# Out of time ordered effective dynamics of a quartic oscillator

### Submission summary

 As Contributors: Soumyadeep Chaudhuri Arxiv Link: https://arxiv.org/abs/1905.08307v5 (pdf) Date accepted: 2019-07-17 Date submitted: 2019-07-11 Submitted by: Chaudhuri, Soumyadeep Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approach: Theoretical

### 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.

### Ontology / Topics

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Published as SciPost Phys. 7, 013 (2019)

Dear Editor,

We thank the referee for his/her comments.

We will begin by addressing the referee's question on whether we could have obtained the generalised fluctuation-dissipation relation between the quartic Schwinger-Keldysh couplings without including the OTO couplings in the analysis. Our response is as follows:

The generalised fluctuation-dissipation relation essentially relies on certain relations between the bath's correlators due to a combination of microscopic reversibility and thermality. Therefore, in principle, one could use these relations between the bath's correlators to derive the generalised fluctuation-dissipation relation without introducing the OTO couplings. However, as we have argued in our paper, studying these relations between the correlators of the bath requires one to include the bath's OTOCs in the analysis. As the OTO couplings of the particle encode the effects of the bath's OTOCs on the particle's dynamics, introducing them simplifies the analysis. As we have shown, these OTO couplings allow one to study the constraints imposed by the bath's microscopic reversibility and thermality separately. Combining these constraints, one can easily get the generalised fluctuation-dissipation relation.

Now, we would like to mention that we have fixed the typos pointed out by the referee in his/her report. In addition, we noticed two more minor errors in the previous version which we enumerate below:

1) In equations (5), (7) and (9), we made a mistake while writing the quartic combinations of the couplings between the particle and the bath oscillators which contribute to the cumulants of the 4-point correlators of O (the bath operator that couples to the particle). We have corrected them in the current version. One can verify that these corrected combinations indeed contribute to the cumulants of the 4-point correlators of O by looking at the Feynman diagrams given in Figure 9 of appendix A.2.

2) We also made a mistake while showing the positions of the insertions in figure 3. We have corrected this in the current version. It is now consistent with the correlator mentioned in the caption of the same figure.

Sincerely,
the Authors

### List of changes

1) In the last line of the first paragraph of the introduction, we have replaced the phrase "degrees of freedom of the system" by "infrared (low-frequency) degrees of freedom of the system".

2) In the line just below equation (19) in page 10, we have replace the phrase "Plugging these expression" by "Plugging these expressions".

3) In the the caption of figure 5, we have replaced "t?0" by "t>0".

4) In equations (5), (7) and (9), we have corrected the combination of the quartic couplings between the particle and the bath oscillators which contribute to the cumulants of the 4-point correlators of O (the bath operator that couples to the particle).

5) In figure 3, we have corrected the positions of the insertions.

6) We have updated the references.

### Submission & Refereeing History

Resubmission 1905.08307v5 on 11 July 2019
Submission 1905.08307v3 on 14 June 2019