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),

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

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. Commented 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. Commented Jan 13, 2022 at 12:14

1 Answer

It can certainly be done in ITensor, but I've had the most luck easily constructing PEPS Hamiltonians in QUIMB. For example, here is the 2D ising model if you don't want to construct it yourself.

At the very least you could use the easy functions in QUIMB to see what the tensors look like, then translate that PEPO back to ITensor. It's certainly harder to intuitively come up with the right PEPO off the bat, as it's harder to visualize slices of six-dimensional objects.

• I once looked up QUINMB and it will give me a hint for my question by referring to the source code. However, I am also considering the generalized operators such as many body interaction like a plaquette operator which acts on the four points on the plane. Commented Jan 13, 2022 at 7:24