A degeneracy bound for homogeneous topological order

TL;DR

This paper introduces a new concept of homogeneous topological order, providing a universal bound on ground state degeneracy for such systems on arbitrary manifolds, applicable to various known topological phases including fractons.

Contribution

It defines homogeneous topological order based on ground state subspace properties and derives a universal degeneracy bound applicable to any closed Riemannian manifold.

Findings

Derived a bound:   c  (L/a)^{d-2} for ground state degeneracy.

Bound is saturated by known topological phases.

Applicable to fracton phases and quantum spin systems.

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.