Tensor Renormalization Group Approach to Two-Dimensional Classical Lattice Models

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 the density matrix renormalization group method. We demonstrate the method—which we call the tensor renormalization group method—by computing the magnetization of the triangular lattice Ising model.

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 $\Psi (\{{\sigma }_{a_n}\},\{{\sigma }_{b_n}\},\{{\sigma }_{c_n}\})$ of the boundary spins.

Figure 5

The tensor renormalization group transformation can be viewed as a two-step change in the triangulation of $R$.

Figure 6

The magnetization of the triangular lattice Ising model as a function of $\alpha =e^{-2\beta J}$, obtained using the TRG method.