting out the offending terms from the effective Lagrangian. Equivalently, onecan achieve the same effect to the same order in perturbation theory by redefining thecoupling constant in terms of its bare value. A calculation of this sort (for the 0(N)Chapter 1. Introduction and Background 14model) is done in reference [10]. The renormalized coupling constant becomesg(L)1— lnL(1.47)With g0 = 2/s, we now see that the coupling constant is of order unity for length scalese6 (1.48)Keeping in mind that this is a Lorentz invariant’ theory, there must be a correspondingmass scale, :oc (1.49)There are other similar heuristic calculations that suggest a mass gap; none are ironclad, but the sum of them together makes for strong evidence that indeed the s = 1 1-DHeisenberg antiferromagnet is disordered at all temperatures and is well described bythe NLo- model. Better justification comes from exact S-matrix results and numericalsimulations. The exact S-matrix results are due to work by Zamolodchikov and Zamolodchikov [11], and Karowski and Weisz [12]. The 0(3) invariance of the NLu model allowsfor an infinite number of conservation laws. These imply strong constraints on S-matrixelements and, consequently, on on-shell Green’s functions. One characteristic of such anS-matrix is factorizability. This means that N-particle scattering can be expressed asproducts of 2-particle scattering matrix elements. The simplest such S-matrix consistentwith the symmetries of the NLu model has a triplet of massive soliton states with aneffective repulsive local interaction. This conjecture has been checked in perturbationtheory in 1/N (for the 0(N) NLu model [12]) to order 1/N2.Numerical results have been pursued since Haldane made his conjecture in 1983 [13,14, 15, 16]. They have all essentially confirmed Haldane’s picture and the validity ofthe NLo- model. To date, the best numerical work has been due to White’s method ofChapter 1. Introduction and Background 15the density matrix renormalization group [14] and recent exact diagonalization [15]. Theformer predicts a gap = .41050(2)J, while the latter has L = .41049(2)J. Numericalinvestigations of the spin operator structure factor [16], S(k), for the isotropic chainshow remarkable agreement with the ‘exact’ S-matrix result for two magnon productionover a region larger than expected; two magnon production is known to dominate at lowmomenta, k < .3ir, from numerical studies [16]. This can be probed in neutron scatteringexperiments [17]. For higher momenta, one must include one magnon contributions whichdominate as k —* K. The intermediate region in momentum space, .3K k .8K, is notexpected to be well represented by the NLu model; this is because the fields, and fdescribe low energy (and therefore large wavelength) excitations about k = K and k = 0,respectively. The same study also determined the correlation length, = 6.03(1), thevelocity, v/J ‘..‘ 2.5 and the coupling constant, g ‘-.‘ 1.28 in rough agreement with the 1/sexpansion result [18] g “.‘ 1.44 and the value derived above, g = 2/s = 2.This ends the introductory discussion of the NLu model. A more in-depth approachwill be taken when we consider anisotropies and develop the necessary tools to calculatethe NMR relaxation rate in Chapter 2.1.4 Boson ModelAlthough the NLu model is convincingly accurate in describing the low energy physicsof the Heisenberg 1-D antiferromagnet, it has several deficiencies. First, off-shell Green’sfunctions are not known; and second, anisotropies are not easily tractable within theframework of the model (the 5- matrix is no longer factorizable, as earlier discussed, sinceone loses the infinite number of conservation laws). A happy compromise which containsall of the qualitative aspects of the NLu model and yet allows for more computabilityChapter 1. Introduction and Background 16and generalization is the Landau-Ginsburg boson model [20]= vj2 + (.)2 ++ (1.50)where the constraint = 1 has been relaxed in the Lagrangian of the NLu model, and ainteraction has been added for stability. The Hamiltonian, (1.50), possesses the correctsymmetries, three massive low energy excitations, and a repulsive weak interaction. Aswith the NLu model, the field acts on the ground state to produce the triplet of massiveexcitations or magnons. We note that this model becomes exact in taking the N -4 colimit of the 0(N) NLa model [10] (recall that N is the number of components of thefield ). As in Eqn. (1.37), the generator of rotational symmetry (the uniform part ofthe spin operator) is(1.51)where 11 (we absorbed the coupling constant,g, into the definition of in (1.50).)Expanding in terms of creation and annihilation operators, we see that 1 acts as a twomagnon operator producing or annihilating a pair, or else flipping the polarization of asingle magnon. This picture is obvious in this simpler model, whereas the same analysis isonly confirmed by the exact S-matrix results and the gratifying agreement with numericalwork in the case of the NLu model. The gap, Li can be phenomenologically fitted toexperiments such as neutron scattering as can be the velocity of light’, v. Includingon-site anisotropyHaniso = (D(sfl2 + E((S)2— (S)2)) (1.52)simply amounts to introducing three phenomenological masses. This will be discussed inmore detail in Chapter 2.Chapter 1. Introduction and Background 17On comparison of the predictions of both models one finds overall qualitative agreement in studies of form factors [16, 191. As one moves away from zero wave vector theagreement between the models weakens. This makes for one of the disadvantages of thebosons. Also, in attempting to calculate certain Green’s functions, such as the staggeredfield correlation function, one is forced to rely on perturbation theory in ). Althoughcan be phenomenologically fitted, there is much ambiguity in choosing the interaction term. One can equally put in by hand any positive polynomial term in Thisis because the fields carry no mass dimension making all polynomial interaction termsrelevant. It should be understood that this model is phenomenological and is introducedfor its simplicity. In the final analysis, justification for its use must come from numericaland real experiments.1.5 Fermion ModelBefore introducing the next model, we would like to begin by apologizing for the crypticdescription of the concepts to be mentioned in this section. A deeper understandingwould require a diversion into conformal field theory tangential to the main lines of thethesis. Instead, the reader is invited to investigate the literature.There is another model exhibiting some of the desirable properties of the boson model.This is an analogue of the Landau-Ginsburg model but phrased in terms of a triplet ofrelativistic fermions:1—. d — 1-.. d —7-1(x) = ,bLzv—— bR+X )- (Rx R) (1.53)Chapter 1. Introduction and Background 18The fields ‘ are Majorana (Hermitean) fermions with equal time anticommutation relations{‘/4(x),’z/,(y)}= 6ss’ö1 6(x—y) S,S’ = L,R (1.54)The L and R label left and right moving fields, respectively. This model is not triviallyrelated to either of the models described above; it was first introduced by Tsvelik [21]to achieve better agreement with experimental data on the anisotropic Haldane Gapmaterial NENP. The motivation comes from a model sitting on the boundary betweenthe Haldane phase and a spontaneously dimerized phase [22], with the HamiltonianH = J> i: i+1 — ( g. g)2] (1.55)This Bethe Ansatz integrable Hamiltonian features a gapless spectrum and has a continuum limit equivalent to a k = 2 Wess-Zumino-Witten (WZW) NLu model. This, inturn, is a conformal field theory [23] equivalent to three decoupled critical Ising models.The well known mapping of the critical Ising model to a massless free Majorana fermion[24] brings us to write (1.55) ast(x)=(’L.-1L_ (1.56)Reducing the biquadratic coupling in (1.55) moves the Ising models away from theircritical point. Symmetry allows the addition of interactions corresponding to mass andfour fermi terms, as in Eqn. (1.53). The four fermi term proportional to ) is theonly marginal one allowed by 0(3) symmetry. It will generally be ignored or treatedperturbatively, in a similar phenomenological spirit to that of the Landau-Ginsburg bosonmodel (ultimate justification for this, as for the boson model, comes from numerical andreal experiments). For weak interactions (which is the case assumed) all Green’s functionswill have simple poles at the phenomenological masses and will be trivial on-shell. TheChapter 1. Introduction and Background 19off-shell behaviour depends on the interaction terms chosen and is therefore very muchmodel dependent.It can easily be checked that (1.54) gives the right commutation relations for theSU(2) algebra, [l(x), 13(y)] = i6(x — y)euidl(x), withr=j(LxL+RxR) (1.57)This allows us to identify rwith the generator of global rotations or the uniform part ofthe spin, . Expanding ‘z].R and 1];’L in terms of creation and annihilation operators, wesee that, here too, ris quadratic in such operators. Notice that this representation forr does not couple left and right movers. This is in sharp contrast to the boson or NLamodels (where one can write the boson operator as a sum of left and right moving parts).We will later see that this point can potentially give experimental predictions which willcontrast between the models.The particles created by the fields, ‘, are identified with massive magnons. Themasses can be fixed by hand to agree with the experimental dispersions so that on-site anisotropy terms coming from Eqn. (1.52) can be easily parametrized, as in theboson model. Other interaction terms which might arise from breaking the symmetryare usualy ignored for ease of calculation. As always, ultimate justification for this isfound in numerical and real experiment.As mentioned above, fis again a two magnon operator. It is also possible to representthe staggered magnetization (the analogue of ) in this approach, but it is considerablymore complicated (one can use bosonization techniques [23]). Near the massless point,this operator reduces to the fundamental field of the WZW model, or equivalently toproducts of the order and disorder fields, b and ji, of the three Ising models [25, 26].These operators are highly non-local with respect to the fermion fields. The correspondingcorrelation functions can be expressed in terms of products of Painlevé functions [24,Chapter 1. Introduction and Background 2027]. They exhibit poles at the fermion masses together with additional structure athigher energy. Unlike the free boson model, a simple interpretation of the staggeredmagnetization density as a single magnon operator doesn’t hold. This complicates theuse of this model.One way to justify the use of the fermion model without resorting to complicatedexplanations is to notice that in the long wave length limit of the 0(3) symmetric case, allmodels are in agreement (see Chapter 2). For smaller wave lengths, the different modelscorrespond to different continuum representations of the lattice model. 0(3) symmetry isbroken differently in each model (for example, see the different results for matrix elementsof fin Chapter 2). The idea is that we have three (two, for lesser symmetry) workabledescriptions whose ultimate merits can only be decided phenomenologically.1.6 Nuclear Magnetic Relaxation RateExperiments on condensed matter systems typically measure observables which are directly related to Green’s functions. This is no surprise since most such experimentsmeasure the response of the system to an external probe. This is in contrast with particle physics experiments which usualy examine the nature of scattering into asymptoticstates. Formally the difference is that particle physicists measure time-ordered Green’sfunctions while their friends in condensed matter physics measure retarded Green’s functions. The nuclear magnetic relaxation rate, 1/T, measures the local correlations of thesystem at low frequency. The probe is the nucleus of some atom in the sample whichhas a non-zero nuclear maglietic moment weakly coupled to the system of interest. Inthe case of the Heisenberg l-D antiferromagnet, we assume that in addition to the spinHamiltonian, H5, there is also Zeeman coupling, H, to a uniform magnetic field, .&,by both the nuclear and Heisenberg spins, and that there is a hyperfine coupling betweenChapter 1. Introduction and Background 21the two systems, HHyper. We also assume for simplicity that the nuclear spins do notdirectly couple to each other.HTot = H8 + Hz + HHyper H0 + HHyper=HSBftGe a-1zNfl.GN i;+ i; (1.58)Ge and GN are the gyromagnetic tensors for the electron and nuclear spins, respectively.A3 is the hyperfine tensor coupling the nuclear spin on site j to the electronic spin onsite i. We now define the characteristic frequencyWNjNH (1.59)In nuclear magnetic resonance (NMR) experiments one strives to temporarily achievea non-equilibrium population difference between nuclear spins with different spin eigenvalues along the uniform field direction. This is normally achieved with pulses of RFelectromagnetic radiation possessing ac magnetic fields perpendicular to the externallyapplied uniform magnetic field. As is well known, a resonance phenomenon occurs atRF-frequencies near WN (in reality it is easier to tune the uniform field to resonate witha fixed RF field).In the presence of a non-equilibrium occupation of states, the nuclear spins “relax”towards an equilibrium configuration by making transitions between states of differentspin eigenvalues. This would not normally be possible if the nuclear spins were completelyfree. Coupling to another system is necessary in order to conserve energy during thetransitions. The energy given off or absorbed must induce a corresponding transition intoa different energy state in the system which couples to the nuclear spins. Let us illustratethe situation with an s = 1/2 nuclear spin. In the absence of hyperfine interactions, weassume that I is a good quantum number (where z is the direction of the static magneticfield), and that GN is isotropic (these are generally good assumptions). The rate equationChapter 1. Introduction and Background 22for the number of nucleii, N, with I + isd Ndt =—N÷Q+_+NL (1.60)Where the transition probabilities per unit time are given by We can rewrite thisin terms of the total number of spins, N and the population difference, n:= N(L÷—— n(÷ + (1.61)Now, if we define(-+ + +-) (1.62)then we see that in the limit that the transition rates only depend on time scales muchshorter than those characteristic of the experimental probe, and in the limit of linearresponse (ie. Q is independent of n) the solution to (1.61) isn(t) = o + a e_t’T1 (1.63)Where n0 is the equilibrium population difference (at finite temperature, states with anenergy difference will necessarily have a population difference). We see that the relaxationrate, 1/T, describes the evolution of the nuclear system towards thermal equilibrium, orlikewise, the decay of the population inversion magnetization achieved by RF pulses inNMR.We now derive an expression for the rate, 1/T. For a system with more generalI, 1/T for a transition from an initial state with P = m to one with IZ = m + 1 isnormalized by the factor 1(1 + 1) — m(m + 1). To begin, we need an expression forthe transition rate m—*m+1 describing a nuclear spin at site j starting in the state withI’ = m and ending up with I = m + 1. It does not matter which nuclear spin we pickif we assume translational invariance; since there is no nuclear spin-spin coupling4,the“In NENP, the dipolar nuclear spin-spin couplings are roughly 200 times smaller than the hyperfinecoupling.Chapter 1. Introduction and Background 23relaxation rate for one is the relaxation rate for the whole system. Let us assume thatthe initial and final Heisenberg spin state are given by the labels n and n’, respectively.Then Fermi’s Golden Rule gives=e_Ef1T2irI <1 = (m + 1), fl’HHyperII. = m, fl> 126(EI (m--1),n! — Eirm,n) z (1.64)Notice that we multiplied the normal expression for the Golden Rule by the Boltzmannprobability that the Heisenberg spin system is in the initial state, I >. The only partof HHyper which will contribute is SrAI:=e_/T2ir(I(I + 1) — m(m + 1)) <n’ISflri> 2ö(E — E — UN) (1.65)i,yThe analogous expression for isIJrr(m+i),n’—*IJ=m,n =-E,/T2ir(I(I + 1) — m(m + 1))IA <nSfln’> 26(E — — UN) (1.66)Since Af = (A_)* we get for the relaxation rate(e_En/T + e_/T)= 2irI >A <n’ISIn>I26(E’ — — UN) (1.67)We now sum over all possible transitions to arrive at 1/Ti:1 f —E,/T + —E/T- =21r>Af <n’IS:1> 26(E —E—wN) e (1.68)1Chapter 1. Introduction and Background 24Since the sum