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 SA in a sub-region A is proportional to log left({GSD}_{tilde{A}}right) , where {GSD}_{tilde{A}} is the ground state degeneracy of a particular restriction of the full model to A. The quantity {GSD}_{tilde{A}} can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂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.
Highlights
Ground states of gapped systems, since they often follow an area law [1,2,3]
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
In the present work we look at the topological entropy and how it depends on the topology of the subregion A
Summary
We would like to point out that figure 2 is not a commuting diagram When this happens, f ∈ hom(C, G)0 is called a chain map and, as we will see, the corresponding gauge configuration is gauge equivalent to the trivial one. An important observation is that the sequence of groups hom(C, G)p can be made into a co-chain complex This is achieved by considering maps δp : hom(C, G)p → hom(C, G)p+1, defined by:. The expression above shows only the part of the sequence that is relevant for the present application, please refer to [19] for a more detailed account Associated to this co-chain complex, there are the so-called Brown cohomology groups [34]. · · · ←−−δ−−1− hom(C, G)−1 ←−−δ0− hom(C, G)0 ←−−δ1− hom(C, G)1 ←−−δ2− · · ·
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