We explore the critical dynamics of driven interfaces propagating through a two-dimensional disordered medium with long-range spatial correlations, modeled using fractional Brownian motion (FBM). Departing from conventional models with uncorrelated disorder, we introduce quenched noise fields characterized by a tunable Hurst exponent \(H\) , allowing systematic control over the spatial structure of the background medium. The interface evolution is governed by a quenched Kardar–Parisi–Zhang (QKPZ) equation modified to account for correlated disorder, namely QKPZ \(_H\) . Through analytical scaling analysis, we uncover how the presence of long-range correlations reshapes the depinning transition, alters the critical force \(F_c\) , and gives rise to a family of critical exponents that depend continuously on \(H\) . Our findings reveal a rich interplay between disorder correlations and the non-linearity term in QKPZ \(_H\) , leading to a breakdown of conventional universality and the emergence of nontrivial scaling behaviors. The exponents are found to change by H in the anticorrelation regime ( \(H<0.5\) ), while they are nearly constant in the correlation regime ( \(H>0.5\) ), suggesting a robust-universal behavior for the latter. By a comparison with the quenched Edwards-Wilkinson model, we study the effect of the non-linearity term in the QKPZ \(_H\) model.
