The High-Temperature Expansion of the Hierarchical Ising Model: From Poincaré Symmetry to an Algebraic Algorithm

Abstract

We show 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, we show 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 2-adic fractions to label the sites. We then apply the new algebraic methods to the calculation of quantities having a random walk interpretation. In particular, we show that the probability of returning at 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.

arXiv comment24 Pages, includes 2 short Mathematica programs appended after "/end" uses phyzzx.tex

References in corpus (2)

• A Numerical Study of the Hierarchical Ising Model: High Temperature Versus Epsilon Expansion
• Duality in Long-Range Ising Ferromagnets

Cited by in corpus (6)

• Nonlinear Aspects of the Renormalization Group Flows of Dyson's Hierarchical Model
• Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model
• A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory
• The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis
• Non-Universal Quantities from Dual RG Transformations
• Small numerators canceling small denominators of the high-temperature scaling variables: a systematic explanation in arbitrary dimensions