We compare simulation data from different sources on two canonical three-dimensional Ising spin glasses (ISGs): the binomial $\ifmmode\pm\else\textpm\fi{}J$ near-neighbor interaction ISG and the Gaussian interaction ISG. We allow for the possibility of corrections to finite size scaling and estimate the correction exponent w. Consistent estimates for the critical temperatures
Tg and for the critical exponents for each system are obtained. The data strongly indicate that critical exponents in the two systems are significantly different from each other. These results thus confirm a breakdown of standard universality rules in Ising spin glasses.
For the Ising model with Gaussian random coupling of average
J0 and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter
J0 can be studied via the size-
L-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension
d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.
We compare numerical estimates from different sources for the ordering temperature
Tg and the critical exponents of the Ising spin glass in dimension 3 with binomial $(\ifmmode\pm\else\textpm\fi{}J)$ interactions. Corrections to finite-size scaling turn out to be important especially for parameters such as the Binder cumulant. For nonequilibrium parameters it is easier to approach the large-size limit and to allow for corrections to scaling. Relying principally on such data, a crossing point defines the freezing temperature
Tg; the possibility that the ordering temperature is zero can definitively be excluded. We estimate an ordering temperature
Tg=1.195(15), with associated estimates of the critical exponents for which corrections to finite-size scaling are well under control. Among the parameters evaluated is the leading dynamic correction to scaling exponent w.
We have studied numerically the dynamics of spin glasses with Ising and
XY symmetry (gauge glass) in space dimensions 2, 3, and 4. The nonequilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ and the nonequilibrium energy per spin,
ene(tw,T), of samples of large size
Lb are measured as a function of anneal time
tw after a quench to temperatures
T. The two observables are compared to the equilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{eq}}(L,T)$ and the equilibrium energy
eeq(L,T), respectively, measured as functions of temperature
T and system size
L for a range of system sizes. For any time and temperature a nonequilibrium time-dependent length scale
L∗(tw,T) can be defined by writing ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)={\ensuremath{\chi}}_{\mathrm{eq}}({L}^{*},T)$ (or the equivalent expression for the energy). Our analysis shows that for all systems studied, an ``effective dynamical critical exponent'' parametrization
L∗(tw,T)=A(T)t1∕z(T) fits the data well at each temperature within the whole temperature range studied, which extends from well above the critical temperature
Tc to near
T=0 for dimension 2 or to well below
Tc for the other space dimensions studied. In addition, the data suggest that the dynamical exponent
z varies smoothly when crossing the transition temperature.
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.
We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.