Large-$N$ limit of the gradient flow in the 2D $O(N)$ nonlinear sigma model

TL;DR

This paper analytically solves the gradient flow in the 2D $O(N)$ nonlinear sigma model at large $N$, confirming the correct normalization of the energy--momentum tensor and reproducing thermodynamic quantities.

Contribution

It provides an analytical solution to the gradient flow in the large-$N$ limit and demonstrates the proper normalization of the energy--momentum tensor in the continuum limit.

Findings

Reproduces thermodynamic quantities via the gradient flow

Confirms correct normalization of the energy--momentum tensor

Validates the gradient flow approach in non-perturbative systems

Abstract

The gradient flow equation in the 2D $O(N)$ nonlinear sigma model with lattice regularization is solved in the leading order of the $1/N$ expansion. By using this solution, we analytically compute the thermal expectation value of a lattice energy--momentum tensor defined through the gradient flow. The expectation value reproduces thermodynamic quantities obtained by the standard large-$N$ method. This analysis confirms that the above lattice energy--momentum tensor restores the correct normalization automatically in the continuum limit, in a system with a non-perturbative mass gap.