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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 HALDANE GAP MATERIALS: AN ILLUSTRATION OF THE USE OF (1+1)-DIMENSIONAL FIELD THEORY TECHNIQUES By Jacob. S. Sagi B. Sc. (Physics) University of Toronto, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  August 1995  ©  Jacob. S. Sagi, 1995  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Physics The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada V6T 1Z1  Date:  Abstract A comprehensive theory of nuclear magnetic relaxation in S  =  1 Haldane gap ma  terials is developed using nonlinear-a, boson and fermion models. We find that at tem peratures much smaller than the lowest gap the dominant contribution to the relaxation rate comes from two magnon processes with T’  ‘-  e_m/T,  where  Lm  is the smallest  gap corresponding to a polarization direction perpendicular to the field direction. As the gap closes, we find that the dominant contribution comes from one magnon pro cesses, and the result depends on the symmetry of the Hamiltonian. Overall the models agree qualitatively, except near the critical regime, where the fermion model is shown to be the best description. We include a thorough discussion of the effects of interchain couplings, nearest neighbour hyperfine interactions and crystal structure, and introduce a new theory of impurities corresponding to broken chain ends weakly coupled to bulk magnons. The work is then applied to recent measurements on NENP. We find overall fair agreement between available T’ data and our calculations. We finish by suggesting further experimental tests of our conclusions.  II  Table of Contents  Abstract  ii  Table of Contents  iii  List of Figures  vi  1  2  Introduction and Background  1  1.1  Introduction  1  1.2  Spin-wave Theory  3  1.3  Non-Linear a (NLu) Model  7  1.4  Boson Model  15  1.5  Fermion Model  17  1.6  Nuclear Magnetic Relaxation Rate  20  Details of the Models 2.1  25  NLa Model: Temperature and Field Dependence of the Spectrum; Exact Results  2.2  25  2.1.1  Temperature Dependence of the Gap  25  2.1.2  Field Dependence of the Gap  29  2.1.3  Exact Results  34  The Free Boson and Fermion Models  37  2.2.1  37  Diagonalization  111  2.2.2  Discussion: Comparison of Spectra and Spin Operator Matrix El ements  3  47  Model Predictions for Tj’  53  3.1  T’ for h << h  58  3.1.1  Isotropic Symmetry  62  3.1.2  Axial Symmetry  63  3.1.3  2 x Z 2 Symmetry 2 x Z Z  66  3.2  Close to the Critical Field  67  3.3  Above the Critical Field  70  3.4  Summary  71  4 Material Properties and Possible Effects on Experiment  5  6  7  72  4.1  Hyperfine Tensor  72  4.2  Impurities  74  4.3  Interchain Couplings  84  4.4  Crystal Structure  86  NENP: Direct Comparison with Experiment  88  5.1  The Structure of NENP and Experimental Ramifications  88  5.2  Analysis of the Data  95  Suggested Experiments and Curious Predictions  109  6.1  Experimentally Testable Conflicts Between Models  109  6.2  Measuring Small Anisotropies  113  6.3  ESRforNENP  115  Concluding Remarks  119  iv  Bibliography  120  V  List of Figures  2.1  Free boson and free fermion dispersions with the gap parameters of NENP. Top graph:  2.2  flIIb;  bottom graph: flWc  Corrected free boson (t  =  z(H)) and free fermion dispersions with the  gap parameters of NENP. Top graph: 3.1  32  ilIb;  bottom graph:  ftIc  33  First non-vanishing contribution to relaxation due to the staggered part of the spin  54  3.2  Inter- vs Intrabranch transitions  56  3.3  The gap structure for 0(3) symmetry  62  3.4  The gap structure for U(1) symmetry. (a) /.S >  4.1  Impurity level diagram when D’  4.2  Impurity level diagrams for D’  5.1  NENP  5.2  Local and crystallographic axes projected onto the ac-plane in NENP  5.3  A projection of the NENP unit cell onto the ac-plane showing two chains  =  zj; (b) LS <z  .  64  .  0  76  0  78 89 .  90  per unit cell  92  5.4  Boson F(h, T) for fields along the b and c chain directions  97  5.5  Fermion F(h, T) for fields along the b and c chain directions  98  5.6  Boson F(h, T) for fields along the b and a chain directions  99  5.7  Fermion F(h, T) for fields along the b and a chain directions  100  5.8  Theoretical (lines) vs. experimental data (circles and squares)  102  vi  5.9  Theoretical (lines) vs. experimental data (circles and squares)  103  5.10 Relaxation rate for field along the b-axis  106  5.11 F(T)—the width of the impurity resonance  107  6.1  Dispersions for the two chain conformations and sample resonant transi tions for a uniform field placed 60° from the crystallographic c-axis.  6.2  .  .  .  116  Resonant field vs. field orientation in the ac-plane for .19 meV transitions. 117  vii  Chapter 1  Introduction and Background  Introduction  1.1  In 1983, Haldane derived his famous result stating that integer spin one dimensional Heisenberg antiferromagnets featured a gap in their low energy excitation spectrum [1]. Since then, much effort has been devoted to further exploration of such systems, both experimentally and theoretically. The purpose of this work is to develop a theoretical framework for the understanding of low energy experiments on one dimensional Haldane gap 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, it is hoped that the reader can become familiar with the tools used to understand integer spin Heisenberg antiferromagnetic chains with anisotropies, and can apply these tools to the analysis of real systems. There are, essentially, three important models that have so far been used to describe the system. In the later sections of this chapter we review the competing descriptions of S  =  1 antiferromagnetic spin-chains, paying some attention to their advantages and  shortcomings. We start by outlining the traditional spin-wave theory’ used to model antiferromagnetism in higher dimensions. After illustrating the deficiencies in this ap proach, we describe the Nonlinear a (NLa) model in some detail. This is followed by an analysis of a simplified yet closely related Landau-Ginsburg boson model. Last, we discuss a free fermion model used recently to successfully treat the case of anisotropic 1  [2]  for a comprehensive discussion of this topic 1  Chapter 1. Introduction and Background  2  spin-chains. We end the chapter with background on the nuclear magnetic relaxation rate, 1/T , for nuclear spins coupled to the spin-chain through hyperfine interactions. 1 Chapter Two focuses on the details of the models, building the tools necessary for a detailed analysis. We discuss the temperature and magnetic field dependence of the NLu model and its possible relevance to the spectrum of the boson model, as well as cite some exact results available in cases of high symmetry. We also diagonalize the free boson and fermion field theories, including on-site anisotropy effects. We derive matrix elements of the uniform part of the spin operator (fourier modes near wave vector zero) between one particle states of magnetic excitations. These are used to compare the different models. Chapter Three explicitly describes the calculation of NMR T , considering various 1 symmetries of the Hamiltonian. We identify the leading mechanisms for low temperature relaxation in the presence of a magnetic field. We discuss three regimes corresponding to different magnitudes of the applied external magnetic field, giving expressions for the rate in each case. We discover that at temperatures much lower than the smallest gap the uniform part of the spin operator contributes most to the relaxation rate; in the absence of interactions, this corresponds to two magnon processes. The rate is found to be T ’ 1  e_m/T,  where  Lm  is the smallest gap corresponding to a polarization  direction perpendicular to the magnetic field. As the externally applied magnetic field approaches a critical value, one of the gaps closes and we find the dominant process to be one 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, we show that the fermion model is the best description and that the expression for T 1 depends on the symmetry of the Hamiltonian. Chapter Four deals with intrinsic effects which must be taken into account when analyzing experimental data. We discuss nearest neighbour hyperfine interactions; we show that these will contribute to order A/A, the ratio of the nearest neighbour coupling  Chapter 1. Introduction and Background  3  to the local coupling. We also consider interchain couplings and show that they introduce a natural infrared cutoff to the diverging density of states at the gap; for sufficiently long chains, 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-i chain end degrees of freedom. We find that the states formed by such end spins in the gap, 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, 4 N N 8 H Ni(C ( 2 ) C10 0 (NENP). We take a close look at the crystal structure of NENP and identify possible terms which may be present in the Hamiltonian. We also note the fact, 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. We find 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. The impurity theory is used to model low field data with qustionable results. The final chapter proposes further experimental tests of the theoretical predictions of this work. We suggest elastic neutron, electron spin resonance and further NMR studies to verify our own.  1.2  Spin-wave Theory  The Heisenberg Hamiltonian describing the isotropic antiferromagnetic spin-chain is H=J  •  Ii+1  J>O  (1.1)  This arises naturally from the Hubbard model for insulators at ‘half’ filling [3]. To understand where this might come from, we follow the case where there is a triplet of possible spin states per site. On each site there are a number of valence electrons (eight Ni+ for example); the degenerate electronic levels , valence electrons in the d shell of 2  Chapter 1. Introduction and Background  4  2 and the symmetry of the crystal fields are split in a way determined by Hund’s rules 2 in a field with octahedral surrounding the ion. In some special cases (as with Ni+ symmetry), a degenerate triplet of states lies lowest. The ensuing low energy physics can be essentially described using effective spin 1 operators [4]. By ‘half’ filling, we mean that there is an effective S  =  1 triplet of states for every site in the chain (ie. there are two  singly 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 comes from allowing a small amplitude for nearest neighbour hopping which is highly suppressed by coulomb repulsion from the electrons already occupying the site. In the quantum case, the spin operators have the commutation relations: [S,S] where  =  j5  fabc c 5 ,  =  .  is the Kronecker Delta Function and  s(s + 1)  (1.2)  is the completely antisymmetric Levi  Civita symbol. It is easy to see that the classical Néel ground state with alternating spins is not the quantum ground state. To this end we write the Hamiltonian in terms of raising and lowering spin operators: S  H  =  j  [s:s7+ + 1  Sx±iSY  +1 S1S ) ]  The Néel ground state is composed of spins alternating in quantum numbers  (1.3) SZ  between  sites. INéel>=s=+1,s=—1,s=+1,...,4=—1>  (1.4)  Hd’s rules maximize the total electronic spin and the total angular momentum of the electrons in 2 the valence shell.  Chapter 1. Introduction and Background  5  This state is clearly not an eigenstate of the above Hamiltonian since upon acting on it, the StS 1 terms in the Hamiltonian generate states with m  =  0. To proceed in  understanding the low energy properties one usually assumes that the ground state is approximately Néel with quantum fluctuations. The picture is that of zero point motion about the positions of the classical Néel spins. What we will shortly see is that the assumption of small fluctuations breaks down in one dimension. The conventional approach makes use of the Holstein-Primakov transformation. One begins by dividing the chain into two sublattices, “A” and “B”, with adjacent sites on separate sublattices. On sublattice “A” one defines S=s—aa,  (1.5)  S1=a/2s—aaj  On sublattice “B” we have 1 S  =  —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 relations and 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, we assume that s is large. This is equivalent to a semi-classical approximation since for s  —+  oo the commutator of the spins will have much smaller eigenvalues than the square  of the spin variables [Sa, Sb]  =  lc 5 fabc  = 0(s) <<0(s ) 2  (1.8)  We expand the spin operators to give S=aV’,  =b/ 1 Sj  (1.9)  Chapter 1. Introduction and Background  6  To leading order, the Hamiltonian reduces to H  =  J  [_s  2 + 2bb 2 + ab + b_ a + ba + ab_ 1 ) + 0(1)] 1 + s(2aa  (1.10)  Fourier transforming: N  >  3 = a  (1.11)  ei 7 2 Nak  with a similar expression for b; N is the number of sites on each sublattice. Ignoring the constant term, we rewrite (1.10) as H  =  2Js>  [aLak + l4bk + (1 +  2a is the sublattice spacing and k  =  te 2 ha)(akb_k  irn/Na for n e  + b)aL)]  [—j, ].  (1.12)  We now make the Bogoli  ubov transformation, Ck  =  ukak  dk  =  Ukb_k  —  Vkb!k  —  vka  (1.13)  where, (1 + csc(ka))’ 12  Uk —ika/2  Vk=  The d’s and with SZ  =  c’s  112) (—1+csc(ka)  (1.14)  are spin wave operators corresponding to magnetic excitations (magnons)  ±1 respectively. This transformation preserves the commutation relations and  the Hamiltonian can now be written as H=2Jssin(ka)  (cck+c4cik)  (1.15)  We see that this low energy description implies the spins are in some coherent state of a’s and b’s built on the Néel state, but there are long wavelength Goldstone modes  Chapter 1. Introduction and Background  with dispersion relation  7  = 2JsIka which allow each of the sublattice magnetization  vectors to make long wavelength rotations. To ensure consistency, we now look at the expectation value of the magnetization (say, on the “A” sublattice), hoping to see that we get the semi-classical result < S >= s  —  0(1). We invert the Bogoliubov transformation  to get (1.16)  ak =ukck+vkdk  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 spin wave occupation number. The problem is now apparent: low wavelength modes cause <  s  —  SZ > to diverge logarithmically. The semi classical picture of a Néel-like ground  state is completely off. This is special to 1 dimension and actually arises as a general consequence of Coleman’s theorem [5]; it states that in (1 + 1) dimensions, infrared divergences associated with Goldstone bosons will always wash out the classical value of the order parameter rendering spontaneous symmetry breaking of continuous symmetries impossible. One is therefore forced to look elsewhere in order to describe the low energy physics of the Heisenberg model.  1.3  Non-Linear a (NLa) Model  The 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 is also based on a large s approximation. One introduces two fields:  —  corresponding to  Chapter 1. Introduction and Background  the Néel order parameter, and  r  8  the uniform magnetization. The spin variable, S is  defined 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 limit is taken in the semi-classical large  approximation to arrive at the NLu model Hamil  s  tonian [6]. There are several problems with this approach. First, parity is broken in intermediate steps and is eventually restored in the continuum limit. Second, and more importantly, the crucial topological term which is found in the continuum Hamiltonian is derived without clear notions of how 1/s corrections may be made. A much more elegant approach will be reviewed here. We will make use of path integral formalism based on independent derivations by Haldane [7] and Fradkin and Stone [8]. These were motivated by similar questions about topological terms in 2-D quantum models thought useful in attempting to describe high temperature superconductors. One begins by defining a coherent basis of states for the S  =  s  representation of  SU(2) [9]:  n> where +  s  + s>  > is the eigenstate of 5Z with eigenvalue  vector perpendicular to both ñ direction (ie. i  >=  and . We see that  i  sñ  (1.19)  cos 6 and (xñ) is a unit  s,  ñ  I>  is the state with spin  =  s  in the  >.) This basis is over-complete and not orthogonal. We  now make use of two identities: <nun2 >= 1  =  2s±  f  eis  ñ 3 d  1,2)  .  (1 +  —  fli  ) 2 n  1)In><  (1.20) (1.21)  We note that the derivation in reference [8] is somewhat in error. We correct their mistakes using a 3 similar derivation due to Ian Affleck (unpublished).  Chapter 1. Introduction and Background  , 1 (n ,  ) 2 n  9  is the area enclosed by the geodesic triangle on the sphere with vertices at  2 and the north pole. There is an ambiguity of 4ir in this definition, but this makes n  little difference when exponentiated since  8 = 4 e  identity is most easily proved by using the  1 for >  , ...,  s  integer or half-integer. The first  representation of SU(2) while the  second follows from the first. We can now use these states to write the partition function or path integral of the system:  rf  =  [N_i  2s± 1d3()]  A1IT  With zXT  —+  =  /3,  Tm  Tm_i  —  =  T,  (ñ(Tm),(Tml))  ei  (1 +  (Tm)ñ(Tm_i)  In the limit N  (Tm)  8 (Tm_i))  —*  =  TN  <(Tm)IHI(Tm)  >  LT  T  (1.23)  cc, the path integral can be written,  f  Tre’ cc  [<(Tm)IHI(Tm)  =  1 T  > — <fl(Tm)IHfl(Tm) >  —  S  (1.22)  0, we expand the exponential to order O(z.T) to get  <ñ(Tm)Ie_THI1i(Tm_i) >< =  <(Tm)Ie_T(Tm_i)> m=2  1=1  >  T  —  [Dñ(T)] e_S  (1((Tm)(Tm_i)  (1.24) 1 +ln(  +(Tm)  .  (Tm_l)))]  m=2  The last term can be written to second order in SZT  ——i—  f  d-(i-)  This vanishes in taking the limit enclosed by the vector Parametrizing  ‘i  ñ(T)  Ofl() tT —÷  ST =  —i--.  LT  f  as dT  2 (ôñ(T))  (1.25)  0. The sum over the phases is just the area  as it traces its periodic path on the surface of the sphere.  