A Numerical Study of the Hierarchical Ising Model: High Temperature Versus Epsilon Expansion

TL;DR

This study numerically investigates the magnetic susceptibility of the hierarchical Ising model above the critical temperature, revealing that a simple power law fits the data well across all temperatures and highlighting discrepancies with epsilon-expansion predictions.

Contribution

The paper provides a comprehensive numerical analysis of the hierarchical Ising model's susceptibility, comparing high-temperature and epsilon-expansion methods, and uncovers notable differences.

Findings

Numerical susceptibility fits a power law across all temperatures.

Values of gamma agree with high-temperature expansion within a few percent.

Significant discrepancies observed with epsilon-expansion results.

Abstract

We study numerically the magnetic susceptibility of the hierarchical model with Ising spins ($\sigma =\pm 1$) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly, using recursive methods which exploit the symmetries of the model. Lattices with up to $2^18$ sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form $(1- \beta /\beta _c )^{- \gamma} $for the {\it whole} temperature range. The numerical values for $\gamma $ agree within a few percent with the values calculated with a high-temperature expansion but show significant discrepancies with the epsilon-expansion. We would appreciate comments about these results.