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.
By an exact calculation of the ground states for the ±J Edwards–Anderson spin glass, one can extrapolate the ground state energy to infinite system sizes. We calculate the exact ground states for the three-dimensional spin glass with free boundaries for system sizes up to 10 and fit different finite-size functions. We cannot decide, only from the quality of the fit, which fitting function to choose. Relying on the literature values for the extrapolated energy, we find the finite-size corrections to vary as 1/L.
The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for
L×L square lattices with
L≤20, and
x = 0.25 and 0.5, where
x is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy
Edw equal to 0 or 4. The probability distribution of the entropy,
Sdw, is found to depend strongly on
Edw. The results for
Sdw when
Edw=4 agree with the prediction of the droplet model. Our results for
Sdw when
Edw=0 agree with those of Saul and Kardar. In addition, we find that the distributions do not appear to be Gaussian in that case. The special role of
Edw=0 domain walls is discussed, and the discrepancy between the prediction of Amoruso, Hartmann, Hastings and Moore and the result of Saul and Kardar is explained.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}
106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-
J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
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.