Numerical solution of large scale Hartree–Fock–Bogoliubov equations

Lin, Lin; Wu, Xiaojie

doi:10.1051/m2an/2020074

Numerical solution of large scale Hartree–Fock–Bogoliubov equations

Journal Article · Wed May 05 00:00:00 EDT 2021 · Mathematical Modelling and Numerical Analysis

DOI:https://doi.org/10.1051/m2an/2020074· OSTI ID:1813387

Lin, Lin ^[1]; Wu, Xiaojie ^[2]

Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Univ. of California, Berkeley, CA (United States)

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 $$\mathcal{O}$$(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. Finally, this enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: US Air Force Office of Scientific Research (AFOSR); USDOE Office of Science (SC)

Grant/Contract Number:: AC02-05CH11231; SC0017867

OSTI ID:: 1813387

Journal Information:: Mathematical Modelling and Numerical Analysis, Journal Name: Mathematical Modelling and Numerical Analysis Journal Issue: 3 Vol. 55; ISSN 0764-583X

Publisher:: EDP SciencesCopyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
Hartree-Fock-Bogoliubov
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Numerical solution of large scale Hartree–Fock–Bogoliubov equations

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