Numerical solution of large scale Hartree-Fock-Bogoliubov equations

TL;DR

This paper introduces an efficient computational method using PEXSI for solving large-scale Hartree-Fock-Bogoliubov equations, significantly reducing computational costs and enabling simulations of large quantum systems under magnetic fields.

Contribution

The paper demonstrates that PEXSI can efficiently solve large-scale HFB equations with reduced computational complexity, outperforming traditional diagonalization methods.

Findings

PEXSI achieves at most O(N_b^2) complexity for large HFB systems.

Successfully simulated a 2D Hubbard-Hofstadter model with over 2.88 million basis functions.

Simulation time for large systems is under 100 seconds using 17280 CPU cores.

Abstract

The Hartree-Fock-Bogoliubov (HFB) theory is the starting point for treating superconducting systems. However, the computational cost for solving large scale HFB equations can be much larger than that of the Hartree-Fock equations, particularly when the Hamiltonian matrix is sparse, and the number of electrons $N$ is relatively small compared to the matrix size $N_{b}$. We first provide a concise and relatively self-contained review of the HFB theory for general finite sized quantum systems, with special focus on the treatment of spin symmetries from a linear algebra perspective. We then demonstrate that the pole expansion and selected inversion (PEXSI) method can be particularly well suited for solving large scale HFB equations. For a Hubbard-type Hamiltonian, the cost of PEXSI is at most $\Or(N_b^2)$ for both gapped and gapless systems, which can be significantly faster than the standard cubic scaling diagonalization methods. We show that PEXSI can solve a two-dimensional Hubbard-Hofstadter model with $N_b$ up to $2.88\times 10^6$, and the wall clock time is less than $100$ s using $17280$ CPU cores. This enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.