Low-temperature spectrum of correlation lengths of the XXZ chain in the antiferromagnetic massive regime

Abstract

We consider the spectrum of correlation lengths of the spin-1 2 XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy finite sets of algebraic equations, the so-called higher-level Bethe Ansatz equations. The behaviour of these equations, if we send the temperature T to zero, is different for zero and non-zero magnetic field h. If h is zero the situation is much like in the case of the usual transfer matrix. Non-trivial higher-level Bethe Ansatz equations remain which determine certain complex excitation parameters as functions of hole parameters which are free on a line segment in the complex plane. If h is non-zero, on the other hand, a remarkable restructuring occurs, and all parameters which enter the description of the quantum transfer matrix eigenvalues can be interpreted entirely in terms of particles and holes which are freely located on two curves when T goes to zero.

Thermal form factor approach to the ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime

“…* ¶ In analogy with conformal field theory the spin of an operator is defined by the value it changes the spin of a state it is acting on. * This contour is different from the contour C in [16] as it is only half as wide. This choice turns out to be more suited for the analysis of the T → 0+ limit of the form factors.…”

“…As we can see, the massive regime is distinguished from the massless regime by the opening of a 'band gap' at the critical field h . The main result of our work [16] was an explicit formula for all correlation lengths, or rather all eigenvalue ratios, in the low-temperature regime. At low enough temperatures all excitations are parameterized by solutions of the higher-level Bethe Ansatz equations (11).…”

“…Thus, instead of ρ n we shall rather write ρ = ρ {x i } n h i=1 , {y j } np j=1 |k . With this change of notation the eigenvalue ratios at finite magnetic field are expressed as follows [16] the curves are closed. In this regime ε is not given by (6), but is rather defined as a solution of a linear integral equation (see e.g.…”

“…In such a way, one does recover that the zero-temperature correlation functions are field independent. In the presence of a magnetic field the Bethe root patterns for all low-temperature 'excitations' of the quantum transfer matrix are of particle-hole type [16]. In the zero-temperature limit the particle and hole roots become unconstrained and densely fill two curve segments in the complex plane.…”

Exact regimes of collapsed and extra two-string solutions in the two down-spin sector of the spin-1/2 massive XXZ spin chain

“…(iv) Recently, the spectrum in the massive regime of the spin-1/2 XXZ spin chain has attracted much interest. For instance, the low-temperature spectrum of correlation lengths are studied in the antiferromagnetic massive regime of the spin-1/2 XXZ spin chain [42,43]. Since some of the results of the present paper are favorable to the string hypothesis, it should be interesting to examine how far thermal properties obtained by assuming the string hypothesis should be valid.…”