Kramers-Wannier Approximation for 3D Ising Model

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.