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Journal of Physics A Mathematical and General

The four-dimensionalXYspin glass

S. Jain

Published Aug 7, 1996• 6 citations
Publisher

Abstract

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 . The critical exponents are estimated to be and . The results imply that the lower critical dimensionality for the XY spin glass is less than four.

DOI
10.1088/0305-4470/29/15/004

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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 . The critical exponents are estimated to be and . The results imply that the lower critical dimensionality for the XY spin glass is less than four.
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±0.15T_c=0.95\pm 0.15Tc​=0.95±0.15. The critical exponents are estimated to be νsg=0.70±0.10ν_{sg}=0.70\pm 0.10νsg​=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 describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.
We study the XY spin glass by large-scale Monte Carlo simulations for sizes up to 243{24}^{3}243, down to temperatures below the transition temperature found in earlier work. The data for the larger sizes show more marginal behavior than that for the smaller sizes, indicating that the lower critical dimension is close, and possibly equal to three. We find that the spins and chiralities behave in a similar manner. We also address the optimal ratio of ``over-relaxation'' to ``Metropolis'' sweeps in the simulation.
We have studied the three-dimensional Ising spin glass with a $\ifmmode\pm\else\textpm\fi{}J$ distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite-size scaling analysis shows quite strong evidence for a finite transition temperature Tc{T}_{c}Tc​ with ordering below Tc{T}_{c}Tc​. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent $\ensuremath{\nu}$ is somewhat higher. Because there may be (unknown) corrections to finite-size scaling, we do not completely rule out the possibility that Tc=0{T}_{c}=0Tc​=0 or that Tc{T}_{c}Tc​ is finite but with no order below Tc{T}_{c}Tc​. However, from our data, these possibilities seem less likely.
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{T}_{c}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{T}_{c}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{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.