Tensor renormalization group approach to 2D classical lattice models

TL;DR

A simple real space renormalization group technique for two-dimensional classical lattice models that is fundamentally based on the theory of quantum entanglement is described and demonstrated by computing the magnetization of the triangular lattice Ising model.

Abstract

We describe a simple real space renormalization group technique for two dimensional classical lattice models. The approach is similar in spirit to block spin methods, but at the same time it is fundamentally based on the theory of quantum entanglement. In this sense, the technique can be thought of as a classical analogue of DMRG. We demonstrate the method - which we call the tensor renormalization group method - by computing the magnetization of the triangular lattice Ising model.

Figures (6)

• Figure 1: A tensor network model on the honeycomb lattice.
• Figure 2: A TRG transformation on the honeycomb lattice.
• Figure 3: A TRG transformation on the square lattice.
• Figure 4: After (a) dividing $R$ into triangles, (b) the partition function of each triangle can be written as a function $\Psi(\{_{a_n}\},\{_{b_n}\},\{_{c_n}\})$ of the boundary spins.
• Figure 5: The tensor renormalization group transformation can be viewed as a two step change in the triangulation of $R$.