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Journal of Applied Physics

Critical exponents for Ising spin glasses through high-temperature series analysis

Rajiv R. P. Singh, Sudip Chakravarty

Published Apr 15, 1987• 5 citations
Publisher

Abstract

We present estimates of the critical exponents ν and η for the three-dimensional ±J distribution Ising spin glass based on high-temperature series analysis. The method of star graph expansion is used to obtain the series for two different susceptibilities to order W17 (W=tanh2 J/kT). The series are analyzed through first-order inhomogeneous differential approximants. Assuming Tc =1.175, we find the exponents to be ν=1.3±0.2 and η=−0.25±0.2.

DOI
10.1063/1.338539

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We present estimates of the critical exponents ν and η for the three-dimensional ±J distribution Ising spin glass based on high-temperature series analysis. The method of star graph expansion is used to obtain the series for two different susceptibilities to order W17 (W=tanh2 J/kT). The series are analyzed through first-order inhomogeneous differential approximants. Assuming Tc =1.175, we find the exponents to be ν=1.3±0.2 and η=−0.25±0.2.
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
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(=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.
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.
High temperature series expansion for the critical exponents of the Ising model are reanalysed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent γ in three dimensions, close to 1.245, therefore lower than the quoted value 1.250, and much closer to the renormalization group (R.G.) value 1.241. The exponent is analysed in two ways : in one method γ is estimated directly while in the other one Tc, is calculated first. With these new values of Tc, the exponent α is recalculated and found to be very close to the R.G. value 0.110. The value of ν is not modified (0.638) and is therefore still a problem for hyperscaling, and is in disagreement with the R.G. value (0.630).
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.
We develop high-temperature expansions for the Edwards-Anderson susceptibility of the \ifmmode\pm\else\textpm\fi{}J three-state Potts spin glasses on d-dimensional hypercubic lattices. For d\ensuremath{\ge}6, d log Pad\'e approximant analysis yields a second-order phase transition with an exponent \ensuremath{\gamma} consistent with the mean-field value of unity. In lower dimensions the extrapolations are not convergent enough to extract the critical behavior. In d=3, the series are analyzed assuming various finite- and zero-temperature transitions and their consistency with available Monte Carlo data is studied.