Vortex condensation in a model of random $\phi^{4}$-graphs

TL;DR

This paper analyzes a solvable model of large $\,\phi^{4}$-graphs embedded in a compactified dimension, revealing a vortex condensation transition akin to the Berezinskii-Kosterlitz-Thouless transition, with implications for phase behavior in matrix models.

Contribution

It introduces a semi-classical solution to a large $\,\phi^{4}$-graph model, demonstrating vortex condensation and a second-order phase transition in a finite-temperature setting.

Findings

Identification of a critical temperature for vortex condensation.

Demonstration of a second-order phase transition from $D=1$ to $D=0$ behavior.

Rapid increase in vortex density near the critical point.

Abstract

We consider a soluble model of large $\phi^{4}$-graphs randomly embedded in one compactified dimension; namely the large-order behaviour of finite-temperature perturbation theory for the partition function of the anharmonic oscillator. We solve the model using semi-classical methods and demonstrate the existence of a critical temperature at which the system undergoes a second-order phase transition from $D=1$ to $D=0$ behaviour. Non-trivial windings of the closed loops in a graph around the compactified time direction are interpreted as vortices. The critical point has a natural interpretation as the temperature at which these vortices condense and disorder the system. We show that the vortex density increases rapidly in the critical region indicating the breakdown of the dilute vortex gas approximation at this point. We discuss the relation of this phenomenon to the Berezinskii-Kosterlitz-Thouless transition in the $D=1$ matrix model formulated on a circle.