Kramers-Wannier Approximation for 3D Ising Model

TL;DR

This paper applies the Kramers-Wannier approximation to the 3D Ising model, using a variational approach with an effective 2D model and CTMRG to estimate the critical point, achieving improved accuracy over previous methods.

Contribution

It introduces a variational method with an effective 2D Ising model and CTMRG to approximate the 3D Ising model's critical point, showing enhanced accuracy.

Findings

Estimated critical point $K_c=0.2184$ is within 1.5% of the true value.

The phase transition appears mean-field like.

Method outperforms previous CTTRG approach.

Abstract

We investigate the Kramers-Wannier approximation for the three-dimensional (3D) Ising model. The variational state is represented by an effective 2D Ising model, which contains two variational parameters. We numerically calculate the variational partition function using the corner transfer matrix renormalization group (CTMRG) method, and find its maximum with respect to the variational parameters. The calculated transition point $K_{\rm c} = 0.2184$ is only 1.5% less than the true $K_{\rm c}$; the result is better than that obtained by the corner transfer tensor renormalization group (CTTRG) approach. The calculated phase transition is mean-field like.