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

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.

arXiv comment11 pages with 15 figures, extensively revised

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• Aspect-Ratio Scaling of Domain Wall Entropy for the 2D $\pm J$ Ising Spin Glass