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Physical review. B, Condensed matter

State counting and low-temperature series

Michael J. Creutz

Published May 1, 1991• 41 citations
Publisher

Abstract

I present a simple numerical technique for evaluating the low-temperature expansion for discrete statistical systems. I begin with a recursive procedure on finite lattices to count the states of a given energy. Comparing these numbers on different lattice sizes, I extract coefficients for the infinite-volume series. I test the method with the three-dimensional Ising model, obtaining the expansion of the average energy through terms involving 34 excited bonds.

DOI
10.1103/physrevb.43.10659

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I present a simple numerical technique for evaluating the low-temperature expansion for discrete statistical systems. I begin with a recursive procedure on finite lattices to count the states of a given energy. Comparing these numbers on different lattice sizes, I extract coefficients for the infinite-volume series. I test the method with the three-dimensional Ising model, obtaining the expansion of the average energy through terms involving 34 excited bonds.
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
The finite lattice method of series expansion is generalised to the qqq-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2q=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26}u26 from u20u^{20}u20. The high-temperature series for the zero-field partition function is extended from v18v^{18}v18 to v22v^{22}v22. Subsequent analysis gives critical exponents in agreement with those from field theory.
Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.