Universality in the partially anisotropic three-dimensional Ising lattice

TL;DR

This study confirms the universality of critical exponents in the anisotropic 3D Ising model and shows that certain finite-size scaling amplitude ratios are independent of anisotropy, providing a new quantitative estimate for one of these ratios.

Contribution

It demonstrates the universality of critical exponents and amplitude ratios in the anisotropic 3D Ising model using transfer-matrix and renormalization-group methods, with a novel quantitative estimate for a key ratio.

Findings

Critical exponents $y_t$ and $y_h$ are universal.

Finite-size scaling amplitude ratios are independent of anisotropy.

First quantitative estimate of $Y_2$ as approximately 2.013.

Abstract

Using transfer-matrix extended phenomenological renormalization-group methods the critical properties of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths ${\vec J}=(J',J',J)$ are studied. Universality of both fundamental critical exponents $y_t$ and $y_h$ is confirmed. It is shown that the critical finite-size scaling amplitude ratios $U=A_{\chi^{(4)}}A_\kappa/A_\chi^2$, $Y_1=A_{\kappa^{''}}/A_\chi$, and $Y_2=A_{\kappa^{(4)}}/A_{\chi^{(4)}}$ are independent of the lattice anisotropy parameter $\Delta=J'/J$. By this for the last above invariant of the three-dimensional Ising universality class we give the first quantitative estimate: $Y_2\simeq2.013$ (shape $L\times L\times\infty$, periodic boundary conditions in both transverse directions).