Large and small Density Approximations to the thermodynamic Bethe Ansatz

TL;DR

This paper derives analytical solutions to thermodynamic Bethe ansatz equations in different density regimes, extending previous results to non-simply laced affine Toda theories and validating them with specific models.

Contribution

It extends large and small density approximations of TBA equations to non-simply laced affine Toda theories and improves these approximations using Y-systems.

Findings

Analytical solutions for TBA equations in large and small density limits.

Extension of results to non-simply laced affine Toda field theories.

Validation of approximations with Sinh-Gordon and other affine Toda models.

Abstract

We provide analytical solutions to the thermodynamic Bethe ansatz equations in the large and small density approximations. We extend results previously obtained for leading order behaviour of the scaling function of affine Toda field theories related to simply laced Lie algebras to the non-simply laced case. The comparison with semi-classical methods shows perfect agreement for the simply laced case. We derive the Y-systems for affine Toda field theories with real coupling constant and employ them to improve the large density approximations. We test the quality of our analysis explicitly for the Sinh-Gordon model and the $(G_2^{(1)},D_4^{(3)})$-affine Toda field theory.