Nontrivial Thermodynamics in 't Hooft's Large-$N$ Limit

TL;DR

This paper investigates the finite volume and temperature correlation functions of the (1+1)-dimensional SU(N) principal chiral sigma model in the large N limit, revealing nontrivial thermodynamic behavior despite simplified S-matrix and TBA equations.

Contribution

It demonstrates that correlation functions at finite volume and temperature are nontrivial and cannot be derived solely from the TBA, using the Leclair-Mussardo formula and exact form factors.

Findings

Correlation functions differ from free theory predictions.

TBA equations are insufficient for finite-volume correlators.

Analytical results for energy-momentum tensor and field operator.

Abstract

We study the finite volume/temperature correlation functions of the (1+1)-dimensional ${\rm SU}(N)$ principal chiral sigma model in the planar limit. The exact S-matrix of the sigma model is known to simplify drastically at large $N$, and this leads to trivial thermodynamic Bethe ansatz (TBA) equations. The partition function, if derived using the TBA, can be shown to be that of free particles. We show that the correlation functions and expectation values of operators at finite volume/temperature are not those of the free theory, and that the TBA does not give enough information to calculate them. Our analysis is done using the Leclair-Mussardo formula for finite-volume correlators, and knowledge of the exact infinite-volume form factors. We present analytical results for the one-point function of the energy-momentum tensor, and the two-point function of the renormalized field operator. The results for the energy-momentum tensor can be used to define a nontrivial partition function.