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Physical Review Letters

Critical Behavior of an Ising Spin-Glass

Rajiv R. P. Singh, Sudip Chakravarty

Published Jul 14, 1986• 180 citations
Publisher

Abstract

Critical behavior of Ising spin-glass with $\ifmmode\pm\else\textpm\fi{}J$ distribution is discussed. The results are based on the high-temperature series expansion of Edwards-Anderson susceptibility in powers of $w (=\frac{{tanh}^{2}J}{\mathrm{kT}})$ for 2D, 3D, and 4D "cubic" lattices to orders 19, 17, and 15, respectively. The lengths of the series are comparable to the best available for the pure Ising model. While 3D and 4D systems exhibit transitions at finite temperatures, in 2D a zero-temperature transition with a power-law divergence (in inverse temperature) for the susceptibility is found. The estimates of critical temperatures and exponents are presented.

DOI
10.1103/physrevlett.57.245

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Physical review. B, Condensed matter•Apr 1, 1994•6 citations
Critical behavior of Ising spin-glass with \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ distribution is discussed. The results are based on the high-temperature series expansion of Edwards-Anderson susceptibility in powers of w(=tanh2JkT)w (=\frac{{tanh}^{2}J}{\mathrm{kT}})w(=kTtanh2J​) for 2D, 3D, and 4D "cubic" lattices to orders 19, 17, and 15, respectively. The lengths of the series are comparable to the best available for the pure Ising model. While 3D and 4D systems exhibit transitions at finite temperatures, in 2D a zero-temperature transition with a power-law divergence (in inverse temperature) for the susceptibility is found. The estimates of critical temperatures and exponents are presented.
We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.
A high-temperature series expansion for the Edwards and Anderson spin-glass order-parameter susceptibility is computed for Ising spins on hypercubic lattices with nearest-neighbor interactions. The series is analyzed by Pad\'e approximants with Rudnick-Nelson-type corrections to scaling. The results agree with the first-order $\ensuremath{\epsilon}$ expansion of Harris, Lubensky, and Chen. The critical exponent ${\ensuremath{\gamma}}_{Q}$ increases monotonically with decreasing dimension, ddd, for d<6d<6d<6, and apparently tends to infinity at d=4d=4d=4; however, the critical temperature does not appear to go to zero at d=4d=4d=4.
We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size LLL: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter ppp, p=0.5p=0.5p=0.5 and p=0.7p=0.7p=0.7 (up to L=28L=28L=28 and L=20L=20L=20, respectively), and the bond-diluted bimodal model for bond-occupation probability pb=0.45{p}_{b}=0.45pb​=0.45 (up to L=16L=16L=16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at pb=0.7{p}_{b}=0.7pb​=0.7 and pb=0.35{p}_{b}=0.35pb​=0.35 (up to L=10L=10L=10) and of the Ising spin glass model with Gaussian bond distribution (up to L=8L=8L=8) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.
We calculate high-temperature graph expansions for the Ising spin glass model with 4 symmetric random distribution functions for its nearest neighbor interaction constants Jij. Series for the Edwards-Anderson susceptibility χ are obtained to order 13 in the expansion variable (J/(kBT)) EA 2 for the general d-dimensional hyper-cubic lattice, where the parameter J determines the width of the distributions. We explain in detail how the expansions are calculated. The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, leads to estimates for the critical threshold (J/(kBTc)) 2 and for the critical exponent γ in dimensions 4, 5, 7 and 8 for all the distribution functions. In each dimension the values for γ agree, within their uncertainty margins, with a common value for the different distributions, thus confirming universality. PACS: 05.70.Jk Critical point phenomena – 75.10.Nr Spin-glass and other random models 1
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​.
We use high-temperature and low-concentration series to treat the dilute spin glass within a model with nearest-neighbor interactions which randomly assume the values +J, 0, -J with probabilities p/2, 1-p, p/2, respectively. Using the Harris no-free-end diagrams scheme in general spatial dimension, we obtained 15th-order series for ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ as a function of temperature for arbitrary dilution, 14th-order series for ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ as a function of dilution for selected temperatures, and 11th-order series for two higher derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ with respect to the ordering field, where ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ is the Edwards-Anderson spin-glass susceptibility. Analysis of these series yields values of TSG{\mathit{T}}_{\mathrm{SG}}TSG​(p), the critical temperature as a function of the dilution p or the analogous critical concentration pSG{\mathit{p}}_{\mathrm{SG}}pSG​(T). Thus we determine a critical line, separating the spin-glass phase from the paramagnetic phase in the T-p plane. We find values of the critical exponent \ensuremath{\gamma} and universal amplitude ratios along the critical line. Universal amplitude ratios and dominant exponents along the critical line are identical to those of the pure spin glass for a wide range of dilution, indicating the same critical behavior as that of the pure spin glass.