A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion

Abstract

We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (σ=±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 2¹⁸ sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form (1-β/β ₀)^g for thewhole temperature range considered. This approximate law implies a simple approximate formula for the coefficients of the high-temperature expansion, and, more importantly, approximate relations among the coefficients themselves. We found that some of these approximate relations hold with errors less then 2%. On the other hand,g differs significantly from the critical exponent γ calculated with the epsilon expansion, even when the fit is restricted to intervals closer toβ ^c. We discuss this discrepancy in the context the renormalization group analysis of the hierarchical model.