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Low rank compression in the numerical solution of the nonequilibrium Dyson equation

by Jason Kaye, Denis Golež

Submission summary

As Contributors: Jason Kaye
Arxiv Link: https://arxiv.org/abs/2010.06511v3 (pdf)
Date accepted: 2021-03-09
Date submitted: 2021-02-26 02:48
Submitted by: Kaye, Jason
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Numerical Analysis
  • Condensed Matter Physics - Computational
Approach: Computational

Abstract

We propose a method to improve the computational and memory efficiency of numerical solvers for the nonequilibrium Dyson equation in the Keldysh formalism. It is based on the empirical observation that the nonequilibrium Green's functions and self energies arising in many problems of physical interest, discretized as matrices, have low rank off-diagonal blocks, and can therefore be compressed using a hierarchical low rank data structure. We describe an efficient algorithm to build this compressed representation on the fly during the course of time stepping, and use the representation to reduce the cost of computing history integrals, which is the main computational bottleneck. For systems with the hierarchical low rank property, our method reduces the computational complexity of solving the nonequilibrium Dyson equation from cubic to near quadratic, and the memory complexity from quadratic to near linear. We demonstrate the full solver for the Falicov-Kimball model exposed to a rapid ramp and Floquet driving of system parameters, and are able to increase feasible propagation times substantially. We present examples with 262144 time steps, which would require approximately five months of computing time and 2.2 TB of memory using the direct time stepping method, but can be completed in just over a day on a laptop with less than 4 GB of memory using our method. We also confirm the hierarchical low rank property for the driven Hubbard model in the weak coupling regime within the GW approximation, and in the strong coupling regime within dynamical mean-field theory.

Published as SciPost Phys. 10, 091 (2021)



List of changes

We thank the referees for their useful comments. In response, we have made the following revisions.

- Figs. 8 and 9 have been merged, and the surrounding discussion has
been modified accordingly.

- We have added a discussion on pg. 10 concerning the discrepancy in
online and offline ranks.

- We have added a remark in the fourth bullet point in the
conclusion addressing the origin and naturalness of the HODLR
compressibility.

Submission & Refereeing History

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Resubmission 2010.06511v3 on 26 February 2021

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