Coercivity Landscape Characterizes Dynamic Hysteresis

Hysteresis, with rich dynamical behaviors-especially in interacting systems-has drawn broad research interest. Yet its dynamic scalings across timescales 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}≡H(⟨ϕ⟩=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 quasistatic limits, namely, lim_{σ→0}lim_{v_{H}→0}H_{c}=0, and lim_{v_{H}→0}lim_{σ→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}∼σ^{2} and (H^{*}-H_{P})∼σ^{4/3} by utilizing renormalization-group theory. Our Letter provides a panoramic view of finite-time scalings of the hysteresis and offers new insights into the interplay between finite-time and finite-size effects in nonequilibrium systems.