With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.
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 having probability
p of ${J}_{ij}=\ensuremath{-}1$ and $(1\ensuremath{-}p)$ of
Jij=+1 are studied. We are able to consider large samples of up to
3202 spins by using sophisticated matching algorithms. We study $L\ifmmode\times\else\texttimes\fi{}L$ systems, but we also consider $L\ifmmode\times\else\texttimes\fi{}M$ 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 $({p}_{c}\ensuremath{\leqslant}p\ensuremath{\leqslant}1\ensuremath{-}{p}_{c})$ are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusplike singularity at
pc. 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 $\ensuremath{\omega}\ensuremath{\approx}2∕3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R\ensuremath{\rightarrow}0$: the distribution of the domain-wall energies converges to a Gaussian for $R\ensuremath{\rightarrow}0$, although the domain walls of neighboring subsystems of size $L\ifmmode\times\else\texttimes\fi{}L$ are not independent.
We study the three-dimensional (3D) bond-diluted Edwards-Anderson (EA) model with binary interactions at a bond occupation of 45% by Monte Carlo (MC) simulations. Using an efficient cluster MC algorithm we are able to determine the universal finite-size scaling (FSS) functions and the critical exponents with high statistical accuracy. We observe small corrections to scaling for the measured observables. The critical quantities and the FSS functions indicate clearly that the bond-diluted model for dilutions above the critical dilution
p∗, at which a spin glass (SG) phase appears, lies in the same universality class as the 3D undiluted EA model with binary interactions. A comparison with the FSS functions of the 3D site-diluted EA model with Gaussian interactions at a site occupation of 62.5% gives very strong evidence for the universality of the SG transition in the 3D EA model.
The systematic Weiss-like and Bethe-like approximations based on the mean-field transfer-matrix method are used to investigate the asymptotic behavior of the induced magnetization on a semi-infinite square lattice, and to investigate the wave-number dependence of the susceptibility in a nonuniform external field. The critical exponents ν, ν', η i and η are estimated following the general CAM prescription. A new scaling relation ν·η i =γ is obtained in the framework of the finite-degree-of-approximation scaling. Together with previous papers, all the static critical exponents have been estimated by the CAM, and are shown to satisfy the well-known scaling relations.
In the droplet picture of the scaling theory for the short-range Ising spin glass, the largest eigenvalue of the correlation matrix diverges algebraically with the Edwards-Anderson susceptibility, the exponent being d/(d-\ensuremath{\theta}), where \ensuremath{\theta} is the stiffness exponent. The spectral density for the correlation matrix has a bump near the lower edge of the spectrum. Comparing these predictions with results from numerical studies of the spectrum might provide a unique test of the droplet picture.