Statistical physics of nonlinear wave interaction

TL;DR

This paper explores the thermodynamic behavior of vector models with four-body interactions, revealing complex phenomena like non-equipartite condensation and continuous transitions, with implications for nonlinear optics.

Contribution

It introduces a novel Monte Carlo algorithm to study dense topologies, uncovering new phenomenology in diluted models and the effects of topological inhomogeneities.

Findings

Spherical model condenses non-equipartitely below a dilution threshold.

XY model exhibits a continuous transition with unbroken O(2) symmetry.

Topological inhomogeneities lead to symmetry conservation and two-timescale dynamics.

Abstract

The thermodynamic properties of vector (O(2) and Complex Spherical) models with four-body interactions are analyzed. When defined in dense topologies, these are effective models for the nonlinear interaction of scalar fields in the presence of a stochastic noise, as has been well established for the case of the mode locking laser formation in a closed cavity. With the help of a novel efficient Monte Carlo algorithm we show how beyond the fully connected case novel and rich phenomenology emerges. Below a certain dilution threshold, the spherical model condensates in a non-equipartite way, while in the XY model the transition becomes continuous and the O(2) symmetry remains unbroken, we attribute this fact to the invariance under local gauge transformations. The introduction of topological inhomogeneities in the network of quadruplets induces novel features: again symmetry conservation; the vanishing of two-point correlators; and a dynamical correlation function presenting two timescales, the large one being related to the transition between different degenerated configurations, connected by nonlocal gauge transformations. We discuss possible experimental implications of these results in the context of nonlinear optics.