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Physical review. B, Condensed matter

Phase diagram of the dilute Ising spin glass in general spatial dimension

Shye Shapira, Lior Klein, Joan Adler, Amnon Aharony A, A. Brooks Harris

Published Apr 1, 1994• 6 citations
Publisher

Abstract

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 ${\mathit{T}}_{\mathrm{SG}}$(p), the critical temperature as a function of the dilution p or the analogous critical concentration ${\mathit{p}}_{\mathrm{SG}}$(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.

DOI
10.1103/physrevb.49.8830

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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 \ensuremathχEA{\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}\ensuremathχEA as a function of temperature for arbitrary dilution, 14th-order series for \ensuremathχEA{\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}\ensuremathχEA as a function of dilution for selected temperatures, and 11th-order series for two higher derivatives of \ensuremathχEA{\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}\ensuremathχEA with respect to the ordering field, where \ensuremathχEA{\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}\ensuremathχEA 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.
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 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, 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.
The Edwards-Anderson model of a spin glass is studied by position-space renormalization-group techniques, using an inhomogeneous generalization of Migdal's approximate recursion relation. We treat the spin-1/2 Ising model with independently random nearest-neighbor interactions in dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific-heat anomaly.
An Ising (spin- 1 / 2 ) model with random positive, negative and zero nearest-neighbour exchange coefficients is considered at zero temperature. The authors define two 'spin glass' susceptibilities, through average sums over ((s i s j ) 2 ) av and ( mod (s i s j ) mod ) av , where ( ) means averaging over degenerate ground states and ( ) av means averaging over the random bond distribution. These susceptibilities are expanded in powers of the concentrations of the non-zero bonds, for the triangular (seven terms), square (nine terms) and d-dimensional hypercubic (eight terms) lattices. All the series show a singularity, which is associated with the paramagnetic to spin-glass phase transition. The phase boundary between these phases, and the corresponding critical exponents, are estimated.
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.