Topological Entanglement Entropy in \(d\)-dimensions for Abelian Higher Gauge Theories

Abstract

We compute the topological entanglement entropy for a large set of lattice models in $d$-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of $d$-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy $S_A$ in a sub-region $A$ is proportional to $\log (GS{D}_{\tilde{A}})$, where \(GSD_{\tilde{A}}\) is the ground state degeneracy of a particular restriction of the full model to \(A\). The quantity $GS{D}_{\tilde{A}}$ can be further divided into a contribution that scales with the size of the boundary $\partial A$ and a term which depends on the topology of $\partial A$. There is also a topological contribution coming from $A$ itself, that may be non-zero when $A$ has a non-trivial homology. We present some examples and discuss how the topology of $A$ affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension $d$. The result is in agreement with entanglement calculations for known topological models.

44 pages, 6 figures

References in corpus (7)

• Criticality, the area law, and the computational power of PEPS
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Ground state entanglement and geometric entropy in the Kitaev's model
• Entanglement renormalization and topological order
• Entanglement and topological entropy of the toric code at finite temperature
• Topological order in a 3D toric code at finite temperature
• On 2-form gauge models of topological phases

Cited by in corpus (7)

• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Fracton-like phases from subsystem symmetries
• Non-Gaussianity of Entanglement Entropy and Correlations of Composite Operators
• Novel quantum phases on graphs using abelian gauge theory
• A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints
• Entanglement Spectrum Resolved by Loop Symmetries