Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory

TL;DR

The paper develops a bulk Chern-Simons framework for interfaces between Abelian topological phases, showing that topological boundary conditions select a gauge-invariant Ishibashi-like state in an extended Hilbert space and that the interface contributes a universal subleading term to entanglement entropy that depends on the boundary condition. It identifies an effective K-matrix, K_eff, governing the interfacial degrees of freedom and demonstrates, via extended Hilbert space methods and replica-trick calculations, that the topological entanglement entropy across the interface is S_EE = −(1/2) log|det K_eff|. The results reproduce and generalize previous microscopic findings (Cano:2014pya) and provide a bulk, continuum explanation for the left-right Ishibashi/spatial entanglement correspondence when the phases are identical. The work also offers two complementary derivations (gauge-invariant gluing in CS and replica/WZW approaches) and sets the stage for extensions to more complex interfaces and non-Abelian topological orders.

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 \cite{Cano:2014pya}. 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.

Figures (7)

• Figure 1: Two topological phases separated a codimension one defect $\Sigma$. The time dimension has been suppressed here.
• Figure 2: (Left) Generically an interface will support global $U(1)^N$ charges (depicted here as Wilson lines ending on $\Sigma$. (Right) The TBCs provide an identification of the gauge group across $\Sigma$ and therefore describe Wilson lines that can permeate the interface.
• Figure 3: (a) The hypersurface $\mathcal{R}$ intersects the interface $\Sigma$ transversely. In the "coupled wire construction," $\mathcal{R}$ is foliated by one dimensional wires each hosting a bosonic theory. In the continuum, $\mathcal{R}$ supports a connection that breaks up into components normal and tangent to $\Sigma$. (b) $\mathcal{R}$ can possess noncontractible cycles and correspondingly the bosonic theory will contain winding modes.
• Figure 4: (A) The spatial 2-sphere partitioned into two discs $D$ and $\bar{D}$. (B) The 2-sphere with an anyon $q$ in $D$ and $-q$ in $\bar{D}$.
• Figure 5: (Left) A cartoon of the reduced density matrix after tracing out the $K^{(R)}$ phase. Regions with level matrix $K^{(L)}$ are denoted by pink shading in this and all other figures, and regions with level matrix $K^{(R)}$ are denoted by blue shading. The introduction of the regulator results in a "keyhole" region in the reduced density matrix. (Middle) $\mathrm{Tr}(\hat{\rho}^n)$ is obtained by gluing $n$ copies of the first figure together cyclically. The figure represents this pictorially for $n=2$. In these illustrations, a transverse dimension, which can be interpreted as Euclidean time, has been suppressed. (Right) This construction is conformally equivalent to a path integral on $S^1\times \mathbb H_2$, which can also be viewed as a solid torus. Top and bottom of this subfigure are identified.