as =  (sinOcosq, sinOsinq5, cosO)  (1.26)  Chapter 1. Introduction and Background  10  we 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 therefore extend the path integral to all sites. This is done by indexing each of the coherent states with a position label, x, and making the substitution (Tm)  >+  ®  (1.28)  fi(Tm,3J)>  Note that  <ñ(T,x)I (x)n(r,x) It is useful to write  I(r, x)  >=  (1.29)  sñ(’r,x)  in terms of a staggered and a uniform part which are slowly  varying in the limit of large s: i(r,x)  (—1)(r,x) +  (1.30)  tr,x)/s  To leading order in 1/s and derivatives of the slowly varying fields, this produces the constraints (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 a posteriori), we find that the leading contribution of the Hamiltonian to the action in the continuum limit is ffdxdr  (() 4 2 r/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  11  Because A is an oriented area with respect to its argument, changing the sign of the argument also changes the sign of the area. Eqn. (1.33) can be written, -isA  =  -f f  dx (A [(r, x) +  where to leading order, ö(r, x) —isA  =  = —-  -f f =  f  dx dr ((r x)  If we compactify =  so that  ó(T, x)j  -  A [(r, x)})  (1.34)  O(r, x) + 2 f/s. This then gives,  f  ((r, x) x S(T, x))  dx  dx dr (r, x). (6(T, x) x 8(r, x))  (ô(T,  —*  x) x  ôT(T,  constant for  x))  -  2 2 + -r 1x  2 ((r, x) x  —÷  8(T,  x))) (1.35)  oc, and maintain the constraint  1 (valid to 1/s ), one can recognize the integral 2 Q=_-fdtdxc.(84x8)  (1.36)  as measuring the winding number of the sphere onto the sphere. The integrand is the Jacobian for the change of variables from compactified coordinates on the plane to those on i-space (also a sphere).  Q  is an integer corresponding to one of the countably many  topologically 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 possible topological 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 between partition functions with even and odd Q’s. It is important to stress that this is a purely quantum mechanical result which has no analogue in the 2-D finite temperature classical Heisenberg model (there is a well known equivalence between (d, 1)-dimensional quantum field theory and d + 1-dimensional finite temperature classical statistical mechanics [10]).  Chapter 1. Introduction and Background  12  A 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.  f  We can solve the equations of motion for i  —.  /—*  —*\  (1.37)  gv  Not surprisingly, ris the generator of rotations. After integrating out the rfields, the final action is, S= —2irisQ+ ffdxdr  (8)2+  !ff’drdx  2 (Ovrc5)  (1.38)  Where we now define, v  =  2Js  g  =  (1.39)  -,  The action can be written S  =  2irisQ + -  f  dx dr 8D’  (140)  It is clear how 1/s corrections entered into the calculation of the topological term. More over, 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 the  topological term in the action, as discussed, and consider the nonlinear u-model: £  2 Js =  2 =  1  (1.41)  We now motivate the idea that, contrary to spin-wave theory, this model features a gap in its low energy spectrum. We first do this in the spirit of reference [10]. We can deal with the constraint by incorporating it into the path integral as a Lagrange multiplier: Z cc  J  -.  DebVAe  _ij-fd2x (o+A(;_1))  (1.42)  Chapter 1. Introduction and Background  13  The constraint is now enforced by the equation of motion for ). The  fields can be  integrated out in the usual way to give cc  (1.43)  where N is the number of components of . As N  —*  cc the path integral is concentrated  near the smallest value of the argument of the exponential. Minimizing this argument with respect to ). we solve for the saddle point, ): Nv  2 Js  —=—<xI 2 2 using  1  (1.44)  the standard rules for functional differentiation. The RHS of the above is simply  the Green’s  function  for a boson field with mass v\/;  <xl <1  1  •1 - >  1 k 2 rd Ix >= ir (2K) kPk+), 2 =  (1.45)  2 A 1 —log-= 4K A  where A is an ultraviolet cutoff, d k = dk dw/v, and kJLk = k 2 2+2 /v Solving for the . square of  the mass, A: 2 A = A2e_46  (1.46)  Another way to see the presence of a mass gap is to integrate out ultraviolet modes and apply the renormalization group. We start with the Lagrangian Eqn. (1.41) and parametrize the fluctuations in terms of slow and fast modes. One then integrates out the fast fields. This calculation is logarithmically infrared divergent. One then renormalizes by subtracting out the offending terms from the effective Lagrangian. Equivalently, one can achieve the same effect to the same order in perturbation theory by redefining the coupling constant in terms of its bare value. A calculation of this sort (for the 0(N)  Chapter 1. Introduction and Background  14  model) is done in reference [10]. The renormalized coupling constant becomes g(L) With g 0  =  (1.47)  1— lnL  2/s, we now see that the coupling constant is of order unity for length scales 6 e  (1.48)  Keeping in mind that this is a Lorentz invariant’ theory, there must be a corresponding mass scale, : (1.49)  oc  There are other similar heuristic calculations that suggest a mass gap; none are iron clad, but the sum of them together makes for strong evidence that indeed the s  =  1 1-D  Heisenberg antiferromagnet is disordered at all temperatures and is well described by the NLo- model. Better justification comes from exact S-matrix results and numerical simulations. The exact S-matrix results are due to work by Zamolodchikov and Zamolod chikov [11], and Karowski and Weisz [12]. The 0(3)  invariance of  the NLu model allows  for an infinite number of conservation laws. These imply strong constraints on S-matrix elements and, consequently, on on-shell Green’s functions. One characteristic of such an S-matrix is factorizability. This means that N-particle scattering can be expressed as products of 2-particle scattering matrix elements. The simplest such S-matrix consistent with the symmetries of the NLu model has a triplet of massive soliton states with an effective repulsive local interaction. This conjecture has been checked in perturbation theory in 1/N (for the 0(N) NLu model [12]) to order 1/N . 2 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 of the NLo- model. To date, the best numerical work has been due to White’s method of  Chapter 1. Introduction and Background  15  the density matrix renormalization group [14] and recent exact diagonalization [15]. The former predicts a gap  =  .41050(2)J, while the latter has L  =  .41049(2)J. Numerical  investigations of the spin operator structure factor [16], S(k), for the isotropic chain show remarkable agreement with the ‘exact’ S-matrix result for two magnon production over a region larger than expected; two magnon production is known to dominate at low momenta, k < .3ir, from numerical studies [16]. This can be probed in neutron scattering experiments [17]. For higher momenta, one must include one magnon contributions which dominate as k —*  K.  The intermediate region in momentum space,  .8K, is not  k  .3K  expected to be well represented by the NLu model; this is because the fields, describe low energy (and therefore large wavelength) excitations about k respectively. The same study also determined the correlation length, velocity, v/J  ‘..‘  2.5 and the coupling constant, g  expansion result [18] g  “.‘  ‘-.‘  = K =  and  and k  =  f 0,  6.03(1), the  1.28 in rough agreement with the 1/s  1.44 and the value derived above, g  =  2/s  =  2.  This ends the introductory discussion of the NLu model. A more in-depth approach will be taken when we consider anisotropies and develop the necessary tools to calculate the NMR relaxation rate in Chapter 2.  1.4  Boson Model  Although the NLu model is convincingly accurate in describing the low energy physics of the Heisenberg 1-D antiferromagnet, it has several deficiencies. First, off-shell Green’s functions are not known; and second, anisotropies are not easily tractable within the framework of the model (the 5- matrix is no longer factorizable, as earlier discussed, since one loses the infinite number of conservation laws). A happy compromise which contains all of the qualitative aspects of the NLu model and yet allows for more computability  Chapter 1. Introduction and Background  16  and generalization is the Landau-Ginsburg boson model [20] (.)2 =  where the constraint  =  2+ vj  +  +  (1.50)  1 has been relaxed in the Lagrangian of the NLu model, and a  interaction has been added for stability. The Hamiltonian, (1.50), possesses the correct symmetries, three massive low energy excitations, and a repulsive weak interaction. As with the NLu model, the field  acts on the ground state to produce the triplet of massive  excitations or magnons. We note that this model becomes exact in taking the N -4 co limit of the 0(N) NLa model [10] (recall that N is the number of components of the field  ).  As in Eqn. (1.37), the generator of rotational symmetry (the uniform part of  the spin operator) is (1.51) where 11 Expanding  (we absorbed the coupling constant,g, into the definition of  in (1.50).)  in terms of creation and annihilation operators, we see that 1 acts as a two  magnon operator producing or annihilating a pair, or else flipping the polarization of a single magnon. This picture is obvious in this simpler model, whereas the same analysis is only confirmed by the exact S-matrix results and the gratifying agreement with numerical work in the case of the NLu model. The gap, Li can be phenomenologically fitted to experiments such as neutron scattering as can be the velocity of light’, v. Including on-site anisotropy Haniso  =  (D(sfl2 +  2 E((S)  —  (S)2))  (1.52)  simply amounts to introducing three phenomenological masses. This will be discussed in more detail in Chapter 2.  Chapter 1. Introduction and Background  17  On comparison of the predictions of both models one finds overall qualitative agree ment in studies of form factors [16,  191.  As one moves away from zero wave vector the  agreement between the models weakens. This makes for one of the disadvantages of the bosons. Also, in attempting to calculate certain Green’s functions, such as the staggered field correlation function, one is forced to rely on perturbation theory in ). Although can be phenomenologically fitted, there is much ambiguity in choosing the interac tion term. One can equally put in by hand any positive polynomial term in  This  is because the fields carry no mass dimension making all polynomial interaction terms relevant. It should be understood that this model is phenomenological and is introduced for its simplicity. In the final analysis, justification for its use must come from numerical and real experiments.  1.5  Fermion Model  Before introducing the next model, we would like to begin by apologizing for the cryptic description of the concepts to be mentioned in this section. A deeper understanding would require a diversion into conformal field theory tangential to the main lines of the thesis. 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 of relativistic fermions: 7-1(x)  =  1—. d ,bLzv—  1-.. d  —  —  bR+  —  X  )-  (Rx  R)  (1.53)  Chapter 1. Introduction and Background  The fields  ‘  18  are Majorana (Hermitean) fermions with equal time anticommutation rela  tions {‘/4(x),’z/,(y)}  =  1 6(x—y) 6ss’ö  S,S’  =  L,R  (1.54)  The L and R label left and right moving fields, respectively. This model is not trivially related 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 Gap material NENP. The motivation comes from a model sitting on the boundary between the Haldane phase and a spontaneously dimerized phase [22], with the Hamiltonian H  =  i:  J>  i+1  —  ( g.  g)2]  (1.55)  This Bethe Ansatz integrable Hamiltonian features a gapless spectrum and has a con tinuum limit equivalent to a k  =  2 Wess-Zumino-Witten (WZW) NLu model. This, in  turn, 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) as t(x)=(’L.-1L_  (1.56)  Reducing the biquadratic coupling in (1.55) moves the Ising models away from their critical point. Symmetry allows the addition of interactions corresponding to mass and four fermi terms, as in Eqn.  (1.53).  The four fermi term proportional to ) is the  only marginal one allowed by 0(3) symmetry. It will generally be ignored or treated perturbatively, in a similar phenomenological spirit to that of the Landau-Ginsburg boson model (ultimate justification for this, as for the boson model, comes from numerical and real experiments). For weak interactions (which is the case assumed) all Green’s functions will have simple poles at the phenomenological masses and will be trivial on-shell. The  Chapter 1. Introduction and Background  19  off-shell behaviour depends on the interaction terms chosen and is therefore very much model dependent. It can easily be checked that (1.54) gives the right commutation relations for the SU(2) algebra, [l(x), 13(y)]  =  i6(x  —  y)euidl(x),  with  r=j(LxL+RxR)  (1.57)  This allows us to identify rwith the generator of global rotations or the uniform part of the spin,  .  Expanding  ‘z].R  and  1];’L  in terms of creation and annihilation operators, we  see that, here too, ris quadratic in such operators. Notice that this representation for  r does not couple left and right movers.  This is in sharp contrast to the boson or NLa  models (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 will contrast between the models. The particles created by the fields,  ‘,  are identified with massive magnons. The  masses can be fixed by hand to agree with the experimental dispersions so that onsite anisotropy terms coming from Eqn. (1.52) can be easily parametrized, as in the boson model. Other interaction terms which might arise from breaking the symmetry are usualy ignored for ease of calculation. As always, ultimate justification for this is found in numerical and real experiment. As mentioned above, fis again a two magnon operator. It is also possible to represent the staggered magnetization (the analogue of  )  in this approach, but it is considerably  more complicated (one can use bosonization techniques [23]). Near the massless point, this operator reduces to the fundamental field of the WZW model, or equivalently to products 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 corresponding correlation functions can be expressed in terms of products of Painlevé functions [24,  Chapter 1. Introduction and Background  20  27]. They exhibit poles at the fermion masses together with additional structure at higher energy. Unlike the free boson model, a simple interpretation of the staggered magnetization density as a single magnon operator doesn’t hold. This complicates the use of this model. One way to justify the use of the fermion model without resorting to complicated explanations is to notice that in the long wave length limit of the 0(3) symmetric case, all models are in agreement (see Chapter 2). For smaller wave lengths, the different models correspond to different continuum representations of the lattice model. 0(3) symmetry is broken differently in each model (for example, see the different results for matrix elements of fin Chapter 2). The idea is that we have three (two, for lesser symmetry) workable descriptions whose ultimate merits can only be decided phenomenologically.  1.6  Nuclear Magnetic Relaxation Rate  Experiments on condensed matter systems typically measure observables which are di rectly related to Green’s functions. This is no surprise since most such experiments measure the response of the system to an external probe. This is in contrast with par ticle physics experiments which usualy examine the nature of scattering into asymptotic states. Formally the difference is that particle physicists measure time-ordered Green’s functions while their friends in condensed matter physics measure retarded Green’s func tions. The nuclear magnetic relaxation rate, 1/T , measures the local correlations of the 1 system at low frequency. The probe is the nucleus of some atom in the sample which has a non-zero nuclear maglietic moment weakly coupled to the system of interest. In the case of the Heisenberg l-D antiferromagnet, we assume that in addition to the spin Hamiltonian, H , there is also Zeeman coupling, H, to a uniform magnetic field, 5  .&,  by both the nuclear and Heisenberg spins, and that there is a hyperfine coupling between  Chapter 1. Introduction and Background  21  the two systems, HHyper. We also assume for simplicity that the nuclear spins do not directly couple to each other. HTot  =  8 + Hz + HHyper H  =HSBftGe a-1zNfl.GN Ge  0 + HHyper H  i;+  i;  (1.58)  and GN are the gyromagnetic tensors for the electron and nuclear spins, respectively.  3 is the hyperfine tensor coupling the nuclear spin on site A  j  to the electronic spin on  site i. We now define the characteristic frequency WNjNH  (1.59)  In nuclear magnetic resonance (NMR) experiments one strives to temporarily achieve a non-equilibrium population difference between nuclear spins with different spin eigen values along the uniform field direction. This is normally achieved with pulses of RF electromagnetic radiation possessing ac magnetic fields perpendicular to the externally applied uniform magnetic field. As is well known, a resonance phenomenon occurs at RF-frequencies near  WN  (in reality it is easier to tune the uniform field to resonate with  a 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 different spin eigenvalues. This would not normally be possible if the nuclear spins were completely free. Coupling to another system is necessary in order to conserve energy during the transitions. The energy given off or absorbed must induce a corresponding transition into a different energy state in the system which couples to the nuclear spins. Let us illustrate the situation with an s  =  1/2 nuclear spin. In the absence of hyperfine interactions, we  assume that I is a good quantum number (where z is the direction of the static magnetic field), and that GN is isotropic (these are generally good assumptions). The rate equation  Chapter 1. Introduction and Background  for the number of nucleii, N, with I  22  + is  dN =—N÷Q+_+NL dt  (1.60)  Where the transition probabilities per unit time are given by  We can rewrite this  in terms of the total number of spins, N and the population difference, =  N(L÷ —  —  n(÷  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 much shorter than those characteristic of the experimental probe, and in the limit of linear response (ie. Q is independent of  n)  the solution to (1.61) is  n(t) =  Where  0 n  o+  (1.63)  a e_t’T1  is the equilibrium population difference (at finite temperature, states with an  energy difference will necessarily have a population difference). We see that the relaxation rate, 1/T , describes the evolution of the nuclear system towards thermal equilibrium, or 1 likewise, the decay of the population inversion magnetization achieved by RF pulses in NMR. We now derive an expression for the rate, 1/T . For a system with more general 1 I, 1/T 1 for a transition from an initial state with P normalized by the factor 1(1 + 1) — the transition rate I’  = m  m—*m+1  m(m  = m  = m  = m  + 1 is  + 1). To begin, we need an expression for  describing a nuclear spin at site  and ending up with I  to one with IZ  j  starting in the state with  + 1. It does not matter which nuclear spin we pick  if we assume translational invariance; since there is no nuclear spin-spin coupling , the 4 “In NENP, the dipolar nuclear spin-spin couplings are roughly 200 times smaller than the hyperfine  coupling.  