Numerical solution of large scale Hartree–Fock–Bogoliubov equations

¹ Department of Mathematics, University of California, Berkeley, CA 94720, USA
² Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

^* Corresponding author: xiaojiewu@berkeley.edu

Received: 7 January 2020
Accepted: 2 November 2020

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 𝒪(N_b²) 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 × 10⁶, and the wall clock time is less than 100 s using 17 280 CPU cores. This enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.