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(p), the critical temperature as a function of the dilution p or the analogous critical concentration
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.
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
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,
d, for
d<6, and apparently tends to infinity at
d=4; however, the critical temperature does not appear to go to zero at
d=4.
We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling
gr or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$
gr=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.
We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$
〉T〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T]h and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T]h, where 〈
〉T indicates a canonical average at temperature T for an arbitrary configuration of random fields and [
]h indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each
hi=\ifmmode\pm\else\textpm\fi{}
h0 and a Gaussian distribution in which each
hi has variance
h02. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$
an(g,d)(J/T
)n, where J is the exchange constant and the coefficients
an(g,d) are polynomials in g\ensuremath{\equiv}
h02/
J2 and in d. We assume that as T approaches its critical value,
Tc, one has \ensuremath{\chi}\ensuremath{\sim}(T-
Tc${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-
Tc${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(
T2)/(
gJ2). We find that A approaches a constant value as T\ensuremath{\rightarrow}
Tc (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.
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(=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.