Ground-State and Domain-Wall Energies in the Spin-Glass Region of the 2D $\pm J$ Random-Bond Ising Model

TL;DR

This study analyzes the ground-state and domain-wall energies of the 2D $ ext{±}J$ random-bond Ising model, revealing scaling behaviors, finite-size effects, and aspect-ratio scaling in the spin-glass region using large-scale computational methods.

Contribution

It provides a detailed analysis of energy scaling and fluctuations in the 2D $ ext{±}J$ Ising model, including the effects of aspect ratio and finite-size corrections, with new insights into the behavior within the spin-glass phase.

Findings

Fluctuations show a cusp at the critical point $p_c$.

Average domain-wall energy converges to a finite value in the thermodynamic limit.

Distribution of domain-wall energies becomes Gaussian as aspect ratio approaches zero.

Abstract

The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds of probability $p$ of $J_{ij}= -1$ and $(1-p)$ of $J_{ij}= +1$ are studied. We are able to consider large samples of up to $320^2$ spins by using sophisticated matching algorithms. We study $L \times L$ systems, but we also consider $L \times M$ samples, for different aspect ratios $R = L / M$. We find that the scaling behavior of the ground-state energy and its sample-to-sample fluctuations inside the spin-glass region ($p_c \le p \le 1 - p_c$) are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusp-like singularity at $p_c$. Inside the spin-glass region the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding $R$ fixed. Here, large finite-size effects are visible, which can be explained for all $p$ by a single exponent $\omega\approx 2/3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R \to 0$: the distribution of the domain-wall energies converges to a Gaussian for $R \to 0$, although the domain walls of neighboring subsystems of size $L \times L$ are not independent.