# What's the tensor network representation for local ground state?

It's well known that many topological phases can be represented using matrix product state and PEPS. Example, toric code, $H=-\sum_{v}A_v-\sum_p B_p$. My question is:

What's the tensor network representation for local ground state?

Here Local ground state means the ground state $H'$ on whole lattice which only contains some terms of $H$ that has nonzero support contained in a sub-region $B$.

The following is a concrete example.

Only the terms in the green boxed are not omitted. My main concern is how to deal with the bond indexes on the boundary.

• Can you be more precise what you mean with "local ground state"? – Norbert Schuch Dec 27 '17 at 16:48
• @NorbertSchuch The ground state of a new Hamiltonian that only contain some terms of original Hamiltonian (on a local patch) – Sirui Lu Jan 8 '18 at 0:52
• Sure, but which terms exactly? Which are omitted? Could you please be more precise? I'm pretty sure there is a succint answer to your question if you ask it in a succinct way, but otherwise, it's just too much work to write it in the utmost generality. – Norbert Schuch Jan 8 '18 at 1:01
• Thanks for your patience! I have did some updates. My main concern is how to deal with the bond indexes on the boundary. – Sirui Lu Jan 8 '18 at 12:48
• Well, the local ground state is not unique. Thus, any choice of boundary conditions (i.e,. of the bond indices at the boundary) gives a valid ground state. See e.g. arxiv.org/abs/1309.4596. – Norbert Schuch Jan 8 '18 at 17:02