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Physica A Statistical Mechanics and its Applications

High-temperature expansion for the longitudinal susceptibility of the spin- model

M. Howard Lee

Published May 1, 1983• 1 citation
Publisher
DOI
10.1016/0378-4371(83)90105-x

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To avoid the difficulties with the usual type of (canonical) high-temperature series which arise when T<Tc and H=O, it is suggested that a high-temperature expansion be accomplished using the microcanonical ensembles of systems defined as involving only those states having values of the order parameter equal to the temperature-averaged one. The method is applied for the Ising and Heisenberg model utilising the molecular-field picture, and for the Heisenberg model using the spin-wave picture. In the latter case, the spin-wave energy is expanded according to the high-temperature parameter. It is shown that (m,m+1)-type Pade approximants are good at all temperatures. It is found that the longitudinal susceptibility chi L, diverges like H-1/2 at T<Tc; consequently, the critical exponent gamma ' no longer holds. A new critical exponent, gamma is defined as scaling the reduced susceptibility, chi 0=H1/2chi L. Deriving the scaling-law relating gamma with beta and delta , gamma is estimated as 0.18.
We calculate the thermodynamics of the one-dimensional spin-1/2 Ising model in the presence of a constant skew magnetic field. We obtain the high-temperature expansion of its Helmholtz free energy (HFE), for the ferromagnetic and antiferromagnetic cases, up to order beta7. This expansion permits us to obtain the behaviour of the model for $ |J| \beta \stackrel {<}{_\sim}} 1 $, when it cannot be described by its classical version. Among the calculated thermodynamical functions of the model, we have the diagonal elements of the magnetic susceptibility tensor for the transverse and logitudinal Ising models, obtained by taking the limits h z -> 0 and h y -> 0, respectively, of the beta-expansion of the HFE. The y-component of the magnetization and the chiyy component of the magnetic susceptibility tensor are almost the same for the antiferro- and ferromagnetic models, at least for $ |J| \beta \stackrel {<}{_\sim}} 1 $; and, c yy is practically independent of the direction of the external magnetic. We also show that, in this region of temperature, the thermodynamics of the Ising model with skew magnetic field and that of an XXZ model with longitudinal magnetic field are not similar.