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arXiv:cond-mat/0010486 · Nagoya International Workshop on Physics and Combinatorics

SIMPLIFICATION OF THERMODYNAMIC BETHE-ANSATZ EQUATIONS

Minoru Takahashi

Published Apr 1, 2001• 18 citations
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Abstract

Thermodynamic Bethe ansatz equations for XXZ model at $|Δ|\ge 1$ is simplified to an integral equation which has one unknown function. This equation is analytically continued to $|Δ|<1$.

DOI
10.1142/9789812810007_0013

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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.
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.
We present a computational framework not based on the string hypothesis for the thermodynamics of statistical and quantum field theory models solvable by the Bethe ansatz. In the cases of XXZ Heisenberg chain and the sine-Gordon quantum field theory we derive a single nonlinear integral equation which determines the free energy (or ground-state scaling function). Our approach is very effective at high temperature, and correctly reproduces the low-temperature central charge and the analytic structure of the corrections.
We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0T=0T=0. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5