SciPost Submission Page
Linear response and exact hydrodynamic projections in Lindblad equations with decoupled Bogoliubov hierarchies
by Patrik Penc, Fabian H. L. Essler
Submission summary
| Authors (as registered SciPost users): | Fabian Essler · Patrik Penc |
| Submission information | |
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| Preprint Link: | scipost_202509_00037v1 (pdf) |
| Date submitted: | Sept. 19, 2025, 5:20 p.m. |
| Submitted by: | Patrik Penc |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider a class of spinless-fermion Lindblad equations that exhibit decoupled BBGKY hier- archies. In the cases where particle number is conserved, their late time behaviour is characterized by diffusive dynamics, leading to an infinite temperature steady state. Some of these models are Yang-Baxter integrable, others are not. The simple structure of the BBGKY hierarchy makes it pos- sible to map the dynamics of Heisenberg-picture operators on few-body imaginary-time Schr¨odinger equations with non-Hermitian Hamiltonians. We use this formulation to obtain exact hydrody- namic projections of operators quadratic in fermions, and to determine linear response functions in Lindbladian non-equilibrium dynamics.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- Presentation of an innovative approach to calculate the evolution of time-dependent correlation functions in dissipative, non-Gaussian systems 2- The technique presented in this work clearly opens a new pathway to investigate the non-trivial dynamics of a broad class of physically important dissipative systems 3- Even the most involved calculations are clearly presented and easy to follow 4- The technique presented in this work is of broad interested for the theoretical community interested in the dynamics of correlated quantum systems
Weaknesses
1- Even though the paper is very well written, it seems to be mainly addressed to the scientific community comfortable with jargon and concepts currently used to describe the hydrodynamic behavior of one-dimensional dissipative and integrable systems. I think that the techniques and results presented in this paper are of much broader scientific interest. Some additional paragraphs of explanation and context would make the paper accessible to a broader community, as it deserves. 2- There is some lack of physical discussion of the results and lack of motivation for some of the chosen models
Report
The important fact is that such models are governed by non-Gaussian Lindblad equations, which are notoriously important for correlated out-of-equilibrium physics and difficult to solve. Specifically, such models describe the Lindblad evolution of free fermions under the action of Hermitian and quadratic jump operators. Even though such models are non-Gaussian, a double commutator structure does not generate higher-order correlation functions, leading to a close set of equations of motions for the operators or, equivalently, decoupling the BBGKY hierarchy. The authors show that a well-suited vectorization procedure maps these equations of motion onto effective few-body problems (two-body for quadratic operators, four-body for quartic operators, ...). The authors provide exact solutions for the two-body problem in various models, even without particle conservation. In the specific case of dephasing processes, a mapping over a complex version of the spinful Hubbard model in one-dimension, allows to rely on a Bethe-Ansatz solution to extend results beyond quadratic operators.
The authors provide a nice demonstration of the effectiveness and importance of their approach, by deriving exactly the long-time hydrodynamic and diffusive behavior of quadratic operators associated to conserved charges (such as the particle number). This is equivalent to finding the so-called "hydrodynamic projections" of such operators, which is an important result for the quantum transport community. Furthermore, the authors ingeniously extend their approach to exactly derive correlation functions between quadratic operators at different times, notoriously important to study dynamical linear responses. They focus on the case of density-density response functions, normally probed in quantum transport and pump-and-probe experiments, and illustrate some effects of the dissipation.
The methods and results illustrated in this work are of great interest for the broad physics community interested in the quantum transport properties of correlated and dissipative systems. For this reason, I strongly recommend publication in SciPost Phyisics. However, as mentioned in the "Weakenesses" section, I think this work deserves to be made more accessible to a broader community by further explaining some concepts and jargon, including also some additional motivation and explanation concerning the physical systems and phenomena which could be explored based on the developed approach (see requested changes below).
Requested changes
[Not ranked in order of importance, but in order of my annotations to the text]
1- After Eq. (2): "Assuming the jump operators are Hermitian, L†α = Lα , or come in adjoint pairs, L†α = Lβ". Definitively correct. However, the last case falls back on the Hermitian one when we redefine couples of jump operators (Lα + L†α)/\sqrt2 and (Lα - L†α)/(i\sqrt2), leading to the same Lindblad equation. Maybe an additional word to clarify this aspect may be useful.
2- As the authors state, Models III and IV are elaborations of Models I and II respectively. Is there a specific reason why the authors chose these elaborations ?
3- Eq. (14), should the label "r" be replaced by the label "s" in the very last exponent of (-1)?
4- After Eq. (19) the authors state: "A key feature of this Hamiltonian is that in all
terms there are at least as many fermion annihilation operators as there are creation operators." Maybe I am missing a key point here, but this does not seem to be the case for the last two lines of Eq. (19). This further conflicts with a statement in page 13: "The vectorized Hamiltonian (19) has the key property that its interaction terms in H contain more annihilation that creation operators"
5-"more annihilation that creation" -> "more annihilation thaN creation"?
6- After equation (22): "non-positive" -> "negative" would simplify the reading
7- What about the \sigma index for n_j in the last line of Eq. (19)?
