Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model

TL;DR

This paper investigates phase transitions in the fluctuations of time-integrated observables in non-interacting Ising spin systems with all-to-all dynamics, revealing a phase transition in the cumulant generating function linked to the random energy model.

Contribution

It establishes a connection between dynamical fluctuations in non-interacting spin systems and the spectral properties of the quantum REM, providing exact results and finite-size corrections.

Findings

Identifies a phase transition in the cumulant generating function for large N.

Connects dynamical fluctuations to spectral analysis of the quantum REM.

Provides numerical validation of finite N corrections.

Abstract

We study the fluctuations of time-additive random observables in the stochastic dynamics of a system of $N$ non-interacting Ising spins. We mainly consider the case of all-to-all dynamics where transitions are possible between any two spin configurations with uniform rates. We show that the cumulant generating function of the time-integral of a normally distributed quenched random function of configurations, i.e., the energy function of the random energy model (REM), has a phase transition in the large $N$ limit for trajectories of any time extent. We prove this by determining the exact limit of the scaled cumulant generating function. This is accomplished by connecting the dynamical problem to a spectral analysis of the all-to-all quantum REM. We also discuss finite $N$ corrections as observed in numerical simulations.