Memory-induced long-range order in dynamical systems

Time non-locality, or memory, is a non-equilibrium property shared by all physical systems. Here, we show that memory is sufficient to induce a phase of spatial long-range order (LRO) even if the system's primary dynamical variables are coupled locally. This occurs when the memory degrees of freedom have slower dynamics than the primary degrees of freedom. In addition, such an LRO phase is non-perturbative, and can be understood through the lens of a correlated percolation transition of the fast degrees of freedom mediated by memory. When the two degrees of freedom have comparable time scales, the length of the effective long-range interaction shortens. We exemplify this behavior with a model of locally coupled spins and a single dynamic memory variable, but our analysis is sufficiently general to suggest that memory could induce a phase of LRO in a much wider variety of physical systems.