over the states n and m’ is a trace over states in the Heisenberg spin system,we can conveniently restrict ourselves to that system only and write1 0O I ‘1= f dt e’ < > AS’(t), AS(O) > (1.69)1°°m, )where <> denotes a thermal average. This is the famous expression derived in [28]. Aspromised, when A3 is well localized, we see that 1/T is related to the low frequencylocal correlation function.Chapter 2Details of the ModelsIn this chapter we discuss in detail the three models introduced in the last chapter. Wewill derive the necessary tools to calculate the relaxation rate 1/Ti and mention somepertinent issues which can be important in investigating 1-D Heisenberg antiferromagnets(1DHAF’s) using other means.2.1 NLu Model: Temperature and Field Dependence of the Spectrum; ExactResults2.1.1 Temperature Dependence of the GapIn this section and the next we will discuss how the excitation energies of the lowestmodes change with varying parameters. This is especially important when one choosesto perform calculations using the Landau-Ginzburg boson model. Since this model implicitly adopts the gap parameters from the NLu model, any dependence of the gaps onmagnetic field or temperature must first be calculated within the framework of the NLumodel. The results can become useful in interpreting experimental data using the simpleboson model.We would like to begin by extending some recent work by Jolicur and Golinelli [29]on the temperature dependence of the low energy spectrum. It may seem strange oreven contradictory at first sight to speak of a spectrum as being temperature dependent.What one must keep in mind is that the low energy description of the NLu model as three25Chapter 2. Details of the Models 26massive bosons with relativistic dispersion, is an effective one. The true excitations ofthe model are collective, and if we insist on maintaining a single particle description weshould not be surprised that the effective single particle interactions will be temperaturedependent (as, therefore, will be the effective one-particle spectrum). A similar approachis taken in BCS theory where the BCS gap has a temperature dependence arising froma consistency condition.In Chapter One we introduced a consistency equation, Eqn. (1.44), for the classicalor saddle point value of the Lagrange multiplier field, A, in the NLu model. This resultalways holds to lowest order in the fluctuating field A(x), regardless of the value of N inthe large N expansion. Of course, it only becomes exact for N —* cc. We can also lookat the consistency equation as a constraint equation guaranteeing that the two pointfunction is unity when evaluated at the origin; when N = 3,1 =< (x) (x) >= G2(O)= (2 kk+ (2.1)where we’ve assumed a renormalized mean value for A [10]. Notice, also, that we’rechoosing to work in Euclidean space. One can likewise see that the constraint equationis nothing more than a minimization of the zero point energy of the system with respectto the fluctuating field A:Eo=3fwk — AJ (2.2)where = vkILk, + A. If we choose to add on-site anisotropies to the model, asin Eqn. (1.52), then the contribution to the Lagrangian (modulo irrelevant terms whichalso break ‘Lorentz invariance’) is— E((S)2 — (Sfl2) —* D((x))2— E((q(x))2— (Y(x))2)) (2.3)Chapter 2. Details of the Models 27We can always find some axes, xyz, so that the addition to the Lagrangian is in theabove form. We can read off the new constraint equation on the Green’s functionv2 ,rdkdw( 11= I (2 w2 + v2k + 2 +1 1+2+v2k+ 2 ++w2 + v2k + 2—(2.4)Where we’ve once more assumed renormalized masses with the correspondence, v2A +2vD ++ L2 + L and v2A ± 2vE / ± . As usual, making this model temperature dependent consists of replacing the integral over w by a corresponding sum overMatsubara frequencies. Eqn. (2.4) becomes,T2 dkf 11= 2ir + v2k + L2(T) + L1 1+2+ v2k +2(T) + + w +v2k+2(T)—(2.5)In summing over the frequencies we use,1 1 1 cot(’-)w + v2k + m2 = 2 iwk=—_(i+ 2 (2.6)2wk \. e/3’k — 1)where Wk = s/v2k+ m2. We also need the following two integrals1 2 e_m_2/2f dk — (ek— 1) 2f dk k2 += (2.7)rd/c 2J — —log(2Av/m) (2.8)Wk VChapter 2. Details of the Models 28where A is an ultraviolet cutoff and we made the approximation /3m>> 1. Gathering allof the above, we can set Eqn. (2.4) equal to Eqn. (2.5) to arrive at/A2frr\A2 1m\A2fm\ —/T—I+/T —I_/Tj-ok’) e e elog z ) 2v” + + (2.9)where + L and z = 4%/± z. Eqn. (2.9) implies a cubic equation forthe square of the temperature dependent gap, L2 (T). Once more making the approximation, j3 >> 1 we can linearize the equation and solve for L(T):/7f -1/T -/T -t_/T 11 1 1- + +- -)In the 0(3) symmetric case this reduces to the formula derived in [29]. We would like tosay a few things about Eqn. (2.10) before going on to the next section. First, notice thatwe implicitly assumed that only the expectation value of the fluctuating field A acquireda temperature dependence. The renormalized values of the anisotropies, Li and donot. This is because renormalization occurs at T = 0 first. At non-zero temperatures thefree energy may acquire a term linear in the fluctuations of A; the constraint equation,Eq. (2.1), amounts to cancelling that contribution in the Lagrangian. The exponentialterms logically appear as a result of calculating<A >= Tr [e”A] (2.11)and then subtracting < A > from the Lagrangian. We would also like to pointout that the validity condition for this analysis, /31-sm >> 1, where -m is the smallestgap, is more robust than seems. It is well known that for a value of the anisotropy,D J [30, 18], the lower gap closes and the system goes through a critical point, intoa phase with a new singlet ground state (the order parameter is a non-local operatorin spins, and in fact, this transition is not reproduced correctly via the NLu model)and a gap. At large negative values, D — .4J, the system goes through an IsingChapter 2. Details of the Models 29transition into an antiferromagnetically ordered phase. The bottom line is that D mustbe small in comparison with z. Similar, but more obvious, cautionary remarks applyto E. Moreover, as discussed in [29], the NLu model is not expected to remain validat temperatures of order twice the gap. This is because the model does not exhibit amaximum in the heat capacity and in the magnetic susceptibility as shown in numericalstudies at these temperatures.Finally, notice that the difference in gaps will close as T increases. This is no surprisesince at high temperatures the mass scales are irrelevant and we expect a restoration of0(3) symmetry.2.1.2 Field Dependence of the GapWe will be interested in adding a magnetic field term to the Hamiltonian. To do soconsistently, we must couple the magnetic field to the generator of rotations (the totalspin operator), > .. In terms of the continuum fields, we couple the magnetic field, fi,to lvia,—gepB -U . J dx ix), and add this to the NLu Hamiltonian. The correspondingEuclidean space Lagrangian is£ =- (Ia/ot + x + v2(8/8x)2 - 2vD() - 2vE(() - (q5Y)2)) (2.12)—.2751where h = ge/LB ii. In the 0(3) and U(1) symmetric cases, where the field really couplesto a conserved charge, no other terms are allowed in (2.12). In the case of lower symmetry,we retain this as the simplest form, realizing that other, more complicated terms mayarise. This time, when we integrate out the fields, the eigenvalues of the propagatorare not as trivial. However, if we assume that the field is placed along a direction ofChapter 2. Details of the Models 30symmetry, say the z-direction, then the Tr log will be over eigenvalues of the matrixw2 + v2k + z2 — h2 + 2wh 00 (12.13)0 0 w2+v2k2+2+}Where, again, we’ve assumed renormalized values for the masses. The eigenvalues, rj,are773 = w2 + v2k +L2(h) + LX= w2 + v2k — h2 +z2(h) ± iJ_4h2ô2 + $ (2.14)We can write the constraint equation asd2kI 1 1 11 (2)2 w2 + v2k + 2 + + w2 + v2k + 2 + + w2 + v2k + 2 —rd2k/1 1 12 I (2.15).i (2ir) \773 77+ ?7_JThe integrand on the right hand side has poles at the negative solutions of the equationsof motion=(k2+z2(h)+)= (k2 + h2 + L2(h) ± 4h2(k2 +z2(h)) + (2.16)Integrating over these poles gives(_L + — k2 — z2(h) + h2 — — k2 — z2(h) + h2 (2 17)J 2K 2w3 w+(c4_w) w_(w—w)Before continuing, we mention that the T dependence can easily be worked in by multiplying the terms in the integrand above by 1 +e—1’ respectively. It is possible tosimplify Eqn. (2.17) further to readrdk/1 1 1 4h2I—(—+—+—— I (2.18)J 4K w3 w w_Chapter 2. Details of the Models 31where it is now clear that (2.15) is satisfied for h —* 0. This can be shown to be the sameresult obtained by minimizing the zero point energy. To solve for z(h), one must decideon a sufficiently large ultraviolet cutoff and resort to a numerical root finding routine.In Figure 2.1 we plot the energy gaps of Eqn. (2.16) unconstrained by Equ. (2.15). Tocompare, we also plot the field dependent gaps of the fermion model which are generallyconsidered in agreement with experiment [47] (at least for the material NENP). The zerofield gaps are fitted to the gaps found in NENP: za = 1.17meV, Lb = 2.52meV and= 1.34meV. The subscripts, ‘a, b’ and ‘c’ refer to appropriate crystal axes of NENP.The dispersions differ most at higher fields, and for large L. In Figure 2.2 we replot thegaps but this time correct for the constraint implied by Eqn. (2.17). The lower branchesshow good agreement right up to fields close to critical. There is, however, a greaterdiscrepancy in the gap corresponding to the field direction.We now turn our attention to a seeming infrared catastrophe which occurs as happroaches the critical value given by= 2(h)—(2.19)this is where the lower gap closes and the integral (2.18) diverges logarithmically in thethermodynamic limit. At first sight one may hope that for k = 0, the last two termsin (2.18) conspire to eliminate the divergence for some value of h and LSh) satisfying(2.19). This, however, requires that= &(h) ( + 1+ 16)) (2.20)be simultaneously satisfied; this is impossible unless E = 0. In fact, for E = 0, z(h)is independent of h. This can be seen directly from Eqn. (2.18) or by understandingthat the variation of the zero point energies w = ‘/v2k+ zS ± h with respect to L.2is independent of h. Let’s try to get a deeper feeling as to what’s happening. InsteadChapter 2. Details of the Models 32H YbI I3.02.55 2.0 zzzzzzCD>2 1.5CDBoson Modelw1.0::EZZZtE’EM000 2.0 4.0 6.0 8.0 10.0H II c4.0 I • I I I3.53.0Boson Model5 2.5—— Fermion ModelCI)1.5__________________1.00.50.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0H (Tesla)Figure 2.1: Free boson and free fermion dispersions with the gap parameters of NENP.Top graph: flub; bottom graph: fljjc.Chapter 2. Details of the Models 333.02.55 2.0E2’ 1.5ci)w1. 2.5cDE>.. 2.02’ci)Cw1.00.50.0H fibFigure 2.2: Corrected free boson (t = z(H)) and free fermion dispersions with the gapparameters of NENP. Top graph: flub; bottom graph: flIc.HI/cH (Tesla)Chapter 2. Details of the Models 34of starting with a finite E we can place the magnetic field near the U(1) critical value,= z(h), and turn on the anisotropy. We writew(k = 0) = I((h) — h)2—(2.21)We now see that the limits h —+ h and E —+ 0 don’t commute after taking the derivative,Regardless of the limit at which we start, we can expect trouble when(2.22)This is true even for h = 0, which is the simplest case where this problem appears.Essentially the trouble arises because, returning to the NLu action, we integrated outlow energy modes. In the U(1) case the day was saved by symmetry which prevented thevariation of < A > with magnetic field. There is no such luck in the Z2 x Z2 x Z2 scenarioand for a sufficiently large value of E, it is no longer correct to integrate out all the q fieldseven for large N. In the case of large N, one can integrate out all modes but the gaplessone to arrive at a critical theory. This should be done when log (-) N/g. ForN = 3 this criteria may be too restrictive, and instead one can adhere to E << L(h) — has the regime where the analysis of this section is valid.2.1.3 Exact ResultsIn this section we note some important results which will be useful in the calculationsof Chapter 3. As was mentioned in the last chapter, the 0(3) model possesses specialproperties that allow for some integrability. In particular, at long wavelengths one cansay much about the matrix elements of the spin operator even after the 0(3) symmetryis broken to U(1) by a magnetic field.As will become clear in the next chapter, we are mostly concerned with matrix elements < k, a 82 (O)q, b>, where < k, a denotes a single magnon state created by theChapter 2. Details of the Models 35staggered magnetization operator, q (this acts as a free boson operator in the large Nlimit) with norm<k, alq, b >= 27röabS(k — q) (2.23)a and b denote polarizations of the magnon states. Clearly, < k + q, aI(O)Ik, b >= 0.This is obvious when the bosons are completely free; interactions do not change thispicture since they must all be even in bosonic operators. The two magnon operator,1i(o), is expected to contribute.1 The matrix element is given by the Karowski and Weiszansatz [12],<k,a1(0)q,b >= ie+G(&) (2.24)where cJJk = /k2v+ z2 and the rapidity variable, 0, is defined viasinh(0’— iir) = ksinh(8”— iir) = q0 = 6’ — 0” (2.25)withG(6) = exp (2 [ dx (e_2x — 1)sin2[x( ir — 0)/2ir] (2.26)Jo x (ex+1)sinh(x) )This ansatz is believed to be exact for the 0(3) NLu model, but is only approximatelytrue for the s = 1 Heisenberg model; however, numerical simulations [16] are in excellentagreement with this form at least half way through the Brillouin zone. Since we willlargely be interested in 1k — qv < L <<irv, this ‘exact’ expression is more than sufficient.speaking, since ris qnadratic in , we expect it to be a two magnon operator only in thelarge N limit.Chapter 2. Details of the Models 36Some particular limits of interest are k q and k —q, corresponding to forwardand back scattering, respectively; in the former case, 8 ilr while in the latter, 0i-ir + 2kv/.G(iir + 2kv/Z) 1- (- + 2 (2.27)This expression is a different result than for free bosons and reflects the effects of interactions. We will later see how this might affect experimental predictions.The above results change when a D type anisotropy is added. Essentially, the difference is that the function, G(0) changes and the gap in the energy of states on theSZ= 0 branch will be shifted. There are no exact results for this model which is why thephenomenological boson and fermion models are important. We can, however, give someuniversal (ie. model independent) results which only depend on the conservation of totalspin in the z-direction, when the momentum exchange is small, v(q— kI << Li. The onlypart of Sz(0) which will contribute will be, essentially, S_k f dx S’(x). Since this isa conserved operator, we can write down the one-particle matrix elements immediately:<k,sZ=±1Sz(O)q,sZ=+1> ±1 (2.28)Also true for all values of vlq — are the following< >= 0< k,sZ=1sZ()q,sZ=_1 >= 0 (2.29)In Chapter 3 we will show that it is largely this universal behaviour which determinesthe relaxation rate, 1/T in anisotropic media.Note that adding a magnetic field to the 0(3) or U(1) system will break the symmetryin the 0(3) case (we naturally assume that in the U(1) case, the field is placed parallel tothe U(1) axis), but will hardly change any other results in either model. This is becauseChapter 2. Details of the Models 37in adding a magnetic field all we have done is add a term to the Hamiltonian proportionalto the conserved charge l= f dx 1’ (x). Since l commutes with the Hamiltonian, theycan be simultaneously diagonalized with all low lying states labeled by their l quantumnumber, +1, 0 or —1. Matrix elements of operators can only differ, at most, by somecumulative phase which corresponds to turning on the field sometime in the past.2.2 The Free Boson and Fermion ModelsWe now turn our attention to the phenomenological models introduced in Chapter 1.After introducing the formalism which we will require to perform calculations, we willcompare and contrast the energy spectra and fundamental matrix elements.2.2.1 DiagonalizationTo do the necessary calculations we need to have a basis of eigenstates for the non-interacting Hamiltonian and know the expansion of the field operators in terms of creation/annihilation operators for these states.Bosons—.2Relaxing the constraint çb = 1 in(2.12) we see that we seek to diagonalize2 1 )] (2.30)For now we assume that the mass and gyromagnetic tensors, D and G respectively,are simultaneously diagonalizable and work in this diagonal basis (this is rigorously truewhen the crystal field symmetry is no lower than orthorhombic— a sketch of a proof isChapter 2. Details of the Models 38found on p. 750 of [41)Li 0 0 9i 0 00 zi 0 G= 0 92 0 (2.31)0 0 L 0 0Also, we mention that we’ve set fl = 1 = a, where a is the lattice spacing. This has theeffect of measuring energy in units of inverse seconds or inverse mass:[E] “..