Chapter 1. Introduction and Background  23  relaxation rate for one is the relaxation rate for the whole system. Let us assume that the initial and final Heisenberg spin state are given by the labels n and n’, respectively. Then Fermi’s Golden Rule gives =  2irI <1  =  (m + 1), fl’HHyperII.  e_Ef1T =  m, fl> 1 6(EI 2  (m--1),n!  —  Eirm,n)  z  (1.64)  Notice that we multiplied the normal expression for the Golden Rule by the Boltzmann probability that the Heisenberg spin system is in the initial state,  I  >. The only part  of HHyper which will contribute is SrAI: =  2ir(I(I + 1)  —  m(m  <n’ISflri>  + 1))  e_/T  ö(E 2  —  E  —  (1.65)  UN)  i,y  The analogous expression for  is IJrr(m+i),n’—*IJ=m,n =  -E,/T  2ir(I(I + 1)  Since Af  =  —  m(m + 1))IA <nSfln’> 2 6(E  — —  (1.66)  UN)  (A_)* we get for the relaxation rate =  2irI > A <n’ISIn> I 6(E’ 2  (e_En/T —  —  UN)  +  e_/T)  (1.67)  We now sum over all possible transitions to arrive at 1/Ti: 1 -  1  =21r>Af  1n> 1 <n’IS:  f —E,/T  6(E 2  —E—wN)  +  e —E/T  (1.68)  Chapter 1. Introduction and Background  24  Since 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 write 1  1 =  f  I  0O  dt °°  e’  <  >  ‘1 AS(O) >  AS’(t), m,  (1.69)  )  where <> denotes a thermal average. This is the famous expression derived in [28]. As promised, when A 3 is well localized, we see that 1/T 1 is related to the low frequency local correlation function.  Chapter 2  Details of the Models  In this chapter we discuss in detail the three models introduced in the last chapter. We will derive the necessary tools to calculate the relaxation rate 1/Ti and mention some pertinent 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; Exact Results  2.1.1  Temperature Dependence of the Gap  In this section and the next we will discuss how the excitation energies of the lowest modes change with varying parameters. This is especially important when one chooses to perform calculations using the Landau-Ginzburg boson model. Since this model im plicitly adopts the gap parameters from the NLu model, any dependence of the gaps on magnetic field or temperature must first be calculated within the framework of the NLu model. The results can become useful in interpreting experimental data using the simple boson 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 or even 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 three  25  Chapter 2. Details of the Models  26  massive bosons with relativistic dispersion, is an effective one. The true excitations of the model are collective, and if we insist on maintaining a single particle description we should not be surprised that the effective single particle interactions will be temperature dependent (as, therefore, will be the effective one-particle spectrum). A similar approach is taken in BCS theory where the BCS gap has a temperature dependence arising from a consistency condition. In Chapter One we introduced a consistency equation, Eqn. (1.44), for the classical or saddle point value of the Lagrange multiplier field, A, in the NLu model. This result always holds to lowest order in the fluctuating field A(x), regardless of the value of N in the large N expansion. Of course, it only becomes exact for N —* cc. We can also look at the consistency equation as a constraint equation guaranteeing that the  two point  function is unity when evaluated at the origin; when N = 3, 1 =< (x) (x) >= G (O) 2  (2.1)  (2 kk+  =  where we’ve assumed a renormalized mean value for A [10]. Notice, also, that we’re choosing to work in Euclidean space. One can likewise see that the constraint equation is nothing more than a minimization of the zero point energy of the system with respect to the fluctuating field A: Eo=3fwk  where  =  —  AJ  (2.2)  vkILk, + A. If we choose to add on-site anisotropies to the model, as  in Eqn. (1.52), then the contribution to the Lagrangian (modulo irrelevant terms which also break ‘Lorentz invariance’) is  —  2 — (Sfl E((S) ) 2  —*  2 D((x))  —  2 E((q(x))  —  (Y(x))2))  (2.3)  Chapter 2. Details of the Models  We can always find some axes,  27  xyz,  so that the addition to the Lagrangian is in the  above form. We can read off the new constraint equation on the Green’s function  ,rdkdw(  2 v  1  I  =  (2  w2  1 +2  k+ 2 +v  2  +  +  1 k+ 2 +v  2  1 2+v w k+ 2  +  (2.4)  2 —  Where we’ve once more assumed renormalized masses with the correspondence, 2vD ++ L 2 + L and  ± 2vE  A 2 v  / ±  A 2 v  +  As usual, making this model temper  .  ature dependent consists of replacing the integral over  by a corresponding sum over  w  Matsubara frequencies. Eqn. (2.4) becomes, 2 T  1  dkf 2ir  =  +2  1 v+2 k (T) + +2  1 (T) + L 2 v+L k +2  +  1 (T) w +v k+2 2  (2.5) —  In summing over the frequencies we use, 1  1 w+v 2 k +m 2  =  1 cot(’-) 2 iwk  =—_(i+ wk \. 2  where  Wk  =  2 e/ ’ 3 k  —  1)  (2.6)  . We also need the following two integrals 2 k+m 2 s/v  f  dk  1 —  e_m_ / 2 2m  2 (ek  —  1)  2f dk  2+ k  =  rd/c  J  Wk  —  2 —log(2Av/m) V  (2.7)  (2.8)  Chapter 2. Details of the Models  28  where A is an ultraviolet cutoff and we made the approximation /3m>> 1. Gathering all of the above, we can set Eqn. (2.4) equal to Eqn. (2.5) to arrive at /A2frr\A2 1m\A2fm\  -ok’)  zj  log where  + L and z  )  e —/T  2v”  =  4%/±  +  e —I+/T  +  e —I_/T  (2.9)  z. Eqn. (2.9) implies a cubic equation for  the square of the temperature dependent gap, L 2 (T). Once more making the approxi mation, j3 >> 1 we can linearize the equation and solve for L(T):  f 7 /  -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 to say a few things about Eqn. (2.10) before going on to the next section. First, notice that we implicitly assumed that only the expectation value of the fluctuating field A acquired a temperature dependence. The renormalized values of the anisotropies, not. This is because renormalization occurs at T  =  Li  and  do  0 first. At non-zero temperatures the  free 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 exponential terms logically appear as a result of calculating <A >= Tr [e”A] and then subtracting  <  A  >  (2.11)  from the Lagrangian. We would also like to point  out that the validity condition for this analysis,  /31-sm  >> 1, where  -m  is the smallest  gap, 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, into  a phase with a new singlet ground state (the order parameter is a non-local operator in 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 Ising  Chapter 2. Details of the Models  29  transition into an antiferromagnetically ordered phase. The bottom line is that D must be small in comparison with z. Similar, but more obvious, cautionary remarks apply to E. Moreover, as discussed in [29], the NLu model is not expected to remain valid at temperatures of order twice the gap. This is because the model does not exhibit a maximum in the heat capacity and in the magnetic susceptibility as shown in numerical studies at these temperatures. Finally, notice that the difference in gaps will close as T increases. This is no surprise since at high temperatures the mass scales are irrelevant and we expect a restoration of 0(3) symmetry. 2.1.2  Field Dependence of the Gap  We will be interested in adding a magnetic field term to the Hamiltonian. To do so consistently, we must couple the magnetic field to the generator of rotations (the total In terms of the continuum fields, we couple the magnetic field, fi, > —gepB -U J dx ix), and add this to the NLu Hamiltonian. The corresponding  spin operator),  ..  to lvia,  .  Euclidean space Lagrangian is £  =  -  (Ia/ot +  x  +  v2(8/8x)2  -  2 2vD()  -  2 2vE(()  -  (q5Y)2))  (2.12)  —.2  751 where h  =  ge/LB  ii.  In the 0(3) and U(1) symmetric cases, where the field really couples  to 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 may arise. This time, when we integrate out the  fields, the eigenvalues of the  propagator  are not as trivial. However, if we assume that the field is placed along a direction of  Chapter 2. Details of the Models  30  symmetry, say the z-direction, then the Tr log will be over eigenvalues of the matrix  v+ k 2+2 w  2 z  2+ h  —  2wh  0 0  0  (12.13)  w2+v2k2+2+}  0  Where, again, we’ve assumed renormalized values for the masses. The eigenvalues, rj, are 773  =  = w 2  2+v w k 2  (h) + 2 k+L 2 +v  (h) ± 2 2+z h  —  LX  iJ_4h2ô2  +  $  +  1 2+v w k+ 2  (2.14)  We can write the constraint equation as  1  1 kI 2 d (2)2 w2 + v k+ 2  2  +  +  1 2+v w k+ 2  rd k 2 /1 .i (2ir) 2\773  1 77+  2  + 1 I  2  —  (2.15)  ?7_J  The integrand on the right hand side has poles at the negative solutions of the equations of motion =  =  (k2+z2(h)+)  (k2 + h (h) ± 4h2(k2 + z 2 2+L (h)) + 2  (2.16)  Integrating over these poles gives  J  (_L +  2K 2w3  —  2 k  —  z ( 2 h) + h 2  —  w+(c4_w)  —  2 k  —  z ( 2 h) + h 2  w_(w—w)  (2 17)  Before continuing, we mention that the T dependence can easily be worked in by mul tiplying the terms in the integrand above by 1 +  e—1’  respectively. It is possible to  simplify Eqn. (2.17) further to read rdk/1 1 1 I—(—+—+——  J 4K w 3  w  w_  2 4h  I  (2.18)  Chapter 2. Details of the Models  31  where it is now clear that (2.15) is satisfied for h —* 0. This can be shown to be the same result obtained by minimizing the zero point energy. To solve for z(h), one must decide on 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). To compare, we also plot the field dependent gaps of the fermion model which are generally considered in agreement with experiment [47] (at least for the material NENP). The zero field 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 the gaps but this time correct for the constraint implied by Eqn. (2.17). The lower branches show good agreement right up to fields close to critical. There is, however, a greater discrepancy in the gap corresponding to the field direction. We now turn our attention to a seeming infrared catastrophe which occurs as h approaches the critical value given by =  2(h) —  (2.19)  this is where the lower gap closes and the integral (2.18) diverges logarithmically in the thermodynamic limit. At first sight one may hope that for k  =  0, the last two terms  in (2.18) conspire to eliminate the divergence for some value of h and LSh) satisfying (2.19). This, however, requires that =  &(h)  (  1+ +  (2.20)  16))  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 understanding that the variation of the zero point energies w  =  k + zS ± h with respect to L. 2 ‘/v 2  is independent of h. Let’s try to get a deeper feeling as to what’s happening. Instead  __________________  Chapter 2. Details of the Models  32 H Yb  I  I  3.0  zzzzzz  2.5  5CD  2.0  >  2 1.5 CD  Boson Model  w 1.0  E ::EZZZ M 0 tE’E 00  4.0  2.0  6.0  10.0  8.0  H II c  4.0  I  •  I  I  I  3.5 3.0  5CI)  2.5  ——  Boson Model Fermion Model  1.5 1.0 0.5 0.0 0.0  2.0  4.0  6.0  H  (Tesla)  8.0  10.0  12.0  14.0  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  33 H fib  3.0 2.5  5 2.0 E 2’ 1.5 ci)  w 1.0 0.5 0.0 HI/c  4.0 3.5 3.0  5cD 2.5 E  >..  2’  2.0  ci) C  w  1.0 0.5 0.0 H (Tesla)  Figure 2.2: Corrected free boson (t = z(H)) and free fermion dispersions with the gap parameters of NENP. Top graph: flub; bottom graph: flIc.  Chapter 2. Details of the Models  34  of 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 write w(k  We now see that the limits h  =  —+  0)  =  I((h)  h and E  —+  —  2 h)  (2.21)  —  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 out low energy modes. In the U(1) case the day was saved by symmetry which prevented the variation of < A  >  with magnetic field. There is no such luck in the Z 2xZ 2xZ 2 scenario  and for a sufficiently large value of E, it is no longer correct to integrate out all the q fields even for large N. In the case of large N, one can integrate out all modes but the gapless one to arrive at a critical theory. This should be done when log N  =  (-)  N/g. For  3 this criteria may be too restrictive, and instead one can adhere to E << L(h)  —  h  as the regime where the analysis of this section is valid. 2.1.3  Exact Results  In this section we note some important results which will be useful in the calculations of Chapter 3. As was mentioned in the last chapter, the 0(3) model possesses special properties that allow for some integrability. In particular, at long wavelengths one can say much about the matrix elements of the spin operator even after the 0(3) symmetry is broken to U(1) by a magnetic field. As will become clear in the next chapter, we are mostly concerned with matrix el ements < k, a 82 (O)q, b>, where < k, a denotes a single magnon state created by the  Chapter 2. Details of the Models  35  staggered magnetization operator, q (this acts as a free boson operator in the large N limit) with norm <k, alq, b >= 27röabS(k  —  (2.23)  q)  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 this picture since they must all be even in bosonic operators. The two magnon operator, i1 (o),  is expected to 1 contribute. The matrix element is given by the Karowski and Weisz  ansatz [12], <k,a1(0)q,b where  cJJk =  v+ 2 /k  >=  ie+G(&)  (2.24)  2 and the rapidity variable, 0, is defined via z sinh(0’ sinh(8”  0 = 6’  iir) = k  —  iir) = q  —  0”  —  (2.25)  with G(6) = exp (2  x 2 dx (e_ [ Jo x  [x(iir 0)/2ir] 2 1)sin (ex+1)sinh(x) )  —  —  (2.26)  This ansatz is believed to be exact for the 0(3) NLu model, but is only approximately true for the s = 1 Heisenberg model; however, numerical simulations [16] are in excellent agreement with this form at least half way through the Brillouin zone. Since we will largely be interested in 1k  —  qv < L <<irv, this ‘exact’ expression is more than sufficient.  speaking, since ris qnadratic in large N limit.  ,  we expect it to be a two magnon operator only in the  Chapter 2. Details of the Models  36  Some particular limits of interest are k  q and k  —q, corresponding to forward  and back scattering, respectively; in the former case, 8  ilr while in the latter, 0  i-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 inter actions. We will later see how this might affect experimental predictions. The above results change when a D type anisotropy is added. Essentially, the dif ference is that the function, G(0) changes and the gap in the energy of states on the SZ  =  0 branch will be shifted. There are no exact results for this model which is why the  phenomenological boson and fermion models are important. We can, however, give some universal (ie. model independent) results which only depend on the conservation of total spin in the z-direction, when the momentum exchange is small, v(q part of Sz(0) which will contribute will be, essentially, S_k  f dx  —  kI <<  Li.  The only  S’(x). Since this is  a conserved operator, we can write down the one-particle matrix elements immediately: <k,sZ=±1Sz(O)q,sZ=+1 > ±1 Also true for all values of  vlq  —  are the following  < <  (2.28)  k,sZ=1sZ()q,sZ=_1  >=  0  >=  0  (2.29)  In Chapter 3 we will show that it is largely this universal behaviour which determines the relaxation rate, 1/T 1 in anisotropic media. Note that adding a magnetic field to the 0(3) or U(1) system will break the symmetry in the 0(3) case (we naturally assume that in the U(1) case, the field is placed parallel to the U(1) axis), but will hardly change any other results in either model. This is because  Chapter 2. Details of the Models  37  in adding a magnetic field all we have done is add a term to the Hamiltonian proportional to the conserved charge l  =  f dx  1’ (x). Since l commutes with the Hamiltonian, they  can be simultaneously diagonalized with all low lying states labeled by their l quantum number, +1, 0 or —1. Matrix elements of operators can only differ, at most, by some cumulative phase which corresponds to turning on the field sometime in the past. 2.2  The Free Boson and Fermion Models  We now turn our attention to the phenomenological models introduced in Chapter 1. After introducing the formalism which we will require to perform calculations, we will compare and contrast the energy spectra and fundamental matrix elements. 2.2.1  Diagonalization  To do the necessary calculations we need to have a basis of eigenstates for the noninteracting Hamiltonian and know the expansion of the field operators in terms of cre ation/annihilation operators for these states. Bosons —.2  Relaxing the constraint çb  =  1 in(2.12) we see that we seek to diagonalize 2  )]  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 true when the crystal field symmetry is no lower than orthorhombic  —  a sketch of a proof is  Chapter 2. Details of the Models  found on p. 750 of  38  [41) Li  0  0  0  zi  0  0  0  L  Also, we mention that we’ve set fl  =  G=  1  =  9i  0  0  0  92  0  0  0  (2.31)  a, where a is the lattice spacing. This has the  effect of measuring energy in units of inverse seconds or inverse mass: [E]  “..‘  [M]’  [s]’  []2  [v]  ‘-.‘  (2.32)  Diagonalizing (2.30) is tedious (especially when the field does not lie in a direction of symmetry, for then all the branches mix) but the idea is to find the right Bogoliubov transformation. Working in momentum space, we define  -.  