# Tensor product operator of 2D ( for Tensor Network)

I have difficulty representing Tensor Product Operators (TPO) of 2D in a concrete form. For example in 1D case, according to the tutorial in ITensor , the TPO of the Hamiltonian for the simple Ising model in 1D is given by the following 3 and 4-legged tensors (consisting of 2 physical link and 1 or 2 virtual link),

$$\begin{equation} \mathcal{H}=J\sum_{i=1}^{N}S^{z}_{i}\cdot S^{z}_{i+1}=H_{a_1}^{\sigma^\prime_1\sigma_1} \left(\prod_{i=2}^{N-1}H_{a_ia_{i+1}}^{\sigma^\prime_i\sigma_i}\right) H_{a_N}^{\sigma^\prime_N\sigma_N} \end{equation}$$

where $$H_{a_1}^{\sigma^\prime_1\sigma_1}= \begin{bmatrix} 0 & JS_1^z &I\\ \end{bmatrix},\ H_{a_ia_{i+1}}^{\sigma^\prime_i\sigma_i}= \begin{bmatrix} I & 0 &0\\ S_i^z& 0 &0 \\ 0&JS_i^z &I \end{bmatrix},\ H_{a_N}^{\sigma^\prime_N\sigma_N}= \begin{bmatrix} I\\ S_N^z\\ 0 \end{bmatrix}$$

$$I$$ is identity and $$S_i^z$$ is the $$z$$ component of spin matrix at site $$i$$.

I tried to apply it to 2D version $$H=J\sum_{\langle i,j \rangle} S_i^z \cdot S^z_j$$ decomposing into the 6-legged tensor product (4 virtual link and 2 physical link) but I couldn't. Do you have any idea or references about it?

• I found a good reference of this problem ( link ). This formulation is based on the idea of automata and it can be applied to higher dimensions. Jan 13, 2022 at 10:48
• Wow! I love that cellular automata approach. I had skimmed that paper but I didn't check the appendix. Jan 13, 2022 at 12:14