Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation: numerical study

TL;DR

This study numerically investigates the fluctuations of a static 1D interface modeled as a directed polymer in a disordered medium, revealing temperature-dependent regimes and the influence of disorder correlations on free-energy fluctuations and interface roughness.

Contribution

The paper introduces a new numerical scheme to analyze the KPZ equation for a 1D interface with finite disorder correlation length, exploring fluctuation regimes and validating analytical predictions.

Findings

Identification of a crossover to a low-temperature regime below T_c(ξ).

Quantification of the amplitude and correlation length of free-energy fluctuations.

Correlation between free-energy fluctuation amplitude and interface roughness regimes.

Abstract

We study numerically the geometrical and free-energy fluctuations of a static one-dimensional (1D) interface with a short-range elasticity, submitted to a quenched random-bond Gaussian disorder of finite correlation length $\xi>0$, and at finite temperature $T$. Using the exact mapping from the static 1D interface to the 1+1 Directed Polymer (DP) growing in a continuous space, we focus our analysis on the disorder free-energy of the DP endpoint, a quantity which is strictly zero in absence of disorder and whose sample-to-sample fluctuations at a fixed growing `time' $t$ inherit the statistical translation-invariance of the microscopic disorder explored by the DP. Constructing a new numerical scheme for the integration of the Kardar-Parisi-Zhang (KPZ) evolution equation obeyed by the free-energy, we address numerically the `time'- and temperature-dependence of the disorder free-energy fluctuations at fixed finite $\xi$. We examine on one hand the amplitude $\tilde{D}_{t}$ and effective correlation length $\tilde{\xi}_t$ of the free-energy fluctuations, and on the other hand the imprint of the specific microscopic disorder correlator on the large-`time' shape of the free-energy two-point correlator. We observe numerically the crossover to a low-temperature regime below a finite characteristic temperature $T_c(\xi)$, as previously predicted by Gaussian-Variational-Method (GVM) computations and scaling arguments, and extensively investigated analytically in [Phys. Rev. E, 87 042406 (2013)]. Finally we address numerically the `time'- and temperature-dependence of the roughness $B(t)$, which quantifies the DP endpoint transverse fluctuations, and we show how the amplitude $\tilde{D}_{\infty}(T,\xi)$ controls the different regimes experienced by $B(t)$ -- in agreement with the analytical predictions of a DP `toymodel' approach.