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Adaptive Numerical Solution of Kadanoff-Baym Equations

by Francisco Meirinhos, Michael Kajan, Johann Kroha, Tim Bode

Submission summary

As Contributors: Johann Kroha · Francisco Meirinhos
Arxiv Link: (pdf)
Code repository:
Date submitted: 2021-10-12 11:08
Submitted by: Meirinhos, Francisco
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


A time-stepping scheme with adaptivity in both the step size and the integration order is presented in the context of non-equilibrium dynamics described via Kadanoff-Baym equations. The accuracy and effectiveness of the algorithm are analysed by obtaining numerical solutions of exactly solvable models. We find a significant reduction in the number of time-steps compared to fixed-step methods. Due to the at least quadratic scaling of Kadanoff-Baym equations, reducing the amount of steps can dramatically increase the accessible integration time, opening the door for the study of long-time dynamics in interacting systems. A selection of illustrative examples is provided, among them interacting and open quantum systems as well as classical stochastic processes. An open-source implementation of our algorithm in the scientific-computing language Julia is made available.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2110.04793v1 on 12 October 2021

Reports on this Submission

Anonymous Report 1 on 2021-11-26 (Invited Report)


-- Clear presentation, including a concise summary of the theory background.
-- Potentially useful technical developments for approaching many-body dynamics of quantum and classical problems by an adaptive choice of time steps.


-- Worked examples not clearly demonstrating the advantage of the new method (so the potential use is not fully backed/proven, from the examples).
-- Seems to be limited to special classes of models, where the interaction can be treated perturbatively (in the spirit of self-consistent Born, no perspective on inclusion of vertex corrections given).


I would recommend publication of the paper after the following points of criticism have been addressed:

Introduction: Why should it be enough to study the two-time correlator? This obviously will require approximations truncating the full Dyson-Schwinger hierarchy. Some comments/anticipation would be useful, already in the introduction.

Reduction Schwinger-Keldysh --> MSR: The example of a free theory is correctly worked out, but a word of caution should be placed in what concerns this reduction in the case of interacting problems: There, it can only be achieved in the frame of further approximations, neglecting terms with 'quantum' fields entering at order higher than two.

3.3.2, memory truncation: should be re-emphasized that the Green's function are connected, therefore one has clustering properties and decay as a function of the relative time coordinate. Interesting would be a discussion of the difference between algebraic (expected in the presence of conservation laws and / or quantum problems) and exponential decay in that coordinate, how does this affect the scaling of the method?

Choice of example models: What is the reason to choose the excitation transfer model? It looks to me that choosing a more generlic (e.g. Bose-Hubbard) model would likewise be suitable to do the benchmark calculations? Later it becomes clear (avoiding hopping in the single-particle sector), but the logic of choice of models should be anticipated.

In the same vein, the paper would benefit very much from more impressive examples. The breadth of examples is an asset, but they are all constrained to non-interacting systems or interacting ones on very few sites (often times, 2). This does not make a convincing case for a new method. Is it not possible to consider, say, and interacting scalar field theory in spatial continuum, see e.g. the cited works by the Berges group, and compare the new technique to previous ones?
The applicability of the method to generic situations (i.e. beyond perturbation theory in the interactions) is not clear, and not even an effort is made to comment on that. This is key in non-trivial out-of-equilibrium dynamics, such as the description of non-thermal fixed points. Some comparison to the so far successfully developed techniques to tackle such problems would be useful.
An answer could be that the latter rely on an effectively classical description due to high mode occupations. Could the present technique also benefit from that?

A more precise / extensive discussion of the advantage obtained from the adaptive method would be helpful, and a service to practitioners who might want to use the method. The only statement I could find about that comparison is in the first example but is very short and qualitative. Are there no scaling arguments available?

Requested changes

see above

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

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