# Shape dependence of entanglement entropy in conformal field theories

## Abstract

Here, we study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R ^{1,d--1}. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We also show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C _{T }appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient σ/CT=π ^{2}/24 in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.

- Authors:

- Univ. of Illinois, Urbana, IL (United States), Dept. of Physics

- Publication Date:

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

- Sponsoring Org.:
- USDOE; National Science Foundation (NSF)

- OSTI Identifier:
- 1395545

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

- Resource Type:
- Journal Article: Accepted Manuscript

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

- Additional Journal Information:
- Journal Volume: 2016; Journal Issue: 4; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT correspondence; field theories in higher dimensions

### Citation Formats

```
Faulkner, Thomas, Leigh, Robert G., and Parrikar, Onkar.
```*Shape dependence of entanglement entropy in conformal field theories*. United States: N. p., 2016.
Web. doi:10.1007/JHEP04(2016)088.

```
Faulkner, Thomas, Leigh, Robert G., & Parrikar, Onkar.
```*Shape dependence of entanglement entropy in conformal field theories*. United States. doi:10.1007/JHEP04(2016)088.

```
Faulkner, Thomas, Leigh, Robert G., and Parrikar, Onkar. Thu .
"Shape dependence of entanglement entropy in conformal field theories". United States. doi:10.1007/JHEP04(2016)088. https://www.osti.gov/servlets/purl/1395545.
```

```
@article{osti_1395545,
```

title = {Shape dependence of entanglement entropy in conformal field theories},

author = {Faulkner, Thomas and Leigh, Robert G. and Parrikar, Onkar},

abstractNote = {Here, we study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R1,d--1. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We also show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient CT appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient σ/CT=π2/24 in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.},

doi = {10.1007/JHEP04(2016)088},

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

issn = {1029-8479},

number = 4,

volume = 2016,

place = {United States},

year = {2016},

month = {4}

}

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