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Modern Physics Letters B

Extension of the High Temperature Susceptibility Expansion for General Ising Systems

Agoes Soehianie, J. Oitmaa

Published Jun 20, 1997• 4 citations
Publisher

Abstract

We have obtained the graphs and weights for the twelfth and thirteenth terms of the high temperature multigraph expansion for the susceptibility of general Ising systems. As applications we have computed the coefficients through thirteenth order for the square lattice with first and second neighbor interactions and first and third neighbor interactions.

DOI
10.1142/s021798499700075x

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We have obtained the graphs and weights for the twelfth and thirteenth terms of the high temperature multigraph expansion for the susceptibility of general Ising systems. As applications we have computed the coefficients through thirteenth order for the square lattice with first and second neighbor interactions and first and third neighbor interactions.
The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.
The high-temperature expansion of the partition function of an Ising model in zero magnetic field is calculated through terms of ${T}^{\ensuremath{-}9}$ for a fcc lattice and variable first-, second-, and third-neighbor interactions (J1{J}_{1}J1​, J2{J}_{2}J2​, and J3{J}_{3}J3​). Values of the transition temperature Tc{T}_{c}Tc​ and of the first-, second-, and third-neighbor correlation coefficients at T=TcT={T}_{c}T=Tc​ are given for J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ and J3J1\frac{{J}_{3}}{{J}_{1}}J1​J3​​ each varying between 0 and 2.
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.
A new method of deriving high temperature series expansions for the susceptibility of the Ising and classical vector spin models is presented. The method involves generating the expansion for the inverse of the susceptibility, in which case only contributions from star graphs need to be considered. An additional term (the thirteenth) has been added to the Ising high temperature susceptibility series for the FCC lattice. Analysis of the extended series does not indicate that revision of previous critical estimates is required. In a test of the capabilities of the method for other systems, seven terms of the susceptibility series of the FCC Heisenberg and planar vector models have been reproduced. Techniques used in generating the graph data are discussed.