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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

Submission summary

As Contributors: Patryk Kubiczek
Arxiv Link:
Date submitted: 2019-04-30
Submitted by: Kubiczek, Patryk
Submitted to: SciPost Physics
Domain(s): Theor. & Comp.
Subject area: Condensed Matter Physics - Theory


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.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Submission 1904.12582v1 on 30 April 2019

Reports on this Submission

Anonymous Report 1 on 2019-5-10 Invited Report


1—Novel and interesting methodological idea to improve convergence in an important class of computational physics problems.

2—Shown to work in practice, and the resulting improvement is well understood.

3—Possibly compatible with other recent ideas in the field.

4—Expected to be competitive with existing methods near the crossover to Falicov–Kimball.


1—The method was not tested in a regime where it should be advantageous. This is a minor weakness, and it is acceptable to leave this for a later paper focusing on the physics.

2—The frequency resolution of the lead density of states is sacrificed compared to competing methods.


The manuscript reports on a novel modification of the real time continuous time Monte Carlo algorithm in the hybridization expansion (CT-HYB) for impurity models. The main idea, implemented here for a single impurity Anderson model, is to expand in the hybridization of only one spin species rather than both. One then obtains dressed but effectively noninteracting (Falicov-Kimball) dynamics for the other spin, the action of which can be efficiently evaluated. This is reminiscent of the bold-line Monte Carlo in references [11] and [12] in both spirit and performance, but is clearly distinct.

Two ways of evaluating the effective local action over the remaining spin are proposed. One of these requires discretization of the bath degrees of freedom and is similar in form to what appears in auxiliary field Monte Carlo methods. This was found to be more efficient here. The other requires a time discretization and is more expensive computationally, but has the advantage that it can be extended to cases with retarded local interactions. This could potentially be very important for e.g. GW+DMFT.

In practice, the average expansion order is reduced by a factor of 2, effectively doubling the reachable timescale when compared to standard CT-HYB. This factor is shown to be largely unaffected by a combination of a magnetic field and chemical potential, but is expected to decrease for multiorbital impurity models.

The method is interesting, novel and has some promising immediate applications (the crossover between the Falicov–Kimball and Hubbard models is mentioned). The manuscript is clear and the theory and results are well presented. I therefore recommend publication after some minor modifications listed below.

Requested changes

1—What is the theoretical computational scaling of the method with the relevant physical and numerical parameters, in both approaches to obtaining the dressed local weight? Is the $O(N^{3})$ result in figure 8 understood?

2—Where the summation over Keldysh indices is mentioned in the introduction, please see also arxiv:1904.11969 and arXiv:1903.11646.

3—Where bold-line CT-HYB is mentioned in the introduction, please see also 10.1103/PhysRevB.82.075109, 10.1103/PhysRevB.84.085134 and 10.1103/physrevlett.116.036801 (currently reference 8).

4—More work that should probably be mentioned in the same context as points 2 and 3 is the literature on iterative summation of path integrals by D. Segal and M. Thorwart.

5—Since the method presented here is essentially orthogonal to e.g. the bold, inchworm and (possibly) Keldsyh summation ideas, it may be combined with these other advances. If so, that would be a major point in favor of the manuscript. Could the authors briefly comment on whether this is thought to be feasible?

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

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