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Physical review. B, Condensed matter

Three-state Potts-model spin glasses on hypercubic lattices

Rajiv R. P. Singh

Published Mar 1, 1991• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.43.6299

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Physical Review Letters•Jul 14, 1986•180 citations
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.
The three-state antiferromagnetic Potts model on d-dimensional hypercubic lattices is studied, where d is 2, 3, 4, 5 and 6. The Hamiltonian is expressed by the spin Hamiltonian with spin 1. The staggered susceptibility is calculated by the high temperature series expansion. By use of the ratio method and the Padé approximation, the critical temperature and the critical exponent γ are obtained. It is concluded that the three-state antiferromagnetic Potts model exhibits the second-order transition when d ≥3.
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 expansions of the staggered susceptibility of the q -state antiferromagnetic Potts model (AFPM) on a d -dimensional hypercubic lattice are calculated. For arbitrary values of q and d , the transition temperatures and the critical exponents γ are determined by use of the ratio method and the Padé approximation. It is shown that the 3- and 4-state AFPM exhibit a second-order phase transition when d ≥3, and the 5- and 6-state AFPM do when d ≥4, etc. In the limit of infinite d , the transition temperature T c is obtained rigorously as k T c / J ={log [2 d /(2 d - q )]} -1 ( k : the Boltzmann constant, J : the interaction parameter), and the ordered phase exists when d > q /2.
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 study the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.
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.