Entanglement-enabled symmetry-breaking orders

- Clear and simple
- Interesting idea

- Not application found
- Missing a classification

The paper is interesting but I think SciPost core fits more for the publication.

The definition of EESBO should be highlight. Maybe talk about the lack of a basis that enable to write the ground states in a product form. Maybe also put an example of not EESBO and long range like the GHZ state from the 2-fold degenerate ground state of the Ising model.

I have some questions about the mechanism of EESBO. The paper considers SPT phases that are not conventional but I think this is not required. Let us consider an ordinary G0= Z2x(ZnxZn) and go to G=ZnxZn; breaking the Z2 symmetry that could be internal and not a lattice symmetry. With the regular SPT classification the 2 ground states could be in n different SPT phases, n-1 of them non-trivial and therefore not able to be written as a product state because of the entanglement given by the projective representation. Why are not those EESBO? If they are, why are they not considered?

I guess that the family of states can be generalized to break T to T^m and have m-degenerate ground states.

Most readers won't be familiar with these unconventional SPT phases. I am familiar with the standard understanding of projective representations at the bonds giving rise to standard SPT phases (also with MPSs) and the RGFP cartoon picture of entangled pairs between sites. Is there an easy characterization or cartoon picture of these unconventional SPT phases? I think it would help a lot if the paper describes them.

See report. Clarify definition and SPT cases

Interesting idea about classifying symmetry breaking.

Not written very clearly.

The manuscript "Entanglement-Enabled Symmetry-Breaking Order" seems
interesting to me. As I understand it, the strategy is to first
specify a symmetry of a full hilbert space (or define a symmetry),
then consider how this symmetry might break. If the broken symmetry
can never be a local product state, then it is declared to be a EESBO.

What is not clear to me is whether this manner of classifying broken
symmetries is going to turn out to be useful or not.

There appears to be no requirement that the state in question is
gapped, or even that it is the ground state of any Hamiltonian. This
work is simply making statements about wavefunctions with some
symmetry that live within a Hilbert space with some symmetry. If this
is so, is this a feature or a bug? Does it suggest that the
classification may end up classifying many things that are not phases
of matter in any sense?

Overall I nonetheless think the paper is interesting and should be
published. I'm not sure there will be overwhelming interest, so I
suggest SciPost core rather than the flagship SciPost.

To give some suggestions, I do have to say that the paper was
extremely hard for me to read. While there is a small community who
is very familiar with these types of arguments, I fear that most even
well-educated condensed matter theorists will find much of the
arguments to be very obtuse.

The examples seem needlessly difficult to think about. For example,
section 5.2 seems the simplest example (at least to me) because one
can take a limit fo a=b=c=1 and then the wavefunction is super-easy to
describe and you can just look at it and see what is going on. Trying
to do it in generality makes it completely impossible to understand.
Similarly, the entire example in section 4 is insanely arcane.

Even in section 5, the authors insist on doing a 3d example,
presumably to evade Mermin-Wagner. But there is no point in this.
Since we didn't ever specify a Hamiltonian, why not just assume it is
a long-ranged Hamiltonian, so that Mermin-Wagner does not apply.
(Indeed, the Hamiltonian doesn't matter anyway!) Then you can just
talk about spin chains (Am I wrong about this?).

I would recommend that before publication, the authors try very hard
to simplify much the discussion, clarify the writing, and put all the
simple examples up front. Yes, I know this is hard to do, but it
really will pay off in the end. If you write a paper that only a tiny
fraction of the community can bother to understand, then it won't have
much impact.

Also note, the paragraph where the EESBO is defined in section 2 gives the
definition almost as a side comment. One can read the paragraph and
not even realize that there has just been a definition made. Please
state it clearly and precisely, not so casually, so people know what
you are talking about! And flag it clearly

"DEFINITION: GIven X,Y, Z, we say that a wavefunction is EESBO if P,
Q, R"

See report. Please simplify examples and write more clearly for a bigger audience.