Simulation of the Zero Temperature Behavior of a 3-Dimensional Elastic Medium

TL;DR

This study uses numerical simulations and combinatorial optimization to analyze the ground state properties of a 3D elastic medium with quenched disorder, revealing power-law structure factors, fractal domain walls, and scaling behaviors indicative of a stable Bragg glass phase.

Contribution

It provides the first numerical evidence of power-law divergences in the structure factor and characterizes the fractal nature of domain walls in the 3D elastic medium with disorder.

Findings

Power law divergence in structure factor $S(k) \\sim A k^{-3}$.

Fractal dimension of domain walls $d_f=2.60(5)$.

Scaling exponent $\\zeta=0.385(40)$.

Abstract

We have performed numerical simulation of a 3-dimensional elastic medium, with scalar displacements, subject to quenched disorder. We applied an efficient combinatorial optimization algorithm to generate exact ground states for an interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\sim A k^{-3}$. We have found numerically consistent values of the coefficient $A$ for two lattice discretizations of the medium, supporting universality for $A$ in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. Our results indicate that the domain walls formed by this change are highly convoluted, with a fractal dimension $d_f=2.60(5)$. We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. As in other disordered systems, perturbations of relative strength $\delta$ introduce a new length scale $L^* \sim \delta^{-1/\zeta}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed scaling analysis of the response of the ground state to the perturbations and obtain $\zeta = 0.385(40)$. This value is consistent with the scaling relation $\zeta=d_f/2- \theta$, where $\theta$ characterizes the scaling of the energy fluctuations of low energy excitations.