Thermodynamic Bethe Ansatz past turning points: the (elliptic) sinh-Gordon model

TL;DR

This paper investigates the Thermodynamic Bethe Ansatz for elliptic sinh-Gordon models, revealing how bound states and resonances influence turning points and UV behavior, and introduces new UV complete integrable theories.

Contribution

It extends TBA analysis to elliptic sinh-Gordon models with resonances, and uncovers new UV complete integrable theories related to non-unitary minimal models.

Findings

Turning points correspond to finite Hagedorn temperatures.

Complex conjugate solutions minimize effective central charge below the turning point.

UV effective central charge approaches zero as the number of resonances increases.

Abstract

We analyze the Thermodynamic Bethe Ansatz (TBA) for various integrable S-matrices in the context of generalized $T\bar T$ deformations. We focus on the sinh-Gordon model and its elliptic deformation in both its fermionic and bosonic realizations. We confirm that the determining factor for a turning point in the TBA, interpreted as a finite Hagedorn temperature, is the difference between the number of bound states and resonances in the theory. Implementing the numerical pseudo-arclength continuation method, we are able to follow the solutions to the TBA equations past the turning point all the way to the ultraviolet regime. We find that for any number $k$ of resonances the pair of complex conjugate solutions below the turning point is such that the effective central charge is minimized. As $k\to\infty$ the UV effective central charge goes to zero as in the elliptic sinh-Gordon model. Finally we uncover a new family of UV complete integrable theories defined by the bosonic counterparts of the S-matrices describing the $\Phi_{1,3}$ integrable deformation of non-unitary minimal models $\mathcal M_{2,2n+3}$.