The Elusive Asymptotic Behavior of the High-Temperature Expansion of the Hierarchical Ising Model

TL;DR

This paper develops a recursive method to compute high-temperature expansion coefficients for the hierarchical Ising model, revealing slow convergence of critical exponent estimates and discussing the potential for models with exact Padé approximants.

Contribution

It introduces a differential formulation for calculating high-temperature expansion coefficients of the hierarchical Ising model, enabling precise analysis of asymptotic behavior.

Findings

Departure from Padé approximation grows slowly with order

Critical exponent estimates converge very slowly with more coefficients

Large number of coefficients needed for accurate critical exponents

Abstract

We present a differential formulation of the recursion formula of the hierarchical model which provides a recursive method of calculation for the high-temperature expansion. We calculate the first 30 coefficients of the high temperature expansion of the magnetic susceptibility of the Ising hierarchical model with 12 significant digits. We study the departure from the approximation which consists of identifying the coefficients with the values they would take if a $[0,1]$ Pad\'e approximant were exact. We show that, when the order in the high-temperature expansion increases, the departure from this approximation grows more slowly than for nearest neighbor models. As a consequence, the value of the critical exponent $\gamma $ estimated using Pad\'e approximants converges very slowly and the estimations using 30 coefficients have errors larger than 0.05. A (presumably much) larger number of coefficients is necessary to obtain the critical exponents with a precision comparable to the precision obtained for nearest neighbor models with less coefficients. We also discuss the possibility of constructing models where a $[0,1]$ Pad\'e approximant would be exact.