Unified Approach to Thermodynamic Bethe Ansatz and Finite Size Corrections for Lattice Models and Field Theories

TL;DR

This paper develops a unified non-linear integral equation framework to analyze finite size and temperature effects in lattice models and field theories, simplifying calculations of free energy and corrections.

Contribution

It introduces a single NLIE approach for various models, including the six-vertex, XXZ chain, and sine-Gordon-massive Thirring model, unifying their finite size and temperature analyses.

Findings

Derived NLIE for multiple models with twisted boundary conditions and external fields.

Solved NLIE iteratively in different temperature regimes, obtaining leading behaviors.

Presented a Riemann-Hilbert formulation for higher-order correction calculations.

Abstract

We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a {\bf single} non-linear integral equation (NLIE). [A system of NLIE appears for nested BA]. We derive the NLIE for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain in an external magnetic field $h_z$ and c) the sine-Gordon-massive Thirring model (sG-mT) in a periodic box of size $\b \equiv 1/T $ using the light-cone approach. This NLIE is solved by iteration in one regime (high $T$ in the XXZ chain and low $T$ in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present.