Universal scaling laws for dynamical-thermal hysteresis

Dynamic hysteresis, the rate-dependent lagged response of materials to external fields, underpins applications from energy-efficient transformers to gas storage systems. A fundamental yet unresolved question is how the hysteresis loop area $A$ scales with the field sweep rate $R$. Here, we reveal that a competition between the field sweep and thermal fluctuations governs a universal crossover between two scaling regimes: $A-A_0\propto R^{1\mathrm{/}3}$ for $R<R^{\ast }$ and $A-A_0\propto R^{2\mathrm{/}3}$ for $R>R^{\ast }$, where $A_0$ is the quasi-static area and the crossover rate $R^{\ast }\propto T\mathrm{/}{T}_c$ depends on the temperature $T$ and the material’s critical temperature $T_c$. We demonstrate these scaling laws universally across experiments of magnetic materials, simulations of Ising and metal-organic framework models, and analytical solutions of a stochastic Langevin equation. This framework not only resolves the long-standing non-universality of reported scaling exponents but also provides a direct design principle for the application of dynamic hysteresis.