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Exact real-time dynamics of single-impurity Anderson model from a single-spin hybridization-expansion

by Patryk Kubiczek, Alexey N. Rubtsov, Alexander I. Lichtenstein

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Submission summary

Authors (as registered SciPost users): Patryk Kubiczek
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-07-26
Date submitted: 2019-06-24 02:00
Submitted by: Kubiczek, Patryk
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


In this work we introduce a modified real-time continuous-time hybridization-expansion quantum Monte Carlo solver for a time-dependent single-orbital Anderson impurity model: CT-1/2-HYB-QMC. In the proposed method the diagrammatic expansion is performed only for one out of the two spin channels, while the resulting effective single-particle problem for the other spin is solved semi-analytically for each expansion diagram. CT-1/2-HYB-QMC alleviates the dynamical sign problem by reducing the order of sampled diagrams and makes it possible to reach twice as long time scales in comparison to the standard CT-HYB method. We illustrate the new solver by calculating an electric current through impurity in paramagnetic and spin-polarized cases.

Author comments upon resubmission

We thank the reviewer for their positive evaluation of our article and for the valuable comments, which we hereafter address.

1 - We have added a detailed discussion of computational complexity into section 3.4. It reads O(<k>^2 N^3) for discretized bath and O(<k> N_t^3) for discretized time approach.

2 - We have added the proposed references. arXiv:1904.11969 appeared shortly before the paper has been sent and we have not had a chance to look into it. arXiv:1903.11646 reports on extension of the Keldysh summation method to previously unattainable parameter regions by conformal transformations of expansion variable. Since this development has far-reaching consequences for the applicability of the Keldysh summation method, we agree it should be cited.

3 - We have included the recommended references. Even though 10.1103/PhysRevB.82.075109 does not deal with real-time dynamics it sets the basis of the bold QMC method. 10.1103/PhysRevB.84.085134 provides bold QMC calculations of impurity current which is of great relevance to our work. We also agree that 10.1103/physrevlett.116.036801 should be mentioned as an application of the bold-line method.

4 - Although iterative path-integral summation methods are not Monte Carlo methods per se, we do agree they share many common features since the motivation behind them was to overcome the sign problem through a deterministic summation of diagrams. As you suggest, we have mentioned iterative summation of path-integrals as a method alternative to QMC in the introduction, citing the following papers: 10.1002/pssb.201349187 and 10.1103/PhysRevB.82.205323.

5 - We support the idea of discussing compatibility of CT-1/2-HYB with bold and inchworm methods. In principle, combining CT-1/2-HYB with bold or inchworm method is non-trivial, since the latter use many-body propagators and the former uses single-particle propagators. Nevertheless, inchworm algorithm around propagators dressed by spin-up processes in the spirit of CT-1/2-HYB is possible. However, in contrast to pure CT-1/2-HYB some spin-up diagrams will still be missed in the inchworm procedure and will have to be sampled separately. We have added a relevant discussion to the conclusion part of the paper.

List of changes

1 - mentioned iterative summation of path integrals (section 1)
2 - added references: arXiv:1904.11969, arXiv:1903.11646, 10.1103/PhysRevB.82.075109, 10.1103/PhysRevB.84.085134, 10.1002/pssb.201349187, 10.1103/PhysRevB.82.205323 (section 1)
3 - corrected typo: −U↑↓(t,t′)n↓(t)n↓(t') was changed to −U↑↓(t,t′)n↓(t) (section 2.3)
4 - extended and reorganized the discussion of the computational complexity (section 3.4)
5 - added a discussion of compatibility of CT-1/2-HYB with inchworm and bold algorithms (section 4)

Published as SciPost Phys. 7, 016 (2019)

Reports on this Submission

Anonymous Report 2 on 2019-7-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1904.12582v2, delivered 2019-07-23, doi: 10.21468/SciPost.Report.1068


1) New idea/computational approach for dynamics of quantum impurity models, relevant topics in condensed matter physics/mesoscopic physics/material science etc.
2)Longer time scales (by a factor 2) can be reached as compared to standard Hybridization Expansion CTQMC
3)Method is clearly explained in the theory part and in the application, including the limitations, the computational complexity, benchmarks etc..
4)The basic idea (expanding around one single spin flavour) could lead to interesting developments both conceptual and practical, related to Falikov Kimball physics..


1)The need for discretizing the bath to evaluate the diagrams for the up spin can be rather severe, expecially to study transport and nonequilibrium effects. It is not completely clear whether the gain in longer time scales accessible is not cancelled off by the need of finite size baths, particularly in interesting and challenging (Kondo) regimes.
2)The gain of a factor 2 with respect to Hyb-CTQMC is good, at the same time recent developments in QMC has resulted in new algorithms that surpass the standard approach (Inchworm, resummation techniques,..).
This is however a minor weakness since in principle the idea of this new approach could be potentially combined with other methods, as the author comment toward the end..


In this work, the authors present a new QMC approach to solve the real-time dynamics of quantum impurity models, a relevant subject of research in condensed matter physics several with potential applications.

The basic idea is to reduce the number of diagrams which are stochastically sampled (and which lead to the infamous sign problem) by evaluating (resumming) certain diagrams semi-analytically. Specifically, the authors consider a spinful Anderson Model and expand the Keldysh partition function only in the down spin. The result is that each diagram is slightly less trivial to evaluate but there are factor 2 less diagrams to sample on average.

The main outstanding challenge that this paper raises in my opinion is how to evaluate efficiently and accurately those diagrams, which correspond to Falikov-Kimball like impurity problems.
The authors propose two methods (discretization of bath size or discretization of time domain) and test only one of them, obtaining good results.

It would be interesting to explore more the second approach: I suspect that using the segment picture and the fact that for QMC sampling only the ratio of weights in Eq.18 between adjacent configuration is needed, one could further simplify the algebra. In addition, Falikov Kimball impurity problems allow an analytical solution (at least for certain non equilibrium problems) and it could be beneficial to explore this direction as well.

To conclude, in my opinion the paper presents an interesting and promising method for a tough and relevant condensed matter problem. The paper is clear and well written, explaining advantages and limitations of the current implementation and could therefore lead to further developments in the community. I am in favor of publication on SciPost without further changes.

  • validity: top
  • significance: high
  • originality: top
  • clarity: top
  • formatting: perfect
  • grammar: excellent

Author:  Patryk Kubiczek  on 2019-07-24  [id 573]

(in reply to Report 2 on 2019-07-23)

We thank the reviewer for their positive feedback on our work. We believe the report accurately identified how the community could benefit from the new method (combination with other approaches, further developments using discretization of time domain), as well as the method's weaknesses (possibly no access to Kondo physics, relatively small improvement factor in comparison to other methods).

Anonymous Report 1 on 2019-6-24 (Invited Report)


The revised manuscript has been substantially improved, especially by the more complete discussion of computational complexity and by the interesting discussion regarding connections to other methods. I recommend publication with no further changes.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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