# A degeneracy bound for homogeneous topological order

### Submission summary

 As Contributors: Jeongwan Haah Arxiv Link: https://arxiv.org/abs/2009.13551v3 (pdf) Date accepted: 2021-01-13 Date submitted: 2021-01-09 00:28 Submitted by: Haah, Jeongwan Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

We introduce a notion of homogeneous topological order, which 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 $\log \mathcal D \le c \mu (L/a)^{d-2}.$ 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 $\mu$ is a constant that only depends on the density of degrees of freedom. If $d=2$, the constant $c$ is the (demi)genus of the space manifold. This bound is saturated up to constants by known examples.

Published as SciPost Phys. 10, 011 (2021)

I would like to thank the referees for their generous comments. One of the referees had specific questions, which I answer as follows.

1) The referee asked to include discussions for models with local order parameters. I restructured the section on nonexamples to discuss such models.

2) The referee asked to include proofs for simple specific fracton models that they have homogeneous topological order. I wrote a new subsection to give a full discussion for topologically ordered (in a conventional sense) translation invariant Pauli stabilizer codes.

3) The referee wondered to what extent the technique in the manuscript can constrain shape of logical operators. This is an important question and deserves deeper look.

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?

--- This deserves deeper investigation.

### List of changes

- Strengthened the statement of the theorem for d=2.
- Mentioned hyperbolic surface of constant curvature.
- Gave one more reference for the asymptotic volume formula of $a$-neighborhood of $(d-2)$-skeleton.
- Restructured the section on nonexamples, which now includes $\Pi$ with local observables.
- Wrote a new subsection 3.1 to show that translation invariant exact code Hamiltonians (a certain class of Pauli stabilizer code) have our homogeneous topological order.