Dyson instability for 2D nonlinear O(N) sigma models

TL;DR

This paper investigates the complex singularities and zero distributions of the average energy in 2D nonlinear O(N) sigma models at large N, revealing unexpected features in the non-perturbative structure and implications for gauge theories.

Contribution

It provides a detailed analysis of the complex singularities and zeros of the partition function in the large-N limit, highlighting novel features distinct from typical models.

Findings

Absence of a cut along the negative real axis in the complex lambda plane.

Zeros of the partition function are confined within a clover-shaped region.

Dispersive representations show the non-perturbative nature of the discontinuity.

Abstract

For lattice models with compact field integration (nonlinear sigma models over compact manifolds and gauge theories with compact groups) and satisfying some discrete symmetry, the change of sign of the bare coupling g_0^2 at zero results in a mere discontinuity in the average energy rather than the catastrophic instability occurring in theories with integration over arbitrarily large fields. This indicates that the large order of perturbative series and the non-perturbative contributions should have unexpected features. Using the large-N limit of 2-dimensional nonlinear O(N) sigma model, we discuss the complex singularities of the average energy for complex 't Hooft coupling lambda= g_0^2N. A striking difference with the usual situation is the absence of cut along the negative real axis. We show that the zeros of the partition function can only be inside a clover shape region of the complex lambda plane. We calculate the density of states and use the result to verify numerically the statement about the zeros. We propose dispersive representations of the derivatives of the average energy for an approximate expression of the discontinuity. The discontinuity is purely non-perturbative and contributions at small negative coupling in one dispersive representation are essential to guarantee that the derivatives become exponentially small when lambda -> 0^+ We discuss the implications for gauge theories.