The high-temperature expansion of the hierarchical Ising model: From Poincaré symmetry to an algebraic algorithm

Source abstract

It is shown that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar Poincaré groups. Using these symmetries, it is shown that the intricate sums appearing in the calculation of the high-temperature expansion of the magnetic susceptibility can be performed, at least up to the fourth order, using elementary algebraic manipulations which can be implemented with a computer. These symmetries appear more clearly if we use the two-adic fractions to label the sites. Then the new algebraic methods are applied to the calculation of quantities having a random walk interpretation. In particular, it is shown that the probability of returning to the starting point after m steps has poles at D=−2,−4,...,−2m, where D is a free parameter playing a role similar to the dimensionality in nearest neighbor models.