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arXiv:hep-lat/9206020 · Physical Review Letters

Low temperature expansion for the Ising model

Gyan V Bhanot, Michael J. Creutz, Jan Lacki

Published Sep 28, 1992• 58 citations
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Abstract

On simple cubic lattices, we compute the low-temperature expansion for the energy of the Ising model through 50 excited bonds in three dimensions and 44 excited bonds in four dimensions. We also give the magnetization through 42 excited bonds. Our method is a recursive enumeration of states with given energies on a set of finite lattices with generalized helical boundary conditions. A linear combination of such lattices cancels finite volume effects.

DOI
10.1103/physrevlett.69.1841

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For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.

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On simple cubic lattices, we compute the low-temperature expansion for the energy of the Ising model through 50 excited bonds in three dimensions and 44 excited bonds in four dimensions. We also give the magnetization through 42 excited bonds. Our method is a recursive enumeration of states with given energies on a set of finite lattices with generalized helical boundary conditions. A linear combination of such lattices cancels finite volume effects.
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.
I present a modification of the shadow-lattice technique, which allows one to derive low temperature series for discrete spin models to high orders. Results are given for the 3-d Ising model up to 64 excited bonds, for the 4-d Ising model up to 96 excited bonds and the 3-d Potts model up to 56 excited bonds. Series expansions are a major tool to obtain critical properties of statistical systems and to test Monte Carlo programs. Usually series are derived using graphical techniques, but recently a method using direct enumeration of the relevant configurations on the lattice has been proposed [1]. Whereas the diagrammatic techniques have the disadvantage that they are hard to port to parallel or vector machines, the direct enumeration relies heavily on fast machines, taking full advantage of parallel architecture, but is computationally too demanding to be implemented on smaller machines. One advantage of direct enumeration is that it is done automatically, i.e. once the program is set up it can in principle calculate the series to any given order. In this paper I shall sketch a method which is a variant of the shadow method (see for
For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.