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

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 spatial directions. It turns out that the spatial anisotropy depends very little on the dilution or crystal-field 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 $\ensuremath{\nu}=0.629\phantom{\rule{0.16em}{0ex}}98(5)$ and $\ensuremath{\eta}=0.0362\phantom{\rule{0.16em}{0ex}}84(40)$, 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.

Topics & Concepts