Yet another way to obtain low temperature expansions for discrete spin systems

Abstract

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