# Universality of quantum information in chaotic CFTs

## Abstract

We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the *ETH density matrix*, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix.

- Authors:

- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
- Univ. of Kentucky, Lexington, KY (United States); Skolkovo Inst. of Science and Technology, Moscow, (Russia)

- Publication Date:

- Research Org.:
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1498701

- Grant/Contract Number:
- SC0012567

- 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: 2018; Journal Issue: 3; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal Field Theory; Integrable Hierarchies; Thermal Field Theory

### Citation Formats

```
Lashkari, Nima, Dymarsky, Anatoly, and Liu, Hong. Universality of quantum information in chaotic CFTs. United States: N. p., 2018.
Web. doi:10.1007/jhep03(2018)070.
```

```
Lashkari, Nima, Dymarsky, Anatoly, & Liu, Hong. Universality of quantum information in chaotic CFTs. United States. doi:10.1007/jhep03(2018)070.
```

```
Lashkari, Nima, Dymarsky, Anatoly, and Liu, Hong. Mon .
"Universality of quantum information in chaotic CFTs". United States. doi:10.1007/jhep03(2018)070. https://www.osti.gov/servlets/purl/1498701.
```

```
@article{osti_1498701,
```

title = {Universality of quantum information in chaotic CFTs},

author = {Lashkari, Nima and Dymarsky, Anatoly and Liu, Hong},

abstractNote = {We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix.},

doi = {10.1007/jhep03(2018)070},

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

number = 3,

volume = 2018,

place = {United States},

year = {2018},

month = {3}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Virasoro conformal blocks and thermality from classical background fields

journal, November 2015

- Fitzpatrick, A. Liam; Kaplan, Jared; Walters, Matthew T.
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
Thermalization and its mechanism for generic isolated quantum systems

journal, April 2008

- Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim
- Nature, Vol. 452, Issue 7189

##
A general transformation formula for conformal fields

journal, April 1994

- Gaberdiel, Matthias
- Physics Letters B, Vol. 325, Issue 3-4

##
Eikonalization of conformal blocks

journal, September 2015

- Fitzpatrick, A. Liam; Kaplan, Jared; Walters, Matthew T.
- Journal of High Energy Physics, Vol. 2015, Issue 9

##
Implications of Conformal Invariance in Field Theories for General Dimensions

journal, May 1994

- Osborn, H.; Petkou, A.
- Annals of Physics, Vol. 231, Issue 2

##
Eigenstate thermalization hypothesis in conformal field theory

journal, March 2018

- Lashkari, Nima; Dymarsky, Anatoly; Liu, Hong
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, Issue 3

##
Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz

journal, April 1996

- Bazhanov, Vladimir V.; Lukyanov, Sergei L.; Zamolodchikov, Alexander B.
- Communications in Mathematical Physics, Vol. 177, Issue 2

##
Thermality of eigenstates in conformal field theories

journal, August 2017

- Basu, Pallab; Das, Diptarka; Datta, Shouvik
- Physical Review E, Vol. 96, Issue 2

##
Recursive representation of the torus 1-point conformal block

journal, January 2010

- Hadasz, Leszek; Jaskólski, Zbigniew; Suchanek, Paulina
- Journal of High Energy Physics, Vol. 2010, Issue 1

##
Relative entropy and holography

journal, August 2013

- Blanco, David D.; Casini, Horacio; Hung, Ling-Yan
- Journal of High Energy Physics, Vol. 2013, Issue 8

##
Thermal fluctuations in quantized chaotic systems

journal, February 1996

- Srednicki, Mark
- Journal of Physics A: Mathematical and General, Vol. 29, Issue 4

##
Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

journal, December 2017

- He, Song; Lin, Feng-Li; Zhang, Jia-ju
- Journal of High Energy Physics, Vol. 2017, Issue 12

##
Remarks on thermalization in 2D CFT

journal, December 2016

- de Boer, Jan; Engelhardt, Dalit
- Physical Review D, Vol. 94, Issue 12

##
Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT

journal, August 2017

- He, Song; Lin, Feng-Li; Zhang, Jia-ju
- Journal of High Energy Physics, Vol. 2017, Issue 8

##
On short interval expansion of Rényi entropy

journal, November 2013

- Chen, Bin; Zhang, Jia-ju
- Journal of High Energy Physics, Vol. 2013, Issue 11

##
Bulk emergence and the RG flow of entanglement entropy

journal, May 2015

- Faulkner, Thomas
- Journal of High Energy Physics, Vol. 2015, Issue 5

Works referencing / citing this record:

##
Rényi entropy at large energy density in 2D CFT

journal, August 2019

- Guo, Wu-zhong; Lin, Feng-Li; Zhang, Jiaju
- Journal of High Energy Physics, Vol. 2019, Issue 8