Renormalization group maps for Ising models in lattice gas variables

TL;DR

This paper develops a high-accuracy method to compute the coefficients of renormalized Hamiltonians for Ising models in lattice gas variables, revealing exponential decay with a slow rate and sensitivity to truncation methods.

Contribution

It introduces a novel method for accurately calculating renormalized Hamiltonian coefficients in lattice gas variables for Ising models, enhancing understanding of their decay properties.

Findings

Coefficients decay exponentially but slowly.

Coefficients in spin variables are sensitive to truncation.

Over 1,000 coefficients computed for the critical Ising model.

Abstract

Real space renormalization group maps, e.g., the majority rule transformation, map Ising type models to Ising type models on a coarser lattice. We show that each coefficient of the renormalized Hamiltonian in the lattice gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.