4(k,t=0)=  fl(k,t  =  0)  [ai, aV]  =  =  1  i/[  —k  27r66(k  —  —  k’)  ak]  (2.33)  ak]  (2.34)  (2.35)  0 is an arbitrary quantity with the dimensions of energy. We need such a quantity to w represent the fields  and  out that when one writes  ]i  in terms of creation and annihilation operators. It turns  k  and ilk in terms of the creation/annihilation operators  which diagonalize the Hamiltonian, the dependence on w 0 disappears. Furthermore, the eigenvalues of the Hamiltonian are also independent of w , as might be expected. We 0 will now restrict ourselves to the case where the field lies in a direction of symmetry. This leaves (2.30) with Z 2 x Z 2 symmetry. Now only the excitations transverse to the  Chapter 2. Details of the Models  39  direction of the applied field mix and we need only solve a (4 x 4) set of equations for the diagonalizing creation/annihilation operators. Without loss of generality, we take the field to lie in the 3-direction, and set g 3  1. All told, the Hamiltonian, Eqn. (2.30), for  =  the mixed states is p00  H=J  Hkdk  -00  =  where  2 o  Ak +  akA  a_k  +  + akB a_k  (2.36)  A=I+—ho 2 4 0 4w 2  (2.37)  B=-I+-4 0 4w  (2.38)  is the usual Pauli matrix, and (zS+v2k2  0  K=I  (2.39)  I  0  ) k 2 i+v  )  The momentum space Hamiltonian can be written in terms of a single matrix M:  Hk=  (,  a_k)M  (  (2.40) 0)_k)  As discussed in [31], we seek the eigenvectors of the (non-hermitian) matrix jM, where  iM=  (A \\_B*  (io\  B\  I _A*)  =  I  \0  I —1)  (2.41)  This comes from requiring the new diagonal creation/annihilation operators to have the standard commutation relations. This also imposes the unusual normalization condition on the eigenvectors:  =  1.  Chapter 2. Details of the Models  40  Summarizing the above, we need to solve:  0  =  (  — —  +  \  4  2  +  4  4(4)0  (2.42)  —I—--— h 1 o 2 —I 4 4w 2 2  4w,j  which can be manipulated to give  ((wo  0  —  w)I  —  2 hcT  2 KIho.  —(wo + w)I  —  2 h  (  )  2 +ôI+ho  The eigenvalues of r M are already known, as they are the solutions to the classical 1 equations of motion and come in pairs ±w. These are naturally the same frequencies given in Eqn.(2.16) with the proper substitutions made for the gaps w=k + 2 LS  =  2+2 k h+  2)  ± 4h2(k2 +  +  (;  2)2  Furthermore, we need only work to find one eigenvector of each pair because if  (2.44)  Ix  (y* is a right-eigenvector of iiM with eigenvalue w, then  x*)J  is a right-eigenvector of  M with eigenvalue —w (X and Y are themselves two-component vectors). The bottom rows of (2.43) give the following set of equations  0  =  (z + 2 v ) k ±,i  0  =  (L + 2 ), k v  —  —  2 , 0 1 + ihw wow±&,  (2.45)  2 w 0 w ,  (2.46)  —  1 , 0 ihw  Chapter 2. Details of the Models  41  where it turns out to be convenient to work with  X + Y and  x  X  —  Y. The top  rows can be manipulated to give 2 (h  o =  o =  —  2 —(h  —  — —  (2.47)  ihwo&,i + wow±&, 2  2 , 0 ihw  —  (2.48)  1 wow±&,  These can then be worked to give (z  k+w 2 +v  2 (z  v+w k +2  2+L+v (h k 2  2+ (h  zi  —  —  )±,i 2 h  )±, 2 h  k 2 +v  —  (2.49)  2 w 0 2w ,  (2.50)  2 , 0 —2ih  (2.51)  1 , 0 2ihw  (2.52)  =  =  —  1 w 0 2cL. ,  =  w)Xth,2  =  Note that if we fix the phase of Xi to be real then  must also be real as X2 and  be pure imaginary. The normalization condition, XtX  —  YY  =  2  must  1, now allows us to solve  for the eigenvectors which form the columns of the transformation matrix between the old and diagonal bases of creation/annihilation operators. In terms of X±,i&,i  —  X±,2’±,2 =  x  1  and  this is (2.53)  The solution is  X±,i  —  —  (  2+ wow±(h  k+w 2 (? + v  —  —  1/2  k 2 +v  —  )(h + 2 h  ?+v + k 2 w 2wow±  w)  k 2 +v  —  2 h  (2.54) —  wi))  (2.55)  ________  Chapter 2. Details of the Models  X±,2  =  42  2zhw 0 k 2 2 + Ls + v h  (2.56) —  2 2i2 i j..2 2 j A 2-” -rW±1& c±,2 =  2wow±  X±,2  With the inclusion of the trivially diagonal unmixed component (ie. the three com ponent), we can define the three by three matrices by the eigenvalues V!+v2k2  and  x  with the columns labeled  and  (+, —, 3) and the rows labeled by the original masses (1, 2, 3): X33 =  V  v2 3 k  One easily verifies that  .  ak=  [xk+k  k+(X-)  (2.58)  bk]  Where the b’s are the operators which diagonalize H. Equations (2.54)—(2.58) are the main results of this section. Before ending, we give some limiting forms for the limit h  —*  and  .  In  0,  / V /+vk Lp  =  While in the limit L 2  —*  /wp  —1f  —  0 (2.59) I+v2k2  , +v 2.f k V2  — —  0  :  x=  X  x  wp —i / k 2 V 21&+v 0  /  wp  -fv 2v’ k V2 Wp / +v 2./ k V2 0  0 Wp  V  \/+v2k2  Fermions We would now like to repeat the diagonalization procedure for the fermion model. The free Hamiltonian with minimal coupling to the magnetic field (ie. coupling only to  Chapter 2. Details of the Models  43  and simplest parametrization of the mass terms (corresponding to anisotropies and giving zero field dispersion branches  ) is v 2 +k  =  =  VãxL  —  3  i with, setting  v  —  =  +i  L,iR,i)  (  X L  +  R X R)]  (2.61)  1,  =  I)L =  f  °°dk _  f°°  (e_itãL,k  {a, aV}  + et_,k)  (2.62)  +  (2.63)  2ir6(k  =  et4,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 becomes Hk  =  c4Mk  (2.65)  k  where IIk—iix  ii  M=I  —Ik—iix  —i,  -*  =  (i  aR,k”\  \  aLk  i  I  (2.66)  )  (2.67)  The 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, we  Chapter 2. Details of the Models  44  assume the field is in a direction of symmetry so that we need only diagonalize a 4 x 4 matrix. Given that the field is in the 3 direction, the eigenvalues are: + 2 w=k  =  2+h k 2+  (1 ±2)  ± 4h2(k2 +  +  (;  2)2  (2.68)  It may be more illuminating to write out M in a basis that is more natural to the U(1) problem. Using  (ak  ‘  aJ,J  )  /  2  it\  aL,k  (i  1  —  1  —  1 —i  /1  a)-i  1.  ‘  (4,k  )  \  a  +t\ \/ aL,k  (2 70)  i)a)  In this basis, M becomes  M=  (  kI  —  1 —iLo  Where L  = 12  and ó  = 12•  3 ho —  1 + iSI i1o  iSI —kI + her 3  I )  (2.71)  The equations for the components of the eigenvectors  possess the symmetries 4 + 1 u , 2 + 3 , h+->—h u +u -u  (2.72)  (2.73)  Chapter 2. Details of the Models  where  w  45  is the eigenvalue. After some algebra, 2iL(k  =  w) w2+2_(k+h)2_62U1 —  (2.74)  2i6(k—w)  U3  (2.75)  (w_h)2_k2+2_2U1  U2 =  2iL(k+w) (k + h) 2 + 2  2 w  —  3 u 62  (2.76)  —  Using the normalization condition, Iu  =  1  (2.77)  we set the phase of u 1 to be real for positive eigenvalues; the above symmetries allow us the freedom to choose a convenient phase for the ui’s corresponding to negative eigenvalues. U’  =  +(k+h) 26(k+w)(w — — 2 ) LS ± 6 [48 ( 2 k+  ((w 2 w)  +  —  (k + h) 2  —  62)2  (2.78) (k 2 + 4/  —  ) 2 w)  +  — ((w+h) + — ( ) + 2 — (w S z k i (k+h)2_62)2+4/X2(k+w)2)]  We define the 6 x 6 diagonalizing matrix with columns given by the eigenvectors i4 as  =  , uL+, uL, u 3 ) 3 (L ui,, u,  (2.79)  (2.80)  Chapter 2. Details of the Models  1 U=— = 1  46  11  0  000  i  —i  0  0  0  0  0  0  0  0  0  (v  ) 1 o vu  o  0  1  1  0  0  0  0  —i  i  0  0  0  0  0  0  v’  2 V  The diagonal operators,  k 13  0  (2.81)  are defined as: ICk’  -.  I  Ik  \  I  (2.82)  dk)  Our freedom in choosing the phases for the eigenvectors corresponding to negative eigenvalues allows us to write  IR T”\ X= T R)  (2.83)  Each index of this matrix runs over six states; the first and last three correspond to right and left movers respectively. In the case of U(1) symmetry or higher, each set would correspond to states of definite spin. The d’s and c’s correspond to left and right moving fermions, respectively. This becomes clear in the limit  i  —f  2 —* 0. Some limiting forms of R and T are: z  /0  $  R(h-0)=  /EE  \/EE  0  0  (2.84)  0 V  W3  Chapter 2. Details of the Models  T(h  —  0)  47  o  _/k  =  o  o  (2.85)  _i\/2z  o  o o  o  0  0  (2.86)  o  o o  T(ö_÷0)=  2.2.2  o  (2.87)  o  Discussion: Comparison of Spectra and Spin Operator Matrix Ele ments  The 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 only for low magnetic field, h << Min(z , 1  /.2).  The difference is most significant at the critical  field where the lower gap vanishes. The boson model predicts h the fermion model gives h  =  2 “LiL  =  ,2 1 Min(L L ) , while  (see Fig. 2.1 and 2.2). Experimental evidence  seems to favour the fermion model, but there are some subtleties which have previously been ignored. The data supporting the fermion dispersion comes from neutron scattering and NMR relaxation rate experiments performed on the anisotropic 1DHAF material, NENP. In analyzing the data, however, crucial structural properties were neglected in the interpretation (namely, the fact that the local chain axes did not coincide with the  Chapter 2. Details of the Models  48  crystallographic axes). This, we believe, also led to a seeming contradiction with ESR data on the same substance which seemed to side with the boson dispersion  [47].2  Aside  from material properties, the possible temperature and field dependence of the mass parameters,  zj, has also been ignored so far. Since the boson model derives from the  NLu model, one should incorporate such field and temperature dependence into these basic parameters. We showed that considering field dependent masses brought closer agreement 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 quadratic one possible when the magnetization density is no longer conserved. It is possible to construct a modified boson Hamiltonian including extra terms designed to reproduce gaps identical with the fermion model [32]. The only constraint on such terms is that they do not mix the sz  =  0 modes corresponding to the degree of freedom parallel to the  field. It is not obvious, however, what justifies such a modification other than a more convenient 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, the phase transition is in the two dimensional zy universality class. The lowest lying mode of the Landau-Ginsburg boson model can be reduced to a single free boson (a phase field corresponding to the Goldstone mode), but the parameters of the resulting low energy Lagrangian 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 correctly describes 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 simply that of free fermions multiplied by a sign function to correct for the statistics. Second, For more details on these matters, please see sections 5.1 and 6.3. 2  Chapter 2. Details of the Models  49  the U(1) fermionic modes can be represented as particles and holes using a single Dirac fermion with chemical potential h (this can be seen in the matrix equation (2.71)); this means that at h > with SZ  z the ground state will be occupied by fermion states, each  1, and hence non-zero magnetization. The simplicity of the coupling to h  guarantees that interactions will be as important near h as they are near h  =  0. In  particular, they will be negligible in the dilute gas limit. We thus see that interactions become progressively more important close to criticality in a boson theory, while the opposite takes place in an equivalent fermion theory. In the Z 2xZ 2 case we expect an Ising-like transition corresponding to the breaking of one of the Z 2 symmetries remaining. Here things are even clearer. Mean field theory for the boson model is completely hopeless as is evidenced by the unphysical behaviour of the lowest lying gap at h> Min(z ,2 1 z ) . This function always possesses a zero even at nonvanishing k-vectors. Moreover, it is imaginary for fields Jk2v2 + Max(t?, 1  + Min(L,  /.)  <  h <  z). The spectrum for the low lying fermion, in contrast, shows all the  desirable properties, vanishing at h gap, Li  iJ2v2  =  2 L 1 /z  only for k  =  0; in addition, the effective  — hj, is as expected in the Majorana fermion representation of the critical  Ising model and so is the relativistic dispersion for long wavelengths. Finally, when we integrate out the more massive fermionic modes we are left with a strictly non-interacting free Majorana fermion theory regardless of any zero-field interactions in (2.61); this is because all interactions will be polynomial in the one Majorana field left, and will vanish by fermi statistics. Thus we see that in contrast with the boson description, the free fermion theory is actually best near h  =  h.  To summarize, on general grounds, one can expect qualitative agreement between both models up to magnetic fields close to h where the fermion model is expected to be a 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  50  We start by defining, la,b(k, q) < a, k i0)q, b>  x  we use  (2.88)  ii for bosons to write l,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 =  r=  Using,  (L X  + bR x  ‘j)R)  for fermions, we write the analogous expression la,b(k,q)  =  (  Vf V*aiR* 1 RV VR + Tt a Vt Va T 1 RV V*T* + Tta —ii — — I (2.90) ‘ T’V Y2VR + RToiVT EVa RTa V T V*uiR*) T TTVT DV*T* + 1 1 _,  , R and T were all defined in the sections on diagonalizing the models. As can 1 x , V, a  be explicitly checked, the 0(3) free bosons are analogous to the NLa model with the function, 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 N approximation to the NLa model. In case of axial symmetry, one need only substitute the correct gaps into the energy factors: b  <k,a1i(0)q,b>=jfk ‘ 2/ with  w=  /k2v2  +  (2.91)  .  The 0(3) fermion model exhibits a non-trivial G(O)-function. We can use the results from Eqns. (2.86) and (2.87) in Eqn. (2.90) to calculate that G(O) = —sech(O/2) =  [V(wk  —  k)(wq  —  q) +  k)(wq + q)]  Wk  ± Wq  (2.92)  Chapter 2. Details of the Models  51  To 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 gap for backscattering, k  ‘  —q. This is because the  f does not couple left and right moving  fermions while the opposite holds true with the bosons (and NLu model). For small momentum exchange, all the models give universal predictions for matrix elements of 8Z  (0). However, matrix elements of S± (0) at small momentum exchange are somewhat  sensitive to the ratio of the gaps, in the boson model, while not at all so in the fermion model. In Chapter 6 we discuss experiments which might investigate this behaviour further. When the symmetry is orthorhombic there are few conservation laws to restrict the form of matrix elements of spin operators. Furthermore, when a magnetic field is added, Lorentz invariance is explicitly broken. We can, however, say that correlations among spin operators are still diagonal: This is true by virtue of the Z 2xZ 2 symmetry. We also know that the new energy eigenstates, labeled by + and —, are mixtures of eigenstates of S 3 with eigenvalues  SZ =  ±1. This guarantees that <k,  —  >oc:  (2.93)  j3 6  It isn’t terribly illuminating to write down the actual matrix elements. We can say, how ever, 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 of the sickness of the free boson model near criticality. Again we see that interactions are expected to play a crucial role in the boson description. We finish by describing some matrix elements near zero magnetic field. We expect that the intrabranch matrix elements, < ±, 3 kS ( O)I±, q >, vanish at k field. Fork=q=Oandh-+O,  =  q with the  Chapter 2. Details of the Models  52  <—,OI1(O)I—,O >=  h  2  32 2)  (Ls?-: =  2h  bosons fermions  (2.94)  i2  where we’ve assumed, L > Li . 2 exchanging 2 and 1. —÷ /..  The result for < +,  OIl(O)I+, 0  >  is obtained by  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 axially  symmetric case.  Chapter 3  Model Predictions for T ’ 1  Let’s recall the expression for 1/T , Eqn. (1.69), derived in Chapter 1: 1  =  f  dte_rt  where h is taken to be in the  <  {S’(t),s(o)}>  (3.1)  direction. As discussed in (2.2.2), only diagonal compo  nents of the spin correlation function will contribute. We also assume that the hyperfine tensor, A , 2 3 is local  , 2 A  =  Aö,  (3.2)  Thus we can write =  2 A  f  dt  e_t  <  {V(  =  0, t), SL(o)}>  (33)  where we’ve used translational invariance to evaluate the correlation function at the origin. We can now take a step back to Eqn. (1.68) and write the above as 1 = 1  I <n!ISV(0)In> 2 2 IAI I 6 (E’  2  ‘-.‘  E  (e —E,,/T —  WN)  + e —E/T  (3.4)  fl,Th’,L’  We will concern ourselves largely with the limit, for H  —  <<T << min (note that 1.5 T), so that the last factor in (3.4) can be set to 2e_EfIT. WN  Consider now the operator in question, (O)  =  WN  lmK  (0) + (0) x ti(0). We wish to  investigate 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 analyze 53  Chapter 3. Model Predictions for T’  54  a E= 2  Figure 3.1: First non-vanishing contribution to relaxation due to the staggered part of the spin the staggered contributions. This will be 1  Tistagg  cc  <n’I(O)In> 2 I 6 (E’  —  —  wN)eEfhT  (3.5)  n,ni  We 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 below h, and we are therefore well justified in using the boson model (or NLu model) to describe the situation. Since (O) is a single magnon operator in the noninteracting theory, it only 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 are no matrix elements with energy difference,  WN  (which is essentially zero, compared to  the other energy scales around). Including interactions, there will be contributions at finite T. The simplest process is shown in Fig. 3.1. It involves a 4 q type interaction, as might occur in the Landau-Ginsburg or NLa model. The vertical line represents the field There will not be contributions from cross terms between the staggered and uniform fields. These 1 vanish because for< njIm > 0, one needs the number of magnons, n + m, to be odd, while this in turn 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.  