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A degeneracy bound for homogeneous topological order
by Jeongwan Haah
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Submission summary
| Authors (as registered SciPost users): | Jeongwan Haah |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2009.13551v1 (pdf) |
| Date submitted: | 2020-09-30 06:58 |
| Submitted by: | Haah, Jeongwan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We introduce a notion of homogeneous topological order, that is obeyed by most, if not all, known examples of topological order including fracton phases on quantum spins (qudits). The notion is a condition on the ground state subspace, rather than on the Hamiltonian, and demands that given a collection of ball-like regions, any linear transformation on the ground space be realized by an operator that avoids the ball-like regions. We derive a bound on the ground state degeneracy $\mathcal D$ for systems with homogeneous topological order on an arbitrary closed Riemannian manifold of dimension $d$, which reads \[ \log \mathcal D \le c (L/a)^{d-2}.\] Here, $L$ is the diameter of the system, $a$ is the lattice spacing, and $c$ is a constant that only depends on the isometry class of the manifold and the density of degrees of freedom. This bound is saturated up to constants by known examples.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2020-11-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2009.13551v1, delivered 2020-11-17, doi: 10.21468/SciPost.Report.2200
Strengths
1- Extremely clear
2- Proves a useful result
3- The proof involves a nice idea
Weaknesses
1- Does not solve all the problems of physics and mathematics.
Report
The result proved in this paper -- that under reasonable assumptions, the log of the groundstate degeneracy of gapped topological phases grows as $L^{d-2}$ or slower in $d$ dimensions -- is not a great surprise to experts in the field (I have seen it used as a heuristic to decide if a solvable lattice model exhibits a non-topological degeneracy). But it is nice to understand why it is true, under clear assumptions. I expect that the notion of "homogeneous topological order" will have other uses. I also expect that the proof technique will be useful for other questions.
Requested changes
None.
Report #1 by Anonymous (Referee 1) on 2020-10-29 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2009.13551v1, delivered 2020-10-29, doi: 10.21468/SciPost.Report.2127
Strengths
1) The paper introduces a new notion of homogeneous topological order and clearly demonstrates its generality and usefulness.
2) The main theorem is a highly nontrivial result, and the paper makes a strong case that the required assumptions are relatively minor.
3) The proof of said theorem is concise and elegant.
4) The paper is clearly written and well-motivated.
Weaknesses
1) The paper provides relatively little intuition for what systems' ground state subspaces fail to have homogeneous topologically order.
Report
This work introduces a new, sharply defined notion called homogeneous topological order, and uses it to provide an elegant proof of an important bound on the ground state degeneracy of a wide variety of systems. It has long been believed that there is no fracton topological order in d=2, and this bound is a powerful and very general step towards that result. The bound also strongly constrains possible fracton-like topological orders in d>2. The fact that the homogeneous topological order condition can be used to concisely prove such a bound strongly suggests that it is a broadly useful condition and merits significant further investigation.
This is an excellent paper. It contains elegant and highly nontrivial new physics, is well-motivated, and is clearly written. I have a few relatively minor requested changes (see below), but I strongly recommend publication.
Requested changes
1) As written, it is not intuitively clear when one should expect the ground state subspace of a system to be a nonexamples, so it is unclear whether the given nonexamples are obvious ones or surprising ones. It would be helpful to elaborate on this. This might even involve mentioning some fairly trivial nonexamples, such as when \Pi contains locally distinguishable states, to emphasize the connection to prior notions of topological order.
2) If feasible, an appendix demonstrating how to prove that an example fracton model (say, X-cube) has homogeneous topological order would help the present paper be more self-contained.
3) A suggestion either for an additional comment in the paper (if the answer is straightforward) or followup work (if not): if a homogeneously topologically ordered subspace is used as a topological error-correcting code, to what extent can the technique used to prove the main theorem be used to restrict the form of logical operators?