# (Non) equilibrium dynamics: a (broken) symmetry of the Keldysh generating functional

### Submission summary

 As Contributors: Camille Aron Arxiv Link: http://arxiv.org/abs/1705.10800v3 (pdf) Date accepted: 2018-01-21 Date submitted: 2017-12-21 01:00 Submitted by: Aron, Camille Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

We unveil the universal (model-independent) symmetry satisfied by Schwinger-Keldysh quantum field theories whenever they describe equilibrium dynamics. This is made possible by a generalization of the Schwinger-Keldysh path-integral formalism in which the physical time can be re-parametrized to arbitrary contours in the complex plane. Strong relations between correlation functions, such as the fluctuation-dissipation theorems, are derived as immediate consequences of this symmetry of equilibrium. In this view, quantum non-equilibrium dynamics -- e.g. when driving with a time-dependent potential -- are seen as symmetry-breaking processes. The symmetry-breaking terms of the action are identified as a measure of irreversibility, or entropy creation, defined at the level of a single quantum trajectory. Moreover, they are shown to obey quantum fluctuation theorems. These results extend stochastic thermodynamics to the quantum realm.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 4, 008 (2018)

We are much grateful to both Referee 1 and Referee 2 for the time they spent writing a thorough, positive, review of our work. The appreciations of both Referees being quite similar, we answer them in a single reply.

While both Referees acknowledged the significance and the originality of our work, they also pointed out that, given the rather formal nature of our approach, additional efforts could be made to better present the physical findings. We fully agree with the Referees, and we thank them for their suggestions in that respect, which we followed (see the List of changes).

We hope that the referees will find the revised version of our manuscript suitable for publication in SciPost without further delay.

Camille Aron
Giulio Biroli
Leticia F. Cugliandolo

### List of changes

- We revised the introductory Section 1.1 "Motivations" to better appeal to the community of quantum thermodynamics by articulating one of our main findings, namely the identification of a quantum version of the irreversible entropy production rate, in the context of what has already been achieved in the classical realm (stochastic thermodynamics), and of the questions that are still debated in the quantum realm.

- We revised the introductory Section 1.3 "Main results", paragraph "Out-of-equilibrium dynamics", by announcing a later discussion on a concrete and simple model where our findings can be worked out and discussed explicitly, numerically or even experimentally.

- We added the corresponding discussion in Sect. 4.2 "Symmetry breaking term", where we consider a two-level system (TLS) which is driven through an avoided crossing at different rates. Despite its simplicity, the time-dependent driven TLS has two virtues:
(i) it belongs to a different `class' of systems than the bosonic and fermionic fields that are treated in the rest of the manuscript. The TLS is an SU(2) symmetric object with a more cumbersome path-integral representation, however the quantum thermodynamic operators that were derived using bosonic and fermionic fields are readily applicable to any quantum mechanical system,
(ii) the excitations created when ramping across the avoided crossing at a finite rate are of purely quantum mechanical origin (contrary to the case of, say, a driven quantum harmonic oscillator which comes with a 'classical' intuition/phenomenology). While the evolution is purely unitary, it is clear that these tunnel excitations can be associated to the production of irreversibility in the system.
The novel quantity $\langle \Sigma \rangle$ is studied numerically in different regimes, and compared to the work $\langle \mathcal{W} \rangle$. This simple model is used again in Section 4.3 "Quantum fluctuation theorems", paragraph "Jarzynski equality", to illustrate our proposed definition of irreversibility production, $\mathcal{S}_{irr}$, given in Eqs. (100) and (104).

### Submission & Refereeing History

Resubmission 1705.10800v3 on 21 December 2017
Submission 1705.10800v2 on 19 July 2017

## Reports on this Submission

### Anonymous Report 1 on 2017-12-21 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1705.10800v3, delivered 2017-12-21, doi: 10.21468/SciPost.Report.305

### Report

The authors have seriously taken into consideration my advice contained in the previous report. I am glad to assess that now the paper is an interesting and complete piece of science, with both technical and practical aspects of interest for the community working in non-equilibrium many body physics.

• validity: good
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: excellent