# Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model

## Abstract

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $$E{\approx}0$$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $$E{\approx}0$$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $$E{\approx}0$$, the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $$E{\approx}0$$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $$1/{t}^{2}$$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.

- Authors:

- Publication Date:

- Research Org.:
- Stony Brook Univ., NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1436006

- Alternate Identifier(s):
- OSTI ID: 1505151

- Grant/Contract Number:
- FG02-88ER40388

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 10; 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; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; quantum chaos; random matrix theory; string theory techniques in condensed matter

### Citation Formats

```
García-García, Antonio M., Jia, Yiyang, and Verbaarschot, Jacobus J. M. Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.106003.
```

```
García-García, Antonio M., Jia, Yiyang, & Verbaarschot, Jacobus J. M. Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model. United States. doi:10.1103/PhysRevD.97.106003.
```

```
García-García, Antonio M., Jia, Yiyang, and Verbaarschot, Jacobus J. M. Fri .
"Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model". United States. doi:10.1103/PhysRevD.97.106003.
```

```
@article{osti_1436006,
```

title = {Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model},

author = {García-García, Antonio M. and Jia, Yiyang and Verbaarschot, Jacobus J. M.},

abstractNote = {We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E{\approx}0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E{\approx}0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E{\approx}0$, the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E{\approx}0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/{t}^{2}$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.},

doi = {10.1103/PhysRevD.97.106003},

journal = {Physical Review D},

number = 10,

volume = 97,

place = {United States},

year = {2018},

month = {5}

}

DOI: 10.1103/PhysRevD.97.106003

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