Abstract
The topological entanglement entropy is used to measure long-range quantum correlations in the ground space of topological phases. Here we obtain closed form expressions for the topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized {mathcal {S}}-matrices. We conjecture a general property of these {mathcal {S}}-matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.
Highlights
The classification of topological phases is fundamental to the study of modern condensed matter physics [1,2,3,4]
These models can be defined in terms of an input unitary fusion category, and their ground states by superpositions of string diagrams labeled by objects from the category
In (3+1)-dimensions, the input category must be equipped with a premodular braiding, leading to a Walker-Wang model [10]
Summary
The classification of topological phases is fundamental to the study of modern condensed matter physics [1,2,3,4]. Particular classes of WalkerWang models [10] have been shown to behave differently using the same diagnostics Modular examples of these models demonstrate zero bulk topological entanglement entropy [27,28], even though, at their boundary, they realize quasiparticle excitations with non-trivial braid statistics [27]. We obtain our results by evaluating the entanglement entropy of various regions of ground states of Levin-Wen and Walker-Wang models. This requires careful analysis of various string diagrams, such. In the bulk of (3+1)-dimensional models, we conjecture, and prove in many cases, that the entropy is the logarithm of the total quantum dimension of the particle content of the theory, extending the results of Ref. To the best of our knowledge, there are no results concerning boundary entropies of Walker-Wang models beyond the (3+1)-dimensional toric code [29]
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