‘ [s]’ [M]’ ‘-.‘ [v] []2 (2.32)Diagonalizing (2.30) is tedious (especially when the field does not lie in a direction ofsymmetry, for then all the branches mix) but the idea is to find the right Bogoliubovtransformation. Working in momentum space, we define-. 14(k,t=0)= ak] (2.33)fl(k,t = 0) = i/[—k — ak] (2.34)[ai, aV] = 27r66(k — k’) (2.35)w0 is an arbitrary quantity with the dimensions of energy. We need such a quantity torepresent the fields and ]i in terms of creation and annihilation operators. It turnsout that when one writes k and ilk in terms of the creation/annihilation operatorswhich diagonalize the Hamiltonian, the dependence on w0 disappears. Furthermore, theeigenvalues of the Hamiltonian are also independent of w0, as might be expected. Wewill now restrict ourselves to the case where the field lies in a direction of symmetry.This leaves (2.30) with Z2 x Z2 symmetry. Now only the excitations transverse to theChapter 2. Details of the Models 39direction of the applied field mix and we need only solve a (4 x 4) set of equations forthe diagonalizing creation/annihilation operators. Without loss of generality, we take thefield to lie in the 3-direction, and set g3 = 1. All told, the Hamiltonian, Eqn. (2.30), forthe mixed states isp00H=J Hkdk-00= Ak + akA a_k + + akB a_k (2.36)A=I+—ho2 (2.37)4 4w0 2B=-I+-- (2.38)4 4w0where o2 is the usual Pauli matrix, and(zS+v2k2 0K=I I (2.39)0 i+v2k))The momentum space Hamiltonian can be written in terms of a single matrix M:Hk= (, a_k)M ( 0)_k) (2.40)As discussed in [31], we seek the eigenvectors of the (non-hermitian) matrix jM, where(A B\ (io\iM= I = I I (2.41)\\_B* _A*) \0 —1)This comes from requiring the new diagonal creation/annihilation operators to have thestandard commutation relations. This also imposes the unusual normalization conditionon the eigenvectors: = 1.Chapter 2. Details of the Models 40Summarizing the above, we need to solve:0 = (+ — — 2 4 + 4(4)0 (2.42)—I—--—1ho —I\ 4 4w,j 4 4w 2 2 2which can be manipulated to give0((wo — w)I — hcT2 —(wo + w)I — h2 (KIho.2 +ôI+ho2 )The eigenvalues of r1M are already known, as they are the solutions to the classicalequations of motion and come in pairs ±w. These are naturally the same frequenciesgiven in Eqn.(2.16) with the proper substitutions made for the gapsw=k2+LS= k2 + h2 + ± 4h2(k2 + 2) + ( ; 2)2 (2.44)IxFurthermore, we need only work to find one eigenvector of each pair because if(y*is a right-eigenvector of iiM with eigenvalue w, then J is a right-eigenvector ofx*)M with eigenvalue —w (X and Y are themselves two-component vectors).The bottom rows of (2.43) give the following set of equations0=(z +v2k)±,i — wow±&,1+ ihw0,2 (2.45)0=(L +v2k),—w0,2— ihw0,1 (2.46)Chapter 2. Details of the Models 41where it turns out to be convenient to work with x X + Y and X — Y. The toprows can be manipulated to giveo = (h2—— ihwo&,i + wow±&,2 (2.47)o = —(h2—— ihw0,2— wow±&,1 (2.48)These can then be worked to give(z + v2k + w—h2)±,i = 2cL.0w,1 (2.49)(z2 + v2k + w—h2)±, = 2w0w, (2.50)(h2 + L + v2k—= —2ih0,2 (2.51)(h2 + zi + v2k— w)Xth,2 = 2ihw0,1 (2.52)Note that if we fix the phase of Xi to be real then must also be real as X2 and 2 mustbe pure imaginary. The normalization condition, XtX — YY = 1, now allows us to solvefor the eigenvectors which form the columns of the transformation matrix between theold and diagonal bases of creation/annihilation operators. In terms of x and this isX±,i&,i — X±,2’±,2 = 1 (2.53)The solution is— ( wow±(h2+ + v2k — w) 1/2X±,i—(? + v2k + w — h2)(h + + v2k — wi)) (2.54)?+v2k+w — h2— (2.55)2wow±Chapter 2. Details of the Models 422zhw0X±,2= h2 + Ls + v2k —(2.56)A2 j 2i2 i 2 j..22-” -rW±1&c±,2 = X±,22wow±With the inclusion of the trivially diagonal unmixed component (ie. the three component), we can define the three by three matrices x and with the columns labeledby the eigenvalues (+, —, 3) and the rows labeled by the original masses (1, 2, 3): X332kV!+v2k2and= V 3v . One easily verifies thatak= [xk+k k+(X-) bk] (2.58)Where the b’s are the operators which diagonalize H. Equations (2.54)—(2.58) are themain results of this section. Before ending, we give some limiting forms for x and . Inthe limit h —* 0,/ Lp 0 0V /+vkx == :________(2.59)I+v2k2While in the limit L2 —* ,/wp / wpV 2.f+vk V 2v’-fvk——1f— —i / wp / Wp 0X — V 21&+vk V 2./+vk0 0 WpV \/+v2k2FermionsWe would now like to repeat the diagonalization procedure for the fermion model. Thefree Hamiltonian with minimal coupling to the magnetic field (ie. coupling only toChapter 2. Details of the Models 43and simplest parametrization of the mass terms (corresponding to anisotropies and givingzero field dispersion branches = + k2v) is= VãxL —3i— L,iR,i) + i ( X L + R X R)] (2.61)with, setting v = 1,°°dk= f _ + et_,k) (2.62)I)L= f°° (e_itãL,k + et4,k) (2.63){a, aV} = 2ir6(k — k’) (2.64)Notice that we coupled the magnetic field to the generator of global rotations,f dx ix), given by (1.57). the Hamiltonian density in k-space becomesHk = c4Mk k (2.65)whereIIk—iix iiM=I I (2.66)—i, —Ik—iix )-* ( aR,k”\= i i (2.67)\ aLkThe idea now is to diagonalize this matrix and find the eigenvalues and eigenvectors.In other words, find the unitary transformation which diagonalizes H. Once more, weChapter 2. Details of the Models 44assume the field is in a direction of symmetry so that we need only diagonalize a 4 x 4matrix. Given that the field is in the 3 direction, the eigenvalues are:w=k2+= k2 + h2 + ± 4h2(k2 + (1 ±2) + ( ; 2)2 (2.68)It may be more illuminating to write out M in a basis that is more natural to theU(1) problem. Using(ak ‘— 1 (i 1 ‘ (4,kaJ,J ) 2 —i ) \ a/ it\ /1 \/ +t\aL,k— 1 1. aL,k (2 70)a)-i i)a)In this basis, M becomes( kI — ho3 i1o + iSIM= I (2.71)—iLo1— iSI —kI + her3 )Where L = 12 and ó = 12• The equations for the components of the eigenvectorspossess the symmetriesu1+-u2,u3+4h+->—h (2.72)(2.73)Chapter 2. Details of the Models 45where w is the eigenvalue. After some algebra,2iL(k — w)= w2+2_(k+h)2_62U1 (2.74)2i6(k—w)U3 (w_h)2_k2+2_2U1 (2.75)2iL(k+w)U2= w2 + 2 — (k + h)2 — 62u3 (2.76)Using the normalization condition,Iu = 1 (2.77)we set the phase of u1 to be real for positive eigenvalues; the above symmetries allow us thefreedom to choose a convenient phase for the ui’s corresponding to negative eigenvalues.U’ = 26(k+w)(w+LS—(k+h)6)± (2.78)[482(k + w)2((w + — (k + h)2 — 62)2 + 4/2(k — w)2) +((w+h)—kSi(z(k+h)2_62)2+4/X2(k+w)2)]We define the 6 x 6 diagonalizing matrix with columns given by the eigenvectors i4as= (L ui,, u,3, uL+, uL,u3) (2.79)(2.80)Chapter 2. Details of the Models 4611 0 000i —i 0 0 0 01 0 0 0 0 0 (v 0U=—1 (2.81)V2 o 0 1 1 0 o vu1)0 0 0 —i i 00 0 0 0 0 v’The diagonal operators, 13k are defined as:-.ICk’Ik I I (2.82)\ dk)Our freedom in choosing the phases for the eigenvectors corresponding to negativeeigenvalues allows us to writeIR T”\X= (2.83)T R)Each index of this matrix runs over six states; the first and last three correspondto right and left movers respectively. In the case of U(1) symmetry or higher, each setwould correspond to states of definite spin.The d’s and c’s correspond to left and right moving fermions, respectively. Thisbecomes clear in the limit i —f z2 —* 0. Some limiting forms of R and T are:$ /0R(h-0)= /EE \/EE 0 (2.84)0 0V W3Chapter 2. Details of the Models 47T(h— 0) = _/k o (2.85)o o _i\/2zo oo o (2.86)o0 0T(ö_÷0)= o o (2.87)o o2.2.2 Discussion: Comparison of Spectra and Spin Operator Matrix ElementsThe spectra for the boson and fermion models are given by Eqns. (2.44) and (2.68),respectively. In the case of U(1) symmetry, L= z, the two sets of formulae agree.However, with the lower orthorhombic symmetry, the two models are in agreement onlyfor low magnetic field, h << Min(z1,/.2). The difference is most significant at the criticalfield where the lower gap vanishes. The boson model predicts h = Min(L1,L2), whilethe fermion model gives h = “LiL2 (see Fig. 2.1 and 2.2). Experimental evidenceseems to favour the fermion model, but there are some subtleties which have previouslybeen ignored. The data supporting the fermion dispersion comes from neutron scatteringand NMR relaxation rate experiments performed on the anisotropic 1DHAF material,NENP. In analyzing the data, however, crucial structural properties were neglected inthe interpretation (namely, the fact that the local chain axes did not coincide with theChapter 2. Details of the Models 48crystallographic axes). This, we believe, also led to a seeming contradiction with ESRdata on the same substance which seemed to side with the boson dispersion [47].2 Asidefrom material properties, the possible temperature and field dependence of the massparameters, zj, has also been ignored so far. Since the boson model derives from theNLu model, one should incorporate such field and temperature dependence into thesebasic parameters. We showed that considering field dependent masses brought closeragreement on the lower branch dispersion between the models up to fields given by Eqn.(2.22).We mention in passing that the Hamiltonian, Eqn. (2.30), is not the oniy quadraticone possible when the magnetization density is no longer conserved. It is possible toconstruct a modified boson Hamiltonian including extra terms designed to reproducegaps identical with the fermion model [32]. The only constraint on such terms is thatthey do not mix the sz = 0 modes corresponding to the degree of freedom parallel to thefield. It is not obvious, however, what justifies such a modification other than a moreconvenient spectrum which replicates the fermion model at low energies.The fermion model is expected to become more accurate close to the critical field.The nature of the critical point was established in Ref. [33]. With U(1) symmetry, thephase transition is in the two dimensional zy universality class. The lowest lying modeof the Landau-Ginsburg boson model can be reduced to a single free boson (a phase fieldcorresponding to the Goldstone mode), but the parameters of the resulting low energyLagrangian must be renormalized to give the correct critical exponents of the xy-model.One does not have to resort to such lengths with the fermion model which correctlydescribes the transition without interactions. This is expected on several grounds. First,the many body ground state wave-function for a dilute gas of repulsive bosons is simplythat of free fermions multiplied by a sign function to correct for the statistics. Second,2For more details on these matters, please see sections 5.1 and 6.3.Chapter 2. Details of the Models 49the U(1) fermionic modes can be represented as particles and holes using a single Diracfermion with chemical potential h (this can be seen in the matrix equation (2.71));this means that at h > z the ground state will be occupied by fermion states, eachwith SZ 1, and hence non-zero magnetization. The simplicity of the coupling to hguarantees that interactions will be as important near h as they are near h = 0. Inparticular, they will be negligible in the dilute gas limit. We thus see that interactionsbecome progressively more important close to criticality in a boson theory, while theopposite takes place in an equivalent fermion theory.In the Z2 x Z2 case we expect an Ising-like transition corresponding to the breaking ofone of the Z2 symmetries remaining. Here things are even clearer. Mean field theory forthe boson model is completely hopeless as is evidenced by the unphysical behaviour of thelowest lying gap at h> Min(z1,z2). This function always possesses a zero even at non-vanishing k-vectors. Moreover, it is imaginary for fields iJ2v2 + Min(L, /.) < h <1Jk2v2 + Max(t?, z). The spectrum for the low lying fermion, in contrast, shows all thedesirable properties, vanishing at h = /z1L2 only for k = 0; in addition, the effectivegap, Li— hj, is as expected in the Majorana fermion representation of the criticalIsing model and so is the relativistic dispersion for long wavelengths. Finally, when weintegrate out the more massive fermionic modes we are left with a strictly non-interactingfree Majorana fermion theory regardless of any zero-field interactions in (2.61); this isbecause all interactions will be polynomial in the one Majorana field left, and will vanishby fermi statistics. Thus we see that in contrast with the boson description, the freefermion theory is actually best near h = h.To summarize, on general grounds, one can expect qualitative agreement betweenboth models up to magnetic fields close to h where the fermion model is expected to bea better description of the system.We now wish to look at some important matrix elements as phrased in the two models.Chapter 2. Details of the Models 50We start by defining,la,b(k, q) < a, k i0)q, b> (2.88)we use x ii for bosons to writel,b(k, q) = — (t(k) x(q) + x(k) (q)) (2.89)where we define the cross product matrix with the Levi-Civita symbol by E =Using, r= (L X + bR x ‘j)R) for fermions, we write the analogous expressionla,b(k,q) =( RV VR + TtaVt Va1T RV V*T* + Tta1Vf V*aiR*—ii — —_, I (2.90)‘ T’V Y2VR + RToiVT EVa1T TTVT DV*T* + RTa1VT V*uiR*)x , V, a1, R and T were all defined in the sections on diagonalizing the models. As canbe explicitly checked, the 0(3) free bosons are analogous to the NLa model with thefunction, G(O), defined in Eqn. (2.24), set to one. This is the general result for the 0(N)model for large N, and makes sense, since the Landau- Ginsburg model is a large Napproximation to the NLa model. In case of axial symmetry, one need only substitutethe correct gaps into the energy factors:b<k,a1i(0)q,b>=jfk‘ (2.91)2/with w = /k2v2 + .The 0(3) fermion model exhibits a non-trivial G(O)-function. We can use the resultsfrom Eqns. (2.86) and (2.87) in Eqn. (2.90) to calculate thatG(O) = —sech(O/2) = [V(wk — k)(wq — q) + k)(wq + q)]Wk ± Wq(2.92)Chapter 2. Details of the Models 51To obtain the U(1) results we, again, make the gap substitutions as done in Eqn. (2.91).This result is quite different than the boson prediction. It, in fact, vanishes with the gapfor backscattering, k ‘ —q. This is because the fdoes not couple left and right movingfermions while the opposite holds true with the bosons (and NLu model). For smallmomentum exchange, all the models give universal predictions for matrix elements of8Z (0). However, matrix elements of S± (0) at small momentum exchange are somewhatsensitive to the ratio of the gaps, in the boson model, while not at all so in the fermionmodel. In Chapter 6 we discuss experiments which might investigate this behaviourfurther.When the symmetry is orthorhombic there are few conservation laws to restrict theform of matrix elements of spin operators. Furthermore, when a magnetic field is added,Lorentz invariance is explicitly broken. We can, however, say that correlations amongspin operators are still diagonal: This is true by virtue of the Z2 x Z2 symmetry. We alsoknow that the new energy eigenstates, labeled by + and —, are mixtures of eigenstatesof S3 with eigenvalues SZ = ±1. This guarantees that<k, — >oc: 6j3 (2.93)It isn’t terribly illuminating to write down the actual matrix elements. We can say, however, that in the boson model, for h —+ h, all matrix elements of form, <—, kIlz(0)Ib, q >,which are not zero by arguments given above, diverge at k = 0 as fractional powers of(h — h). Everything is nice and finite with the fermions. This is another symptom ofthe sickness of the free boson model near criticality. Again we see that interactions areexpected to play a crucial role in the boson description.We finish by describing some matrix elements near zero magnetic field. We expectthat the intrabranch matrix elements, < ±,kS3(O)I±, q >, vanish at k = q with thefield. Fork=q=Oandh-+O,Chapter 2. Details of the Models 52h 2 32<—,OI1(O)I—,O >= (Ls?-:2) bosons2h= fermions (2.94)i2where we’ve assumed, L > Li2. The result for < +, OIl(O)I+, 0 > is obtained byexchanging 2 and 1. Notice that this limit does not commute with the U(1) limit,—÷ /.. This is to be expected since these matrix elements are constant in the axiallysymmetric case.Chapter 3Model Predictions for T1’Let’s recall the expression for 1/T, Eqn. (1.69), derived in Chapter 1:= f dte_rt < {S’(t),s(o)}> (3.1)where h is taken to be in the direction. As discussed in (2.2.2), only diagonal components of the spin correlation function will contribute. We also assume that the hyperfinetensor, A23, is localA2, = Aö, (3.2)Thus we can write= A2f dt e_t < {V( = 0, t), SL(o)}> (33)where we’ve used translational invariance to evaluate the correlation function at theorigin. We can now take a step back to Eqn. (1.68) and write the above as1 ( —E,,/T + —E/T= 2 IAI2 <n!ISV(0)In>I26(E’ — E — WN) e e (3.4)1 fl,Th’,L’We will concern ourselves largely with the limit, WN <<T << min (note that WN lmKfor H ‘-.