Nonlinear $XY$ and $p$-clock models on sparse random graphs: mode-locking transition of localized waves

TL;DR

This paper investigates nonlinear XY and p-clock models on sparse random graphs, analyzing their phase transitions and mode-locking behavior relevant to laser physics, and assesses the accuracy of discretized models in simulating continuous spins.

Contribution

It introduces a statistical mechanics analysis of nonlinear XY and p-clock models on bipartite sparse graphs, exploring their phase transitions and mode-locking phenomena, and evaluates the discrete approximation's effectiveness.

Findings

Unmagnetized phase-locking occurs at the phase transition.

Linear phase dependence on frequency in broad regimes.

Discretized p-clock models approximate XY models with specific limits.

Abstract

A statistical mechanic study of the $XY$ model with nonlinear interaction is presented on bipartite sparse random graphs. The model properties are compared to those of the $p$-clock model, in which the planar continuous spins are discretized into $p$ values. We test the goodness of the discrete approximation to the XY spins to be used in numerical computations and simulations and its limits of convergence in given, $p$-dependent, temperature regimes. The models are applied to describe the mode-locking transition of the phases of light-modes in lasers at the critical lasing threshold. A frequency is assigned to each variable node and function nodes implement a frequency matching condition. A non-trivial unmagnetized phase-locking occurs at the phase transition, where the frequency dependence of the phases turns out to be linear in a broad range of frequencies, as in standard mode-locking multimode laser at the optical power threshold.