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 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.

45 pages, 15 figures; Changes: Estimate of $\eta$ corrected to 0.036284(40) [Before 0.036244(40) due to trivial but annoying mistake]. Text rewritten at several places. New appendix B, several Figures changed. Core results unchanged

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