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Physical review. B, Condensed matter

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

R. N. Bhatt, A. P. Young

Published Apr 1, 1988• 301 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.37.5606

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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.
The three-dimensional Ising spin-glass in zero field with nearest-neighbor interactions having \ifmmode\pm\else\textpm\fi{}J distribution is studied by Monte Carlo simulations for samples of linear dimension L with 3\ensuremath{\le}L\ensuremath{\le}20. Results for the probability distribution, PL{\mathrm{P}}_{\mathrm{L}}PL​(q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at Tc{\mathrm{T}}_{\mathrm{c}}Tc​${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.
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 explore the behavior of the nearest-neighbor Ising spin glass with discrete bonds (± J ) on the face-centered cubic lattice using both Monte Carlo simulations and high-temperature expansions. A phase transition to a spin glass phase is found in zero magnetic field ( H ) at a nonzero temperature ( T = T c ), with critical exponents close to that for the simple cubic lattice. In the ( H, T )-plane, the contours of constant spin glass susceptibility show an upturn towards higher H at the lowest T , consistent with the possibility of a de Almeida-Thouless line of phase transitions in a field. Further, the upturn is similar to those on the simple cubic lattice and less pronounced than on the four-dimensional hypercubic lattice, despite the higher coordination number of the f.c.c. lattice, suggesting that the upturn is a dimension-related property.
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.
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...
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 zzz 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.