Trajectory phase transitions in noninteracting spin systems

We show that a collection of independent Ising spins evolving stochastically can display surprisingly large fluctuations toward ordered behavior, as quantified by certain types of time-integrated plaquette observables, despite the underlying dynamics being noninteracting. 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 noninteracting 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 and (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 noninteracting nature of the original dynamics.