# Interface contributions to topological entanglement in abelian Chern-Simons theory

## Abstract

Here, 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.

- Authors:

- Univ. of Illinois, Urbana, IL (United States)
- Univ. of Illinois, Urbana, IL (United States); Univ. of California, Santa Barbara, CA (United States)
- Univ. of Pennsylvania, Philadelphia, PA (United States)

- Publication Date:

- Research Org.:
- Univ. of Illinois, Urbana (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1465511

- Grant/Contract Number:
- [FG02-13ER42001; SC0009932]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- [Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 9]; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Chern-Simons Theories; Topological Field Theories; Gauge Symmetry; Topological States of Matter

### Citation Formats

```
Fliss, Jackson R., Wen, Xueda, Parrikar, Onkar, Hsieh, Chang -Tse, Han, Bo, Hughes, Taylor L., and Leigh, Robert G. Interface contributions to topological entanglement in abelian Chern-Simons theory. United States: N. p., 2017.
Web. doi:10.1007/JHEP09(2017)056.
```

```
Fliss, Jackson R., Wen, Xueda, Parrikar, Onkar, Hsieh, Chang -Tse, Han, Bo, Hughes, Taylor L., & Leigh, Robert G. Interface contributions to topological entanglement in abelian Chern-Simons theory. United States. doi:10.1007/JHEP09(2017)056.
```

```
Fliss, Jackson R., Wen, Xueda, Parrikar, Onkar, Hsieh, Chang -Tse, Han, Bo, Hughes, Taylor L., and Leigh, Robert G. Thu .
"Interface contributions to topological entanglement in abelian Chern-Simons theory". United States. doi:10.1007/JHEP09(2017)056. https://www.osti.gov/servlets/purl/1465511.
```

```
@article{osti_1465511,
```

title = {Interface contributions to topological entanglement in abelian Chern-Simons theory},

author = {Fliss, Jackson R. and Wen, Xueda and Parrikar, Onkar and Hsieh, Chang -Tse and Han, Bo and Hughes, Taylor L. and Leigh, Robert G.},

abstractNote = {Here, 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.},

doi = {10.1007/JHEP09(2017)056},

journal = {Journal of High Energy Physics (Online)},

number = [9],

volume = [2017],

place = {United States},

year = {2017},

month = {9}

}

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