# Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators

## Abstract

Here, the out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh nonlinear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs. The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the Altshuler-Aronov-Khmelnitskii dephasing rate in interacting metals, but here it involves two replicas of the system (two “worlds”). The intraworld contributions reproduce the Altshuler-Aronov-Khmelnitskii dephasing (that would correspond to a decay of the correlator), while the interworld terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound, is not a reliable indicator of many-body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an importantmore »

- Authors:

- Univ. of Maryland, College Park, MD (United States)

- Publication Date:

- Research Org.:
- Univ. of Maryland, College Park, MD (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1482111

- Alternate Identifier(s):
- OSTI ID: 1482151

- Grant/Contract Number:
- SC0001911; DESC0001911

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 98; Journal Issue: 20; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Liao, Yunxiang, and Galitski, Victor.
```*Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators*. United States: N. p., 2018.
Web. doi:10.1103/PhysRevB.98.205124.

```
Liao, Yunxiang, & Galitski, Victor.
```*Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators*. United States. doi:10.1103/PhysRevB.98.205124.

```
Liao, Yunxiang, and Galitski, Victor. Wed .
"Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators". United States.
doi:10.1103/PhysRevB.98.205124.
```

```
@article{osti_1482111,
```

title = {Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators},

author = {Liao, Yunxiang and Galitski, Victor},

abstractNote = {Here, the out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh nonlinear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs. The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the Altshuler-Aronov-Khmelnitskii dephasing rate in interacting metals, but here it involves two replicas of the system (two “worlds”). The intraworld contributions reproduce the Altshuler-Aronov-Khmelnitskii dephasing (that would correspond to a decay of the correlator), while the interworld terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound, is not a reliable indicator of many-body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an important general question of the physical meaning of the OTOCs often used in calculations and proofs. We briefly discuss possible connections of the OTOCs to observables in quantum interference effects and level statistics via a generalization of the Bohigas-Giannoni-Schmit conjecture to many-body chaotic systems.},

doi = {10.1103/PhysRevB.98.205124},

journal = {Physical Review B},

number = 20,

volume = 98,

place = {United States},

year = {Wed Nov 14 00:00:00 EST 2018},

month = {Wed Nov 14 00:00:00 EST 2018}

}