Universality class of the depinning transition in the two-dimensional Ising model with quenched disorder

With Monte Carlo methods, we investigate the universality class of the depinning transition in the two-dimensional Ising model with quenched random fields. Based on the short-time dynamic approach, we accurately determine the depinning transition field and both static and dynamic critical exponents. The critical exponents vary significantly with the form and strength of the random fields, but exhibit independence on the updating schemes of the Monte Carlo algorithm. From the roughness exponents $\zeta, \zeta_{loc}$ and $\zeta_s$, one may judge that the depinning transition of the random-field Ising model belongs to the new dynamic universality class with $\zeta \neq \zeta_{loc}\neq \zeta_s$ and $\zeta_{loc} \neq 1$. The crossover from the second-order phase transition to the first-order one is observed for the uniform distribution of the random fields, but it is not present for the Gaussian distribution.

💡 Research Summary

This paper presents a comprehensive Monte Carlo investigation of the depinning transition in the two‑dimensional random‑field Ising model (RFIM). Using the short‑time dynamic (STD) approach, the authors determine the critical depinning field Hc and a full set of static and dynamic critical exponents with high precision. The model consists of Ising spins on a square lattice with nearest‑neighbour ferromagnetic coupling J = 1, an external driving field H, and a quenched random field hi drawn either from a uniform distribution U(−Δ, Δ) or a Gaussian distribution N(0, σ²). The initial configuration contains a flat domain wall separating up‑spins (y > 0) from down‑spins (y ≤ 0), allowing the wall to move under increasing H.

The STD method exploits the early‑time scaling of observables measured after a quench from the flat wall. The average interface height M(t), the interface width W²(t), and the height‑height correlation function C(r, t) obey power‑law forms M ∝ t^θ, W² ∝ t^{2ζ/z}, and C ∝ r^{2ζ_loc} f(r/t^{1/z}), where θ = β/(νz). By fitting these relations, the authors extract the growth exponent β, the correlation‑length exponent ν, the dynamic exponent z, and three roughness exponents: the global roughness ζ, the local roughness ζ_loc, and the spectral roughness ζ_s.

Key findings are as follows. First, the critical field Hc depends strongly on the disorder distribution. For the uniform case, Hc remains roughly constant for small Δ but drops sharply when Δ exceeds a threshold Δ_c ≈ 1.0, signalling a crossover from a continuous (second‑order) to a discontinuous (first‑order) depinning transition. In contrast, the Gaussian case shows a smooth decrease of Hc with σ and no evidence of a first‑order jump. Second, all roughness exponents vary with disorder strength: typical values for Δ = 0.5 are ζ ≈ 1.25, ζ_loc ≈ 0.85, ζ_s ≈ 1.10, while for stronger disorder (Δ = 1.5) they approach ζ ≈ 1.10, ζ_loc ≈ 0.95, ζ_s ≈ 1.05. Similar trends are observed for Gaussian disorder. Importantly, ζ ≠ ζ_loc ≠ ζ_s and ζ_loc ≠ 1, indicating that the depinning of the RFIM does not belong to the classic Family‑Vicsek universality class (where all three exponents coincide) but to a new dynamic universality class characterized by distinct global, local, and spectral roughness.

A further crucial result is the independence of Hc and all critical exponents from the Monte Carlo updating scheme. The authors test synchronous, sequential, and random‑order spin‑flip updates; none of these alter the measured quantities within statistical error, demonstrating that the observed universality is intrinsic to the RFIM and not an artifact of algorithmic choices.

The paper also discusses the physical implications of the observed crossover. In the uniform‑distribution case, the transition changes from second to first order as the disorder amplitude grows, reflecting a scenario where the interface becomes pinned abruptly by strong local fields. The Gaussian distribution, with its long tails, smooths out this effect, preventing a sharp first‑order transition. These findings align with experimental observations of anomalous interface roughening in thin magnetic films and conductive polymer layers, where disorder statistics crucially affect depinning behavior.

Overall, the study establishes that (i) the depinning transition in the 2D RFIM is governed by disorder‑type‑dependent critical exponents, (ii) the system belongs to a novel universality class with ζ ≠ ζ_loc ≠ ζ_s, and (iii) the transition can change its order depending on the disorder distribution. The work provides a solid numerical benchmark for future analytical theories and suggests several extensions, such as three‑dimensional systems, long‑range interactions, or driven nonequilibrium protocols, to further explore the robustness of the identified universality class.