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arXiv:1310.2910 · Physical Review B

Critical parameters of the three-dimensional Ising spin glass

Marco Baity‐Jesi, R. A. Baños, Andrés De la Cruz, L. A. Fernández, Jose Miguel Gil-Narvion, A. Gordillo-Guerrero, D. Íñiguez, A. Maiorano, Filippo Mantovani, Enzo Marinari, Víctor Martín-Mayor, Jorge Monforte-Garcia, A. Muñoz Sudupe, Denis Navarro, Giorgio Parisi A, Sergio Perez-Gaviro, Marcello Pivanti, Federico Ricci‐Tersenghi, Juan Jesus Ruiz-Lorenzo, Sebastiano Fabio Schifano, Beatriz Seoane, Alfonso Tarancón, R. Tripiccione, David Yllanes

Published Dec 20, 2013• 116 citations
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Abstract

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 $L=40$ using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain ${T}_{\mathrm{c}}=1.1019(29)$ 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 ${T}_{\mathrm{c}}$.

DOI
10.1103/physrevb.88.224416

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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, \ensuremathν=2.562(42)\ensuremath{\nu}=2.562(42)\ensuremathν=2.562(42) for the thermal exponent, \ensuremathη=\ensuremath−0.3900(36)\ensuremath{\eta}=\ensuremath{-}0.3900(36)\ensuremathη=\ensuremath−0.3900(36) for the anomalous dimension, and \ensuremathω=1.12(10)\ensuremath{\omega}=1.12(10)\ensuremathω=1.12(10) for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield \ensuremathα=\ensuremath−5.69(13)\ensuremath{\alpha}=\ensuremath{-}5.69(13)\ensuremathα=\ensuremath−5.69(13), \ensuremathβ=0.782(10)\ensuremath{\beta}=0.782(10)\ensuremathβ=0.782(10), and \ensuremathγ=6.13(11)\ensuremath{\gamma}=6.13(11)\ensuremathγ=6.13(11). We also compute several universal quantities at Tc{T}_{\mathrm{c}}Tc​.
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I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈Sx{S}_{x}Sx​(0)Sx{S}_{x}Sx​(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.
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