Coercivity Landscape Characterizes Dynamic Hysteresis

Abstract

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.

Figures (3)

• Figure 1: Hysteresis loops and coercivity landscape of the stochastic $\phi^4$ model. Hysteresis loops at (a): $v_H=10^{-5}$(solid), $4\times10^{-4}$(dashed); (b): $v_H=0.01$(solid), $0.02$(dashed), $0.04$(dash-dotted); (c): $v_H=10$(solid), $20$(dashed), $40$(dash-dotted); (d): $v_H=100$(solid), $500$(dashed), $1500$(dash-dotted). (e) Coercivity $H_c$ as a function of driving rate $v_H$ with $\sigma=0.7$, DPT represents dynamic phase transition. (f) Coercivity landscape $H_c(v_H,\sigma)$, where the noise strength from top to bottom is $\sigma=$ 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0. (g) A contour illustration of (f) after interpolation. In this plot, limited driving amplitude $H_m=50$ is applied in (a-e) while (f-g) not limited driving amplitude to ensure that the order parameter is always saturated saturated; $a_2 = -4$, and all quantities are nondimensionalized by $a_4$ and $\lambda$.
• Figure 2: (a) Finite-size scaling of the coercivity plateau. $\sigma$ from 0.1 to 1.0 with interval 0.1. (b) Coercivity reduced by the reference plateau coercivity. Curves from right to left correspond to noise strengths $\sigma =$ 1.0, 0.8, 0.7, 0.5, 0.4, 0.3. (c) Coercivity reduced by the minimum observed coercivity within the simulation timescale shown in Fig. \ref{['fig:phi4-combined']}(f). Curves from left to right correspond to $\sigma =$ 0.3, 0.2, 0.1, 0.
• Figure 3: Finite-time and finite-size scalings for hysteresis in interacting systems. The results for the Curie-Weiss model of $N$ spins are obtained in the companion paper CPregular.