We determine accurate values of ordering temperatures and critical exponents for Ising-spin-glass transitions in four dimensions, using a combination of finite-size scaling and nonequilibrium scaling techniques. We find that the exponents $\ensuremath{\eta}$ and
z vary with the form of the interaction distribution, indicating nonuniversality at Ising-spin-glass transitions. These results confirm conclusions drawn from numerical data for three dimensions.
Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures
βc, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for
βc and the exponent
ν from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable
τ=1−β2/βc2 and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes L. Comparisons between the 4d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents
γ and
ν, depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.
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.
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.
Ising and Heisenberg ferromagnets and spin-glasses are studied at the percolation threshold for power-law bond distributions, P(J)\ensuremath{\sim}\ensuremath{\Vert}J${\ensuremath{\Vert}}^{\ensuremath{\alpha}\mathrm{\ensuremath{-}}1}$ for J\ensuremath{\rightarrow}0. Nonuniversal, \ensuremath{\alpha}-dependent, critical behavior is found. A hierarchical model for the ``backbone'' yields, via finite-size scaling, expressions for the exponent t in the relation \ensuremath{\Upsilon}\ensuremath{\propto}(p-
pc)t for the generalized stiffness coefficient, and for the percolation-to-thermal crossover exponent \ensuremath{\varphi}.
I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈
Sx(0)
Sx(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (
83,
163,
323, and
643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.