The High-Temperature Expansion of the Hierarchical Ising Model: From Poincar\'e Symmetry to an Algebraic Algorithm

TL;DR

This paper reveals a Poincaré symmetry in the hierarchical Ising model, enabling algebraic computation of high-temperature expansions and related quantities, with implications for understanding random walk probabilities.

Contribution

It introduces a novel algebraic method leveraging Poincaré symmetry to compute high-temperature expansions and analyze random walk properties in the hierarchical Ising model.

Findings

Symmetry group decomposes into rotations and translations.

High-temperature susceptibility sums can be computed algebraically up to fourth order.

Return probability poles occur at specific negative even values of D.

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 Poincar\'e 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.