We study the 4D Heisenberg spin-glass model with Gaussian nearest-neighbour interactions. We use finite-size scaling to analyse the data. We find a behaviour consistent with a finite temperature spin-glass transition. Our estimates for the critical exponents agree with the results from the epsilon -expansion.
We study the 4d Heisenberg spin glass model with Gaussian nearestneighbor interactions. We use finite size scaling to analyze the data. We find a behavior consistent with a finite temperature spin glass transition. Our estimates for the critical exponents agree with the results from "-expansion. PACS Numbers 7510N-0270 Preprint Roma1 n o 1054 The lower critical dimension d l of the short range models remains one of the most controversial questions in the spin glass [1]-[3] theory. In the last ten years a consistent number of works has been devoted to studying the Ising model, which seems to be [4, 5] very close to d l in d = 3. In the case of the short range isotropic Heisenberg spin glass, the conclusions of various computer simulations in d = 3 [6, 7] agree that the system is below d l . Using domain wall renormalization group techniques, it was argued [2, 3] that d l = 4 for this model. To our knowledge, there are no previous numerical simulations on the 4d Heisenberg spin glass...
The nearest-neighbour XY spin glass on a hypercubic lattice in four dimensions is studied by Monte Carlo simulations. A finite- size scaling analysis of the data leads to a finite temperature spin glass transition at Tc=0.95±0.15. The critical exponents are estimated to be νsg=0.70±0.10 and η_{sg}=-0.28\pm 0.38$. The results imply that the lower critical dimensionality for the XY spin glass is less than four.
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.
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.
The spin and the chirality orderings of the Heisenberg spin glass in two dimensions with the nearest-neighbour Gaussian coupling are investigated by equilibrium Monte Carlo simulations. Particular attention is paid to the behaviour of the spin and the chirality correlation lengths. In order to observe the true asymptotic behaviour, a fairly large system size L ⪆ 20 ( L the linear dimension of the system) appears to be necessary. It is found that both the spin and the chirality order only at zero temperature. At high temperatures, the chiral correlation length stays shorter than the spin correlation length, whereas at lower temperatures below the crossover temperature T × , the chiral correlation length exceeds the spin correlation length. The spin and the chirality correlation-length exponents are estimated above T × to be ν SG = 0.9 ± 0.2 and ν CG = 2.1 ± 0.3, respectively. These values are close to the previous estimates on the basis of the domain-wall-energy calculation. Discussion is given about the asymptotic critical behaviour realized below T × .