‘ 1.5 T), so that the last factor in (3.4) can be set to 2e_EfIT.Consider now the operator in question, (O) = (0) + (0) x ti(0). We wish toinvestigate whether dominant contributions to (3.4) come from the staggered field, (0),or the uniform part of the spin, (0) x fl(O). Let us first use the boson model to analyze53Chapter 3. Model Predictions for T’ 54Figure 3.1: First non-vanishing contribution to relaxation due to the staggered part ofthe spinthe staggered contributions. This will be1cc <n’I(O)In>I26(E’ — — wN)eEfhT (3.5)Tistagg n,niWe assume, that we are in the regime T << min(”’), where 1min(h) is the lowest(possibly field dependent) gap, or in other words, that the magnetic field is well belowh, and we are therefore well justified in using the boson model (or NLu model) to describethe situation. Since (O) is a single magnon operator in the noninteracting theory, itonly has matrix elements between states whose energies differ by a single magnon energy,w(k, h); since qS is evaluated at the origin, k can be arbitrary. In particular, there areno matrix elements with energy difference, WN (which is essentially zero, compared tothe other energy scales around). Including interactions, there will be contributions atfinite T. The simplest process is shown in Fig. 3.1. It involves aq4-type interaction, asmight occur in the Landau-Ginsburg or NLa model. The vertical line represents the field1There will not be contributions from cross terms between the staggered and uniform fields. Thesevanish because for< njIm > 0, one needs the number of magnons, n + m, to be odd, while this inturn implies <n urn >= 0. Certain types of strnctural perturbations, such as discussed in Chapter 4,may change this analysis at high temperature and/or high fields.aE= 2Chapter 3. Model Predictions for T’ 55çb. The incoming line from the right represents a thermally excited magnon of non-zeromomentum, k, and energy 2z (for simplicity, we use the isotropic model at zero field tomake this argument; extending this to the anisotropic models is straight forward). Thetwo outgoing lines to the left represent magnons at rest (recall that the wave vector, k,is actually shifted by ir, and so the lowest energy antiferromagnetic spin excitations varyspatially as e). This diagram gives a non-zero matrix element proportional to )/&(where ) parametrizes the ‘ interaction). Note, however, that since the initial and finalstate energies must be at least 2z, there will be a Boltzmann suppression factor ofe_2/Tto this contribution. Thusoc A2e_T (3.6)T1 StaggIncluding anisotropy and a finite field will give various contributions of this type. Thegreatest will be suppressed by exp(—2min(h)/T). It is also consistent to interpret thisresult as giving the single magnon a finite width at T 0. This, however, cannot changethe conclusion that there is a double exponential suppression factor contrary to the modelproposed by Fujiwara et. al. [34].Let us now consider the contributions to 1/T from the uniform part of the spin:oc <n’l(0)In> 26(E — E — N)eT (37)TiUnifAs discussed in Chapter 2, the 1-particle matrix elements selected above are non-zeroin general, even in the non-interacting boson or fermion model. This is because ib(o)is a two magnon operator, able to create one magnon and annihilate another. In thepresence of anisotropy and magnetic field, the three magnon branches are split, so wemust distinguish between interbranch and intrabranch transitions (see Fig. 3.2). Thisis possible since contributions may come from all wave vectors. One set of importantprocesses (ie. the ones corresponding to transitions between the lowest energy magnonChapter 3. Model Predictions for T’Magnon Dispersions For Lowest Two Branches56EFigure 3.2: Inter- vs Intrabranch transitionsChapter 3. Model Predictions for T’ 57states) will come from single particle intrabranch transitions along the lowest mixedbranch (intrabranch transitions are not allowed along the branch corresponding to sZ = 0since h is parallel to z). This will be one of the leading effects with Boltzmann suppressionof e- (h)/T, where i (h) is the lowest field dependent gap. It is important to realizethat these will only be present if the hyperfine coupling, A+z, is non-zero. In fact the zerofield gap structure and the choice of direction for placing the magnetic field will affectwhether there are competing transitions. The next contribution, possibly as significantas the one just described, can come from intrabranch transitions along the second lowestbranch and/or interbranch transitions between the lowest mixed branch and one of theother two branches. The Boltzmann suppression factor will, again, favour the lowestenergy processes which occur at the gap to the highest branch involved in the transition.The important point is that the Boltzmann suppression factor for any of these processesis larger than that associated with contributions from the staggered field. To summarize,the dominant contribution to 1/T at T << min’ will come from Eqn. (3.7).As we approach the critical field, the above analysis breaks down. As discussedpreviously, interactions are expected to become large in the boson model. Moreover, asthe gap closes, the Boltzmann factors will fail to discriminate between the contributionsof the uniform and staggered fields to 1/T. Arguments involving the fermion modelare tricky because the staggered field has no simple representation in terms of fermionicoperators. However, we explicitly show later in this chapter that the staggered componentwill dominate sufficiently close to h.Above the critical field, the analysis depends on the symmetry. For U(1) or highersymmetry, the system remains critical and the staggered correlator remains dominant.For lower symmetry, the gap opens up once more, and sufficiently far above the criticalfield, we expect the uniform correlator to dominate once more.Chapter 3. Model Predictions for T’ 583.1 T1 for h <<heIn this section we concern ourselves with the regime discussed above, WN << T <<‘min(1) We calculate the relaxation rate in the isotropic, U(1) and Z2 scenarios.We begin by deriving a general result valid in this regime, and proceed to discuss itsapplication in the different cases of symmetry.Consider a contribution to 1/T coming from transitions between branches r and s.Without loss of generality, we assume that r has a higher or equal gap to s (r, in fact,could be the same branch as s). We call the corresponding contribution to Eqn. (3.4),*rsThis will be a sum over single particle states on r and s:rs= 4K AI2f 6(ws(q) — r(k) — wN)e )/T, <k, r1i(O)q, s> 2 (3.8)Note that there will be a similar contribution with the labels s and r exchanged, if sand r are different branches. This corresponds to scattering an initial particle on thes branch through the hyperfine interaction to a final magnon on the r branch, or thereverse process. We take account of both of these possibilities later. Also, we are keepingWN finite to cut off infrared divergences which crop up in the intrabranch processes. Wenow do the integral over q to get-— 4IAI2 [°°dk(Wr(k)+WN)Tirs j JO 2K Q(k)eT (Il,6k,Q(k))12+ Il8(k, _Q(k))12) () (39)q q=Q(k)where Q(k) is defined by w8(Q(k)) = r(k) + WN, and l is as defined in (2.88). WhenLr(O) >> T, the above integral will be strongly peaked at k = 0. Moreover, we canneglect WN in wr(k) +WN. The only factors in the integrand for which we should retain ak-dependence are the exponential and the possibly infrared divergent denominator, Q(k).Chapter 3. Model Predictions for T’ 59We therefore write-11=2IAw(0) Q(o))12+ I1(0, -Q(o))2)P00 e(k)/I dk Q(k) (3.10)JoIt is not too difficult to show that to first order in small quantities, k2 and N, one has(8w’’ Iaw’\ (3.11)Q2(k) Q2(0) +l )q=Q(O) )q=oWe can also expand the exponent:w(0)/T+ (3.12)r(k)/T( ( k2wr(0)T )The relevant integral over k then becomes() k2g=oe 2wr (O)T/ dk (3.13)i”’ “-‘ kJO Q2 (0) + oq2 ) q=Q(O) 8q2 } q=O(2)k2By changing variables, 2r( —+ k, we can write (3.13) as—k2a’ 00f6w\ (0w\ J dk (3.14)Jq=o )q=o ° yk2+ €rs(T, h)wherec8(T,h)— Q2(o) ()qQ(O) (3.15)— 2wr(0)TThe h-dependence of c will largely come from its dependence on r(0). The integralcan be expressed in terms of special functions:lOw 2eT,2Ko (rs(T, h)/2) (3.16)/ q=Q(O)Chapter 3. Model Predictions for T’ 60where K0 is the zero order modified Bessel Function. When the gap, Wr(0), is very largecompared to the typical momentum, Q(0), exchanged in the transitions, (this is the casefor intrabranch transitions), crs —* 0. In this limite8(T,h’’2Ko (ors(T, h)/2) —+ — log (crs(T, h)/4)—(3.17)where ‘y = 0.577216... is Euler’s constant. We can now summarizeI = 2IA)r(0) (il,(o, Q(°))12+ 1(0, -Q(°))12)eT(0)1TTirs j ir(O4) 2 (4) 2 e(T,Ko (ors(T, h)/2) (3.18)q Q(0) q 0The full expression for the relaxation rate is(3.19)The effect of interchanging s and r in (3.8) is therefore included in the above.An important thing to learn from the above calculation is that contributions fromtransitions between states involving small momentum exchange (Q —+ 0) will dominatedue to the logarithmic divergence in (3.18). This is particularly the case with intrabranchversus interbranch transitions. In intrabranch transitions one is allowed momentum exchanges as small as Q ‘s.’ 2Lr(0)WN/V. This will typically be much smaller than thesmallest allowed interbranch momentum exchange, Q ‘‘ (Lr(0) —8(0))/v. The conclusion is that, unless the branches in question are extremely close to each other, interbranchtransitions will play a secondary role to intrabranch processes, even ignoring the moreobvious suppression due to different Boltzmann factors. Of course, if the hyperfine interaction has high symmetry, one will not see intrabranch transitions at all. This suggeststhat an NMR relaxation study could provide information as to the nature of the hyperfinetensor.Chapter 3. Model Predictions for Tj1 61Before expounding on this result in the individual cases of different symmetry, wewould like to mention the effects of higher temperature, or correspondingly, includingthe k-dependence of the various terms approximated at k = 0. In the more generalZ2 x Z2 situation,we are strictly justified in expanding the exponent in Eqn. (3.12) onlyfor T << (h). For higher temperatures, one expects contributions from k > Lr (h)/v,where the expansion is not convergent; in this case one is better off numerically integrating(3.10) (making the k = 0 approximation for the other terms is still valid, as we will see).In either case, we can estimate the error in neglecting terms of order k2. First, noticethat all gaps are always greater than v2k2 + L — h, where is the smallest zerofield gap. We therefore write—c,.(h,k)/Tf dkk_e f dkk21e_V22_1m/TJk2 + Q2(k)=-— i: dwe_Tw(w2— 2)Th_1 < e_(m_h) (2Max(T 1m))2fl (3.20)The last estimate is actually quite generous, especially for large n. In the worse casescenario of the Haldane phase, /v ‘ 1/4. This allows us to expect an error of at most10% in neglecting the k-dependence of the terms in (3.10).We still have to estimate the error incurred in making the expansion in the exponentialat T << The next term in the expansion is with v= ()qo• This willgive a contribution,‘ kv e_k2/2(0)T Te_(O)/Te_/’T I dk r_______(3 21i 8zT /k2+Q 4LrThis is potentially more serious as T —÷ /.x,. or —* 0. To summarize, Eqn. (3.10)is generally a very good approximation; when T << r, one can safely expand theexponential, while for T , one is better off numerically integrating (3.10).2One is more fortunate in the U(1) case; because of the simplicity of the gaps, the expansion is goodfor k &L, regardless of the value of h.Chapter 3. Model Predictions for T’ 62AIntrabrancil TransitionsInterbranch TransitionsFigure 3.3: The gap structure for 0(3) symmetry.3.1.1 Isotropic SymmetryIn the case of 0(3) symmetry, the field dependent gap structure is as in Fig. 3.3. Thelowest and highest branches correspond to magnons with sZ = ±1 respectively. Themiddle branch corresponds to Z = 0. The interbranch gap is h. Using the result ofthe previous section and that of 2.1.3 we can immediately write down the intrabranchcontributions to 1/Ti:-—= A2±[1og(4T/w)— 7] (e__!T + e_T) (3.22)Tijntra irvwhere in this case, Q = 2wNL/v. It is quite likely that higher dimensional effects maycut off this contribution at energy scales larger than WN. For example, weak interchaincouplings, J1, would replace WN in the above expression by a quantity of order J1.Chapter 3. Model Predictions for T’ 63The interbranch contributions between the lower two and higher two branches can belikewise calculated to give!= (IAI2+ IAI2)e/2TKo((h + + e_T)(3.23)T1 Inter ltV 2Following Fujiwara et. al. [34], we write=--- + ---- F(h,T)eT (3.24)T1 TiJntra TiJnterWe see that the nature of F(h, T) depends largely on the form of the hyperfine coupling. Aless general and somewhat more qualitative version of this formula was given by Jolicceurand Golinelli [29], and by Troyer et. al. [35], independently of our work. Jolicceur andGolinelli discussed the isotropic NLu model and derived only the leading exponentialdependence on temperature; Troyer et. al. considered the Heisenberg ladder problem,which has a low energy one-magnon excitation spectrum identical to that in the isotropicNLu model, and only included the leading interbranch transition in their expression.3.1.2 Axial SymmetryHere we are faced with two possible situations: the sz 0 branch can lie above or belowthe doublet. In the former case, the larger the interbranch gap between the doublet andthe singlet branches, the more suppressed will be the interbranch contributions to l/T1.On the other hand, in the latter scenario, inter- and intrabranch contributions will alwaysbe on the same footing (see Fig 3.4). The expression for the intrabranch transitions willbe essentially identical to the one in the 0(3) case:I= IAI24[log(4T/N) —7](e_(_/T + e_+T) (3.25)Tijntra 7rvwhere zj is the gap to the SZ = ±1 branches and L3 is the gap to the SZ = 0 branch.The corresponding formula for the interbranch transitions is somewhat more subtle andChapter 3. Model Predictions for T’ 64(a) (b)Intrabranch Transitions A3---- Interbranch TransitionsFigure 3.4: The gap structure for U(1) symmetry. (a) / > &L; (b) z3 <TTv77 LUV—LidVJ_93——+Tv)—‘ust{14A{(i((o)t—‘o)lii+((o)b‘o)i)(Io’ö—‘o)liI+1((o)’ö‘oYiI)Tv,(+T)/ 9T)oM/ 9}aJLJOUJtJ+—uo‘qnopoquqiojsisjuisoquq‘saqooqujTV_V<+/)IJ=/+TV(L)=(‘i)iii TV_v>(z/).w=svjT________VV?/V_,+Tv)—TLV_—V—V>—T1t77Tt77 LiiVL—LidVLi——v—i+v)—(9){(?J((o)b—‘o)ii+((o)t‘o)ii))°X1s(i((o)‘—‘o)liI+I((o)’‘o)1i)T)OM/ 9ta}(iJLlOUJTj--(Ifi+V++vI)=—sisjuisoquoquaoqdjozsoqpuatjiqsaijqtraqTpitpuospudp(y)uissqppgjoozisuossjqnopodsiqijuisJouoiisodriospuadp1TjiojsuoparjpoyjChapter 3. Model Predictions for T 66(zS.