## SciPost Submission Page

# Out of time ordered effective dynamics of a quartic oscillator

### by Bidisha Chakrabarty, Soumyadeep Chaudhuri

#### This is not the current version.

### Submission summary

As Contributors: | Soumyadeep Chaudhuri |

Arxiv Link: | https://arxiv.org/abs/1905.08307v3 |

Date submitted: | 2019-06-14 |

Submitted by: | Chaudhuri, Soumyadeep |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | High-Energy Physics - Theory |

### Abstract

We study the dynamics of a quantum Brownian particle weakly coupled to a thermal bath. Working in the Schwinger-Keldysh formalism, we develop an effective action of the particle up to quartic terms. We demonstrate that this quartic effective theory is dual to a stochastic dynamics governed by a non-linear Langevin equation. The Schwinger-Keldysh effective theory, or the equivalent non-linear Langevin dynamics, is insufficient to determine the out of time order correlators (OTOCs) of the particle. To overcome this limitation, we construct an extended effective action in a generalised Schwinger-Keldysh framework. We determine the additional quartic couplings in this OTO effective action and show their dependence on the bath's 4-point OTOCs. We analyse the constraints imposed on the OTO effective theory by microscopic reversibility and thermality of the bath. We show that these constraints lead to a generalised fluctuation-dissipation relation between the non-Gaussianity in the distribution of the thermal noise experienced by the particle and the thermal jitter in its damping coefficient. The quartic effective theory developed in this work provides extension of several results previously obtained for the cubic OTO dynamics of a Brownian particle.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-7-3 Invited Report

### Strengths

1 - This paper is very well written.

2 - It provides a useful introduction and overview of recent progress in formulating consistent effective actions for a Brownian particle which incorporate out of time ordered correlators and is developed up to a quartic order in the field expansion.

3 - It derives genuinely new generalized Onsager and fluctuation-dissipation relations among certain coefficients of the quartic Schwinger-Keldysh couplings.

4 - The authors also illustrate the findings with an instructive explicit example.

### Weaknesses

1. The new generalized Onsager and fluctuation-dissipation relations are among quartic couplings, thus, at this stage, it is not clear how widespread of a role they might play in applications and how one can (experimentally) verify them.

### Report

The paper addresses a difficult question of classifying all possible couplings arising at the quartic order in the field expansion of the Schwinger-Keldysh effective action for a Brownian particle and its out of time ordered generalization.

By carefully analyzing the constraints imposed on this effective action by the time reversibility and thermality of the bath to which the quantum Brownian particle is coupled, the authors are able to identify novel Onsager and fluctuation-dissipation relations on certain quartic couplings.

This result is sufficiently interesting for the paper to be published. Moreover, given that the paper is written very clearly, is well structured and nicely formatted, I can recommend for the paper to be published without any changes apart from some small typos and one question enumerated below.

### Requested changes

Questions:

1 - Have you not included the OTO couplings of the generalized Schwinger-Keldysh effective action, would you have found the generalized fluctuation-dissipation relations among the quartic Schwinger-Keldysh couplings?

Typos:

1 - Introduction: last sentence of the first paragraph:

"... for only the degrees of freedom of the system."

Perhaps: "... for only the INFRARED (LOW-ENERGY) degrees of freedom of the system." ?

2 - Below (19):

"Plugging these expression..."

perhaps "Plugging these expressions..." ?

3 - Caption of Figure 5: "t?0" perhaps $t>0$?