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arXiv:cond-mat/0102027 · Journal of Physics A Mathematical and General

Equivalence of TBA and QTM

Minoru Takahashi, Masahiro Shiroishi, Andreas Klümper

Published Mar 26, 2001• 39 citations
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Abstract

The traditional thermodynamic Bethe ansatz equations for the XXZ model at |Δ|⩾1 are derived within the quantum transfer matrix method. This provides further evidence of the equivalence of both methods. Most importantly, we derive an integral equation for the free energy formulated for just one unknown function. This integral equation differs in physical and mathematical aspects to the established ones. The single integral equation is analytically continued to the regime |Δ|<1.

DOI
10.1088/0305-4470/34/13/105

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The traditional thermodynamic Bethe ansatz equations for the XXZ model at |Δ|⩾1 are derived within the quantum transfer matrix method. This provides further evidence of the equivalence of both methods. Most importantly, we derive an integral equation for the free energy formulated for just one unknown function. This integral equation differs in physical and mathematical aspects to the established ones. The single integral equation is analytically continued to the regime |Δ|<1.
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The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp (1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the nonlinear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation based on the string hypothesis.
Abstract The Thermodynamic Bethe Ansatz (TBA) allows us to study finite size and finite temperature effects of an integrable model. This chapter investigates the integral equations that determine the free energy and gives their physical interpretation. It discusses Casimir energy, Bethe relativistic wave function, the derivation of thermodynamics, the meaning of pseudo-energy (dressed energy and momentum), infrared and ultraviolet limits, the coefficient of bulk energy, the general form of the TBA equations, the thermodynamics of the free field theories, L-channel quantization and the LeClair–Mussardo formula. It also covers the application of the Yang–Lee S-matrix, the magnetic field Ising model, and the tricritical Ising model.
Thermodynamic Bethe ansatz equations for XXZ model at ∣Δ∣≥1|Δ|\ge 1∣Δ∣≥1 is simplified to an integral equation which has one unknown function. This equation is analytically continued to ∣Δ∣<1|Δ|<1∣Δ∣<1.
We derive traditional thermodynamic Bethe ansatz (TBA) equations for the sl ( r + 1) Uimin–Sutherland model from the T -system of the quantum transfer matrix. These TBA equations are identical to the those from the string hypothesis. Next we derive a new family of nonlinear integral equations (NLIEs). In particular, a subset of these NLIEs forms a system of NLIEs which contains only a finite number of unknown functions. For r = 1, this subset of NLIEs reduces to Takahashi's NLIE for the XXX spin chain. A relation between the traditional TBA equations and our new NLIEs is clarified. Based on our new NLIEs, we also calculate the high-temperature expansion of the free energy.
Abstract We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi–Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing first on the one-dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the SU ( 2 ) chiral Gross–Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross–Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples.