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A convenient Keldysh contour for thermodynamically consistent perturbative and semiclassical expansions

by Vasco Cavina, Sadeq S. Kadijani, Massimiliano Esposito, Thomas Schmidt

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Authors (as registered SciPost users): Vasco Cavina
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-04-19 15:20
Submitted by: Cavina, Vasco
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


The work fluctuation theorem (FT) is a symmetry connecting the moment generating functions (MGFs) of the work extracted in a given process and in its time-reversed counterpart. We show that, equivalently, the FT for work in isolated quantum systems can be expressed as an invariance property of a modified Keldysh contour. Modified contours can be used as starting points of perturbative and path integral approaches to quantum thermodynamics, as recently pointed out in the literature. After reviewing the derivation of the contour-based perturbation theory, we use the symmetry of the modified contour to show that the theory satisfies the FT at every order. Furthermore, we extend textbook diagrammatic techniques to the computation of work MGFs, showing that the contributions of the different Feynman diagrams can be added to obtain a general expression of the work statistics in terms of a sum of independent rescaled Poisson processes. In this context, the FT takes the form of a detailed balance condition linking every Feynman diagram with its time-reversed variant. In the second part, we study path integral approaches to the calculation of the MGF, and discuss how the arbitrariness in the choice of the contour impacts the final form of the path integral action. In particular, we show how using a symmetrized contour makes it possible to easily generalize the Keldysh rotation in the context of work statistics, a procedure paving the way to a semiclassical expansion of the work MGF. Furthermore, we use our results to discuss a generalization of the detailed balance conditions at the level of the quantum trajectories.

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Anonymous Report 2 on 2023-7-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2304.03681v1, delivered 2023-07-20, doi: 10.21468/SciPost.Report.7540


This is a very nice work, and I strongly recommend it for publication.


No significant weaknesses.


This is a very nice work, and I strongly recommend it for publication. The search for a proof of thermodynamic consistency within the formalism of Keldysh Green's functions has been a goal of quantum thermodynamics for many years. This work appears to resolve much of the problem. This work is sufficiently deep that it will take me more time to *fully* understand it than is reasonable for a refereeing process. However, I make some suggestions for changes below, based on my current understanding.

Requested changes

I have three suggestions for changes that would make the work easier to follow (based on my current understanding). I suggest that the authors implement those suggestions that make sense.

(1) If I understand correctly this work proves thermodynamic consistency for a work fluctuation theorem in a situation with a single temperature. That would mean that the proof applies to the thermodynamics of situations in which one is doing work on the system (on average), and that work is dissipated as heat (on average). But it does not apply to situations in which a temperature difference is used to create work. If so, I think it would be worth saying this to the reader in the introduction. If I am mistaken about this, then the authors should definitely add a discussion of this point in the manuscript. In any case, this work is a big step forward.

(2) I found it very hard to follow the crucial text between Eq. (35) and (36). This is the text that explains how to identify the diagram that is the "rev" partner of another diagram. I feel a figure with a simple example (or two) would help readers understand quickly. For example, what is the "rev" partner of the diagram in Fig 3 or the dumbbell in Fig 6?
Indeed, it would be great if there is a simple graphical rule to identify the "rev" partner of a given diagram. This would be analogous to Whitney's rule (in the context of a different type of Keldysh method in the manuscript's Ref [31]) in which one simply rotates a diagram by 180 degrees to find its time-reversed partner. However, I would understand if there is no such simple rule here.

(3) VERY MINOR COMMENT: I found the acronyms GF, FT, MGF and CGF to be distracting, and hard to remember while concentrating on following the mathematics. This was particularly so because the G and F are different in GF, FT and MGF. As there is no page limit, I think readers will be pleased if all these acronyms are replaced by the full words throughout the manuscript.

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

Anonymous Report 1 on 2023-7-11 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2304.03681v1, delivered 2023-07-11, doi: 10.21468/SciPost.Report.7488


This work establishes a connection between the fluctuation theorem in the quantum and semiclassical regimes by utilizing the Keldysh non-equilibrium diagrammatic expansion and the Keldysh path-integral approach based on the geometric symmetry of novel modified Keldysh contours.


The current version appears to have several ambiguities. It would be preferable to clarify the connections to previous related works.


In the paper, the authors discuss the thermodynamic consistency of the Keldysh formalism in the perturbative and path-integral approaches. The novel ingredients of their work seem to be the modified contour Fig. 1c and the symmetrized contour Fig. 4c. The modified contour allows for the interpretation of the fluctuation theorem (FT) for the generating function based on the geometrical symmetry, as illustrated in Fig. 2. By performing the perturbative expansion, the authors also highlight that the work statistics can be understood as the summation of Poisson processes, as shown in Eq. (33). If I understand correctly, the symmetrized contour enables the expression of 'stochastic' work in a trajectory-independent form, as demonstrated in Eq. (52), and facilitates the derivation of the detailed balance relation at the level of quantum trajectories, as exemplified by Eq. (66).

The paper provides a sufficiently concise explanation of the theoretical framework. Technical details, along with several explicit examples, are provided in the appendices. The topic of the thermodynamically consistent Keldysh non-equilibrium field theory is important, and the paper aims to offer an interesting interpretation of the FT based on the geometrical symmetry of properly modified Keldysh contours. Therefore, I believe the paper is suitable for publication. However, I have some questions and comments that I hope will contribute to further improving the paper.

Requested changes

1) Is the contour introduced in the present paper in Fig. 1c equivalent to that introduced previously, for example, in Section 8 of H. Umezawa's book, "Advanced Field Theory: Micro, Macro, and Thermal Physics" (AIP, New York, 1993)?
2) The time-dependent Hamiltonian defined at the 'complex time,' e.g., the sinusoidal driving with frequency \Omega, H_1(z) \propto \sin(\Omega z), does not make sense. Therefore, it is better to explain the definition of H_1(z) more carefully.
3) In Sec. 4.2, the authors found that the work statistics become the sum of independent rescaled Poisson processes, Eq. (33). I believe this holds for the lowest-order diagrams. However, when one performs an infinite summation, it is known that the statistics change for the charge full-counting statistics (See, e.g., Eq. (9) of Ref. [26]). I would expect a similar behavior for the work statistics.
4) In Appendix C, the authors calculated the d=1,2,3,4 diagrams in Fig. 3b. It is unclear why the contributions of d=1,2 diagrams vanish. Additionally, the origin of (..)|_{\lambda=0} terms in the d=3,4 diagrams in Eq. (86) is not clear either. I speculate that these terms may originate from the d=1,2 diagrams.
5) The authors should also provide an explicit derivation of the FT at the level of the cumulant generating function using Eqs. (33) and (37). Additionally, it would be helpful to clarify the connection between E_d and E^d introduced above Eq. (32).
6) The authors discovered that the FT holds at every order in the expansion of the generating function. This finding has been utilized in previous literature on the FT based on Keldysh diagrams [e.g., Saito and Utsumi, "Symmetry in Full Counting Statistics, Fluctuation Theorem, and Relations among Nonlinear Transport Coefficients in the Presence of a Magnetic Field," arXiv:0709.4128; Phys. Rev. B 78, 115429 (2008); Utsumi and Saito, "Fluctuation Theorem in a Quantum-Dot Aharonov-Bohm Interferometer," Phys. Rev. B 79, 235311 (2009); Utsumi, Entin-Wohlman, Ueda, Aharony, "Full-counting statistics for molecular junctions: Fluctuation theorem and singularities," Phys. Rev. B 87, 115407 (2013)]. It would be valuable for the authors to comment on the novel developments achieved compared to the previous works.
7) The authors state that the 'stochastic' work becomes trajectory independent in the classical limit (\hbar \to 0) in Eq. (60). However, this contradicts Eq. (56) and the fact that the stochastic work in the stochastic thermodynamics is typically trajectory dependent. A more thorough explanation is necessary to resolve this apparent contradiction.
8) Besides the references by Funo and Quan [22, 23], theFT based on the path integral in the semiclassical limit and the FT in the classical limit based on the Martin-Siggia-Rose-Janssen-deDominicis action have been developed [e.g., Mallick, Moshe, Orland, "A field-theoretic approach to non-equilibrium work identities," J. Phys. A: Math. Theor. 44, 095002 (2011); Utsumi, Golubev, Marthaler, Schön, Kobayashi, "Work fluctuation theorem for a classical circuit coupled to a quantum conductor," Phys. Rev. B 86, 075420 (2012); Aron, Barci, Cugliandolo, González Arenas, Lozano, "Dynamical symmetries of Markov processes with multiplicative white noise," J. Stat. Mech. 053207 (2016)]. It would be enlightening to compare the present work with these previous theories. 
9) In the third line of Eq. (16), there might be an error in the ordering of U^\dagger and U.
10) Before Eq. (18), the authors wrote, "With this in mind, ... of the contour itself as shown in Fig. 3." Should it be Fig. 2 instead of Fig. 3?
11) The definition of the step function in Eq. (27) should be provided.
12) Below Eq. (37), the authors wrote C_W(\lambda,t_f) = C_W^{\rm rev}(\lambda,t_f), which appears to be inconsistent with Eq. (17).
13) In the figure captions and the main text, the authors sometimes use small letters to indicate specific figures, such as Fig. 1a. It would be preferable to use capital letters, as they are used in the figures. Additionally, before Eq. (79), the authors wrote "... the results of App. (B)..." while below Eq. (84), the authors wrote "... in detail in App. D." It would be preferable to maintain consistent notation throughout the paper.
14) In Fig. 2D, t_f might be t_f + I \hbar \lambda.
15) Below Eq. (62), it appears that x_{q \uparrow}(-\hbar \lambda/2) should be x_{q \uparrow} (-i \hbar \lambda/2) based on the definition in Eq. (47).
16) In Eq. (68), the authors introduced the term W_{cl}. Is it the same as W_0(t_f) defined before Eq. (61)?
17) Before Eq. (121), the authors wrote "... and replacing the expressions (120) and (121) ...". This statement may be incorrect.
18) Since there are a lot of equations and I have not checked all of them, it would be advisable for the authors to recheck all derivations.

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

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