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arXiv:hep-lat/0209019 · Nuclear Physics B - Proceedings Supplements

New algorithm of the high-temperature expansion for the Ising model in three dimensions

H. Arisue, T. Fujiwara

Published May 1, 2003• 4 citations
arXivHTMLPDF
DOI
10.1016/s0920-5632(03)01701-8

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It is well known that the partition function of two-dimensional Ising model can be expressed as a Grassmann integral over the action bilinear in Grassmann variables. The key aspect of the proof of this equivalence is to show that all polygons, appearing in Grassmann integration, enter with fixed sign. For three-dimensional model, the partition function can also be expressed by Grassmann integral. However, the action resulting from low-temperature expansion contains quartic terms, which does not allow explicit computation of the integral. We wanted to check - apparently not explored - the possibility that using the high-temperature expansion would result in action with only bilinear terms. (in two dimensions, low-T and high-T expansions are equivalent, but in three dimensions, they differ.) It turned out, however that polygons obtained by Grassman integration are not of fixed sign for any ordering of Grassmann variables on sites. This way, it is not possible to express the partition function of three-dimensional Ising model as a Grassman integral over bilinear action.

arXiv:2309.07718•Sep 14, 2023•0 citations
New algorithm of the finite lattice method is presented to generate the high-temperature expansion series of the Ising model. It enables us to obtain much longer series in three dimensions when compared not only to the previous algorithm of the finite lattice method but also to the standard graphical method. It is applied to extend the high-temperature series of the simple cubic Ising model from beta^{26} to beta^{46} for the free energy and from beta^{25} to beta^{32} for the magnetic susceptibility.
It is well known that the partition function of two-dimensional Ising model can be expressed as a Grassmann integral over the action bilinear in Grassmann variables. The key aspect of the proof of this equivalence is to show that all polygons, appearing in Grassmann integration, enter with fixed sign. For three-dimensional model, the partition function can also be expressed by Grassmann integral. However, the action resulting from low-temperature expansion contains quartic terms, which does not allow explicit computation of the integral. We wanted to check - apparently not explored - the possibility that using the high-temperature expansion would result in action with only bilinear terms. (in two dimensions, low-T and high-T expansions are equivalent, but in three dimensions, they differ.) It turned out, however that polygons obtained by Grassman integration are not of fixed sign for any ordering of Grassmann variables on sites. This way, it is not possible to express the partition function of three-dimensional Ising model as a Grassman integral over bilinear action.