Abstract
We study the entanglement entropy between (possibly distinct) topological phases across an interface using an Abelian Chern-Simons description with topological boundary conditions (TBCs) at the interface. From a microscopic point of view, these TBCs correspond to turning on particular gapping interactions between the edge modes across the interface. However, in studying entanglement in the continuum Chern-Simons description, we must confront the problem of non-factorization of the Hilbert space, which is a standard property of gauge theories. We carefully define the entanglement entropy by using an extended Hilbert space construction directly in the continuum theory. We show how a given TBC isolates a corresponding gauge invariant state in the extended Hilbert space, and hence compute the resulting entanglement entropy. We find that the sub-leading correction to the area law remains universal, but depends on the choice of topological boundary conditions. This agrees with the microscopic calculation of [1]. Additionally, we provide a replica path integral calculation for the entropy. In the case when the topological phases across the interface are taken to be identical, our construction gives a novel explanation of the equivalence between the left-right entanglement of (1+1)d Ishibashi states and the spatial entanglement of (2+1)d topological phases.
Highlights
It is known that the set of gapping interactions that can glue the boundary modes for two given topological phases is not unique
Since Topological boundary conditions (TBCs) are playing the same role as gapping interactions for the topological field theory, it is perhaps unsurprising that these algebraic criteria are equivalent to those classifying gapping interactions
In this paper we have addressed the gapped interfaces between topological phases from a bulk Chern-Simons approach
Summary
We discuss classical interfaces in Abelian Chern-Simons theory with the gauge group U(1)N. We illustrate the fact that K can be glued to itself in the trivial manner (e.g., the gauge field being continuous across the interface, aL = aR, corresponds to the solution v(L) = −v(R) = 1N×N ), but in multiple ways Even in this homogeneous case, the choice of boundary conditions is far from unique. The action in terms of these pure gauge modes is a total derivative and so the path integral is over the U(1)2N -valued fields living on R (see, for instance, [23]) This is the standard reduction of Abelian Chern-Simons theory to U(1) Wess-Zumino-Witten (WZW) on R. I as introduced in [1]
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