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Physical review. B, Condensed matter

Excitation spectrum of planar spin-½ Heisenberg xxz chains

T. Schneider, E. Stoll, Ueli Glaus

Published Aug 1, 1982• 29 citations
Publisher

Abstract

We present analytic weak-coupling and numerical finite-chain results for the dynamic form factor associated with the out-of-plane fluctuations at zero temperature and zero field. The dynamic form factor associated with the in-plane fluctuations is treated numerically only. In the ferromagnetic regime, the dominant features are traced back to particle-hole pair continua and the associated bound states. With the approach toward the isotropic ferromagnet, the resonance associated with the bound state of the particle-hole pairs gradually exhausts the spectrum, corresponding in this limit to the magnon peak. In the antiferromagnetic regime, the spectrum is dominated by the particle-hole pair continuum. The analytic results for the dynamic form factor associated with the out-of-plane fluctuations reproduce and extend the well-known properties of the continuum model to leading order in the coupling constant. As expected from the relation between the six-vertex model and the $\mathrm{xxz}$ chain, these properties include singular behavior of dynamic form factors depending continuously on the coupling constant.

DOI
10.1103/physrevb.26.1321

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arXiv:1003.4557•Journal of Statistical Mechanics Theory and Experiment•May 31, 2011•66 citations
We present analytic weak-coupling and numerical finite-chain results for the dynamic form factor associated with the out-of-plane fluctuations at zero temperature and zero field. The dynamic form factor associated with the in-plane fluctuations is treated numerically only. In the ferromagnetic regime, the dominant features are traced back to particle-hole pair continua and the associated bound states. With the approach toward the isotropic ferromagnet, the resonance associated with the bound state of the particle-hole pairs gradually exhausts the spectrum, corresponding in this limit to the magnon peak. In the antiferromagnetic regime, the spectrum is dominated by the particle-hole pair continuum. The analytic results for the dynamic form factor associated with the out-of-plane fluctuations reproduce and extend the well-known properties of the continuum model to leading order in the coupling constant. As expected from the relation between the six-vertex model and the xxz\mathrm{xxz}xxz chain, these properties include singular behavior of dynamic form factors depending continuously on the coupling constant.
We present analytic weak-coupling and numerical finite-chain results for the specific heat, the longitudinal zero-field susceptibility, and the longitudinal dynamic form factor of the planar ferromagnetic xxz\mathrm{xxz}xxz chain at finite temperatures. It is shown that the "bound state" probed by the dynamic form factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ within the Hartree-Fock approximation has to be interpreted as the first moment of Szz{S}_{\mathrm{zz}}Szz​ of the states lying above the continuum appearing in the isotropic xy\mathrm{xy}xy model.
The excitation spectrum of the ferromagnetic s=12s=\frac{1}{2}s=21​ Ising-Heisenberg chain as probed by dynamic form factors is studied. Exact results are presented for zero temperature, and the modifications introduced by thermally induced transitions are elucidated by finite-chain calculations. A particularly interesting result is the occurrence of a thermally induced resonance around zero frequency for an Ising-like model. Its origin is traced back to transitions involving magnon bound states. The resonance structure of the dynamic form factors associated with a Heisenberg-like system turns out to be less spectacular, because the contributions of different induced transitions nearly overlap on the frequency scale; so that rather broad resonance structures result. Our findings have an important bearing on related models with s>12s>\frac{1}{2}s>21​, such as CsNiF3{\mathrm{F}}_{3}F3​, and on the soliton aspect of the classically treated systems.
Results are presented and interpreted for the zero- and finite-temperature excitation spectrum of ferromagnetic Ising-Heisenberg chains, as probed by dynamic form factors. Finite-chain calculations demonstrate the occurrence of a thermally induced resonance centered around zero frequency, and a broadening of the magnon and two-magnon bound-state resonances at low TTT. The occurrence of bound states in the associated transitions demonstrates the failure of finite-order magnon perturbation theory.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
The ferromagnetically polarized gapped XXZ spin chain is studied at low temperatures. Utilizing only the one- and two-magnon spectrums and focusing on the magnon-creation contribution to the transverse dynamical susceptibility, we represent the latter in the form of the Dyson equation. Then, following the well known correspondence between the imaginary part of magnetic susceptibility and dynamical structure factor, we get the low-temperature formula for the magnon-peak lineshape. The suggested approach is effective only if the processes related to magnon creations and to transitions from magnons to coupled magnon pairs are energetically separated. As it is shown in the paper, such separation is inherent in the easy-axis chains with rather strong anisotropy. We present several plots and discuss their lineshapes. As the supplemental result we obtain integral representations for the temperature-dependent magnon resonance shift and the parameter which is usually associated with the decay rate. The low-temperature behavior of the resonance shift is studied in details. All calculations are performed up to controllable error $o({e}^{\ensuremath{-}\ensuremath{\beta}{E}_{\mathrm{gap}}})$.
We study the thermodynamic limit of the particle–hole form factors of the XXZ Heisenberg chain in the massless regime. We show that, in this limit, such form factors decrease as an explicitly computed power law in the system size. Moreover, each corresponding amplitude can be obtained as a product of a 'smooth' and a 'discrete' part: the former depends continuously on the rapidities of the particles and holes, whereas the latter has an additional explicit dependence on the set of integer numbers that label each excited state in the associated logarithmic Bethe equations. We also show that special form factors corresponding to zero-energy excitations lying on the Fermi surface decrease as a power law in the system size with the same critical exponents as in the long-distance asymptotic behavior of the related two-point correlation functions. The methods that we develop in this paper are rather general and can be applied to other massless integrable models associated with the six-vertex R -matrix and having determinant representations for their form factors.