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Canadian Journal of Physics

High temperature multigraph expansions for general Ising systems

J. Oitmaa

Published Jan 1, 1981• 9 citations
Publisher

Abstract

A high temperature expansion, in terms of connected graphs with single and multiple edges, is developed for general Ising systems with interactions of more than one type. The graphical information obtained is sufficient to derive 11 terms in the expansion of the high temperature zero-field susceptibility and 12 terms in the zero-field free energy for any Ising system. Series to this order are presented for the square lattice with nearest and next nearest neighbour interactions.

DOI
10.1139/p81-002

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A high temperature expansion, in terms of connected graphs with single and multiple edges, is developed for general Ising systems with interactions of more than one type. The graphical information obtained is sufficient to derive 11 terms in the expansion of the high temperature zero-field susceptibility and 12 terms in the zero-field free energy for any Ising system. Series to this order are presented for the square lattice with nearest and next nearest neighbour interactions.
A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.
Extended high-temperature series expansions are presented for the spin-s Ising model on the face-centred cubic lattice. Series coefficients for the zero-field internal energy (U 0 ) and the second moment ( mu 2 ) of the correlation function are derived, for 1 / 2 <or=s<or= 9 / 2 , to order 14 in the high-temperature expansion variable K(=J/kT). For s= 1 / 2 , the fourth field derivative of the free energy is calculated to order K 14 and the magnetisation series is given to order 13 and 27 in the temperature and magnetic field variables respectively. For s=1 to 9 / 2 , the zero-field susceptibility series are derived to order K 14 .
We have calculated the high-temperature series expansion of the zero-field susceptibility of the square-lattice Ising model with first and second neighbour interactions to the 20th order by computer. Our results extend the previous calculation by Hsiao and Lin to two more orders. We use the Padé approximants to estimate the critical exponent γ and the critical temperature. Our result 1.747 < γ < 1.753 supports the universality conjecture that all two-dimensional Ising models have the same critical exponent γ = 1.75.
Ising lattices consisting of n=2,3,4,and5n=2, 3, 4, \mathrm{and} 5n=2,3,4,and5 interacting plane square lattice layers are studied by hightemperature series expansions. Specifically, seven to nine terms of the zero-field susceptibility expansion have been obtained for (a) free-surface boundary conditions (in which each surface spin interacts with only five nearest neighbors); and (b) periodic boundary conditions (in which all spins interact equivalently with six nearest neighbors). Estimates of the critical temperatures Tc(n){T}_{c}(n)Tc​(n) obtained by ratio and Pad\'e approximant techniques are presented. These results are consistent with the conjectures that ${T}_{c}(\ensuremath{\infty})\ensuremath{-}{T}_{c}(n)$ varies with thickness nnn, as ${n}^{\ensuremath{-}\ensuremath{\lambda}}$ with $\ensuremath{\lambda}=1$ in case (a) and $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\nu}}_{c}}\ensuremath{\simeq}1.56$ in case (b); but other, somewhat larger, values of $\ensuremath{\lambda}$ are not excluded.
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.