8- Title of Section IV.A : n_\sigma = \sum_j n_j\sigma ? For pedagogical reasons, stating explicitly, before Section IV.A, that you map the equations of motion for quadratic operators onto a 2-body problem may be helpful to those getting acquainted with the formalism.
9- Eq. (28). If I get it right, here the \sigma label becomes suddenly a space label, while it was associated to the pseudo-spin in the previous section. For clarity, it would be better to change label.
10- Page 6: "This reflects the existence of a diffusive hydrodynamic mode related to the particle number U (1) symmetry of the model." Of course I agree, but this could be quite obscure to the general reader.
11- After Eq. (47) the authors point to the relation between two-particle bound states and diffusive modes. Can the authors elaborate more on such connection? What do we learn from it ?
12- Section IV.B for models with n_\uparrow=2. As the authors state, it is totally expected that the charged operators decay exponentially. Is there something then that we should learn from the exact calculation of the exponential decay, beyond its analytic expression?
13- Section V. If I understood correctly, the authors rely on their wave-function formalism to calculate how quadratic operators decompose on the longest lived eigen-modes of the Lindblad 2-body operator sector. They can so analytically derive their diffusive behavior. According to their terminology, this procedure unveils "the hydrodynamic projection" of such operators on the diffusive modes. This is a very interesting result! However, I had to go through the entire calculation to (re)construct this line of reasoning. Maybe illustrating the philosophy of the calculation with more than a "We now show how to explicitly evaluate (70) for Model I", before starting developing the technicalities (maybe even in the introductory Section I), may help the reader following why some calculations are done. The sentence "this is a key result of our work" after Eq. (91) comes out of the blue and it is not really said why. I had to construct the (very interesting indeed!) answer myself looking at the following equations and plots.
14- Fig. 10. Why exactly does the time evolution of the circles corresponding to (q,d)=(\pi/5,10) initially follow the curve of (0,1) and suddenly change slope ?
15- Section VI. Study of Model V. This is a quite interesting model because it is the only one without charge conservation, but still solvable as decoupling the BBGKY hierarchy. Nonetheless, the authors derive exclusively the 2-body spectrum associated to quadratic operators and derive their expected exponential decay, which is expected. Could the authors briefly elaborate on some more interesting physical effects which could be studied in this model thanks to their analytical solution?
16-The appearance in Fig. 17 of the 2-particle continuum in the Lindblad spectrum is extremely intriguing. Could be this understood based on an analogue of the Lehmann representation of the response function extended to the Lindblad formalism?
17- "simnple" -> "simple" before Eq. (135)
18- Is the correct interpretation of Eq. (135) that Eq. (134) can be understood as the scalar product of the two eigen-functions corresponding to the density operators at time t and time zero ?
19- Conclusions. "This would enable the study of transport properties involving two-particle Green’s functions [67, 68]". This research program has been partially already addressed (to my knowledge) in
https://iopscience.iop.org/article/10.1088/1742-5468/2010/05/L05002
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.144301
https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.013109
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.136301
It may be also important to mention that an alternative self-consistent method to calculate two-point correlation functions was also devised in
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.100301
20- I could not find any reference to the Appendix in the main text
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1- Very. through and detailed 2- Authors go beyond hydrodynamic projection and study linear response
Weaknesses
Report
I find this paper insightful and well written. It will certainly contribute to the literature on Lindbladian hydrodynamics. I strongly recommend publication in SciPost Physics.
Requested changes
I only would like to ask the authors to consider the following suggested optional changes and questions:
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In the introduction (p.2, lhs column): "The presence of a conservation law then implies that certain observables will exhibit hydrodynamic power-law tails at late times, which are related to the existence of diffusive eigenmodes of the Lindbladian" - Is this obvious? Maybe one would need to have add certain caveats?
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"In contrast to unitary quench dynamics [50] integrability in LEs does not imply the existence of conservation laws. This is immediately obvious from the fact that many integrable LEs have unique steady states that are completely mixed, i.e. correspond to infinite temperature density matrices." - one should cite the relevant papers e.g. in [21-31]. For instance, in [26] the unique steady state is all down (up to certain boundary bound states and the formation of domain walls in the thermodynamic limit).
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p. 7, lhs column: In Fig 3 why are there (two) gaps? Is there a physical reason? In fact Fig. 3 does not seem to be referenced in the text.
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p. 10, rhs column: "B. Spatially local operators" - why spatially local? It seems to me to be more like "Spatially local without well defined momentum"? Maybe renaming this would make things clearer.
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In Sec. VI dealing with quadratic operators in model V, it would be possibly interesting to see how the operators exponentially decay? Or is it not interesting?
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p. 17, lhs column: "It would be interesting to pursue a more precise description by treating the ”particles” as interacting and following the logic of Ref. [64]." - This claim I find interesting and would ask that the authors elaborate slightly.
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Is the relevance of momentum $\pi$ for the eigenvalues closing to 0 related to a possible $\eta$-pairing operator (at momentum $\pi$) that appears from "spin up and down" vectorized fermions? Can the authors comment?
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)