Chapter 3. Model Predictions for T’  55  çb. The incoming line from the right represents a thermally excited magnon of non-zero momentum, k, and energy 2z (for simplicity, we use the isotropic model at zero field to make this argument; extending this to the anisotropic models is straight forward). The two 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 vary spatially as e). This diagram gives a non-zero matrix element proportional to (where ) parametrizes the  ‘  interaction). Note, however, that since the initial  )/&  and final  /T 2 state energies must be at least 2z, there will be a Boltzmann suppression factor of e_  to this contribution. Thus  1 Stagg T  oc  e_ 2 A T  (3.6)  Including anisotropy and a finite field will give various contributions of this type. The greatest will be suppressed by exp(—2min(h)/T). It is also consistent to interpret this result as giving the single magnon a finite width at T  0. This, however, cannot change  the conclusion that there is a double exponential suppression factor contrary to the model proposed by Fujiwara et. al. [34]. Let us now consider the contributions to 1/T 1 from the uniform part of the spin:  TiUnif  oc  6(E <n’l(0)In> 2  —  E  —  N)eT  (37)  As discussed in Chapter 2, the 1-particle matrix elements selected above are non-zero in 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 the presence of anisotropy and magnetic field, the three magnon branches are split, so we must distinguish between interbranch and intrabranch transitions (see Fig. 3.2). This is possible since contributions may come from all wave vectors. One set of important processes (ie. the ones corresponding to transitions between the lowest energy magnon  Chapter 3. Model Predictions for T’  56  Magnon Dispersions For Lowest Two Branches  E  Figure 3.2: Inter- vs Intrabranch transitions  Chapter 3. Model Predictions for T’  57  states) will come from single particle intrabranch transitions along the lowest mixed branch (intrabranch transitions are not allowed along the branch corresponding to sZ  =  0  since h is parallel to z). This will be one of the leading effects with Boltzmann suppression of e- (h)/T, where  i  (h) is the lowest field dependent gap. It is important to realize  that these will only be present if the hyperfine coupling, A+z, is non-zero. In fact the zero field gap structure and the choice of direction for placing the magnetic field will affect whether there are competing transitions. The next contribution, possibly as significant as the one just described, can come from intrabranch transitions along the second lowest branch and/or interbranch transitions between the lowest mixed branch and one of the other two branches. The Boltzmann suppression factor will, again, favour the lowest energy 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 processes is larger than that associated with contributions from the staggered field. To summarize, the dominant contribution to 1/T 1 at T << min’ will come from Eqn. (3.7). As we approach the critical field, the above analysis breaks down.  As discussed  previously, interactions are expected to become large in the boson model. Moreover, as the gap closes, the Boltzmann factors will fail to discriminate between the contributions of the uniform and staggered fields to 1/T . Arguments involving the fermion model 1 are tricky because the staggered field has no simple representation in terms of fermionic operators. However, we explicitly show later in this chapter that the staggered component will dominate sufficiently close to h. Above the critical field, the analysis depends on the symmetry. For U(1) or higher symmetry, the system remains critical and the staggered correlator remains dominant. For lower symmetry, the gap opens up once more, and sufficiently far above the critical field, we expect the uniform correlator to dominate once more.  Chapter 3. Model Predictions for T’  58  1 for h <<he T  3.1  In this section we concern ourselves with the regime discussed above,  WN  << T <<  We calculate the relaxation rate in the isotropic, U(1) and Z 2 scenarios.  ) 1 ‘min(  We begin by deriving a general result valid in this regime, and proceed to discuss its application in the different cases of symmetry. Consider a contribution to 1/T 1 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), *rs  This will be a sum over single particle states on r and s: =  4K  rs  2 AI  f  6(ws(q)  —  r(k)  —  )/T,  wN)e  <k, r1i(O)q, s>  Note that there will be a similar contribution with the labels  s  2  (3.8)  and r exchanged, if s  and r are different branches. This corresponds to scattering an initial particle on the s branch through the hyperfine interaction to a final magnon on the r branch, or the reverse process. We take account of both of these possibilities later. Also, we are keeping WN  finite to cut off infrared divergences which crop up in the intrabranch processes. We  now do the integral over q to get [°°dk(Wr(k)+WN)  -—  Tirs  eT  2 JO 4IAI j  2K  Q(k)  (Il, k 6 , Q(k))1 2+8 Il ( k, _Q(k))12)  where Q(k) is defined by w (Q(k)) 8  =  r(k) +  WN,  () q  and l is as defined in (2.88). When  Lr(O) >> T, the above integral will be strongly peaked at k neglect  WN  in wr(k)  +WN.  (39) q=Q(k)  =  0. Moreover, we can  The only factors in the integrand for which we should retain a  k-dependence are the exponential and the possibly infrared divergent denominator, Q(k).  Chapter 3. Model Predictions for T’  59  We therefore write 1  =  -1  2IAw(0)  Q(o))1 + I1(0, -Q(o))2) 2 e(k)/  P00  I Jo  dk  (3.10)  Q(k)  It is not too difficult to show that to first order in small quantities, k 2 and (k 2 Q )  Q ( 2 0) +  We can also expand the exponent: r(k)/T  (  Iaw’\  (8w’’ l  N,  one has (3.11)  )q=Q(O) )q=o  ( w(0)/T+  k  2wr(0)T  )  (3.12)  The relevant integral over k then becomes  () /  JO  Q2  (2)  By changing variables,  wr (O)T 2  e  dk  2r(  “-‘ (0) + i”’ k oq ) q=Q(O) 8q 2 2 } q=O  (3.13)  k2  —+  k, we can write (3.13) as  f6w\ (0w\ Jq=o  k2  g=o  2 —k  a’ 00  )q=o  J  dk  °  yk2+ €rs(T, h)  (3.14)  where —  Q ( 2 o)  c ( 8 T,h) —  The h-dependence of  c  ()qQ(O)  2wr(0)T  (3.15)  will largely come from its dependence on r(0). The integral  can be expressed in terms of special functions:  lOw  2  eT, K 2 o (rs(T, h)/2) / q=Q(O)  (3.16)  Chapter 3. Model Predictions for T’  60  0 is the zero order modified Bessel Function. When the gap, Wr(0), is very large where K  compared to the typical momentum, Q(0), exchanged in the transitions, (this is the case for intrabranch transitions),  crs —*  0. In this limit  e8(T,h’’ K 2 o (ors(T, h)/2) —+ — log (crs(T, h)/4)  where ‘y  =  (3.17)  —  0.577216... is Euler’s constant. We can now summarize  I Tirs  2IA ) 2 r(0)  =  ir  j  (O4)  2  q  Q(0)  (il,(o, Q(°))1 2 + 1(0,  (4)  ) 2 -Q(°))1  eT(0)1T  2  e(T, K 2 o  (ors(T, h)/2)  (3.18)  q 0  The 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 from transitions between states involving small momentum exchange  (Q  —+  0) will dominate  due to the logarithmic divergence in (3.18). This is particularly the case with intrabranch versus interbranch transitions. In intrabranch transitions one is allowed momentum ex changes as small as  Q  ‘s.’  2Lr(0)WN/V.  This will typically be much smaller than the  smallest allowed interbranch momentum exchange,  Q  ‘‘  (Lr(0)  —  8 ( 0))/v.  The conclu  sion is that, unless the branches in question are extremely close to each other, interbranch transitions will play a secondary role to intrabranch processes, even ignoring the more obvious suppression due to different Boltzmann factors. Of course, if the hyperfine inter action has high symmetry, one will not see intrabranch transitions at all. This suggests that an NMR relaxation study could provide information as to the nature of the hyperfine tensor.  _______  Chapter 3. Model Predictions for Tj 1  61  Before expounding on this result in the individual cases of different symmetry, we would like to mention the effects of higher temperature, or correspondingly, including the k-dependence of the various terms approximated at k  =  0. In the more general  2xZ Z 2 situation , we are strictly justified in expanding the exponent in Eqn. (3.12) only 2 for 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 k . First, notice 2 that all gaps are always greater than v2k2 + L  h, where  —  is the smallest zero  field gap. We therefore write  f =  -—  i:  —c,.(h,k)/T  f  dkk_e  Jk2 +  Q ( 2 k)  2 — 2)Th_1 dwe_Tw(w  dkk21e_V22_1m/T  < e_(m_h)  (2Max(T  1m))2fl  (3.20)  The last estimate is actually quite generous, especially for large n. In the worse case scenario of the Haldane phase, /v  ‘  1/4. This allows us to expect an error of at most  10% 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 exponential at T <<  The next term in the expansion is  with v  = ()qo•  This will  give a contribution, e_/’T  ‘  I dk  i  Te_(O)/T  /2(0)T 2 kvr e_k  8zT /k +Q 2  This is potentially more serious as T  —÷  /.x,. or  —*  is generally a very good approximation; when T << exponential, while for T  ,  (3 21  4Lr  0. To summarize, Eqn. (3.10) r,  one can safely expand the  one is better off numerically integrating (3.10).  One is more fortunate in the U(1) case; because of the simplicity of the gaps, the expansion is good 2 &L, regardless of the value of h.  for k  Chapter 3. Model Predictions for T’  62  A Intrabrancil Transitions Interbranch Transitions  Figure 3.3: The gap structure for 0(3) symmetry. 3.1.1  Isotropic Symmetry  In the case of 0(3) symmetry, the field dependent gap structure is as in Fig. 3.3. The lowest and highest branches correspond to magnons with sZ middle branch corresponds to  Z  =  =  ±1 respectively. The  0. The interbranch gap is h. Using the result of  the previous section and that of 2.1.3 we can immediately write down the intrabranch contributions to 1/Ti: -—  =  Tijntra where in this case,  Q  A2±[1og(4T/w) irv =  —  ] (e__!T + e_T) 7  (3.22)  . It is quite likely that higher dimensional effects may 2 2wNL/v  cut off this contribution at energy scales larger than couplings, J , would replace 1  WN  WN.  For example, weak interchain  in the above expression by a quantity of order J . 1  Chapter 3. Model Predictions for T’  63  The interbranch contributions between the lower two and higher two branches can be likewise calculated to give  !  1 Inter T  =  2+ (IAI  IAI2)e/2TKo((h + ltV  +  2  e_T)(3.23)  Following Fujiwara et. al. [34], we write =  1 T  ---  TiJntra  +  T F(h,T)e  ----  TiJnter  (3.24)  We see that the nature of F(h, T) depends largely on the form of the hyperfine coupling. A less general and somewhat more qualitative version of this formula was given by Jolicceur and Golinelli [29], and by Troyer et. al. [35], independently of our work. Jolicceur and Golinelli discussed the isotropic NLu model and derived only the leading exponential dependence on temperature; Troyer et. al. considered the Heisenberg ladder problem, which has a low energy one-magnon excitation spectrum identical to that in the isotropic NLu model, and only included the leading interbranch transition in their expression. 3.1.2  Axial Symmetry  Here we are faced with two possible situations: the sz  0 branch can lie above or below  the doublet. In the former case, the larger the interbranch gap between the doublet and the singlet branches, the more suppressed will be the interbranch contributions to l/T . 1 On the other hand, in the latter scenario, inter- and intrabranch contributions will always be on the same footing (see Fig 3.4). The expression for the intrabranch transitions will be essentially identical to the one in the 0(3) case:  I  Tijntra  where  zj  =  4 [log(4T/N) IAI2  is the gap to the  7rv  SZ  =  —  7 ] (e_(_/T  +  e_+T)  ±1 branches and L 3 is the gap to the  SZ  (3.25) =  0 branch.  The corresponding formula for the interbranch transitions is somewhat more subtle and  Chapter 3. Model Predictions for T’  (a)  64  (b)  3 A  Intrabranch Transitions ----  Interbranch Transitions  Figure 3.4: The gap structure for U(1) symmetry. (a) / >  &L;  z < (b) 3  LU  V  — —  TTv77  Lid  VJ  +  —  v) T  3 _9  —  ‘u  { (i ((o) t— ‘o) lii +  st{ 14A  ((o)b ‘o) i)  9 T)oM/ 9 (Io’ö— ‘o)liI + 1 1((o)’ö ‘oYiI) Tv,(+T)/ }  aJL  JOUJtJ  +  —  uo ‘qnop oq uq ioj  sis  juis oq uq ‘s aqo oq uj  (z/).w=sv j  TV_v>  +/)IJ=/+TV  TV_V<  (L)  V  T  V  T  V —  { (?J ((o) b— ‘o)  ii  ?/  V  V_,+Tv) LV_ —  V >  T t 1 77 L  Lii  (9)  qtraq  uo s  Tpitp  =  (‘i)iii  — —  Tt77 V Li  —  — v—i+v)  —  —  Lid  ((o)t ‘o) ii)  +  s 1 )°X  (i ((o) ‘— ‘o)liI + I ((o)’ ‘o)1i)  9 T)OM/  ta}  lOUJTj  (iJL  --(Ifi+V +  +vI)  =  —  juis oq uoq  sis atjiq saij  ozis  uaoq d  uo spudp (y) ui o dsi  s jqnop  qi  jo zs oq pu  rio spuadp  Jo uoiisod  jo  s sq ppg juis  ioj suoparj poyj  Tj 1  Chapter 3. Model Predictions for T  2rs  M(h)  (zS.—h) — 2 2T  _)  2 (ta+h) 2 —z T 3 2  I  66  — — —  2 Q(O)v  — —  h<  2tT 2 2 Q ( 2 O)v  T 3 2  3  3  3 &L—hM(h)—h h<L—Li  =  (3.29)  -  (Ls = 3 M(h) A seeming catastrophe occurs when two of the branches cross at h  =  I3 —  The  interbranch contribution to 1/T 1 diverges logarithmically. There are essentially two ef fects that would cut off this divergence. Higher dimensional couplings can be counted on once more to replace Q(0) as it approaches zero, with a quantity of order 10J /J as 1 derived in Eqn. (4.40) in Chapter Four. Also, the divergence in the integrand leading to this problem is  1 i/h— I  .  This will be cured by a field with finite width. One  still expects a peak in the relaxation rate, but this will be smoothed by the mentioned effects. 3.1.3  2 x Z Z 2 x Z 2 Symmetry  There 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 analogous expressions from the more symmetric situations.  1 lijntra  =  (l(0o)2W(o)  2 4Aj  ()  (log(4T/wN)  —  y) x  -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, then there 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’  3.2  67  Close to the Critical Field  In this section we give qualitative results on the behaviour of the relaxation rate and in the process prove that the contribution from the staggered correlator becomes crucial as 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 tensor possesses high symmetry (thereby ruling out intrabranch contributions from the uniform spin 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. Long wavelength modes are now expected to play the most important role; we therefore write the dispersion relation of the lowest branch as w(k, h) where v  =  f  (h — h) +  k 2 v  0(3) and u(1) cases  (  and the effective gap is  =  (3.31)  3 case ) 2 (Z = 2(h — 1 h)h/(L +  /2).  Now  that the gap is actually smaller than the temperature, we must include multiparticle processes. This is simply done by replacing the Boltzmann weight by the appropriate occupation factors,  f (w) (1— f (w)).  The derivation is straight forward and can be found  in standard texts on many body physics (for example, see [36]). The uniform contribution to 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 2.2.2 to get 1  —  4iA’  )Unif —  2 71V  2  2 1 z ) Jo 2 (‘ — z  dk  (w + wN)sech () 2 i/(w +  2 WN)  —  3 33  Chapter 3. Model Predictions for Tj’  At criticality we set  /e  68  = 0. We may simply rescale the integration variable to obtain  oc T  (T_1)Ufljf  (3.34)  This is expected from the Ising model where the uniform part of the spin corresponds to the Ising energy density operator , 3 fermions this operator is  =  ‘/L?’R  .  ,  of scaling dimension 1. In terms of Majorana  The correlator of the energy density operator on  the infinite Euclidean plane is known from its scaling dimension and the restrictions of conformal field theory to be [24] < f(z)E(0) >=  2 1/jz  (3.35)  If periodic boundary conditions are placed in the time direction (corresponding to finite temperature), the correlator can be obtained by making a conformal transformation from the Euclidean plane into the cylinder (see [24, 37]),  z  =  e 8 2 T:  irTlye)‘2 1  <f(z)f(O)>  Setting  z  (336)  2 sin(Tirz)1  = it + 6, we can get the contribution to T by integrating over 1  ) Unif  1 d t e -jwNt  f dtet:  2 (irT/ve)  sinh(T7r[t  —  iö])  Changing variables, and assuming the integral is analytic as  2  UN —*  0 (this can actually be  proven by contour techniques), we see that by rescaling the time variable we reproduce Eqn. (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 cases from Ref. [33]. On the infinite Euclidean plane we have <t(z)(O)>zI  <u(z)u(0) >r  U(1)  (338) Z  Ising  The rest of the analogy goes as follows: the magnetic field plays the role of temperature as is obvious 3 from the form of the spectrum; the inverse temperature is analogous to the size of the system in the Euclidean time direction; finally, the staggered magnetization corresponds to the disorder field, ci.  Chapter 3. Model Predictions for TI’  The field  =  69  q + iq’ is the charged U(1) field of the boson model; there are no  problems in using this (in the U(1) case) as long as we account for the interactions. u is 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 only primary operator in the model. Once more, making a conformal transformation into the cylinder of circumference 1/T, <t(z)(0) > <  Once more, setting z  I sin(irzT)I (-T/v  sin(irzT)I  U(1) (339) Ising  it +6 and integrating over time, we get that in the experimentally  =  important limit, T >>  a(z)a(0)  (2T/v  WN,  (TI’)stag cc (2rTL/v2)_ + O(WN)  U(1)  + O(WN)  Ising  (TI’) stag  °  (irT/ve)*  (3.40) For both symmetries, this implies a significantly stronger contribution from the staggered part than the uniform part. In fact, as long as we are sufficiently close to the critical regime, perturbation theory tells us that the above result will only be suppressed by factors of order O(Ih  —  hI/T). In order to observe this behaviour experimentally, one  must have T sufficiently large (having a large anisotropy,  —  2,  would also help), so  that the decrease in relaxation with temperature is obvious. This would require that the experiment be done over a broad range of temperatures so that any constant contributions to 1/T, could be subtracted. In any case, the above should at least serve to clarify that the staggered contribution becomes important in this regime. Farther still from criticality, the analysis breaks down. We do, however, expect the staggered spin contribution to influence TI’ through to the region T  Ih  —  hI.  Chapter 3. Model Predictions for Tj 1  3.3  70  Above the Critical Field  Far above the critical field, h  —  h >> T, the situation becomes even simpler. In the  0(3) and U(1) case the system remains critical. The relaxation rate will be dominated by 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 the relaxation rate since  f  WN  isn’t strictly zero:  dt < 0IS(t,0)0 >< 0S(0,0)0> eLvt  =  ö(wN)/L 21r(M) 2  (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 dependence )’’ 2 cc (2irzT/v  where  (3.42)  is the critical exponent of the staggered spin correlator. Haldane argued  + 0(p), where c  =  I 1I/L  ij  [38]. Thus the field dependence of 1/T 1 is only through  =  i.  When axial symmetry is broken, the gap reappears for h> h. In this case, one can use the fermion model to calculate the relaxation rate. This is made much simpler since the gaps to the two upper branches are presumably much higher than the lower gap (by at least 2L_ (h)). Therefore only intrabranch processes along the lower branch need be considered. The result is 1  ()  =  At sufficiently large magnetic field, h >>  l,_(0, 0)12  (log(4T/wN) 2 l(0,0)  Li  1  —  —÷ 2 2v  2+  , 2 L  h  -  )  (3.43)  some of the expressions simplify:  —  2+z (z )/2 1  /  ‘2+11’\ —  2  )  The rate will drop exponentially with increasing magnetic field.  (3.44)  Chapter 3. Model Predictions for T’  3.4  71  Summary  We would like to summarize the main results of this section. At temperatures much lower than the lowest gap , 4 2 symmetry), T’ Z  Lm  (this can be below the critical field or far above it in the case of  e_m/T.  This is due to the dominance of two magnon intrabranch  relaxation processes. In the case of axial or higher symmetry, the only temperature and field dependence comes from  ---  1 Intra T  cx [log(4T/wN)  —  (3.45)  Given such symmetry, this is a model independent result. Including anisotropies and interbranch 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 contribu tions come from one magnon processes (due to the staggered part of the spin). When the field reaches its critical value, whereupon the gap vanishes, the relaxation rate is given by 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 and eventually becomes large once more.  We refer to the lowest gap corresponding to a polarization direction perpendicular to the field. 4  Chapter 4  Material Properties and Possible Effects on Experiment  4.1  Hyperfine Tensor  Here we briefly discuss the effect of the nature of the Hyperfine tensor, A, on the NMR relaxation rate. As a reminder, u and v are spin indices while i and coupling the nuclear spin at site  j are spatial indices  j to the atomic spin on site i.  Assuming the magnetic field lies in the z direction, if Aj’ is isotropic in its spin indices, only AI 2 will contribute in Eqn.  (1.69).  In particular, there will be no  intrabranch contributions as these require a coupling to S.  Interbranch transitions  will not be limited, but no contribution to them will come from the term proportional to  These statements still hold true for a hyperfine tensor diagonal in the  Heisenberg spin basis.  Note that this implies that for the 0(3) model, intrabranch  transitions are prohibited so long as one assumes that the nuclear gyromagnetic tensor is simultaneously diagonalizable with the hyperfine tensor. In general, especially if the NMR nucleus does not coincide with the magnetic ion giving rise to the effective spin in the 1DHAF, the anisotropies on the spin chain will not be simultaneously diagonalizable in the hyperfine tensor basis. Moreover, if there is more than one nuclear moment per spin contributing to the signal, it is unlikely that the effective  will have the same symmetry as the nuclear Zeeman interaction. Thus  conditions have to be quite convenient for intrabranch transitions to be missing from the rate. This can be important at very low temperatures where we can experimentally  72  Chapter 4. Material Properties and Possible Effects on Experiment  73  distinguish 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 with field. Also, it is possible that the lower gap depends very weakly on the field for certain magnetic field directions. This is true for anisotropic materials where it is difficult to place the field along a direction of symmetry local to the chain. There are two ways to distinguish the transitions in these cases.  The simplest solution is to repeat the  experiment changing the magnetic field direction until one clearly sees the  e_(l)/T  behaviour. Alternatively, one can try to extract information out of the low temperature behaviour, rather than field dependence. For T as log(T/wN)  —  h, F(h,T) in Eqn. (3.24) will behave  ‘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 gap  at zero field. In principle, if one has enough information about the gap structure of the chain, it is possible to deduce the relative values of the hyperfine matrix elements from the relaxation rate. This may be done by comparing the ratios of the rates measured with magnetic field along each of the effective gap directions (or in case of high symmetry, any three perpendicular directions); assuming one has extracted the intrabranch contributions from the measurements of 1/T , one can then work backwards using 1 —  Tilntra  cc A 2 I 3  (4.1)  (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, we  Chapter 4. Material Properties and Possible Effects on Experiment  74  suggest experiments which would distinguish between the models. Finally, we discuss nearest neighbour effects of At’. When these exist, contributions to 1 1/T will come from the correlation function, <S(t)S ( 1 O) >=  <S(t)S”(O)>  where S is the nth fourier mode. Taking N <S(t)S÷ ( 1 O) >=< S(t)S(O) >  +  —+  (4.2)  cc, we write  f  (e  _i) <S(t)Sk(O)>  (4.3)  Expanding in k, the odd contributions in k from the second term will vanish when integrated over all states. From the work done in the last chapter, we know that powers of 2 will give small perturbations of order . k /v We see that including nearest neighbour 2 2L contributions, 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  Impurities  So 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 so there are many chains of varying lengths in each sample. In this section we deal with the fact 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 with Cu) to explore the issue further; we will restrict ourselves to ‘pure’ samples, although our 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 ther  modynamic limit) and a gap  —  has indicated that at the ends of a finite chain there are  Chapter 4. Material Properties and Possible Effects on Experiment  75  free spin-i degrees of freedom. It is expected that the same holds true for finite chains of the Heisenberg antiferromagnet. Indeed, there has been convincing numerical and experimental evidence for this conjecture [40, 41]. The same evidence also supports the notion that these nearly free end spins can interact with each other, essentially exchang ing virtual NLu bosons. The interaction is exponentially decaying with the size of the chain: 1 H Here, the correlation length,  ,  (  i)Le_L/  o 4  (4.5)  SL  is roughly six lattice spaces. For experimentally realizable  lengths, this interaction is negligible and the finite chain has a fourfold degenerate ground state. In a ‘pure’ sample, these finite chains will lie end to end, or be separated by some non-magnetic defect. It is therefore reasonable to presume that two adjacent end spins will interact with an effective exchange coupling, J’, that will vary in strength from zero to something of order J. Recent work [42] has explored this situation in depth. For weak coupling between adjacent end spins the effective Hamiltonian can be written, • 1 H=J’  where  g 2 2 j.  g  (4.6)  is a spin-i operator and c is the projection of a spin-i end spin into a spin  subspace. From numerical work [41, 16], we know that c gives a low lying triplet above a singlet with a gap zE  =  1. This immediately  o J 2 ’ (we assume that J’ is  antiferromagnetic. The ferromagnetic case is expected to give similar results, reversing the order of the singlet and triplet, but numerical work has not yet been done to support the analysis in this limit). This triplet will sit inside the Haldane gap. The triplet corresponds 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 for  Chapter 4. Material Properties and Possible Effects on Experiment  jt  76  >+  lit>  It4>lt>  -  Figure 4.1: Impurity level diagram when D’ is accurate up to JE  ‘-  =  0.  .3J. The embedded states become delocalized and join the  continuum at about J’  .7J. This picture is unchanged up to about J’  “-‘  2J, after  which the triplet returns as a bound state in the Haldane gap to merge with the singlet state as J’ —+ oo. In the type of chain we are considering, J’ is unlikely to become much greater than J; it is much more likely that defects in a pure sample will serve to reduce the effective coupling between sites rather than enhance it. To understand the effect on NMR relaxation we explore the environment of NMR nucleii near the end spins’. Take, for example, the nuclear spin coupled to  .  It sees the  Zeeman split level diagram shown in Fig. 4.1. If we assume that the ‘free end spins’ are not completely free, but are weakly coupled to the magnons on the chain, then relaxation can occur in two ways: when two levels are transition  —  this will happen for a fields h  .‘  WN  apart the chain end spin could make a  J’. Alternatively, a thermal magnon coupled  to the chain end could decay into another magnon with or without the accompaniment of 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 effects of end spin excitations in NMR.  Chapter 4. Material Properties and Possible Effects on Experiment  must be  WN.  77  Marked on the diagram are the transitions that can be induced by  g:  solid  arrows represent possible transitions that potentially do not require coupling to the rest of the chain; dashed arrows represent transitions that could occur only if the impurity is 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, the transition will be broadened by thermal magnons. This implies that the characteristic width will have a temperature dependence exponential in the Haldane gap. There is an additional 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 the following term to the chain-end effective Hamiltonian, Eqn. (4.6), HE  —÷  D 2 HE + ci  ((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 the effective Hamiltonian. The second term is allowed by symmetry, and we presume that it is a consequence of the defect (which arguably, would manifest itself in accordance with the available symmetry). We assume D’  >  —  J’, so that the exchange interaction is still  antiferromagnetic in the z-direction. As a result of this, the two by  —,  while the  SZ =  ±1 levels will be shifted by  .  Z  =  levels will shift  The transitions induced by the  hyperfine interaction are shown in the new level diagrams in Fig. 4.2. It is this more general case for possible transitions which we now carefully analyze (further anisotropic purturbations will not qualitatively change the picture). We begin by characterizing the interaction between the bulk magnons and the end spin. Since we have no information as to the nature of this coupling we will parametrize it in the spirit of Mitra et. al. [43] using free bosons which after scattering with the impurity obtain a phase shift. We make two assumptions: first, that leakage across the impurity site is negligible, and second, that the impurity spin coupling to the boson does  Chapter 4. Material Properties and Possible Effects on Experiment  78  D’>O  k> It  k 144>  itt>  It i,> k  J+D’/2  Itt>  It >— I t>  It >-I t>  Figure 4.2: Impurity level diagrams for D’  #  0.  not allow for exchange of spin (this is consistent so long as J’ is sufficiently weak); bosons on each side of the impurity will have wave-functions of form: C(k) (e  —  e_2i6e_?j  where i refers to the boson branch, k is assumed positive, to the sector of the impurity spin with  SZ =  2 IC±(k)1  (4.8) =  1/2L and ± refers  ±. The boundary condition, q(L)  =  0,  gives k  =  (nir  —  k))/L  (4.9)  We assume that the phase shifts are small and grow with increasing energy. This is tantamount to assuming a large step potential barrier of infinite extent (thus allowing no leakage for states below the barrier). This is certainly true in the limit J’ —+ 0. A  Chapter 4. Material Properties and Possible Effects on Experiment  79  heuristic ansatz which has this behaviour is 6(k)  (4.10)  Notice that the ‘s need not be orthogonal to the q’s, since these states are in separate Hubert Spaces. The energy of one of these bosons at low temperatures is given by the free form 2 2 (ki±) v Before going on we note that  WN  (4.11)  is typically much smaller than the energy level spacings  due to finite size, for typical chain lengths. For example, in NENP, 5E chains L  -‘  .04 Kelvin for  l000a. One may therefore question the validity of boson assisted transitions  when the bosons lack the ability to ‘fine tune’ a transition so that the difference between initial and final states is  WN.  What saves the day, in this case, are higher dimensional  effects. For sufficiently long chains or sufficiently strong interchain couplings, these will densely fill the spacings in energy levels along the chain direction. Assuming this is the case, (we show the conditions for this explicitly in the next section) we will not worry about this point further. Consider the coupling of a nuclear spin to one of the impurity spins, say S (from here on we will implicitly write S  =  S’, for ease of notation). The familiar formula for  the transition rate can be cast as 1 j-11  { + where  < n ; 1  <  ; lIe 1 n  =—j  n,l,n’,l’  t)(HHSILIe_it(J:1+HE)Ir;  lIe_(HHSIn,; 1’  ji = —/2 =  2 I t dte” A1  >< ni,;  lIeit  1’  >< n,;  ô+H)SILIe_it(H  ±, 0. Hb is the free boson Hamiltonian;  l’IS’  mi; I  HE) j 1 ;  1  >  >}  (4.12)  > denotes a state of the 1 Ii; m  impurity spin (ie. an eigenstate of HE), which for brevity, we denote as 1  >; n 1  denotes  Chapter 4. Material Properties and Possible Effects on Experiment  the boson content of the state; in general,  I > will contain  ni;  free boson states that have projections onto the S  =  two  80  different multiparticle  ± subspaces of 1; m 1 > (with  appropriate phase shifts). To elaborate and make things a bit clearer, take the  S’ =  0  state from the triplet. ; 0 ni,  where the states  d:  1,0 >  (I t, > øIn  >  >  +1 t t> ®In>)  correspond to the phase shifted bosons with S’  tively. Making the approximation, a 1 —  =  2 —  ez’) cos(t[E 1  roo  I  IL  —  Ei’])I < IlSILhIl,  >  n,1,n’,l’  1l  <ntn,  Let’s pick a particular transition and work it through. Consider ,u and  It’ >= 1, 0  h  D’/2, we expect strong resonance if D’  ‘  ±, respec  2  —ZWNt  dt  +e  —  =  1, we return to the relaxation rate:  =  —00  11  >.  (4.13)  =  >  I  (4.14)  +, 1 >= 1, 1  >  This transition could be of the type we’ve been discussing where for  I’, i) =  0. The expression for the rate becomes  e_’’/2_  e  Re  >  n+eit(En  L  1 E_!)  2 dt e_tIA_+l <  (4.15)  2  I,.  )  Since the boson multiparticle states,  <  n,  are direct products of symmetrized free N  particle 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’, ) = Re  {eith12ExP  —e(J’/2)  e_+eit+__,)I  1+ (_z n,n,,i,j  L  <n  2 dt etIAI  iIn’;i>  12)  }  (4.16)  Chapter 4. Material Properties and Possible Effects on Experiment  81  where we’ve used the fact that the grand partition function for noninteracting bosons is the exponentiated partition function for a single boson. n and n’ now index single boson states, and i and  j denote boson branches.  The overlap of the boson states can be  calculated from the form given by Eqn. (4.8),  I <;iln’;j>  12  =  (sin(6?fk)±6(k))  8  where we have parametrized the momenta of n and  2 sin(8k)_8(k)))  —  (4.17)  ri’ with k and k’, respectively.  Finally,  we write the one particle partition function (we can ignore the phase shifts for this purpose) as =  <n;iln;i>  e E 3 l+  I <n; ilri; i> I  =  n,i  (4.18)  Combining all of this allows us to write the exponential in Eqn. (4.16) as e_/Tf  4 dkdk’  (i —  2 (k2k)  xe_) / k 2 2i(eit  =  —  elT  8T (i 22 v  —  i)2  2 —k’ (k )v 2 2  00  j  1)  —  dx dx’e  {  1  (4.19)  —  (_  The effect of the phase shifts is contained in the factor,  2 x’)  c\i  —  ] ,\i)2,  (4.20)  e(t)  as seen from Eqn.  (4.10). Corrections to this due to 0(k ) contributions to 6(k) will be suppressed by 3 factors of . 2 T 1 2L /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 time behaviour of  e(t).  Ree(t)  In this limit, the real part of —  —  e(t)  i)2f  becomes: dx dxIe5m(r  —  x’)Tt/2)  Chapter 4. Material Properties and Possible Effects on Experiment  IT i 8 T(  —* _e_j  —  ) t 2 I  82  —F(T)ItI  (4.21)  The expression for the relaxation rate becomes  1 —  <  ‘  =  2 A  (1, 1’, )  dt  e_t  cos (t(D’/2  A—--2  ‘2  e_(J’+UJ’/2_1)  ZE  —  h)) e_ItT)  (4.22)  FI T 1 P ( 2 T) + (h — 2 D’/2)  (4.23)  This is the most important equation of this section. The other transitions can be treated the same way to arrive at analogous results. The key issue to note is that the impurity relaxation rate is an extremely sensitive function of the temperature and field. At tem peratures well below the gap it is essentially a delta-function of h. As the temperature increases and becomes comparable to the gap, the rate broadens rapidly. Before summing up, we discuss the other possible transitions. changing  ,u  First, notice that  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 via S ’ 2 , and therefore no transitions between the singlet alkd the s 2  =  0 state of the triplet.  In other words, the solid circles in Figs. 4.1 and 4.2 are ignorable as are the dashed lines from  Ii, 0 > to  0>. This is expected in all but the most extreme of anisotropic exchange  impurity models. Furthermore, the effect of reversing the spin states on the triplet is the same as reversing the sign of magnetic field: , mj’ —+ —m 1 m , —m 1 ’ 1  +—+  h  —+  —h  (4.25)  Chapter 4. Material Properties and Possible Effects on Experiment  83  Finally, 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 impurity relaxation rate involving all eight possible impurity level transitions is 4  1 --  rp  ““  ‘  11  2  2 A—Ui  D’—4’ ‘—  ,—13Ei  j’—’  i1 f1  ‘—‘E ‘  ‘‘  P ” 2 T’)  T  2fi E  (426  where E 1 denotes one of the four possible initial impurity states, and Ef is the difference in 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 neglect surface effects). Now it can be seen more clearly that all but two of the elements in the sum above will contribute little due to the narrow gaussian form. The important terms are those where the energy in the gaussian is small; this can happen for certain magnetic fields: h  ID’/21 and h  for most defects.  J’ + D’/2. In Heisenberg chains we might expect J’ >>  ID’I  Furthermore, the impurity contribution should be most evident at  lower fields where the gap still lies high. Consequently, in experiment, one expects the h  ID’/21 transition to dominate the picture of impurity contributions to 1/T . 