Roughening of the Anharmonic Elastic Interface in Correlated Random Media

We study the roughening properties of the anharmonic elastic interface in the presence of temporally correlated noise. The model can be seen as a generalization of the anharmonic Larkin model, recently introduced by Purrello, Iguain, and Kolton [Phys. Rev. E {\bf 99}, 032105 (2019)], to investigate the effect of higher-order corrections to linear elasticity in the fate of interfaces. We find analytical expressions for the critical exponents as a function of the anharmonicity index $n$, the noise correlator range $\theta \in[0,1/2]$, and dimension $d$. In $d=1$ we find that the interface becomes faceted and exhibits anomalous scaling for $\theta > 1/4$ for any degree of anharmonicity $n > 1$. Analytical expressions for the anomalous exponents $\alpha_\mathrm{loc}$ and $\kappa$ are obtained and compared with a numerical integration of the model. Our theoretical results show that anomalous roughening cannot exist for this model in dimensions $d > 1$.