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arXiv:cond-mat/0006211 · Physical review. B, Condensed matter

Critical behavior of the three-dimensional Ising spin glass

H.G. Ballesteros, Andrés De la Cruz, L. A. Fernández, Víctor Martín-Mayor, J. Pech, Juan Jesus Ruiz-Lorenzo, Alfonso Tarancón, Pedro Téllez, C.L. Ullod, C. Ungil

Published Dec 1, 2000• 252 citations
arXivHTMLPDF

Abstract

We have simulated, using parallel tempering, the three-dimensional Ising spin glass model with binary couplings in a helicoidal geometry. The largest lattice $(L=20)$ has been studied using a dedicated computer (the SUE machine). We have obtained, measuring the correlation length in the critical region, strong evidence for a second-order finite-temperature phase transition, ruling out other possible scenarios like a Kosterlitz-Thouless phase transition. Precise values for the \ensuremath{\nu} and \ensuremath{\eta} critical exponents are also presented.

DOI
10.1103/physrevb.62.14237

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We have simulated, using parallel tempering, the three-dimensional Ising spin glass model with binary couplings in a helicoidal geometry. The largest lattice (L=20)(L=20)(L=20) has been studied using a dedicated computer (the SUE machine). We have obtained, measuring the correlation length in the critical region, strong evidence for a second-order finite-temperature phase transition, ruling out other possible scenarios like a Kosterlitz-Thouless phase transition. Precise values for the \ensuremath{\nu} and \ensuremath{\eta} critical exponents are also presented.
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 have studied the three-dimensional Ising spin glass with a ± 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, with ordering below Tc. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent, ν, is somewhat higher. Because there may be (unknown) corrections to finite size scaling, we do not completely rule out the possibility that Tc = 0 or that Tc is finite but with no order below Tc. However, from our data, these possibilities seem less likely.
We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to L=40L=40L=40 using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain Tc=1.1019(29){T}_{\mathrm{c}}=1.1019(29)Tc​=1.1019(29) for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at Tc{T}_{\mathrm{c}}Tc​.
The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.
We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.
We perform numerical simulations, including parallel tempering, a four-state Potts glass model with binary random quenched couplings using the JANUS application-oriented computer. We find and characterize a glassy transition, estimating the critical temperature and the value of the critical exponents. Nevertheless, the extrapolation to infinite volume is hampered by strong scaling corrections. We show that there is no ferromagnetic transition in a large temperature range around the glassy critical temperature. We also compare our results with those obtained recently on the ``random permutation'' Potts glass.