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Bulletin of the American Physical Society

Symmetries in the entanglement spectrum and topological phases protected by spatial discrete symmetries

Po-Yao Chang, Shinsei Ryu

Published Mar 21, 2013• 0 citations
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Symmetry-protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension. Here we formulate this phenomenon in the framework of symmetry-resolved entanglement (SRE) using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically SRE via a discrete path integral on multisheet Riemann surfaces with generalized defects. The resulting path integral is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows us to identify universal entanglement properties. Specifically, we demonstrate an equiblock decomposition of the reduced density matrix into symmetry sectors, for all one-dimensional topological phases protected by finite Abelian unitary symmetries.
Modulated symmetries are internal symmetries which do not commute with spatial symmetries; dipolar symmetries are a prime example. We give a general recipe for constructing topological phases protected by modulated symmetries via a defect network construction, generalizing the crystalline equivalence principle to modulated symmetries. We demonstrate that modulated symmetries can be treated identically to unmodulated symmetries in the absence of spatial symmetries, but in the presence of spatial symmetries, some defect networks which are non-anomalous for unmodulated symmetries become anomalous for modulated symmetries. We apply this understanding to classify symmetry-protected topological phases protected by translation symmetry plus either discrete or continuous dipolar symmetries in (1+1)D and (2+1)D and obtain a number of other (1+1)D classification results for modulated symmetry-protected topological phases.