I'm totally new to tensor networks, and I'm currently learning on my own from papers, tutorials, and videos.
Right now, I'm trying to understand how to construct a Matrix Product Operator (MPO) for a very simple spin-chain Hamiltonian.
The Hamiltonian I'm working with is:
$$ H = J \sum_{i=1}^{L-1} \sigma^z_i \sigma^z_{i+1} $$
What I'm trying to understand:
- How to build the MPO tensors $ W^{[i]} $ for this Hamiltonian
- What the structure of each local MPO tensor is
- What bond dimension is needed
- How to define the boundary vectors
- why the structure works (not just the final formula)
I've seen the following MPO structure suggested:
Each local MPO tensor is a $ 3 \times 3 $ matrix whose entries are $2 \times 2 $ operators:
$$ W^{[i]} = \begin{bmatrix} \mathbb{I} & 0 & 0 \\ \sigma^z & 0 & 0 \\ 0 & J\sigma^z & \mathbb{I} \end{bmatrix} $$
What I would like help with:
- Could someone explain or derive this structure?
- Why does this MPO encode the full Hamiltonian correctly?
- How does this representation “build up” each term $ \sigma^z_i \sigma^z_{i+1} $ in the sum?
- What does the MPO actually look like for ( L = 4 ) sites?
- Any references or visual explanations would be appreciated!
I'm trying to build intuition from the ground up, so I really appreciate any help. Thanks in advance!