A degeneracy bound for homogeneous topological order

• URL: http://arxiv.org/abs/2009.13551v3
• Date: Thu, 7 Jan 2021 05:43:54 GMT
• Title: A degeneracy bound for homogeneous topological order
• Authors: Jeongwan Haah
• Abstract summary: We introduce a notion of homogeneous topological order, which is obeyed by most, if not all, known examples of topological order. We derive a bound on the ground state degeneracy $mathcal D$ for systems with homogeneous topological order.
• Score: 0.30458514384586394
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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