—h)2— — Q(O)v2 h < —_) 2T — 2tT 32rs 2 2 2 2(ta+h) —z—Q(O)v —23T 23T 3I &L—hM(h)—h h<L—Li3M(h) = - (3.29)(Ls3=M(h)A seeming catastrophe occurs when two of the branches cross at h= I3 — Theinterbranch contribution to 1/T diverges logarithmically. There are essentially two effects that would cut off this divergence. Higher dimensional couplings can be countedon once more to replace Q(0) as it approaches zero, with a quantity of order 10J/J asderived in Eqn. (4.40) in Chapter Four. Also, the divergence in the integrand leadingto this problem is 1 . This will be cured by a field with finite width. Onei/h— Istill expects a peak in the relaxation rate, but this will be smoothed by the mentionedeffects.3.1.3 Z2 x Z2 x Z2 SymmetryThere isn’t much more to say which would be illuminating in this case. We can, however,easily give the results for intrabranch contributions. These behave as the analogousexpressions from the more symmetric situations.1 4Aj2= (log(4T/wN)— y) xlijntra(l(0o)2W(o) () -1 e(0T + (- +)) (3.30)1_ (0, 0)12 depends on h as per Eqn. (2.94). The formulae for interbranch transitions will,again, depend on the positions of the branches and the relative gaps between branches.Note that if there are intrabranch transitions allowed by the hyperfine coupling, thenthere will also be transitions between the + and — branches. Finally, from Eqn. (2.94),we see that (3.30) vanishes quadratically with the field.Chapter 3. Model Predictions for T’ 673.2 Close to the Critical FieldIn this section we give qualitative results on the behaviour of the relaxation rate andin the process prove that the contribution from the staggered correlator becomes crucialas h —+ h. We assume that we are now in the regime Ih — hi << T. In this limit,intrabranch processes along the lowest branch will dominate. Even if the hyperfine tensorpossesses high symmetry (thereby ruling out intrabranch contributions from the uniformspin operator), we expect intrabranch contributions from the staggered part of the spin.Since the fermion model becomes exact in this limit, we will rely on its predictions. Longwavelength modes are now expected to play the most important role; we therefore writethe dispersion relation of the lowest branch asw(k, h) = f (h — h) + v2k 0(3) and u(1) cases (3.31)( (Z2)3 casewhere v = and the effective gap is = 2(h— h)h/(L1+ /2). Nowthat the gap is actually smaller than the temperature, we must include multiparticleprocesses. This is simply done by replacing the Boltzmann weight by the appropriateoccupation factors, f (w) (1—f (w)). The derivation is straight forward and can be foundin standard texts on many body physics (for example, see [36]). The uniform contributionto the relaxation rate is given by(Ti-’) = 41A iilz(oo)I2fOOdk(w+WN)ff(W)(l - ff(w)) (;)-‘ (3.32)Due to the simple form of the density of states in the isotropic or axially symmetric case,one still obtains logarithmic behaviour for the above formula. In the anisotropic case,things are a bit different. We can combine the last expression with the results from 2.2.2to get— 4iA’ 2 z12 dk(w + wN)sech2(3 331 )Unif— 71V2 (‘ — z2) Jo i/(w + WN)2 —Chapter 3. Model Predictions for Tj’ 68At criticality we set /e = 0. We may simply rescale the integration variable to obtain(T_1)Ufljf oc T (3.34)This is expected from the Ising model where the uniform part of the spin correspondsto the Ising energy density operator3,, of scaling dimension 1. In terms of Majoranafermions this operator is ‘/L?’R = . The correlator of the energy density operator onthe infinite Euclidean plane is known from its scaling dimension and the restrictions ofconformal field theory to be [24]< f(z)E(0) >= 1/jz2 (3.35)If periodic boundary conditions are placed in the time direction (corresponding to finitetemperature), the correlator can be obtained by making a conformal transformation fromthe Euclidean plane into the cylinder (see [24, 37]), z =e28T:1irT ‘2<f(z)f(O)> lye) (336)sin(Tirz)12Setting z = it + 6, we can get the contribution to T by integrating over f dtet:1dt -jwNt (irT/ve)21 ) Unif e sinh(T7r[t— iö]) 2Changing variables, and assuming the integral is analytic as UN —* 0 (this can actually beproven by contour techniques), we see that by rescaling the time variable we reproduceEqn. (3.34).We now turn our attention to the behaviour of the staggered correlator at criticality.We know the form of this function in both U(1) (and therefore 0(3)) and Ising casesfrom Ref. [33]. On the infinite Euclidean plane we have<t(z)(O)>zI U(1)(338)<u(z)u(0) >r Z Ising3The rest of the analogy goes as follows: the magnetic field plays the role of temperature as is obviousfrom the form of the spectrum; the inverse temperature is analogous to the size of the system in theEuclidean time direction; finally, the staggered magnetization corresponds to the disorder field, ci.Chapter 3. Model Predictions for TI’ 69The field = q + iq’ is the charged U(1) field of the boson model; there are noproblems in using this (in the U(1) case) as long as we account for the interactions. uis the disorder operator of the Ising model. It is highly non-local in fermionic language,and aside from its dual, the order operator, and the energy density operator, is the onlyprimary operator in the model. Once more, making a conformal transformation into thecylinder of circumference 1/T,<t(z)(0) > (2T/v U(1)I sin(irzT)I (339)< a(z)a(0) (-T/v Isingsin(irzT)IOnce more, setting z = it +6 and integrating over time, we get that in the experimentallyimportant limit, T >> WN,(TI’)stag cc (2rTL/v2)_ + O(WN) U(1)(TI’) stag ° (irT/ve)* + O(WN) Ising(3.40)For both symmetries, this implies a significantly stronger contribution from the staggeredpart than the uniform part. In fact, as long as we are sufficiently close to the criticalregime, perturbation theory tells us that the above result will only be suppressed byfactors of order O(Ih — hI/T). In order to observe this behaviour experimentally, onemust have T sufficiently large (having a large anisotropy,— 2, would also help), sothat the decrease in relaxation with temperature is obvious. This would require that theexperiment be done over a broad range of temperatures so that any constant contributionsto 1/T, could be subtracted. In any case, the above should at least serve to clarify thatthe staggered contribution becomes important in this regime.Farther still from criticality, the analysis breaks down. We do, however, expect thestaggered spin contribution to influence TI’ through to the region T Ih — hI.Chapter 3. Model Predictions for Tj1 703.3 Above the Critical FieldFar above the critical field, h— h >> T, the situation becomes even simpler. In the0(3) and U(1) case the system remains critical. The relaxation rate will be dominatedby the staggered part of the spin. The fact that develops a vacuum expectation value(or likewise, the non zero magnetization of the ground state) will have no effect on therelaxation rate since WN isn’t strictly zero:f dt < 0IS(t,0)0 >< 0S(0,0)0> eLvt = 21r(M)ö(wN)/L (3.41)where ii is the magnetization. We can therefore say that for the 0(3) and U(1) models,the relaxation rate (assuming WN <<T) has the simple temperature dependencecc (2irzT/v)’’ (3.42)where is the critical exponent of the staggered spin correlator. Haldane argued ij =+ 0(p), where c = I 1I/L [38]. Thus the field dependence of 1/T is only through i.When axial symmetry is broken, the gap reappears for h> h. In this case, one canuse the fermion model to calculate the relaxation rate. This is made much simpler sincethe gaps to the two upper branches are presumably much higher than the lower gap (byat least 2L_ (h)). Therefore only intrabranch processes along the lower branch need beconsidered. The result is1= () l(0,0)2(log(4T/wN) - ) (3.43)At sufficiently large magnetic field, h >> Li—L2, some of the expressions simplify:l,_(0, 0)12 1 —÷ h—(z2 +z1)/22v / ‘2+11’\2 + — 2 ) (3.44)The rate will drop exponentially with increasing magnetic field.Chapter 3. Model Predictions for T’ 713.4 SummaryWe would like to summarize the main results of this section. At temperatures much lowerthan the lowest gap4, Lm (this can be below the critical field or far above it in the case ofZ2 symmetry), T’ e_m/T. This is due to the dominance of two magnon intrabranchrelaxation processes. In the case of axial or higher symmetry, the only temperature andfield dependence comes from--- cx [log(4T/wN) — (3.45)T1 IntraGiven such symmetry, this is a model independent result. Including anisotropies andinterbranch processes, the relaxation rate is given by Eqns. (3.18) and (3.19).When the lowest gap is much smaller than the temperature, the dominant contributions come from one magnon processes (due to the staggered part of the spin). When thefield reaches its critical value, whereupon the gap vanishes, the relaxation rate is givenby Eqn. (3.40). Above the critical field, the system remains critical with axial symmetry,and the rate is then given by Eqn. (3.42). With Ising symmetry, the gap reopens andeventually becomes large once more.4We refer to the lowest gap corresponding to a polarization direction perpendicular to the field.Chapter 4Material Properties and Possible Effects on Experiment4.1 Hyperfine TensorHere we briefly discuss the effect of the nature of the Hyperfine tensor, A, on the NMRrelaxation rate. As a reminder, u and v are spin indices while i and j are spatial indicescoupling the nuclear spin at site j to the atomic spin on site i.Assuming the magnetic field lies in the z direction, if Aj’ is isotropic in its spinindices, only AI2 will contribute in Eqn. (1.69). In particular, there will be nointrabranch contributions as these require a coupling to S. Interbranch transitionswill not be limited, but no contribution to them will come from the term proportionalto These statements still hold true for a hyperfine tensor diagonal in theHeisenberg spin basis. Note that this implies that for the 0(3) model, intrabranchtransitions are prohibited so long as one assumes that the nuclear gyromagnetic tensoris simultaneously diagonalizable with the hyperfine tensor.In general, especially if the NMR nucleus does not coincide with the magnetic iongiving rise to the effective spin in the 1DHAF, the anisotropies on the spin chain willnot be simultaneously diagonalizable in the hyperfine tensor basis. Moreover, if thereis more than one nuclear moment per spin contributing to the signal, it is unlikely thatthe effective will have the same symmetry as the nuclear Zeeman interaction. Thusconditions have to be quite convenient for intrabranch transitions to be missing fromthe rate. This can be important at very low temperatures where we can experimentally72Chapter 4. Material Properties and Possible Effects on Experiment 73distinguish between the processes. First of all, intrabranch transitions along the ‘—‘branch will increase exponentially with field (for example, in the 0(3) symmetric case,the behaviour is e_(_1)/T). This can be a most obvious difference at low temperatures.However, as is clear from the discussion in the last chapter, the—‘ gap can lie above the= 0 gap (for example, if one places the field along the a-chain direction in NENP);thus both inter- and intrabranch processes will feature the same exponential rise withfield. Also, it is possible that the lower gap depends very weakly on the field for certainmagnetic field directions. This is true for anisotropic materials where it is difficult toplace the field along a direction of symmetry local to the chain. There are two waysto distinguish the transitions in these cases. The simplest solution is to repeat theexperiment changing the magnetic field direction until one clearly sees the e_(l)/Tbehaviour. Alternatively, one can try to extract information out of the low temperaturebehaviour, rather than field dependence. For T h, F(h,T) in Eqn. (3.24) will behaveas log(T/wN)—‘y if intrabranch transitions are allowed. If they are prohibited, F(h, T)will more likely behave as /T/(6 + h) where S is roughly the smallest interbranch gapat zero field.In principle, if one has enough information about the gap structure of the chain, it ispossible to deduce the relative values of the hyperfine matrix elements from the relaxationrate. This may be done by comparing the ratios of the rates measured with magneticfield along each of the effective gap directions (or in case of high symmetry, any threeperpendicular directions); assuming one has extracted the intrabranch contributions fromthe measurements of 1/T, one can then work backwards using— cc A3I2 (4.1)Tilntra(3 corresponds to the field direction) to arrive at ratios of the hyperfine matrix elements.This will be, presumably, model dependent even in the low field limit. In Chapter 6, weChapter 4. Material Properties and Possible Effects on Experiment 74suggest experiments which would distinguish between the models.Finally, we discuss nearest neighbour effects of At’. When these exist, contributionsto 1/T will come from the correlation function,<S(t)S1(O) >= <S(t)S”(O)> (4.2)where S is the nth fourier mode. Taking N —+ cc, we write<S(t)S÷1(O) >=< S(t)S(O) >+ f (e _i) <S(t)Sk(O)> (4.3)Expanding in k, the odd contributions in k from the second term will vanish whenintegrated over all states. From the work done in the last chapter, we know that powers ofk2 will give small perturbations of order 2L/v. We see that including nearest neighbourcontributions, the relaxation will be essentially given by making the substitution,—* A + Aj (4.4)in all previous expressions where, A, is the nearest neighbour hyperfine coupling.4.2 ImpuritiesSo far, we have dealt with a single, infinitely long spin chain. In real experiments,however, chains are always finite, and they come in three dimensional crystals, and sothere are many chains of varying lengths in each sample. In this section we deal with thefact that these chains often end or have defects. This is what we mean by impurities.Of course one can introduce doping (for example, replace some Ni sites in NENP withCu) to explore the issue further; we will restrict ourselves to ‘pure’ samples, althoughour treatment can be extended to doped samples.We start by describing a ‘finite’ chain. Exact work on a related S = 1 Hamiltonian— the ‘valence bond solid’ [39], which also features a unique ground state (in the thermodynamic limit) and a gap — has indicated that at the ends of a finite chain there areChapter 4. Material Properties and Possible Effects on Experiment 75free spin-i degrees of freedom. It is expected that the same holds true for finite chainsof the Heisenberg antiferromagnet. Indeed, there has been convincing numerical andexperimental evidence for this conjecture [40, 41]. The same evidence also supports thenotion that these nearly free end spins can interact with each other, essentially exchanging virtual NLu bosons. The interaction is exponentially decaying with the size of thechain:H1 ( i)Le_L/ 4o SL (4.5)Here, the correlation length, , is roughly six lattice spaces. For experimentally realizablelengths, this interaction is negligible and the finite chain has a fourfold degenerate groundstate.In a ‘pure’ sample, these finite chains will lie end to end, or be separated by somenon-magnetic defect. It is therefore reasonable to presume that two adjacent end spinswill interact with an effective exchange coupling, J’, that will vary in strength from zeroto something of order J. Recent work [42] has explored this situation in depth. For weakcoupling between adjacent end spins the effective Hamiltonian can be written,H=J’1•g2j. g (4.6)where is a spin-i operator and c is the projection of a spin-i end spin into a spinsubspace. From numerical work [41, 16], we know that c 1. This immediatelygives a low lying triplet above a singlet with a gap zE = o2J’ (we assume that J’ isantiferromagnetic. The ferromagnetic case is expected to give similar results, reversingthe order of the singlet and triplet, but numerical work has not yet been done to supportthe analysis in this limit). This triplet will sit inside the Haldane gap. The tripletcorresponds to bound states at the chain ends; this has been seen numerically in [42],which also demonstrated that the above first order perturbation theory result forChapter 4. Material Properties and Possible Effects on Experiment 76jt >+lit>- It4>lt>Figure 4.1: Impurity level diagram when D’ = 0.is accurate up to JE ‘- .3J. The embedded states become delocalized and join thecontinuum at about J’ .7J. This picture is unchanged up to about J’ “-‘ 2J, afterwhich the triplet returns as a bound state in the Haldane gap to merge with the singletstate as J’ —+ oo. In the type of chain we are considering, J’ is unlikely to become muchgreater than J; it is much more likely that defects in a pure sample will serve to reducethe effective coupling between sites rather than enhance it.To understand the effect on NMR relaxation we explore the environment of NMRnucleii near the end spins’. Take, for example, the nuclear spin coupled to . It sees theZeeman split level diagram shown in Fig. 4.1. If we assume that the ‘free end spins’ arenot completely free, but are weakly coupled to the magnons on the chain, then relaxationcan occur in two ways: when two levels are WN apart the chain end spin could make atransition — this will happen for a fields h .