The XY model is one of the foundational systems in statistical mechanics — a lattice of planar spins interacting with their neighbors. In two dimensions, it exhibits the Berezinskii-Kosterlitz-Thouless (BKT) transition: not a conventional phase transition with a sharp order parameter, but a topological one where vortex-antivortex pairs unbind at a critical temperature. Below the transition, the system has quasi-long-range order — correlations decay as a power law rather than exponentially, but true long-range order is absent. The Mermin-Wagner theorem forbids it.
Shi, Chaté, and Mahault (2602.22028) subject this system to time-correlated noise instead of the usual white noise. The motivation comes from active crystals — assemblies of self-propelled particles arranged in lattice-like configurations — where the fluctuations driving each particle have memory. A bacterium pushes in one direction for a while before reorienting. The noise is persistent.
The central finding: the system can remain quasi-ordered despite correlations decaying much faster than equilibrium allows. In equilibrium, the BKT phase constrains how fast correlations can decay — the exponent of the power law is bounded. With persistent noise, this bound is violated. The system maintains order even when, by equilibrium standards, it shouldn't.
Yet the transition itself remains BKT. The topology doesn't change — the phase transition is still about vortex unbinding. But the scaling exponents vary continuously with the persistence time of the noise. The universality class is the same; the critical exponents are not.
This is a subtle distinction. Usually in statistical mechanics, the universality class determines the exponents. Systems with the same symmetries and dimensionality share the same critical behavior regardless of microscopic details. Persistent noise breaks this by introducing a new relevant parameter — the persistence time — that isn't present in the equilibrium classification. The transition mechanism is unchanged (topological), but the quantitative details depend on how long the noise remembers its direction.
The connection to active crystals is direct. Recent experiments showed that active crystals can sustain very large deformations without melting. The persistent XY model explains why: the persistence allows the system to absorb fluctuations that would destroy equilibrium order. The crystal is more robust because the noise that would melt it is temporally smooth. Sharp, uncorrelated kicks are more destructive than slow, correlated pushes of the same total magnitude.
There's a broader point about the relationship between temporal correlation in the driving and spatial order in the response. White noise — memoryless, uncorrelated — is maximally disruptive per unit energy because every kick is independent. Persistent noise concentrates its disruption in time, which paradoxically makes it less effective at destroying spatial order. The system has time to respond to and partially accommodate a slow push. It cannot accommodate instantaneous random kicks. Memory in the noise produces robustness in the structure.