Memory-induced long-range order in dynamical systems

TL;DR

The paper demonstrates that time non-locality, manifested as slow memory degrees of freedom, can induce spatial long-range order in systems with locally coupled variables when memory evolves more slowly than the primary dynamics ($g>\gamma$). Using a general two-DOF analysis and a concrete spin-glass–memory model on a 2D lattice, it shows robust, non-perturbative LRO accompanied by scale-free avalanche statistics with $\alpha_\gamma \approx 2$ for a broad range of memory rates. A correlated percolation framework reveals a continuous transition at a nontrivial $p_c$ with exponents distinct from standard 2D percolation, indicating a new universality class arising from memory effects. The results imply wide applicability across dynamical systems with multiple timescales and point to practical implications for memcomputing and neuromorphic platforms, where memory can drive collective behavior without fine-tuning.

Abstract

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.

Figures (3)

• Figure 1: Cartoon of a continuous spin system with memory on a 2D lattice. Nearest-neighbor spins $s_i$ and $s_j$ are coupled by the memory DOF $x_{i j}$. Dynamics from Eq. (\ref{['eq:memory_dynamics']}) are simulated in a square lattice of size $N = 16^2$ with $\gamma = 0.1$. For a randomly chosen site $i$, $s_i(t)$ (blue) and its four $x_{i j}(t)$ (red) are plotted.
• Figure 2: Avalanche size distributions (a-c) for the memory frequency scale $\gamma = 0.15$, $0.25$, and $0.40$ with scale-invariance analyses (d-f) performed in the upper-right insets. For each $\gamma$, distributions are plotted for lattice sizes $N \equiv L^2 = 16^2$, $32^2$, $64^2$, $128^2$, and $256^2$. We detect scale-free (SF) avalanches for all $\gamma \in \{0.15, 0.25, 0.40\}$, corroborated by the simultaneous collapse of all distributions in the scale-invariance analyses, with properly chosen exponents. Each lattice size is simulated in 100 distinct instances over a time $T = 200$. See the SM for more details.
• Figure 3: (a) Visualization of the transition into the LRO phase as a type of dynamic percolation on a $256\times 256$ lattice. In each snapshot, sites are considered occupied (colored) if their corresponding spin flips within $5 \times 10^3$ time steps, where each time step $\Delta t = 0.048\gamma^{-1/3}$. The largest connected cluster is highlighted in red (with periodic boundary conditions), while all smaller clusters are shown in blue. Only in the LRO phase do percolating clusters exist. A visualization of how the percolating clusters dynamically build up can be found in the Supplemental Movies. (b) The density of frozen spins, $\rho$, as a function of $\gamma$. For small $\gamma$, no spins are frozen, and the system is in the percolating phase. As $\gamma$ increases, more spins become locked into quasiperiodic orbits and no longer flip, driving the system toward a phase with short-range correlations and no system-wide clusters. The phase transition is continuous, and $\rho(\gamma)$ is independent of system size $L$. The inset shows $\rho(\gamma)$ in log-log scale. Up to the transition point $\gamma \simeq 0.5$, $\rho(\gamma)$ follows an approximate power-law relationship, $\rho\sim\gamma^{9.5}$.