‘ J’. Alternatively, a thermal magnon coupledto the chain end could decay into another magnon with or without the accompanimentof an end spin transition — again, the energy difference between initial and final states‘We would like to thank D. MacLaughlin for privately communicating his suggestions on the effectsof end spin excitations in NMR.Chapter 4. Material Properties and Possible Effects on Experiment 77must be WN. Marked on the diagram are the transitions that can be induced by g: solidarrows represent possible transitions that potentially do not require coupling to the restof the chain; dashed arrows represent transitions that could occur only if the impurityis coupled to the rest of the chain; solid circles represent spectator transitions which,again, require magnon assistance. It is easy to see that in all the above scenarios, thetransition will be broadened by thermal magnons. This implies that the characteristicwidth will have a temperature dependence exponential in the Haldane gap. There is anadditional mechanism, which we now discuss, that can, in general, affect this picture.If the material is anisotropic, with for example, a D-type anisotropy, we must add thefollowing term to the chain-end effective Hamiltonian, Eqn. (4.6),HE —÷ HE + ci2D ((S1)2 + (SI)2) + D’S’S’ (4.7)The first term is the familiar on-site anisotropy; it will only contribute a c-number to theeffective Hamiltonian. The second term is allowed by symmetry, and we presume that itis a consequence of the defect (which arguably, would manifest itself in accordance withthe available symmetry). We assume D’ > — J’, so that the exchange interaction is stillantiferromagnetic in the z-direction. As a result of this, the two Z = levels will shiftby —, while the SZ = ±1 levels will be shifted by . The transitions induced by thehyperfine interaction are shown in the new level diagrams in Fig. 4.2. It is this moregeneral case for possible transitions which we now carefully analyze (further anisotropicpurturbations will not qualitatively change the picture).We begin by characterizing the interaction between the bulk magnons and the endspin. Since we have no information as to the nature of this coupling we will parametrizeit in the spirit of Mitra et. al. [43] using free bosons which after scattering with theimpurity obtain a phase shift. We make two assumptions: first, that leakage across theimpurity site is negligible, and second, that the impurity spin coupling to the boson doesChapter 4. Material Properties and Possible Effects on Experiment 78Figure 4.2: Impurity level diagrams for D’ # 0.not allow for exchange of spin (this is consistent so long as J’ is sufficiently weak); bosonson each side of the impurity will have wave-functions of form:C(k) (e — e_2i6e_?j (4.8)where i refers to the boson branch, k is assumed positive, IC±(k)12 = 1/2L and ± refersto the sector of the impurity spin with SZ = ±. The boundary condition, q(L) = 0,givesk = (nir — k))/L (4.9)We assume that the phase shifts are small and grow with increasing energy. This istantamount to assuming a large step potential barrier of infinite extent (thus allowingno leakage for states below the barrier). This is certainly true in the limit J’ —+ 0. AD’>Ok >itt>It i,> kJ+D’/2It k144>Itt>It >-I t>It >— I t>Chapter 4. Material Properties and Possible Effects on Experiment 79heuristic ansatz which has this behaviour is6(k) (4.10)Notice that the ‘s need not be orthogonal to the q’s, since these states are in separateHubert Spaces. The energy of one of these bosons at low temperatures is given by thefree formv2(ki±)2 (4.11)Before going on we note that WN is typically much smaller than the energy level spacingsdue to finite size, for typical chain lengths. For example, in NENP, 5E .04 Kelvin forchains L -‘ l000a. One may therefore question the validity of boson assisted transitionswhen the bosons lack the ability to ‘fine tune’ a transition so that the difference betweeninitial and final states is WN. What saves the day, in this case, are higher dimensionaleffects. For sufficiently long chains or sufficiently strong interchain couplings, these willdensely fill the spacings in energy levels along the chain direction. Assuming this is thecase, (we show the conditions for this explicitly in the next section) we will not worryabout this point further.Consider the coupling of a nuclear spin to one of the impurity spins, say S (fromhere on we will implicitly write S = S’, for ease of notation). The familiar formula forthe transition rate can be cast as1j- =—j dte”IA1211 n,l,n’,l’{ < n1; lIe t)(HHSILIe_it(J:1+HE)Ir; 1’ >< n,; l’IS’ mi; I >+ < n1; lIe_(HHSIn,; 1’ >< ni,; lIeit ô+H)SILIe_it(H HE)1j; 1 >} (4.12)where ji = —/2 = ±, 0. Hb is the free boson Hamiltonian; Ii; m1> denotes a state of theimpurity spin (ie. an eigenstate of HE), which for brevity, we denote as 1 >; n1 denotesChapter 4. Material Properties and Possible Effects on Experiment 80the boson content of the state; in general, ni; I > will contain two different multiparticlefree boson states that have projections onto the S = ± subspaces of 1; m1 > (withappropriate phase shifts). To elaborate and make things a bit clearer, take the S’ = 0state from the triplet.ni,0; 1,0 > (I t, > øIn > +1 t t> ®In>) (4.13)where the states d: > correspond to the phase shifted bosons with S’ = ±, respectively. Making the approximation, a = 1, we return to the relaxation rate:1 2 roo— =— I dt —ZWNt 211 —00 ILez’) cos(t[E1— + e— Ei’])I < IlSILhIl, > 1l <ntn, > I (4.14)n,1,n’,l’Let’s pick a particular transition and work it through. Consider ,u = +, 1 >= 1, 1 >and It’ >= 1, 0 >. This transition could be of the type we’ve been discussing where forh ‘ D’/2, we expect strong resonance if D’ > 0. The expression for the rate becomesI’, i) = e_’’/2_ L dt e_tIA_+l2Re e n+eit(En E_!)1 < 2 (4.15)I,. )Since the boson multiparticle states, < n, are direct products of symmetrized free Nparticle states, and since the energy of such a state is the sum of single particle energies,we can write the last equation in terms of single particle states:1’, ) = —e(J’/2) L dt etIAI2Re {eith12ExP (_z1 + e_+eit+__,)I <n iIn’;i> 12) } (4.16)n,n,,i,jChapter 4. Material Properties and Possible Effects on Experiment 81where we’ve used the fact that the grand partition function for noninteracting bosonsis the exponentiated partition function for a single boson. n and n’ now index singleboson states, and i and j denote boson branches. The overlap of the boson states can becalculated from the form given by Eqn. (4.8),I <;iln’;j> 12 = 8 (sin(6?fk)±6(k)) — sin(8k)_8(k)))2 (4.17)where we have parametrized the momenta of n and ri’ with k and k’, respectively. Finally,we write the one particle partition function (we can ignore the phase shifts for thispurpose) as=e3El+ <n;iln;i>= I <n; ilri; i> I (4.18)n,iCombining all of this allows us to write the exponential in Eqn. (4.16) ase_/Tf dkdk’4(k2k)2(i—(k2 —k’2)vxe_)2k/2i(eit 2— 1) (4.19)8T 00 1 —=—elTv22(i — i)2 j dx dx’e { (_ x’)2 ] e(t) (4.20)The effect of the phase shifts is contained in the factor, c\i — ,\i)2, as seen from Eqn.(4.10). Corrections to this due to 0(k3) contributions to 6(k) will be suppressed byfactors of 2L1T/v. The imaginary part of e(t) will shift the resonance from h = D’/2.This shift, at low temperatures, will be negligible. We are interested in the long timebehaviour of e(t). In this limit, the real part of e(t) becomes:Ree(t)—— i)2f dx dxIe5m(r — x’)Tt/2)Chapter 4. Material Properties and Possible Effects on Experiment 82—*_e_j IT8i (—)2tI —F(T)ItI (4.21)The expression for the relaxation rate becomes1 A2— (1, 1’, )< dt e_t cos (t(D’/2 — h)) e_ItT) (4.22)‘ A—--2 F1T=‘2e_(J’+UJ’/2_1) I (4.23)ZE P2(T) + (h — D’/2)2This is the most important equation of this section. The other transitions can be treatedthe same way to arrive at analogous results. The key issue to note is that the impurityrelaxation rate is an extremely sensitive function of the temperature and field. At temperatures well below the gap it is essentially a delta-function of h. As the temperatureincreases and becomes comparable to the gap, the rate broadens rapidly.Before summing up, we discuss the other possible transitions. First, notice thatchanging ,u has the same effect as reversing the sign of h and exchanging 1 and 1’:JL —+ ü —* h —h ÷—* 1 -* 1’ (4.24)Note that in this simple model of boson-impurity coupling there are no transitions viaS2’, and therefore no transitions between the singlet alkd the s2 = 0 state of the triplet.In other words, the solid circles in Figs. 4.1 and 4.2 are ignorable as are the dashed linesfrom Ii, 0 > to 0>. This is expected in all but the most extreme of anisotropic exchangeimpurity models. Furthermore, the effect of reversing the spin states on the triplet is thesame as reversing the sign of magnetic field:m1, mj’ —+ —m1,—m1’ +—+ h —+ —h (4.25)Chapter 4. Material Properties and Possible Effects on Experiment 83Finally, the result of exchanging the 0, 1 > state with the singlet amounts to adding J’to the associated energy factor in the Lorentzian. A final expression for the impurityrelaxation rate involving all eight possible impurity level transitions is1 4 2 A—Ui2-- ““ D’—4’ ,—13Ei ‘‘ (426rp ‘ — ‘- j’—’‘ P2”T’ E211 i1 f1 ‘—‘E ) T fiwhere E1 denotes one of the four possible initial impurity states, and Ef is the differencein energy between the initial state and one of the two possible consequent final states.The factor of two represents the contribution of both end spins on each chain (we neglectsurface effects). Now it can be seen more clearly that all but two of the elements in thesum above will contribute little due to the narrow gaussian form. The important termsare those where the energy in the gaussian is small; this can happen for certain magneticfields: h ID’/21 and h J’ + D’/2. In Heisenberg chains we might expect J’ >> ID’Ifor most defects. Furthermore, the impurity contribution should be most evident atlower fields where the gap still lies high. Consequently, in experiment, one expects theh ID’/21 transition to dominate the picture of impurity contributions to 1/T.In a real sample, the NMR signal from the impurity will be proportional to the densityof the impurities. Moreover, since defects will vary from chain to chain, one would be wiseto average over a random distribution of couplings, J’ and D’. In practice, experimentaldata could be analyzed for the ‘peak’ values of J’ and D’. One could then model thedistribution of couplings with the appropriate peak values. This could, in principle bechecked against low temperature ESR measurements which ought to concur with theimpurity model.A final expression for the relaxation rate due to impurities is(---) = fdJ’ dD’ p(J’ — J’)p(’ — D’)1 (J’,D’) (4.27)T1 Imp T11where ñ is the density of impurities (or inverse length of the average chain); p is somedistribution function.Chapter 4. Material Properties and Possible Effects on Experiment 844.3 Interchain CouplingsIn previous sections we mentioned the effects of interchain couplings on various aspectsof the physics. We now examine these in more detail.Nearest neighbour interchain couplings will enter the Hamiltonian asH—H+J1‘• (4.28)where < i, j > index nearest neighbour spins not on the same chain. We can return to thederivation of the NLu model to see the effect of this additional term. Taylor expandingthe continuum representation for,.,and assuming reflection symmetry about a site,Eqn. (1.38) will change toS = 2irisQ+ 2LxLy fd4x(ö)2+ 2LxLy f dx(2LxLyfd4x(8vT)2 (4.29)where we chose the z-direction to be along the chain, and the vector, , is the displacement vector to the ith nearest neighbour of a spin not on the same chain (again,we assume that is smaller than the correlation length. Note that the correction todynamical part of the Lagrangian will correspond to J —* J + J1. Presumably, Jj << J,meaning that we were justified in ignoring this term. Setting the lattice spaces, x, /y,to 1, we can now write an effective Landau-Ginsburg Hamiltonian to describe the physics;Eqn. (1.50) will read?() V2 V (4.30)The leading relevant interaction terms will always be local. Ignoring these, the resultingequations of motion are=(v8 — + 2J1s ( )2) (4.31)Chapter 4. Material Properties and Possible Effects on Experiment 85The dispersion relation becomes,w2=vk+vi( (4.32)where v± cx %/J7. It is from this last formula which we now extract qualitative information about interchain coupling effects.First, recall that we claimed that for finite chains of certain lengths in real experimental situations, we no longer need to concern ourselves with 1-D finite size effects. Inother words, we said that energy levels arising from interchain couplings will densely fillthe small gaps between magnon energy levels, Let’s calculate this length interms of J and J1. We start by assuming a simple form for the interchain contributionto the dispersion:k’)2=vai(k+k) (4.33)where a± is some typical interchain distance, and expected to be 0(1). The size of theinterchain band will be vo,r2 Setting this equal to the gap in the magnon levels we getL2-=L (434)VL IIn NENP, for example, this corresponds to lengths of approximately 100 lattice units.There is also the issue of cutting off divergent integrals which we discussed in Chapter3. In calculating transition rates, one often encounters integrals such asf dk dq 6(4 — — E)f(k) (4.35)When E is close to the gap between the branches, wL and w, this integral can divergelogarithmically in the infrared. If one introduces interchain couplings, the integral overthe delta function becomes(2ir)4 f dq d2kd2q±6(w — — E)Chapter 4. Material Properties and Possible Effects on Experiment 86_ii2 _i2 2,2 2,2 2 2 2 2d UJj U q±61v I’u Vjj — V q — vq 436j q (4)2a 2, 2L 2/where we have assumed a simple form for the interchain dispersion. For ease of calculation, we now assume that L = /.j z. The integral becomesL dk dqk2 2k 2 282av J vk2 + vk — vq +v— vqj=42vaf dkIv2k2 + vIk (4.37)At low momentum, k, where we need a cutoff, this integral is approximatelyLir (4.38)6v±va±If we write the integral in Eqn. (4.35) as1dk f(k) (439)v2J v7-Cthen the cutoff, C, is seen to be144va (4.40)2V JWe recall that the for intrabranch transitions, Q 2LwN/v. Comparing this to C, wefind that the cutoff becomes important for WN <7OJjaI. For example, in NENP, whereJ1 ‘‘ 25 mK, we expect the cutoff to significantly dominate over the Larmour frequency.4.4 Crystal StructureWhen analyzing experimental data in terms of the idealized Heisenberg model with onsite anisotropies, one must keep in mind that the symmetry of the proposed spin-chainHamiltonian may be constrained by the symmetry of the crystal and the local symmetryChapter 4. Material Properties and Possible Effects on Experiment 87about the magnetic ion. Additional terms may be added or subtracted to accommodatethe structure of the substance, and these can have a great effect on the interpretationof data. Some important questions which must be asked before deciding on a modelHamiltonian for the material are: is the local crystal field symmetry about the magneticion commensurate with the symmetry of the unit cell? Is there more than one chainper unit cell? If so, are all chains identical? Is there more than one magnetic ion of asingle chain per unit cell? If so, is there true translational symmetry from one spin siteto another?In the next chapter, we analyze experiments performed on NENP. In so doing, wewill address such considerations.Chapter 5NENP: Direct Comparison with Experiment5.1 The Structure of NENP and Experimental RamificationsA schematic diagram of Ni(G2H8)0(G104(NENP) is shown in Fig 5.1. Eachchain is comprised of Ethylenediamine-Nickel chelates separated by nitrite groups. Themagnetic ion is Ni2+; experiments indicate that these ions interact antiferromagneticallyalong the chain with coupling J 55K. There is a large single ion anisotropy, D ‘.