A transfer matrix method is used to study the variation with length scale L of the distribution P L (J eff ) of effective couplings in Ising spin glasses at zero temperature. For a gaussian initial distribution the authors find in two dimensions J(L) identical to ( mod J eff mod ) varies as L -1 nu / for 2<or=L<or=12, with nu =3.4+or-0.1, implying a zero-temperature phase transition with correlation length exponent nu . In three dimensions the effective couplings initially increase with length scale, J(L) varies as L -1 nu / for 2<or=L<or=4, with -1/ nu approximately=0.2, suggesting a phase transition at finite temperature. The results are described surprisingly well by the zero-temperature version of the Migdal-Kadanoff renormalisation group scheme.
A transfer matrix method is used to study the variation with length scale L of the distribution P L (J eff ) of effective couplings in Ising spin glasses at zero temperature. For a gaussian initial distribution the authors find in two dimensions J(L) identical to ( mod J eff mod ) varies as L -1 nu / for 2<or=L<or=12, with nu =3.4+or-0.1, implying a zero-temperature phase transition with correlation length exponent nu . In three dimensions the effective couplings initially increase with length scale, J(L) varies as L -1 nu / for 2<or=L<or=4, with -1/ nu approximately=0.2, suggesting a phase transition at finite temperature. The results are described surprisingly well by the zero-temperature version of the Migdal-Kadanoff renormalisation group scheme.
The dependence of the effective coupling J'(L,m) on length scale L and spin dimension m is studied numerically for vector spin glasses using the 'defect energy' method for space dimension d=2,3 (m=2,3) and 1<or=d<or=4 (m= infinity ). Scaling towards weak coupling, J'(L,m) approximately L -1 nu /, L>>L c (m), is found in all cases studied implying a power-law divergence of the correlation length, xi approximately T - nu , for T to 0 but no finite-temperature transition. The characteristic length L c (m), which increases with m, separates a preasymptotic (small L) regime where m= infinity behaviour is observed, J'(L,m) approximately=J'(L, infinity ), from the asymptotic (large-L) regime which gives the true critical behaviour for finite m. Values of nu (m,d) obtained are nu ( infinity ,2) approximately=0.65, nu ( infinity ,3) approximately=1.0, nu ( infinity ,4) approximately=1.5 nu (2,2) approximately=1,1, nu (2.3) approximately=2.2. The m=2 exponents are in good agreement with a Migdal-Kadanoff-like renormalisation group calculation. For d=1 the unexpected results nu approximately=O for m= infinity is obtained if frustration is included. For d=3, anisotropy is expected to induce an Ising-like phase transition at a finite temperature, and a crossover expression for the transition temperature is derived.
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.
We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc is nonzero with a long-range-ordered phase below Tc. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
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.
Critical point scaling in a field H applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3, we have determined the exponents and the critical scaling function f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f(R) in d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of R for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function f(R) at some negative value of R, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.
Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions: (i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and (ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.