# Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole

## Abstract

It has been recently proposed by Maldacena and Qi that an eternal traversable wormhole in a two-dimensional anti–de Sitter space is the gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models coupled by a relevant interaction (which we will refer to as spin operator). We study spectral and eigenstate properties of this coupled SYK model. We find that level statistics in the tail of the spectrum, and for a sufficiently weak coupling, show substantial deviations from random matrix theory, which suggests that traversable wormholes are not quantum chaotic. By contrast, for sufficiently strong coupling, corresponding to the black hole phase, level statistics are well described by random matrix theory. This transition in level statistics coincides approximately with a previously reported Hawking-Page transition for weak coupling. We show explicitly that this thermodynamic transition turns into a sharp crossover as the coupling increases. Likewise, this critical coupling also corresponds to the one at which the overlap between the ground state and the thermofield double state (TFD) is smallest. In the range of sizes we can reach by exact diagonalization, the ground state is well approximated by the TFD only in the strong coupling limit. This is due to themore »

- Authors:

- Publication Date:

- Research Org.:
- State Univ. of New York (SUNY), Albany, NY (United States)

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

- OSTI Identifier:
- 1532437

- Alternate Identifier(s):
- OSTI ID: 1610129

- Grant/Contract Number:
- FG02-88ER40388; PHY-1607611

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 2; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; astronomy & astrophysics; physics; gauge-gravity dualities; quantum aspects of black holes; quantum chaos; quantum gravity; random matrix theory

### Citation Formats

```
García-García, Antonio M., Nosaka, Tomoki, Rosa, Dario, and Verbaarschot, Jacobus J. M. Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.026002.
```

```
García-García, Antonio M., Nosaka, Tomoki, Rosa, Dario, & Verbaarschot, Jacobus J. M. Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole. United States. doi:10.1103/PhysRevD.100.026002.
```

```
García-García, Antonio M., Nosaka, Tomoki, Rosa, Dario, and Verbaarschot, Jacobus J. M. Mon .
"Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole". United States. doi:10.1103/PhysRevD.100.026002.
```

```
@article{osti_1532437,
```

title = {Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole},

author = {García-García, Antonio M. and Nosaka, Tomoki and Rosa, Dario and Verbaarschot, Jacobus J. M.},

abstractNote = {It has been recently proposed by Maldacena and Qi that an eternal traversable wormhole in a two-dimensional anti–de Sitter space is the gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models coupled by a relevant interaction (which we will refer to as spin operator). We study spectral and eigenstate properties of this coupled SYK model. We find that level statistics in the tail of the spectrum, and for a sufficiently weak coupling, show substantial deviations from random matrix theory, which suggests that traversable wormholes are not quantum chaotic. By contrast, for sufficiently strong coupling, corresponding to the black hole phase, level statistics are well described by random matrix theory. This transition in level statistics coincides approximately with a previously reported Hawking-Page transition for weak coupling. We show explicitly that this thermodynamic transition turns into a sharp crossover as the coupling increases. Likewise, this critical coupling also corresponds to the one at which the overlap between the ground state and the thermofield double state (TFD) is smallest. In the range of sizes we can reach by exact diagonalization, the ground state is well approximated by the TFD only in the strong coupling limit. This is due to the fact that the ground state is close to the eigenstate of the spin operator corresponding to the lowest eigenvalue which is an exact TFD at infinite temperature. In this region, the spectral density is separated into blobs centered around the eigenvalues of the spin operator. For weaker couplings, the exponential decay of coefficients in a tensor product basis, typical of the TFD, becomes power law. Finally, we also find that the total Hamiltonian has an additional discrete symmetry which has not been reported previously.},

doi = {10.1103/PhysRevD.100.026002},

journal = {Physical Review D},

number = 2,

volume = 100,

place = {United States},

year = {2019},

month = {7}

}

DOI: 10.1103/PhysRevD.100.026002

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