Restoring isotropy in a three-dimensional lattice model: The Ising universality class

TL;DR

This study investigates a generalized 3D lattice model to understand and restore isotropy, accurately determining critical exponents and fixed point values, thereby advancing the understanding of the 3D Ising universality class.

Contribution

It introduces a method to eliminate spatial anisotropy in a 3D lattice model and precisely determines critical exponents and fixed point quantities for the Ising universality class.

Findings

Spatial anisotropy depends mainly on the ratio of coupling constants.

Critical exponents ν and η are accurately determined and agree with conformal bootstrap results.

Fixed point values of dimensionless quantities like the Binder cumulant are provided.

Abstract

We study a generalized Blume-Capel model on the simple cubic lattice. In addition to the nearest neighbor coupling there is a next to next to nearest neighbor coupling. In order to quantify spatial anisotropy, we determine the correlation length in the high temperature phase of the model for three different directions. It turns out that the spatial anisotropy depends very little on the dilution parameter $D$ of the model and is essentially determined by the ratio of the nearest neighbor and the next to next to nearest neighbor coupling. This ratio is tuned such that the leading contribution to the spatial anisotropy is eliminated. Next we perform a finite size scaling (FSS) study to tune $D$ such that also the leading correction to scaling is eliminated. Based on this FSS study, we determine the critical exponents $\nu=0.62998(5)$ and $\eta=0.036284(40)$, which are in nice agreement with the more accurate results obtained by using the conformal bootstrap method. Furthermore we provide accurate results for fixed point values of dimensionless quantities such as the Binder cumulant and for the critical couplings. These results provide the groundwork for broader studies of universal properties of the three-dimensional Ising universality class.