1 In a real sample, the NMR signal from the impurity will be proportional to the density  of the impurities. Moreover, since defects will vary from chain to chain, one would be wise to average over a random distribution of couplings, J’ and D’. In practice, experimental data could be analyzed for the ‘peak’ values of J’ and D’. One could then model the distribution of couplings with the appropriate peak values. This could, in principle be checked against low temperature ESR measurements which ought to concur with the impurity model. A final expression for the relaxation rate due to impurities is  (---)  1 Imp T  =  fdJ’ dD’ p(J’  —  J’)p(’  —  D’) (J’,D’) 1 11 T  (4.27)  where ñ is the density of impurities (or inverse length of the average chain); p is some distribution function.  Chapter 4. Material Properties and Possible Effects on Experiment  4.3  84  Interchain Couplings  In previous sections we mentioned the effects of interchain couplings on various aspects of the physics. We now examine these in more detail. Nearest neighbour interchain couplings will enter the Hamiltonian as 1 H—H+J  (4.28)  ‘•  where < i, j > index nearest neighbour spins not on the same chain. We can return to the derivation of the NLu model to see the effect of this additional term. Taylor expanding the continuum representation for ,., and assuming reflection symmetry about a site, Eqn. (1.38) will change to S  =  2irisQ  + 2LxLy  f  2 x 4 d (ö)  2LxLy  + 2LxLy  f  dx (  fd4x(8vT)2  (4.29)  where we chose the z-direction to be along the chain, and the vector, ,  is  the dis  placement 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 to  dynamical part of the Lagrangian will correspond to J  —*  J+J . Presumably, Jj << J, 1  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 resulting equations of motion are (v8 =  —  + 2J s 1  (  )2)  (4.31)  Chapter 4. Material Properties and Possible Effects on Experiment  85  The dispersion relation becomes, 2 2 w =v + + k vi( where v± cx  (4.32)  %/J7. It is from this last formula which we now extract qualitative infor  mation about interchain coupling effects. First, recall that we claimed that for finite chains of certain lengths in real experi mental situations, we no longer need to concern ourselves with 1-D finite size effects. In other words, we said that energy levels arising from interchain couplings will densely fill the small gaps between magnon energy levels,  Let’s calculate this length in  terms of J and J . We start by assuming a simple form for the interchain contribution 1 to the dispersion: (4.33)  k’) = 2 vai(k+k)  where a± is some typical interchain distance, and expected to be 0(1). The size of the interchain band will be  vo,r2  Setting this equal to the gap in the magnon levels we get L2-=L VL  (434)  I  In 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 Chapter 3. In calculating transition rates, one often encounters integrals such as  f  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 diverge logarithmically in the infrared. If one introduces interchain couplings, the integral over the delta function becomes 4 (2ir)  f  dq d k 2 2 d q ±6(w  —  —  E)  Chapter 4. Material Properties and Possible Effects on Experiment  _i2  _ii2  j  dq  UJj U  I’u q± 2,2 v 61  2,2  Vjj  2,  4 (4)2a  2 2  q 2L  V  —  —  86  2 2 vq 2/  436  where we have assumed a simple form for the interchain dispersion. For ease of calcula tion, we now assume that L L 82av  z. The integral becomes  = /.j  dk dq  J  =  vk2 + vk  a v 2 4  f  k 2 + v2 k  2  —  2 vqj  vq  —  dkIv2k2 + vIk  (4.37)  At low momentum, k, where we need a cutoff, this integral is approximately Lir  (4.38)  6v±va±  If we write the integral in Eqn. (4.35) as 1 d k  v2J  f(k)  (439)  v7-C  then the cutoff, C, is seen to be 144va J  V 2  (4.40)  We recall that the for intrabranch transitions,  Q  find that the cutoff becomes important for  <7OJjaI. For example, in NENP, where  1 J  ‘‘  4.4  WN  . Comparing this to C, we 2 2LwN/v  25 mK, we expect the cutoff to significantly dominate over the Larmour frequency.  Crystal Structure  When analyzing experimental data in terms of the idealized Heisenberg model with on site anisotropies, one must keep in mind that the symmetry of the proposed spin-chain Hamiltonian may be constrained by the symmetry of the crystal and the local symmetry  Chapter 4. Material Properties and Possible Effects on Experiment  87  about the magnetic ion. Additional terms may be added or subtracted to accommodate the structure of the substance, and these can have a great effect on the interpretation of data. Some important questions which must be asked before deciding on a model Hamiltonian for the material are: is the local crystal field symmetry about the magnetic ion commensurate with the symmetry of the unit cell? Is there more than one chain per unit cell? If so, are all chains identical? Is there more than one magnetic ion of a single chain per unit cell? If so, is there true translational symmetry from one spin site to another? In the next chapter, we analyze experiments performed on NENP. In so doing, we will address such considerations.  Chapter 5  NENP: Direct Comparison with Experiment  5.1  The Structure of NENP and Experimental Ramifications  4 N N 8 H Ni(G ( 2 G10 A schematic diagram of ) 0 (NENP) is shown in Fig 5.1. Each chain is comprised of Ethylenediamine-Nickel chelates separated by nitrite groups. The +; experiments indicate that these ions interact antiferromagnetically 2 magnetic ion is Ni along the chain with coupling J  55K. There is a large single ion anisotropy, D  as well as a small axial symmetry breaking anisotropy E estimated at 1 J / J  ‘.‘  12K,  2K. Interchain couplings are  iO [44].  It is important to realize that two neighbouring Ni 2 along the b-direction are not equivalent; rather, one is related to the other by a ir rotation about the b axis. Also, the angle along the N  —  Ni —0 bond is not exactly r, meaning that the Ni site is not truly  centrosymmetric. Most importantly, the local symmetry axes of each Ni ion are rotated with respect to the abc (crystallographic) axes. To demonstrate this we now note the coordinates of the Nitrogen atoms in the Ethylenediamine chelate surrounding the Nickel (placing the Nickel at the origin): [45]  Atom  a (A)  N(1)  2.053 (3)  .162 (3)  .338 (3)  N(2)  .619 (3)  —.184 (3)  —1.971 (3)  b  88  (A)  c  (A)  Chapter 5. NENP: Direct Comparison with Experiment  89  NENP a  ONi  •c 00  •N  Figure 5.1: NENP  Chapter 5. NENP: Direct Comparison with Experiment  90  C C, N(2)  N(1)  a  a’  •_..  Projection b’ Vector  of the Canting  Figure 5.2: Local and crystallographic axes projected onto the ac-plane in NENP  The other Nitrogen atoms in the chelate can be obtained by reflection through the Nickel. One easily sees that projecting this structure onto the b plane yields symmetry axes (in the b plane) rotated  60° from the ac-axes. This is shown in Fig. 5.2. The  inclination of the local Nickel axes from the abc system can be obtained by taking the cross product of the two Nitrogen vectors (ie. the normal to the plane described by the  Chapter 5. NENP: Direct Comparison with Experiment  91  four Nitrogen atoms in the chelate:  n  =  (—.06 (1), .98 (1), —.11 (1))  The local Ni b’-axis makes a in the ac plane is  -‘  100  (5.1)  angle with the b-axis, while the azimuthal angle  —28° from c. The 10° tilt is roughly about the a’ direction of the  local symmetry axes. One may worry that the NO group may distort the local symmetry axes, but re markably enough, when projected onto the ac plane, the three atoms in the molecular ion sit on the c’ axis. This reinforces our suspicion that the local symmetry axes are indeed the above. Next, we consider the whole space group of NENP. The most recent attempt to solve for the crystal symmetries has concluded that the true space group of the material is a [45]; this is a non-centrosymmetric space group with a screw 2 symmetry about 1 Pri2 the b axis, diagonal glide plane reflection symmetry along the a axis, and an axial glide plane reflection symmetry along c. Experimentally, attempts to solve the structure in a have not been successful; rather, it seems that Pnma gives a better fit. The 1 Pn2 main difference between the two is the presence in Fnma of a mirror plane parallel to b at i-b, centers of symmetry at various locations in the unit cell, and two-fold screw axes separating these centers of symmetry. The reason for the experimental discrepancy is 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 the axial glide planes along a, the diagonal glide planes along c and the 2 screw symmetry about b. These generate a total of 4 Ni sites per primitive cell and two chains through each cell. The two chains are such that the Ni chelates on one are the mirror image of the other. Figure 5.3 shows a projection of this picture onto the ac plane. The presence of the  _a  Chapter 5. NENP: Direct Comparison with Experiment  92  C(1) NJ  C(2)  Ii  N(2)  C  I  2A  Figure 5.3: A projection of the NENP unit cell onto the ac-plane showing two chains per unit cell  Chapter 5. NENP: Direct Comparison with Experiment  93  2 screw symmetry about each chain axis introduces staggered contributions to the local anisotropy and gyromagnetic tensors. This is because, as motivated above, these are not diagonal 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 the same (ie. that at each site they can be simultaneously diagonalized). This is rigorously true when the crystal field symmetry about the magnetic ion is no lower than orthorhom bic (a sketch of a proof is found on p. 750 of [4]). We can get the required parametrization for the g-tensors from high temperature uniform susceptibility data [45]. This is based on the idea that at high temperatures the Ni atoms will behave as an ensemble of uncoupled spins (s  =  1) with the same gyromagnetic tensor as in the antiferromagnetic case. With  this in mind we get  (O) + Gfr sin 2 2 (0) G’ cos  0  0  0  Ga’  0  0  0  (O)+Gesin 2 Gj,cos ( 6)  0  0 sin(O) cos(0)(Gb’  0  0  0  sin(0)cos(O)(Gw—Ge) 0  0  g=  Here 0  ‘-.‘  100,  and Ga’  =  2.24, G / 1  =  2.15, G’  =  (5.3)  —  Ge’) (5.4)  2.20 are the values for the local  G-tensor that give the observed high temperature 9-tensor when averaged over the unit cell. Correspondingly, the anisotropy tensors must have the following form:  D  D’  0  0  0  Da’  0  0  0  , 1 D  (5.5)  Chapter 5. NENP: Direct Comparison with Experiment  d= tan(28)  o  o  0  0  0  D) 0  0  (Db’  —  tan(26)  (Db’  94  —  Do’) (5.6)  The parameters Dat, D, D,, are to be fitted by experiment to the model used to describe the system. The boson Hamiltonian can now be written  fl)+ -.d(x  (5.7)  2 i).gf ) 2] l  The term containing d breaks the Z 2 symmetry along the a’ (lowest mass) direction. It will also renormalize the masses. The second effect can be ignored in the approximation that 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 magnetic field. along  .  gap will always persist. The staggered field term will break the Z 2 symmetry 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. The effect on the relaxation rate will be small, although there may be consequences in other experiments [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 found in a unit cell of NENP ‘chain 1’ and ‘chain 2’ corresponding to the chains in the upper left and lower right corners of Figure 5.3 respectively. The dispersion branches of chain 1 are given by Eqn. (20) of [47] (the expressions are roots of a complicated cubic equation and we feel that citing them will not prove illuminating) only the field is the c’ axis where  —  300  from  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  95  Experiments which average over signals, like susceptibility or NMR T’ measure ments, must consider their results an average of two different measurements (correspond ing 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 each chain. The NMR relaxation calculations performed in Chapter 3 assume the field is placed along one of the crystal axes. In this special case, the dispersions for the two different chains are identical. Although the dispersions will be more complicated as will be the matrix elements, lb(O, 0), we do not expect great qualitative differences between a calculation as done in Chapter 3 and one which accounts for the actual symmetry when the field is placed along a crystal axis. There will also be contributions due to the  i— r  correlator; these are also expected to be small. There are, however, important manifes tations of having two inequivalent chains. These will be discussed in the next chapter when we suggest further experiments. In conclusion, consideration of the crystal structure introduces both symmetry break ing terms and two inequivalent chains per unit cell. The symmetry breaking terms will give small corrections to the relaxation rate. 5.2  Analysis of the Data  By far, the most studied Haldane gap S  =  1 material is NENP. The most recent measure  ments of the relaxation rate, 1/T , on this substance have been made by Fujiwara et. al. 1 [34]. Before directly comparing our results to the data we discuss the expected results on a pure (infinite) system. The three gaps are given by neutron scattering: =  2.52meV and La  =  1.34meV. We use v  =  /a =  1.17meV,  10.9meV, and the generic value of 2.2 for  the electronic g-factor. Since we do not have an accurate description for the hyperfine coupling of the Ni ion to the protons in its surrounding chelate, we assume a uniform  Chapter 5. NENP: Direct Comparison with Experiment  96  value for all the contributing hyperfine matrix elements in a given direction of the applied field. Writing T’  =  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 fields along 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 by appropriate occupation factors in Eqn. (3.18): fb(1+fb) fj’(l  —  f)  =  =  cosech ( 2 j)/4 for bosons, and  sech ( 2 j±)/4 for fermions [36]. Within approximations used, multiparticle  effects amount to multiplying the final expressions by (1 ± e) . At higher tempera 2 tures it is also necessary to include the k-dependence of the integrand past the peak at the origin. We expect that at temperatures T  • and fields h  the numerically  integrated results would differ by about 10 percent. F(h, T) is shown for fields up to 9 Tesla even though the (/3w >> 1) approximation is no longer valid at such fields. This is done to contrast the predictions of the boson and fermion models. It’s easy to see that the boson result for F(h, T) diverges at the critical field, while no such catastrophe is present in the fermion result. This divergence is logarithmic and infrared. It will persist even after account is made for the staggered part 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 again is 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 tran sitions only for small field. In this regime, one must also be careful to include intrabranch transitions in the second lowest branch. All these processes are of the same order. Even though the intrabranch rates vanish at low fields, the interbranch contributions are sup pressed by the absence of low momentum transitions (ie.  Q for the interbranch transitions  C,)  0  CD  Ct’  0  Cl)  CD  5  0 J) 0  Ct’  clq  LI.  -  I—  •0  0.0  10.0  0.0  20.0  30.0  40.0  50.0  2.0  4.0 H (Tesla)  6.0  8.0  Contributions From The Uniform Part of the Spin  Boson F(h,T):  10.0  1H 10 at 1.4K 110 at 2.0 K H 110 H at4.2K HIIb at 1.4 K Hub at 2.0 K HI,b at 4.2K  Chapter 5. NENP: Direct Comparison with Experiment  98  OCOC’ —  —  —  —  —  —  0 I I I  I  I I I  —  I  II  1—0-  (Ls)  q  (j )-j  Figure 5.5: Fermion F(h, T) for fields along the b and c chain directions.  J)  0  CD  I-  Ct’  0  Cl)  CD  0  Cl)  0  CD  aq  I— LL  .  0.0  10.0  0.0  20.0  30.0  40.0  50.0  2.0  4.0 H (Tesla)  6.0  8.0  Contributions From The Uniform Part of the Spin  Boson F(h,T):  10.0  Hiiaatl.4 K Hiia at 2.0 K Hiia at 4.2 K HIIbatl.4K Hub at 2.0 K HIIb at 4.2K  Co CO  rj  CD  U)  0.0  30.0  40.0  50.0  2.0  4.0 H (Tesla)  6.0  8.0  Contributions From The Uniform Part of the Spin  Fermion F(h,T):  10.0  -  Huaatl.4K Hiiaat2.OK H at4.2K HIIbatl.4K H,Ibat2.0K HIIbat4.2K  I.  Chapter 5. NENP: Direct Comparison with Experiment  is 1 O( J L  —  as opposed to  O(WN).)  101  For this case, only i need be calculated.  When the field is along the c direction (corresponding to the middle gap), we restrict ourselves to calculating intrabranch transitions along the lower branch and interbranch ones between the lower and c branch. There are no intrabranch processes along the c axis. Calculating the interbranch transitions amounts to calculating  and  When the field is along the a direction, the calculation proceeds as above. The crossing of the branches provides for the interesting effect mentioned earlier. The peak in 1/T 1 can be used to locate the true ac-axes for the chain Notice that F(h, T) for the field parallel to the b axis is nearly field independent over a large range of the magnetic field. This behaviour is quite easy to understand from the universal results valid in the axially symmetric case, discussed in Chapter 3. When the field is along b, the system is only slightly anisotropic, and so the axially symmetric results 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 other hand, F vanishes quadratically as h symmetry corresponding to z of order L  —  —  =  —*  —*  0). On the  0. Including the small breaking of the axial  2°K, F,, is essentially constant down to low fields  iT, before rapidly decreasing as seen in in the figures.  We now proceed to directly compare our results with those of Pujiwara et. al. Since the 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 a  102  Chapter 5. NENP: Direct Comparison with Experiment  H II b Fit to Fermion Model  100.00  I  I  H=7.1 T H=6.65 T •H=5.45T •H=4.32T — H=7.51 T H=6.65T H=5.45T H=42T  o  C  • •  0 0  10.00  ——  C N  cli  1.00  N  N  N flN  C  .  • N  .  N  S  N  0.8  0.6  0.4  0.2  0.0  Fit to Boson Model  100.00  I  I  •  o %C o  0  0  •  0  •  °°  1.00  H=7.5l T cH=6.65T •H=5.45T •H=4.32T —H=7.51T H=6.65T H=545T 0  ‘  0  ‘  N N  N N  — N N  S  •  .•  N  0.10  I  0.0  0.2  I  0.4 ) 1 1 (K T  0.6  0.8  Figure 5.8: Theoretical (lines) vs. experimental data (circles and squares)  103  ment Chapter 5. NENP: Direct Comparison with Experi  H//C Fit to Fermion Model  100.00  I —  H=7.51 T H=6.23T H=5.45T H=4.32T a H=7.51 T  —  ——  o  0.10 00  0.8  0.6  0.4  0:2  Fit to Boson Model  100.00  I  \  I  •  I  •  —H=7.51T H=6.23T — H=5.45 T — — H=4.32T oH=7.51 H=6.23T• H=5.45 T •H=4.32T  ‘-.,  ——  10.00  \ \\., \‘O  •  N-a  .%  I.—  1.00  4 b • \.%  -  \ \  ‘S‘  S--  0  \ \  %. a’.  \  0  \  0.10  I  0.0  0.2  I  0.4 ) 1 1 (K T  I  0.6  0.8  squares) Figure 5.9: Theoretical (lines) vs. experimental data (circles and  Chapter 5. NENP: Direct Comparison with Experiment  104  field placed along the w-direction:  Ab  1 (  ‘  8.5 MHz fermions 7.0 MHz  (5.9)  bosons  (17.7 MHz fermions  112.0  MHz  (5.10)  bosons  These 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  3 r  2 MHz  (5.11)  Also, 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. (5.12)  H 3 A,uoX  At about 4K, the susceptibility is roughly a fourteenth of its maximum value. Given that XMax  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 accurate evaluation of the hyperfine matrix elements is still unavailable. Overall, the fermion fit is 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  —  HIb  data becomes progressively worse as the field is increased. Fitting to the lower field data seems to give better overall agreement than fitting to the higher field results. This is not the case for the  hIIc data (at least with the fermions). Since the field in the experiment  was not actually placed along the  chain c-axis, we might expect even worse agreement  between this set of data and our calculations! In fact, as mentioned before, we expect a  Chapter 5. NENP: Direct Comparison with Experiment  very weak field dependence for the  105  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 results agree. This implies that the relaxation is largely mediated through thermal bosons and that 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 the density of states. We believe that we have taken account of the obvious mechanisms for relaxation. 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 increase in the relaxation at the mid-field range. Moreover, they would be expected to play a similar role when the field is placed along the c-axis. There are 16 protons in the chelate surrounding each Ni ion. Nuclear dipole-dipole interactions among them are energetically 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, but it’s hard to explain why there would an additional dependence on the magnitude of the field. 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 theory alone. 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 shift constants,  )4,  in Eqn.  (4.10) are 0(1), the impurity resonance width, F, derived in  the 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 and another peaked at D’/2 + J’  =  =  h  h. This means that we expect two bumps in the  relaxation rate due to impurity effects. The width of the bumps should correspond to the width of distribution of impurity couplings. The problem arises when we see that the  Chapter 5. NENP: Direct Comparison with Experiment  106  H//b Experimental Data 0 0  0 0  U  0 10.000 0  0T=1.4K • T=2.O K •T=4.2 K  1.000  . 0 I  0.100  0.0  0  0  5.0 H (Tesla)  10.0  Figure 5.10: Relaxation rate for field along the b-axis.  Chapter 5. NENP: Direct Comparison with Experiment  W(meV)  107  Impunty Resonance Wuith (H=3 T)  T (K)  Figure 5.11: F(T)—the width of the impurity resonance temperature dependence of the low field data is roughly exponential:  e_J’/T,  where  the impurity coupling J’ is about 4.7K. Furthermore, the sharp decrease from zero field relaxation suggests D’  =  0. By analyzing Eqn. (4.26) we see that the second bump  should have little temperature dependence. This means that assuming the first bump sits near h  =  0, the second must be larger and separated by about 3.5T. This is clearly  not the case. Indeed, we would need a complicated distribution of couplings, J’ and D’, 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 should take notice, however, that the data was taken for a field along the b-axis, where other problems were present at mid-field. Finally, we would like to mention some recent NMR data collected on the 1-D S  =  1  spin chain AgVP 6 by Takigawa et. al. [48]. This material is highly one dimensional S 2  Chapter 5. NENP: Direct Comparison with Experiment  with a large gap (z  320K) and very nearly isotropic (3  108  4K). These characteristics  make it ideal for analysis using our results. There are, however, some questions about the properties of the material which would have to be analyzed before an understanding of the NMR results is possible within the framework proposed here. The gap deduced from studies on the Vanadium atom (z  ‘-.‘  410K) conflicts signiflcalltly with those performed on  the phosphorus sites and with neutron scattering data. In addition, the material has very low symmetry (corresponding to the space group P2/a) and very little is known about the possible small E and D terms in the Hamiltonian and their corresponding symmetry. There is fair qualitative agreement between the 31 P NMR data and our theory, and it is possible to explain some of the discrepancies using a temperature dependent anisotropic gap structure, but we feel that not enough is yet understood about gross features of the material to justify such speculation at this time.  Chapter 6  Suggested Experiments and Curious Predictions  We finish by pointing in this final chapter towards further experimental work which could serve to both better understand S = 1 1DHAF’s as well as corroborate and clarify some of the issues raised in this thesis.  6.1  Experimentally Testable Conflicts Between Models  When discussing the matrix elements, < k, aSi(O)q, b >, within the different models, we noticed that there were some discrepancies between predictions. We now examine this hoping to offer experiments that would resolve the issue in favour of one model or another. We start by discussing experiments on isotropic systems. In this case, the major differences between the predictions of the models concern large 0 transitions, where we recall from Chapter 2 that <  k,ai(O)q,b >  jfiabG(O)  cosh(O) =  —  kq)/z 2 v  (6.1)  This is especially dramatic in the case of backscattering. The problem with an experi ment which probes large 0 transitions is that contributions from matrix elements of the staggered part of the spin may be large as well. This can be cured by looking for a low temperature 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 a 1  109  Chapter 6. Suggested Experiments and Curious Predictions  double Boltzmann factor,  e_ / 2 T.  110  A good candidate for such an experiment 1 is elastic  neutron scattering at zero or near zero magnetic field. The cross section is proportional to the spin correlation function; for elastic scattering, this is S(Q, 0) cc  <n (0)m> 2 6(w  —  cc.rn)6(Q  —  k + km)e_m’T  (6.2)  n,m  At sufficiently low temperatures, this expression is simpler than the analogous one for the relaxation rate thanks to the momentum conserving delta function. The energy conserving delta function ensures that only backscattering will contribute to the cross section. Using the results of Chapter 3 we easily integrate this to give 2e_wQ/2/T 2 S(Q,0) cc G(0)2  (6.3)  vQ  The NLa model gives 0 1 G “ ‘I  2  —  —  At large  Q.  64  this will behave as 1/ log 2 (vQ//.S. This is very different from the free boson  Q  prediction of G(0) large  2 tanh(0/2) 4 1 + (0/ir) ir 0/2 2 64I1+(0/2ir)  =  1, and from the free fermion prediction of G(0)  We need to qualify what we mean by ‘large’  Q.  —*  /(vQ) for 2 z  As discussed in Chapter  1, the field theoretic models introduced are expected to be accurate only for zero and  K.  Q  near  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 is  the investigation of the structure function near the border region where the field theories begin 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 expect  that the differences between the models should be discernible in this range. The reason we suggest 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 momentum  transitions  Chapter 6. Suggested Experiments and Curious Predictions  111  backscattering transitions contribute. For nonzero field, interbranch transitions can occur at large momentum which will not necessarily select oniy backscattering events. This will not serve to make the interpretation transparent. The condition for backscattering even in the presence of a magnetic field is  Q>> h/v  (6.5)  In the case of axial symmetry, we can suggest the same technique to investigate the difference between the zero field predictions of the boson and fermion models. Regardless of 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 boson  model prediction only involves the exponential factor. The same comments apply to the case where axial symmetry is broken as well. This is no surprise since at large enough momentum, 0(3) symmetry is effectively restored. Elastic neutron scattering is a good probe for the matrix elements involving large momentum and small energy exchange. Other techniques which explore the opposite regime are electron spin resonance (ESR) or far infrared absorption experiments. In both, one subjects the magnetic system to an external source of electromagnetic radiation (the microwave frequency value of the radiation depends on the transitions one is interested in investigating). The RF field couples to the spins in the same way that a magnetic field does, assuming that the electric dipole moment of the electrons on the magnetic ion is much smaller than the effective spins. 2 The interaction Hamiltonian is therefore  = flRFG. cos(ôt) 1 H  (6.6)  Since the coupling is to the total spin of the system, the resonant transitions implied by Fermi’s Golden Rule will involve energy w and zero momentum exchange. At low A rigorous treatment would try to treat the coupling to the electric dipole moment; this can be done 2 within the spin manifold using the Wigner-Eckart theorem. We will not bother with such a treatment here, but we note that it may be crucial in understanding some experiments on NENP [32]  Chapter 6. Suggested Experiments and Curious Predictions  112  temperatures, the power absorbed when a uniform field is applied to the system will be 1(w) cc n,m  <rilSom> I 6(w(h) 2  —  1< a;k,w(h)ISoIb;k,w(h)>  wm(h)  —  w)S(k  —  km)e_wm/T  2e/Tk  where a and b denote one magnon states and k satisfies, 4(h)  —  (6.7)  w(h) = w. Since  the density of states factor in the above is divergent for k = 0, it stands that 1(w) will have a peak at the value of h for which z(h)  —  LP(h) =  w. (The divergence will be  cured 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 much easier 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. 0 <a;0 I b;0 IS >=  =  f f  dx <a;0jS(x)Ib;0  >=  f  dx  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, preferably large, anisotropy. For example, considering axial symmetry with a large D anisotropy,  Il_(0, 0)12  (z / 3 z+3 L./z +2)  Bosons  2  Fermions  The maximal difference corresponds to  (6.9)  L3/&L  2 which leads to a discrepancy of about  13% between the models. The closer the two branches lie, the better the agreement be tween the models. This suggests the following experiment on highly anisotropic materials (NENP being a prime candidate). One chooses two RF frequencies. The first should cor respond to the large interbranch gap, D, and the peak absorption ought to be measured  Chapter 6. Suggested Experiments and Curious Predictions  113  with a low field placed along the direction of the D anisotropy. The second RF frequency should be 0(E) if the material breaks axial symmetry, or 0(h) if the material is axially symmetric. This should then be used to measure the absorbed ESR power with the posi tions of the uniform and RF fields exchanged. This second transition will involve matrix elements which will be gap independent in both models. The matrix elements from the first transition can be extracted and compared to that of the first. If the boson model is a better description even at these low fields, then the two matrix elements should be identical. One may argue that it is redundant to make both measurements since, if the gaps are known, Eqn. (6.7) should give the correct description. The problem lies in cutting off the infrared divergence at the absorption peak. This will introduce an unknown proportionality constant. This divides out when comparing the two measurements. The ratio of the two measurements would be 1(wi) ) 2 1(w  —  1 1  —  —  where we assume a small field, h <<  e_whIT /3 e2/T  &L,  (  ,  0)  2  ( 6 10 ) .  and small E << D.  To end this section, while on the subject of ESR experiments, we would like to propose additional experiments to test the impurity model presented in Chapter 4. ESR is ideal for such tests. Used in conjunction with T’ measurements on a given sample, it would be possible to characterize the couplings J’ and D’ of the end spins 6.2  Measuring Small Anisotropies  Recall that we expect a peak in T’ whenever two branches cross. Experiments on Haldane Gap materials have yet to look for these. The sharpness of this peak depends on the interchain couplings which cut off the diverging integral in the calculation of the relaxation rate. Often, this will be broad because intrabranch transitions will share  Chapter 6. Suggested Experiments and Curious Predictions  the same cutoff (ie. when J 1 >  &.‘N).  114  However, the bump should be experimentally  observable. We propose that information about the anisotropy tensor can be extracted from this phenomenon. Essentially, one looks for the lowest field at which this bump occurs. 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 seem ingly degenerate doublet unresolved by other experimental techniques, such as neutron scattering. One begins by placing a uniform magnetic field in the plane perpendicular to the axial direction (ie. somewhere in the xy-plane). The magnitude of the field should be h 2 > ( 6 /a  where  —  is the uncertainty in resolving the doublet. One then  proceeds to measure T 1 for different angles in the xy-plane spanning a region of at most 1800. If there is an E type anisotropy, one ought to see some structure to the data as a function of angle. Moreover, if there is such structure, we expect a bump at the angle where the branches cross. Once this angle is found, the experiment is repeated for somewhat lower field. The angle where the new bump should be seen would be greater than the old. There is actually enough information in these two measurements already to determine the anisotropy tensor. The dispersion relations are a function of the angle of the field (relative to some axis), the field magnitude and the gaps. The only unknowns are the absolute angle (or location of the axes of the anisotropy tensor) and the difference in 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 angle until 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 associated with the middle gap. This field also gives the anisotropy:  I  —  =  /h 2 ILD  —  I.  It would be interesting to perform such an experiment on NENP. Presumably, one would find two angles corresponding to the two inequivalent chains in each unit cell.  Chapter 6. Suggested Experiments and Curious Predictions  115  Moreover, one would be able to verify the claims made in the last chapter regarding the positions of the local anisotropy tensor in NENP.  6.3  ESR for NENP  In the last chapter we noted that NENP has two inequivalent chains per unit cell. Fur thermore, their local anisotropy tensor was argued to have symmetry axes which did not correspond to the crystal axes. These facts have important ramifications for ESR ex periments on NENP. Figure 6.1 shows the dispersions for chains 1 and 2 (bold and light lines, 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 the ESR experiment. Figure 6.2 shows the resonance field versus orientation of field in the crystallographic ac-plane. The lower branch denotes transitions in chain 1 while the upper branch cor responds to transitions in chain 2. The transitions were calculated at .19 meV. corre sponding to 47 0Hz. In addition the experimental results of Date and Kindo [49} are represented by the circles. One immediately sees that the data does not compare well with the predictions based on the models we’ve used so far, for instead of following one of the branches, the experimental results lie between them. Furthermore, it seems unlikely that perturbations will cause such a significant shift in the resonance field. One sees that the discrepancy is [49] was also  ‘—‘  ‘.‘  ±1 Tesla. One possible explanation is that since the ESR signal in  ±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 as one single peak. Seen that way the model predictions are in good agreement except for the large field regime. One also has to keep in mind that the high-field boson dispersions are not accurate and therefore the predictions at larger angles could easily be .5 Tesla  __  Chapter 6. Suggested Experiments and Curious Predictions  116  cc Cct .c-c  0  / /  , ,  0 CD  /1  E  /  0  J’  /  /  /  /  /  /  , f  —  f  /  I  (jew) ABJeu3  Figure 6.1: Dispersions for the two chain conformations and sample resonant transitions for a uniform field placed 600 from the crystallographic c-axis.  Chapter 6. Suggested Experiments and Curious Predictions  117  4-  a) ><--  LU  o  P° /  I  •  I  /  I  / /  / /  /  -d cc 0  /  0..  z  LI.]  I  I  I  000  C -  Z  CD  (isei) PIe!d eouuose Figure 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 NENP which specifically look for the double resonance predicted here. To end this discussion, we’d like to elaborate on a previously made statement re garding the assignment of masses to the local Ni symmetry axes. It’s easy to see that switching the masses around is tantamount to a ir/2 shift in Figure 5.2 (the fact that the gyromagnetic constants are not the same in orthogonal directions will not change the 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 at 0 and 90 degrees, where the two chain resonances coincide. This determines the proper labeling of the local symmetry axes.  Chapter 7  Concluding Remarks  With the increasing theoretical interest in low dimensional systems, there has been a proportionate 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 led to predictions pertaining to other types of experiments. It is hoped that our efforts will aid in both extending and clarifying existing knowledge of the subject.  119  Bibliography  [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 a review. [3] P.W. 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