‘ 12K,as well as a small axial symmetry breaking anisotropy E 2K. Interchain couplings areestimated at J1/J iO [44].It is important to realize that two neighbouring Ni2 along the b-direction are notequivalent; rather, one is related to the other by a ir rotation about the b axis. Also, theangle along the N — Ni —0 bond is not exactly r, meaning that the Ni site is not trulycentrosymmetric. Most importantly, the local symmetry axes of each Ni ion are rotatedwith respect to the abc (crystallographic) axes. To demonstrate this we now note thecoordinates of the Nitrogen atoms in the Ethylenediamine chelate surrounding the Nickel(placing the Nickel at the origin): [45]Atom a (A) b (A) c (A)N(1) 2.053 (3) .162 (3) .338 (3)N(2) .619 (3) —.184 (3) —1.971 (3)88Chapter 5. NENP: Direct Comparison with Experiment 89aONi•c00•NNENPFigure 5.1: NENPChapter 5. NENP: Direct Comparison with Experiment 90CaFigure 5.2: Local and crystallographic axes projected onto the ac-plane in NENPThe other Nitrogen atoms in the chelate can be obtained by reflection through theNickel. One easily sees that projecting this structure onto the b plane yields symmetryaxes (in the b plane) rotated 60° from the ac-axes. This is shown in Fig. 5.2. Theinclination of the local Nickel axes from the abc system can be obtained by taking thecross product of the two Nitrogen vectors (ie. the normal to the plane described by theN(2)C,N(1)a’•_.. Projection of the Cantingb’ VectorChapter 5. NENP: Direct Comparison with Experiment 91four Nitrogen atoms in the chelate:n = (—.06 (1), .98 (1), —.11 (1)) (5.1)The local Ni b’-axis makes a -‘ 100 angle with the b-axis, while the azimuthal anglein the ac plane is —28° from c. The 10° tilt is roughly about the a’ direction of thelocal symmetry axes.One may worry that the NO group may distort the local symmetry axes, but remarkably enough, when projected onto the ac plane, the three atoms in the molecularion sit on the c’ axis. This reinforces our suspicion that the local symmetry axes areindeed the above.Next, we consider the whole space group of NENP. The most recent attempt to solvefor the crystal symmetries has concluded that the true space group of the material isPri21a [45]; this is a non-centrosymmetric space group with a screw 2 symmetry aboutthe b axis, diagonal glide plane reflection symmetry along the a axis, and an axial glideplane reflection symmetry along c. Experimentally, attempts to solve the structure inPn21a have not been successful; rather, it seems that Pnma gives a better fit. Themain difference between the two is the presence in Fnma of a mirror plane parallel tob at i-b, centers of symmetry at various locations in the unit cell, and two-fold screwaxes separating these centers of symmetry. The reason for the experimental discrepancyis attributed to disorder in the orientation of the nitrite group, the perchlorate anions,and the existence of a local or pseudo center of symmetry lying very close to the Ni(thousandths of an Angstrom) [45]. A crucial point is that both space groups share theaxial glide planes along a, the diagonal glide planes along c and the 2 screw symmetryabout b. These generate a total of 4 Ni sites per primitive cell and two chains througheach cell. The two chains are such that the Ni chelates on one are the mirror image of theother. Figure 5.3 shows a projection of this picture onto the ac plane. The presence of theChapter 5. NENP: Direct Comparison with Experiment 92C(1)NJ C(2)Ii_aN(2)CI 2AFigure 5.3: A projection of the NENP unit cell onto the ac-plane showing two chains perunit cellChapter 5. NENP: Direct Comparison with Experiment 932 screw symmetry about each chain axis introduces staggered contributions to the localanisotropy and gyromagnetic tensors. This is because, as motivated above, these are notdiagonal in the crystallographic coordinate system. The resulting spin Hamiltonian is(5.2)We make the assumption that the symmetry of the anisotropy and g-tensors is thesame (ie. that at each site they can be simultaneously diagonalized). This is rigorouslytrue when the crystal field symmetry about the magnetic ion is no lower than orthorhombic (a sketch of a proof is found on p. 750 of [4]). We can get the required parametrizationfor the g-tensors from high temperature uniform susceptibility data [45]. This is based onthe idea that at high temperatures the Ni atoms will behave as an ensemble of uncoupledspins (s = 1) with the same gyromagnetic tensor as in the antiferromagnetic case. Withthis in mind we getG’ cos2(O) + Gfr sin2(0) 0 00 Ga’ 0 (5.3)0 0 Gj,cos2(O)+Gesin6)0 0 sin(O) cos(0)(Gb’— Ge’)g= 0 0 0 (5.4)sin(0)cos(O)(Gw—Ge) 0 0Here 0 ‘-.‘ 100, and Ga’ = 2.24, G1/ = 2.15, G’ = 2.20 are the values for the localG-tensor that give the observed high temperature 9-tensor when averaged over the unitcell. Correspondingly, the anisotropy tensors must have the following form:D’ 0 0D 0 Da’ 0 (5.5)0 0 D1,Chapter 5. NENP: Direct Comparison with Experiment 94o o tan(26) (Db’ — Do’)d= 0 0 0 (5.6)tan(28) (Db’ — D) 0 0The parameters Dat, D, D,, are to be fitted by experiment to the model used todescribe the system. The boson Hamiltonian can now be writtenfl)+ (5.7)-.d(x i).gfl2)2]The term containing d breaks the Z2 symmetry along the a’ (lowest mass) direction.It will also renormalize the masses. The second effect can be ignored in the approximationthat the ç term is ignored if we assume the masses are physical. Symmetry breaking,however, leads to the presence of a static staggered field even below a critical magneticfield.. gap will always persist. The staggered field term will break the Z2 symmetryalong c’ or b axis, depending on whether the field is applied in the b or c’ direction,respectively. A static staggered moment will likewise appear due to this term. Theeffect on the relaxation rate will be small, although there may be consequences in otherexperiments [32, 46].We would now like to discuss the effect of having two inequivalent chains per unit cell,with local axes different from the crystallographic axes. We label the two chains foundin a unit cell of NENP ‘chain 1’ and ‘chain 2’ corresponding to the chains in the upperleft and lower right corners of Figure 5.3 respectively. The dispersion branches of chain 1are given by Eqn. (20) of [47] (the expressions are roots of a complicated cubic equationand we feel that citing them will not prove illuminating) only the field is — 300 fromthe c’ axis where is the angle of the field from the crystallographic c-axis. Similarly,the dispersion branches of chain 2 are calculated with the field a— 150° from the c’ axis.Chapter 5. NENP: Direct Comparison with Experiment 95Experiments which average over signals, like susceptibility or NMR T’ measurements, must consider their results an average of two different measurements (corresponding to the two different chains with their relatively different applied field configurations).On the other hand, experiments such as ESR, should show a separate signal for eachchain. The NMR relaxation calculations performed in Chapter 3 assume the field isplaced along one of the crystal axes. In this special case, the dispersions for the twodifferent chains are identical. Although the dispersions will be more complicated as willbe the matrix elements, lb(O, 0), we do not expect great qualitative differences between acalculation as done in Chapter 3 and one which accounts for the actual symmetry whenthe field is placed along a crystal axis. There will also be contributions due to the i— rcorrelator; these are also expected to be small. There are, however, important manifestations of having two inequivalent chains. These will be discussed in the next chapterwhen we suggest further experiments.In conclusion, consideration of the crystal structure introduces both symmetry breaking terms and two inequivalent chains per unit cell. The symmetry breaking terms willgive small corrections to the relaxation rate.5.2 Analysis of the DataBy far, the most studied Haldane gap S = 1 material is NENP. The most recent measurements of the relaxation rate, 1/T, on this substance have been made by Fujiwara et. al.[34]. Before directly comparing our results to the data we discuss the expected results ona pure (infinite) system. The three gaps are given by neutron scattering: /a = 1.17meV,= 2.52meV and La = 1.34meV. We use v = 10.9meV, and the generic value of 2.2 forthe electronic g-factor. Since we do not have an accurate description for the hyperfinecoupling of the Ni ion to the protons in its surrounding chelate, we assume a uniformChapter 5. NENP: Direct Comparison with Experiment 96value for all the contributing hyperfine matrix elements in a given direction of the appliedfield. WritingT’ = F(h, T)e_(1)/T (5.8)we use the results from Chapter 3 to plot F(h,T) for bosons and fermions and for fieldsalong the chain a, b and c directions. The results are shown in Figs. 5.4 - 5.7.We included multiparticle transitions by simply replacing the Boltzmann factor byappropriate occupation factors in Eqn. (3.18): fb(1+fb) = cosech2(j)/4 for bosons, andfj’(l—f) = sech2(j±)/4 for fermions [36]. Within approximations used, multiparticleeffects amount to multiplying the final expressions by (1 ±e)2.At higher temperatures it is also necessary to include the k-dependence of the integrand past the peak atthe origin. We expect that at temperatures T • and fields h the numericallyintegrated results would differ by about 10 percent.F(h, T) is shown for fields up to 9 Tesla even though the (/3w >> 1) approximationis no longer valid at such fields. This is done to contrast the predictions of the bosonand fermion models. It’s easy to see that the boson result for F(h, T) diverges at thecritical field, while no such catastrophe is present in the fermion result. This divergenceis logarithmic and infrared. It will persist even after account is made for the staggeredpart of the correlation function. Multiparticle scattering will in fact worsen the effect,since the bose distribution function diverges as 1/w with vanishing energy w. This againis evidence of the inadequacy of the free boson model close to criticality.In NENP, when the field is along the b direction, we expect relevant interbranch transitions only for small field. In this regime, one must also be careful to include intrabranchtransitions in the second lowest branch. All these processes are of the same order. Eventhough the intrabranch rates vanish at low fields, the interbranch contributions are suppressed by the absence of low momentum transitions (ie. Q for the interbranch transitionsclq Ct’ 0 J) 0 5 CD Cl) 0 Ct’ CD 0 C,)BosonF(h,T):ContributionsFromTheUniformPartoftheSpin•0 I— - LI. 0.0 0.0H110at1.4KH110at2.0KH110at4.2KHIIbat1.4KHubat2.0KHI,bat4.2K2. 5. NENP: Direct Comparison with Experiment 98OCOC’— — — — — —0I I —I II II III1—0- q(Ls) (j )-jFigure 5.5: Fermion F(h, T) for fields along the b and c chain directions.BosonF(h,T):ContributionsFromTheUniformPartoftheSpinrjaq CD 0 Cl) 0 CD Cl) 0 Ct’ I- CD 0 J)I— . LL50. 0.0Hiiaatl.4KHiiaat2.0KHiiaat4.2KHIIbatl.4KHubat2.0KHIIbat4.2K0.,T):50.0ContributionsFromTheUniformPartoftheSpinHuaatl.4KHiiaat2.OKHat4.2K40.0HIIbatl.4KH,Ibat2.0KHIIbat4.2K30.0-CDU)0.010.0H(Tesla) 5. NENP: Direct Comparison with Experiment 101is O(1JL—as opposed to O(WN).) For this case, only i need be calculated.When the field is along the c direction (corresponding to the middle gap), we restrictourselves to calculating intrabranch transitions along the lower branch and interbranchones between the lower and c branch. There are no intrabranch processes along the caxis. Calculating the interbranch transitions amounts to calculating andWhen the field is along the a direction, the calculation proceeds as above. The crossingof the branches provides for the interesting effect mentioned earlier. The peak in 1/Tcan be used to locate the true ac-axes for the chainNotice that F(h, T) for the field parallel to the b axis is nearly field independent overa large range of the magnetic field. This behaviour is quite easy to understand fromthe universal results valid in the axially symmetric case, discussed in Chapter 3. Whenthe field is along b, the system is only slightly anisotropic, and so the axially symmetricresults roughly apply. F,, is roughly independent of field with axial symmetry since l_is nearly h independent (in fact, F,, exhibits a logarithmic divergence as h —* 0). On theother hand, F vanishes quadratically as h —* 0. Including the small breaking of the axialsymmetry corresponding to z—= 2°K, F,, is essentially constant down to low fieldsof order L — iT, before rapidly decreasing as seen in in the figures.We now proceed to directly compare our results with those of Pujiwara et. al. Sincethe hyperfine coupling is not known, we find a best fit to it using the experimental data.This is best done for mid-sized fields: in the low field regime impurities may dominate,and in the high field regime the staggered part of the spin is expected to contribute. Figs.5.8 and 5.9 are such fits to the boson and fermion models.In producing these fits we get different values for A, the hyperfine coupling for aChapter 5. NENP: Direct Comparison with Experiment 102H II bFit to Fermion Model100.00 I I H=7.1 To C H=6.65 T•H=5.45T•H=4.32T— H=7.51 T• 0 H=6.65T• 0 H=5.45T10.00C—— H=42TN cliC1.00 NN.N .flN •NN SN0.0 0.2 0.4 0.6 0.8Fit to Boson Model100.00 I I •‘ 0 H=7.5l T0 cH=6.65To•H=5.45T%Co•H=4.32T0 0 —H=7.51T• 0 H=6.65T• °° H=545T1.00 ‘NNNN —N SN • .•N0.10 I I0.0 0.2 0.4 0.6 0.8T1 (K1)Figure 5.8: Theoretical (lines) vs. experimental data (circles and squares)Chapter 5. NENP: Direct Comparison with Experiment103H//CFit to Fermion Model100.00 — I—H=7.51 TH=6.23TH=5.45T——H=4.32To a H=7.51 T0.10 00 0:2 0.4 0.6 0.8Fit to Boson Model100.00 I • I • I—H=7.51TH=6.23T\ ‘-., — —— H=5.45 T——H=4.32ToH=7.5110.00 H=6.23T-\ \\., • H=5.45 T•H=4.32T• \‘OI.— .% N-a• b41.00 - \.%\‘S S--\ ‘ 0\\ %.\ a’.0\0.10 I I I0.0 0.2 0.4 0.6 0.8T1 (K1)Figure 5.9: Theoretical (lines) vs. experimental data (circles and squares)Chapter 5. NENP: Direct Comparison with Experiment 104field placed along the w-direction:1 8.5 MHz fermionsAb ‘ (5.9)( 7.0 MHz bosons(17.7 MHz fermions(5.10)112.0 MHz bosonsThese values are reasonable for dipolar hyperfine couplings between a nuclear spin (s =1/2) and the Ni spin at a distance of about 2)1:A IN/B 2 MHz (5.11)r3Also, we can get a similar feeling for the size of the hyperfine couplings from Knight shift[51] and magnetic susceptibility [44] data for a field placed along the b-axis.A,uoX3H (5.12)At about 4K, the susceptibility is roughly a fourteenth of its maximum value. Given thatXMax 1/J, and that the Knight shift at large fields is about 10, we get A .‘ 8 MHz.It should be noted, however, that these are order of magnitude estimates; an accurateevaluation of the hyperfine matrix elements is still unavailable. Overall, the fermion fitis the better of the two. This is more obvious at high fields when the anisotropy—is high (ie. when the field is along the c-axis). For both models, the fit to the HIbdata becomes progressively worse as the field is increased. Fitting to the lower field dataseems to give better overall agreement than fitting to the higher field results. This is notthe case for the hIIc data (at least with the fermions). Since the field in the experimentwas not actually placed along the chain c-axis, we might expect even worse agreementbetween this set of data and our calculations! In fact, as mentioned before, we expect aChapter 5. NENP: Direct Comparison with Experiment 105very weak field dependence for the hub data which would result from being close to U(l)symmetry. This was the universal result of Chapter 3.As is evident from the figures, the slope of the hIb data and the calculated resultsagree. This implies that the relaxation is largely mediated through thermal bosons andthat the calculation is off by a T-independent multiplicative factor. For small anisotropy,h >> E, this effect cannot come from the matrix elements for the transition or thedensity of states. We believe that we have taken account of the obvious mechanisms forrelaxation. Terms coming from the structural properties of NENP into the Hamiltonian(as discussed in the last section) are too small to be responsible for such a large increasein the relaxation at the mid-field range. Moreover, they would be expected to playa similar role when the field is placed along the c-axis. There are 16 protons in thechelate surrounding each Ni ion. Nuclear dipole-dipole interactions among them areenergetically negligible, and thus could not be the cause for the increase in relaxation.It is certainly conceivable that the averaged hyperfine coupling is highly anisotropic, butit’s hard to explain why there would an additional dependence on the magnitude of thefield. Perhaps the discrepancy is due to reasons intrinsic to the experiment.Next we attempt to fit to the low field measurements taken for field along the b-axis.We find that for fields less than 4 Tesla, it is not sufficient to consider the bulk theoryalone. The relaxation rate decreases with increasing field in this regime (see Fig. 5.10).We can try to apply the impurity model to explain the data. Assuming the phase shiftconstants, )4, in Eqn. (4.10) are 0(1), the impurity resonance width, F, derived inthe last chapter can be graphed as in Fig. 5.11. As is clear from the plot and Eqn.(4.26), the impurity relaxation rate is essentially one delta-function peaked at D’/2 = hand another peaked at D’/2 + J’ = h. This means that we expect two bumps in therelaxation rate due to impurity effects. The width of the bumps should correspond tothe width of distribution of impurity couplings. The problem arises when we see that theChapter 5. NENP: Direct Comparison with Experiment 106H//bExperimental Data0000U010.00000T=1.4K1.000 • T=2.O K•T=4.2 K.0I 00.100 00.0 5.0 10.0H (Tesla)Figure 5.10: Relaxation rate for field along the b-axis.Chapter 5. NENP: Direct Comparison with Experiment 107W(meV)Impunty Resonance Wuith (H=3 T)T (K)Figure 5.11: F(T)—the width of the impurity resonancetemperature dependence of the low field data is roughly exponential: e_J’/T, wherethe impurity coupling J’ is about 4.7K. Furthermore, the sharp decrease from zero fieldrelaxation suggests D’ = 0. By analyzing Eqn. (4.26) we see that the second bumpshould have little temperature dependence. This means that assuming the first bumpsits near h = 0, the second must be larger and separated by about 3.5T. This is clearlynot the case. Indeed, we would need a complicated distribution of couplings, J’ andD’, to get a proper fit. Adding an E type anisotropy will not change these conclusions.We thus do not have a satisfactory explanation for the low field behaviour. One shouldtake notice, however, that the data was taken for a field along the b-axis, where otherproblems were present at mid-field.Finally, we would like to mention some recent NMR data collected on the 1-D S = 1spin chain AgVP2S6by Takigawa et. al. [48]. This material is highly one dimensionalChapter 5. NENP: Direct Comparison with Experiment 108with a large gap (z 320K) and very nearly isotropic (3 4K). These characteristicsmake it ideal for analysis using our results. There are, however, some questions about theproperties of the material which would have to be analyzed before an understanding ofthe NMR results is possible within the framework proposed here. The gap deduced fromstudies on the Vanadium atom (z ‘-.