Virial expansion and TBA in O(N) sigma models

TL;DR

The paper validates the TBA description for 1+1 dimensional even-$N$ $O(N)$ nonlinear sigma models by computing the finite-temperature free energy and performing a virial expansion. It establishes the UV central charge from the TBA, solves the constant Y-system via the Q-system, and cross-checks the second virial coefficient against S-matrix data. The special case $O(4)$ is analyzed with a reduced $A$-type Dynkin diagram, and the large-$N$ limit is shown to reproduce Luscher’s mass-gap results and agree with virial coefficients. These results reinforce the consistency of the TBA approach for non-linear sigma models and connect finite-volume/finite-temperature observables to conformal data and large-$N$ physics.

Abstract

We study the free energy of the 1+1 dimensional O(N) nonlinear sigma-models for even N using the TBA equations proposed recently. We give explicit formulae for the constant solution of the TBA equations (Y-system) and calculate the first two virial coefficients. The free energy is also compared to the leading large N result.