A degeneracy bound for homogeneous topological order

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 \[ \log \mathcal D \le c μ(L/a)^{d-2}.\] 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.

12 pages, 2 figures (v2) clarified the main theorem for d=2, added some detail for Pauli stabilizer models, (v3) minor corrections

References in corpus (4)

• Local stabilizer codes in three dimensions without string logical operators
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• More Exotic Field Theories in 3+1 Dimensions
• 3-d topological quantum memory with a power-law energy barrier

Cited by in corpus (16)

• Generalized Symmetries in Condensed Matter
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• Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics
• Gapped Lineon and Fracton Models on Graphs
• Bifurcating subsystem symmetric entanglement renormalization in two dimensions
• Fractons on Graphs and Complexity
• Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders
• Layer Codes
• Lift-Connected Surface Codes
• Fractal Subsystem Symmetries, Anomalies, Boundaries, and Effective Field Theory
• The spin-orbital Kitaev model: from kagome spin ice to classical fractons
• Generalized Modulated Symmetries in ${\mathbb{Z}}_2$ Topological Ordered Phases
• Fractons on the edge
• Layer codes as partially self-correcting quantum memories
• Entanglement renormalization of fractonic anisotropic ${\mathbb{Z}}_N$ Laplacian models