# Weaker Formulations of Bulk-boundary Correspondence for Interacting Systems

From this post, it seems that bulk-boundary correspondence does not hold in general for interacting systems. What is meant by bulk-boundary correspondence there appears to be the existence of robust (against local perturbations respecting the symmetry) gapless excitations for a lattice with boundary given the nontrivial SPT order, or perhaps topological order, for a lattice without boundary. The toric code model serves as a counterexample, as Prof. Fu mentioned.

However, there is not a clear-cut definition of bulk-boundary correspondence, and I have been mulling over possible alternative formulations of "correspondence" which have been proved rigorously. As a little motivation, people sometimes say a model is "topological" when there is a "nontrivial" boundary. For example, the AKLT state on a periodic chain belongs to a nontrivial SPT class with respect to the on-site SO(3) symmetry, and it has free spin-$\frac12$'s at the ends of an open chain. Also, in Prof. Wen's paper arXiv:1106.4752v2, it appears that the nontriviality of the bulk SPT order was proved by appealing to the nontriviality of the boundary. This makes me wonder whether any sort of converse to the aforementioned correspondence holds true in general, i.e. whether a "nontrivial" boundary implies a nontrivial bulk SPT or topological order.

More generally, I would like to know if the answer to any of the questions below is known, with modifications if necessary.

For definiteness, let us restrict to spin systems and an on-site symmetry representation. A closed (open) lattice will mean a finite lattice without (respectively with) boundary, and all open lattices are thought of as sub-regions of closed lattices. Given a local Hamiltonian $H_{\rm closed}$ defined on a closed lattice, we can obtain a local Hamiltonian $H_{\rm open}$ defined on an open lattice by throwing away all terms that do not lie entirely within the open lattice. We then allow for new terms that respect the symmetry and are localized near the boundary to be added to $H_{\rm open}$. Any statement below about open lattices should be interpreted as that there exists a modified $H_{\rm open}$ such that the statement holds true. Here is my list of questions:

(1) If $H_{\rm closed}$ has a short-range entangled ground state, does it mean $H_{\rm open}$ has a short-range entangled ground state?

(2) If $H_{\rm open}$ has a short-range entangled ground state, does it mean $H_{\rm closed}$ has a short-range entangled ground state?

(3) If $H_{\rm closed}$ has a gapped short-range entangled ground state, does it mean $H_{\rm open}$ has a gapped short-range entangled ground state?

(4) If $H_{\rm open}$ has a gapped short-range entangled ground state, does it mean $H_{\rm closed}$ has a gapped short-range entangled ground state?

(5) If $H_{\rm closed}$ has the trivial SPT order (i.e. gapped, symmetric, SRE, equivalent to tensor product), does it mean $H_{\rm open}$ has the trivial SPT order?

(6) If $H_{\rm open}$ has the trivial SPT order, does it mean $H_{\rm closed}$ has a trivial SPT order?

(7) If $H_{\rm closed}$ has the trivial SPT order, does it mean $H_{\rm open}$ has a gapped symmetric ground state?

(8) If $H_{\rm closed}$ has the trivial SPT order, does it mean $H_{\rm open}$ has a gapped ground state?

I would be equally interested if you knew the answers to any similar questions, and would appreciate any reference regarding these matters. Also, my sense is that the field theoretic versions of some questions might be better understood, and I would appreciate if you could point me to them. Thanks in advance!

A very late answer, but for symmetry-protected topological (SPT) phases, I believe it is true (certainly, no counterexamples are known) that the boundary is "non-trivial" if and only if the bulk is a non-trivial SPT phase. Here "non-trivial" boundary has a very specific meaning. A boundary is "non-trivial" if there is NO symmetry-respecting terms that we can add to $H_{\mathrm{open}}$ that will lead to the boundary being both short-range entangled and symmetry-respecting. (Thus, we see that "non-trivial" boundary is really a property of $H_{\mathrm{closed}}$, as we do not consider a particular form of $H_{\mathrm{open}}$.) This still admits various possibilities: the boundary could be topologically ordered, or gapless, or spontaneously break the symmetry. All these possibilities have been observed in particular models.
Thus, your list of questions is, I think, a bit too restrictive. The answers would depend on the specific choice of $H_{\mathrm{open}}$. We should be more interested in the "universal" properties of the boundary which are inherited from the bulk.