Numerical studies of Ising spin glasses in two, three, and four dimensions

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 ${T}_{c}$=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 ${T}_{c}$ 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 ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.