Trajectory phase transitions in non-interacting spin systems

TL;DR

This paper demonstrates that non-interacting Ising spin systems can exhibit large fluctuations and phase transitions in trajectory space, with the nature of the transition depending on the observable studied.

Contribution

It reveals the existence of trajectory phase transitions in non-interacting spin systems and provides explicit analytical and numerical analysis of these phenomena.

Findings

Large deviation transitions can be continuous or first order.

A duality in the tilted generator facilitates analysis.

Numerical methods confirm the theoretical predictions.

Abstract

We show that a collection of independent Ising spins evolving stochastically can display surprisingly large fluctuations towards ordered behaviour, as quantified by certain types of time-integrated plaquette observables, despite the underlying dynamics being non-interacting. In the large deviation (LD) regime of long times and large system size, this can give rise to a phase transition in trajectory space. As a non-interacting system we consider a collection of spins undergoing single spin-flip dynamics at infinite-temperature. For the dynamical observables we study, the associated tilted generators have an exact and explicit spin-plaquette duality. Such setup suggests the existence of a transition (in the large size limit) at the self-dual point of the tilted generator. The nature of the LD transition depends on the observable. We consider explicitly two situations: (i) for a pairwise bond observable the LD transition is continuous, and equivalent to that of the transverse field Ising model; (ii) for a higher order plaquette observable, in contrast, the LD transition is first order. Case (i) is easy to prove analytically, while we confirm case (ii) numerically via an efficient trajectory sampling scheme that exploits the non-interacting nature of the original dynamics.