On the negative coupling O(N) model in 2d at high temperature

TL;DR

This paper analyzes a two-dimensional N-component scalar quantum field theory with negative coupling at high temperature, revealing that non-principal saddle points in complex analysis provide a consistent non-perturbative solution.

Contribution

It demonstrates how non-principal Riemann sheet saddle points enable a consistent non-perturbative solution of the 2d negative-coupling O(N) model at all temperatures.

Findings

Saddle points on non-principal Riemann sheets are crucial for the model's consistency.

The model reduces to PT-symmetric quantum mechanics at high temperature.

Matching quantum mechanics solutions with field theory saddle points extends the solution's validity.

Abstract

In this work, I consider N-component scalar quantum field theory in two dimensions interacting with an upside-down quartic potential. Working in the large N limit, the model can be solved non-perturbatively using the saddle-point method for sufficiently strong negative coupling. At high temperature, the O(N) model dimensionally reduces to ${\cal PT}$-symmetric quantum mechanics, for which powerful non-perturbative solution methods exist. It is found that the solution from quantum mechanics can be matched by the saddle-point method in quantum field theory when allowing for saddles beyond the principal Riemann sheet. I show that saddle points on non-principal Riemann sheets lead to a fully consistent solution of the 2d negative-coupling O(N) model for all temperatures.