## SciPost Submission Page

# 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: | https://arxiv.org/abs/1904.12582v2 |

Date submitted: | 2019-06-24 |

Submitted by: | Kubiczek, Patryk |

Submitted to: | SciPost Physics |

Domain(s): | Theor. & Comp. |

Subject area: | Condensed Matter Physics - Theory |

### Abstract

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:

### Author comments upon resubmission

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)

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-6-24 Invited Report

### 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.