Iterative subspace algorithms for finite-temperature solution of Dyson equation

TL;DR

This paper develops and compares iterative acceleration techniques for solving the Dyson equation at finite temperature, demonstrating improved convergence especially for strongly correlated systems in Green's function methods.

Contribution

It introduces a generalized residual concept and applies acceleration methods like DIIS, LCIIS, and KAIN to enhance Dyson equation solutions, enabling convergence in challenging cases.

Findings

Commutator residuals outperform difference residuals in convergence.

Generalized CDIIS and LCIIS methods successfully converge restricted GF2 for strongly correlated systems.

Practical guidelines are provided for achieving convergence in difficult scenarios.

Abstract

One-particle Green's functions obtained from the self-consistent solution of the Dyson equation can be employed in evaluation of spectroscopic and thermodynamic properties for both molecules and solids. However, typical acceleration techniques used in the traditional quantum chemistry self-consistent algorithms cannot be easily deployed for the Green's function methods, because of non-convex grand potential functional and non-idempotent density matrix. Moreover, the inclusion of correlation effects in the form of the self-energy matrix and changing chemical potential or fluctuations in the number of particles can make the optimization problem more difficult. In this paper, we study acceleration techniques to target the self-consistent solution of the Dyson equation directly. We use the direct inversion in the iterative subspace (DIIS), the least-squared commutator in the iterative subspace (LCIIS), and the Krylov space accelerated inexact Newton method (KAIN). We observe that the definition of the residual has a significant impact on the convergence of the iterative procedure. Based on the Dyson equation, we generalize the concept of the commutator residual used in DIIS (CDIIS) and LCIIS, and compare it with the difference residual used in DIIS and KAIN. The commutator residuals outperform the difference residuals for all considered molecular and solid systems within both GW and GF2. The generalized CDIIS and LCIIS methods successfully converged restricted GF2 calculations for a number of strongly correlated systems, which could not be converged before. We also provide practical recommendations to guide convergence in such pathological cases.