‘ 410K) conflicts signiflcalltly with those performed onthe phosphorus sites and with neutron scattering data. In addition, the material has verylow symmetry (corresponding to the space group P2/a) and very little is known aboutthe possible small E and D terms in the Hamiltonian and their corresponding symmetry.There is fair qualitative agreement between the 31P NMR data and our theory, and it ispossible to explain some of the discrepancies using a temperature dependent anisotropicgap structure, but we feel that not enough is yet understood about gross features of thematerial to justify such speculation at this time.Chapter 6Suggested Experiments and Curious PredictionsWe finish by pointing in this final chapter towards further experimental work which couldserve to both better understand S = 1 1DHAF’s as well as corroborate and clarify someof the issues raised in this thesis.6.1 Experimentally Testable Conflicts Between ModelsWhen discussing the matrix elements, < k, aSi(O)q, b >, within the different models,we noticed that there were some discrepancies between predictions. We now examinethis hoping to offer experiments that would resolve the issue in favour of one model oranother.We start by discussing experiments on isotropic systems. In this case, the majordifferences between the predictions of the models concern large 0 transitions, where werecall from Chapter 2 that< k,ai(O)q,b > jfiabG(O) cosh(O) = —v2kq)/z (6.1)This is especially dramatic in the case of backscattering. The problem with an experiment which probes large 0 transitions is that contributions from matrix elements of thestaggered part of the spin may be large as well. This can be cured by looking for a lowtemperature experiment (T << ), where the energy exchanged with the probe is small.As shown in the analysis of 1/T, the staggered contributions will be suppressed by a109Chapter 6. Suggested Experiments and Curious Predictions 110double Boltzmann factor, e_2/T. A good candidate for such an experiment1 is elasticneutron scattering at zero or near zero magnetic field. The cross section is proportionalto the spin correlation function; for elastic scattering, this isS(Q, 0) cc <n (0)m> 26(w—cc.rn)6(Q— k + km)e_m’T (6.2)n,mAt sufficiently low temperatures, this expression is simpler than the analogous one forthe relaxation rate thanks to the momentum conserving delta function. The energyconserving delta function ensures that only backscattering will contribute to the crosssection. Using the results of Chapter 3 we easily integrate this to giveS(Q,0) cc G(0)22e_wQ/2/T (6.3)vQThe NLa model givesG10 2— ir4 1 + (0/ir)2 tanh(0/2) 6 4“ ‘I— 64I1+(0/2ir) 0/2At large Q this will behave as 1/ log2(vQ//.S. This is very different from the free bosonprediction of G(0) = 1, and from the free fermion prediction of G(0) —*z2/(vQ) forlarge Q. We need to qualify what we mean by ‘large’ Q. As discussed in Chapter1, the field theoretic models introduced are expected to be accurate only for Q nearzero and K. If we want to explore the two magnon nature of the structure function,we must be near Q 0. What we mean by ‘large’ momentum elastic scattering isthe investigation of the structure function near the border region where the field theoriesbegin to diverge from numerical simulations [16]; a region which satisfies all the criterion is.27r Q < .4ir. This corresponds to energies three to six times that of the gap. We expectthat the differences between the models should be discernible in this range. The reason wesuggest the experiment be done at zero or nearly zero magnetic field is to ensure that only‘T’ relaxation is not an appropriate tool since the transitions are dominated by small momentumtransitionsChapter 6. Suggested Experiments and Curious Predictions 111backscattering transitions contribute. For nonzero field, interbranch transitions can occurat large momentum which will not necessarily select oniy backscattering events. This willnot serve to make the interpretation transparent. The condition for backscattering evenin the presence of a magnetic field isQ>> h/v (6.5)In the case of axial symmetry, we can suggest the same technique to investigate thedifference between the zero field predictions of the boson and fermion models. Regardlessof the size of D, if one only considers the cross section for scattering with Qv > J,then the fermion model predicts a result that vanishes as while the bosonmodel prediction only involves the exponential factor. The same comments apply to thecase where axial symmetry is broken as well. This is no surprise since at large enoughmomentum, 0(3) symmetry is effectively restored.Elastic neutron scattering is a good probe for the matrix elements involving largemomentum and small energy exchange. Other techniques which explore the oppositeregime are electron spin resonance (ESR) or far infrared absorption experiments. In both,one subjects the magnetic system to an external source of electromagnetic radiation (themicrowave frequency value of the radiation depends on the transitions one is interestedin investigating). The RF field couples to the spins in the same way that a magnetic fielddoes, assuming that the electric dipole moment of the electrons on the magnetic ion ismuch smaller than the effective spins.2 The interaction Hamiltonian is thereforeH1= flRFG. cos(ôt) (6.6)Since the coupling is to the total spin of the system, the resonant transitions impliedby Fermi’s Golden Rule will involve energy w and zero momentum exchange. At low2A rigorous treatment would try to treat the coupling to the electric dipole moment; this can be donewithin the spin manifold using the Wigner-Eckart theorem. We will not bother with such a treatmenthere, but we note that it may be crucial in understanding some experiments on NENP [32]Chapter 6. Suggested Experiments and Curious Predictions 112temperatures, the power absorbed when a uniform field is applied to the system will be1(w) cc <rilSom>I26(w(h) — wm(h) — w)S(k — km)e_wm/Tn,m1< a;k,w(h)ISoIb;k,w(h)> 2e/Tk (6.7)where a and b denote one magnon states and k satisfies, 4(h) — w(h) = w. Sincethe density of states factor in the above is divergent for k = 0, it stands that 1(w) willhave a peak at the value of h for which z(h)— LP(h) = w. (The divergence will becured by higher dimensional effects as discussed previously.) In a typical experiment,one judiciously chooses the RF frequency, w, to be in the vicinity of desirable transitions,and the uniform field is then tuned to the peak in the absorption power. This is mucheasier to do than to fine tune the RF field.Let us now relate the ESR matrix elememts to la,b(0, 0), calculated in Chapter 2.<a;0IS0Ib;0 >= f dx <a;0jS(x)Ib;0 >= f dx= f dx <a;OIS(0)lb;0 > Ll(0,O) (6.8)Interesting conflicts between the models can be seen when there is some kind of, preferablylarge, anisotropy. For example, considering axial symmetry with a large D anisotropy,Il_(0, 0)12 (z3/ + L./z3+2) Bosons (6.9)2 FermionsThe maximal difference corresponds to L3/&L 2 which leads to a discrepancy of about13% between the models. The closer the two branches lie, the better the agreement between the models. This suggests the following experiment on highly anisotropic materials(NENP being a prime candidate). One chooses two RF frequencies. The first should correspond to the large interbranch gap, D, and the peak absorption ought to be measuredChapter 6. Suggested Experiments and Curious Predictions 113with a low field placed along the direction of the D anisotropy. The second RF frequencyshould be 0(E) if the material breaks axial symmetry, or 0(h) if the material is axiallysymmetric. This should then be used to measure the absorbed ESR power with the positions of the uniform and RF fields exchanged. This second transition will involve matrixelements which will be gap independent in both models. The matrix elements from thefirst transition can be extracted and compared to that of the first. If the boson modelis a better description even at these low fields, then the two matrix elements should beidentical.One may argue that it is redundant to make both measurements since, if the gapsare known, Eqn. (6.7) should give the correct description. The problem lies in cuttingoff the infrared divergence at the absorption peak. This will introduce an unknownproportionality constant. This divides out when comparing the two measurements. Theratio of the two measurements would be1(wi) 1 — e_whIT /3026 101(w2)— 1 — e2/T ( , ) ( . )where we assume a small field, h << &L, and small E << D.To end this section, while on the subject of ESR experiments, we would like to proposeadditional experiments to test the impurity model presented in Chapter 4. ESR is idealfor such tests. Used in conjunction with T’ measurements on a given sample, it wouldbe possible to characterize the couplings J’ and D’ of the end spins6.2 Measuring Small AnisotropiesRecall that we expect a peak in T’ whenever two branches cross. Experiments onHaldane Gap materials have yet to look for these. The sharpness of this peak dependson the interchain couplings which cut off the diverging integral in the calculation ofthe relaxation rate. Often, this will be broad because intrabranch transitions will shareChapter 6. Suggested Experiments and Curious Predictions 114the same cutoff (ie. when J1 > &.‘N). However, the bump should be experimentallyobservable. We propose that information about the anisotropy tensor can be extractedfrom this phenomenon. Essentially, one looks for the lowest field at which this bumpoccurs. This would give the direction of the lowest branch and the size of the anisotropy.We now explain this further.We assume the material in question has a well resolved D anisotropy and a seemingly degenerate doublet unresolved by other experimental techniques, such as neutronscattering. One begins by placing a uniform magnetic field in the plane perpendicular tothe axial direction (ie. somewhere in the xy-plane). The magnitude of the field shouldbe h2 > 6(/a—where is the uncertainty in resolving the doublet. One thenproceeds to measure T1 for different angles in the xy-plane spanning a region of atmost 1800. If there is an E type anisotropy, one ought to see some structure to the dataas a function of angle. Moreover, if there is such structure, we expect a bump at theangle where the branches cross. Once this angle is found, the experiment is repeated forsomewhat lower field. The angle where the new bump should be seen would be greaterthan the old. There is actually enough information in these two measurements alreadyto determine the anisotropy tensor. The dispersion relations are a function of the angleof the field (relative to some axis), the field magnitude and the gaps. The only unknownsare the absolute angle (or location of the axes of the anisotropy tensor) and the differencein gaps, I—.The two measurements could be used to solve for these two unknowns.In principle, one could also continue lowering the field and looking for the bump angleuntil it’s clear that signal is being lost when the field is reduced further. At this point,one has located the minimum crossing field which must lie along the direction associatedwith the middle gap. This field also gives the anisotropy: I—=h2/ILD— I.It would be interesting to perform such an experiment on NENP. Presumably, onewould find two angles corresponding to the two inequivalent chains in each unit cell.Chapter 6. Suggested Experiments and Curious Predictions 115Moreover, one would be able to verify the claims made in the last chapter regarding thepositions of the local anisotropy tensor in NENP.6.3 ESR for NENPIn the last chapter we noted that NENP has two inequivalent chains per unit cell. Furthermore, their local anisotropy tensor was argued to have symmetry axes which did notcorrespond to the crystal axes. These facts have important ramifications for ESR experiments on NENP. Figure 6.1 shows the dispersions for chains 1 and 2 (bold and lightlines, respectively) when the field is ?r/3 from the crystallographic c axis in the ac-plane.This is an example of how transitions at two field strengths ought to be possible in theESR experiment.Figure 6.2 shows the resonance field versus orientation of field in the crystallographicac-plane. The lower branch denotes transitions in chain 1 while the upper branch corresponds to transitions in chain 2. The transitions were calculated at .19 meV. corresponding to 47 0Hz. In addition the experimental results of Date and Kindo [49} arerepresented by the circles. One immediately sees that the data does not compare wellwith the predictions based on the models we’ve used so far, for instead of following one ofthe branches, the experimental results lie between them. Furthermore, it seems unlikelythat perturbations will cause such a significant shift in the resonance field. One sees thatthe discrepancy is ‘.‘ ±1 Tesla. One possible explanation is that since the ESR signal in[49] was also ‘—‘ ±1 Tesla in width and symmetric (in conflict with the predictions of [47]),the signal from the resonances in both chains was somehow smeared and interpreted asone single peak. Seen that way the model predictions are in good agreement except forthe large field regime. One also has to keep in mind that the high-field boson dispersionsare not accurate and therefore the predictions at larger angles could easily be .5 TeslaChapter 6. Suggested Experiments and Curious Predictions 116ccCct.c-c0/ , 0/ , CD__/1E / J’0—/ ,/ f/ f/ // I/(jew) ABJeu3Figure 6.1: Dispersions for the two chain conformations and sample resonant transitionsfor a uniform field placed 600 from the crystallographic c-axis.Chapter 6. Suggested Experiments and Curious Predictions 1174-a)><--LU P ° I • Io /I ////-d/ cc/ 0/0..z ILI.] I 000 CI-Z CD(isei) PIe!d eouuoseFigure 6.2: Resonant field vs. field orientation in the ac-plane for .19 meV transitions.Chapter 6. Suggested Experiments and Curious Predictions 118(or more) off the mark. We propose that further ESR experiments be done on NENPwhich specifically look for the double resonance predicted here.To end this discussion, we’d like to elaborate on a previously made statement regarding the assignment of masses to the local Ni symmetry axes. It’s easy to see thatswitching the masses around is tantamount to a ir/2 shift in Figure 5.2 (the fact thatthe gyromagnetic constants are not the same in orthogonal directions will not changethe ESR resonance graph much since the ratio of the gyromagnetic constants is 0.98).Redrawing Figure 6.2 with this geometry misses the experimental results by 4 Tesla at0 and 90 degrees, where the two chain resonances coincide. This determines the properlabeling of the local symmetry axes.Chapter 7Concluding RemarksWith the increasing theoretical interest in low dimensional systems, there has been aproportionate increase in the number of both realizable physical systems and experiments.This work offers a comprehensive analysis of NMR relaxation in Haldane gap materials,taking account of anisotropy and other material properties. As well, our analysis has ledto predictions pertaining to other types of experiments. It is hoped that our efforts willaid in both extending and clarifying existing knowledge of the subject.119Bibliography[1] F.D.M. Haldane, Phys. Lett. 93A, 464 (1983). For a review see I. Affleck, J.Phys.:Condensed Matter 1, 2047 (1989).[2] See C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963) Ch. 4, for areview.[3] P.W. Anderson in Solid State Physics, ed. F. Seitz and D. Turnbull (Academic Press,New York, 1963) Vol. 14, p. 99.[4] See A. Abragam and B. 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