Coercivity Landscape Characterizes Dynamic Hysteresis

Hysteresis, with rich dynamical behaviors-especially in interacting systems-has drawn broad research interest. Yet its dynamic scalings across time scales lack a unified description, and their transitions remain unclear. Here, we study the stochastic $ϕ^4$ model driven periodically by an external field $H$. For large systems with small noise strength $σ$, we find the coercivity $H_c \equiv H(\langleϕ\rangle=0)$ sequentially exhibits distinct behaviors with increasing driving rate $v_H$: $v_H$-scaling increase, stable plateau ($v_H^0$), $v_H^{1/2}$-scaling increase, and abrupt decline to disappearance. The plateau reflects the competition between thermodynamic and quasi-static limits, namely, $\lim_{σ\to 0}\lim_{v_H\to 0}H_c = 0$, and $\lim_{v_H\to 0}\lim_{σ\to 0}H_c=H^$. Here, $H^$ is exactly the field-driven first-order phase transition point. In the post-plateau regime, $(H_{c} - H_{P})$ scales with $(v_{H} - v_{P})^{2/3}$ with $v_{P}$ and $H_{P}$ being the reference points of the plateau. Moreover, we reveal a finite-size scaling for the coercivity plateau as $v_{P}\simσ^{2}$ and $(H^*-H_P)\simσ^{4/3}$ by utilizing renormalization-group theory. Our work provides a panoramic view of finite-time scalings of the hysteresis and offers new insights into finite-time/finite-size effect interplay in non-equilibrium systems.

💡 Research Summary

The paper investigates dynamic hysteresis by focusing on a newly defined observable, the coercivity (H_c), in a periodically driven stochastic (\phi^4) model. The model’s free‑energy density (f_4(\phi,H)=\frac12 a_2\phi^2+\frac14 a_4\phi^4-H\phi) together with a Langevin dynamics (\partial_t\phi=-\lambda\partial_\phi f_4+\zeta(t)) (Gaussian white noise of strength (\sigma)) captures a Z(_2)‑symmetric double‑well potential for (a_2<0). When the external field (H) reaches the spinodal value (H^*=\pm\sqrt{-4a_2^3/(27a_4)}), the metastable minimum disappears, signalling a first‑order phase transition (FOPT).

Coercivity is defined as the field at which the ensemble‑averaged order parameter crosses zero, (H_c\equiv H(\langle\phi\rangle=0)). By deriving the associated Fokker‑Planck equation and an exact moment equation, the authors obtain a compact evolution relation \