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Physical review. B, Condensed matter

Series-expansion analysis of critical-temperature shifts of finite Ising lattices with "self-consistent" boundary conditions

H. C. Bolton, C. H. Law

Published Jan 1, 1979• 1 citation
Publisher

Abstract

Finite Ising $N\ifmmode\times\else\texttimes\fi{}N$ ($N=1\ensuremath{-}5$) and $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ ($N=1\ensuremath{-}8$) systems with one of the three self-consistent extended mean-field boundary conditions all around and on both sides, respectively, are studied. The boundary conditions used are: ($a$) extended mean-field boundary condition using average, abbreviated EA (in which each boundary spin takes some average value); ($b$) extended mean-field boundary condition using probability, abbreviated EP (in which each boundary spin takes the values \ifmmode\pm\else\textpm\fi{}1 with certain probabilities) and ($c$) extended Bethe-Peierls, EBP (in which each boundary spin is assumed to be acted on by some effective field). The self-consistent equations are then derived by letting the average magnetization of the internal spins of the system be equal to that of the boundary spins. The finite systems considered therefore represent some high-order systematic generalization of the familiar molecular-field and Bethe-Peierls approximations. The equations for the critical temperature are formulated graphically in terms of some high-temperature expansion series which are calculated to the ninth and tenth orders. Two of the series in the equations of $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ are directly related to the layer and local susceptibilities considered by Binder and Hohenberg for the Ising infinite half plane. Critical temperatures obtained numerically with the use of truncation and extrapolation techniques. Results on the systems with EA and EP indicate that the shifts of their critical temperatures can be described by a power law ${N}^{\ensuremath{-}\ensuremath{\lambda}}$ where $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\gamma}}_{1}}$ and ${\ensuremath{\gamma}}_{1}(\ensuremath{\approx}\frac{11}{8})$ is the critical exponent of the layer susceptibility. This is different from the value $\ensuremath{\lambda}=1$ which is believed true for two-dimensional finite systems with periodic or free boundary conditions. No conclusive remark can be made to systems with EBP through they suggest some even higher value (\ensuremath{\ge} 2).

DOI
10.1103/physrevb.19.388

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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.
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