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$$. We first provide a concise and relatively self-contained review of the HFB theory for general finite sized quantum systems, with special focus on the treatment of spin symmetries from a linear algebra perspective. We then demonstrate that the pole expansion and selected inversion (PEXSI) method can be particularly well suited for solving large scale HFB equations. For a Hubbard-type Hamiltonian, the cost of PEXSI is at most for both gapped and gapless systems, which can be significantly faster than the standard cubic scaling diagonalization methods. We show that PEXSI can solve a two-dimensional Hubbard-Hofstadter model with N$$ 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.
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DOI : 10.1051/m2an/2020074
Keywords: Hartree–Fock–Bogoliubov, pole expansion and selected inversion, superconductivity, Hubbard-Hofstadter
@article{M2AN_2021__55_3_763_0,
author = {Lin, Lin and Wu, Xiaojie},
title = {Numerical solution of large scale {Hartree{\textendash}Fock{\textendash}Bogoliubov} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {763--787},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2020074},
mrnumber = {4253163},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020074/}
}
TY - JOUR AU - Lin, Lin AU - Wu, Xiaojie TI - Numerical solution of large scale Hartree–Fock–Bogoliubov equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 763 EP - 787 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020074/ DO - 10.1051/m2an/2020074 LA - en ID - M2AN_2021__55_3_763_0 ER -
%0 Journal Article %A Lin, Lin %A Wu, Xiaojie %T Numerical solution of large scale Hartree–Fock–Bogoliubov equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 763-787 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020074/ %R 10.1051/m2an/2020074 %G en %F M2AN_2021__55_3_763_0
Lin, Lin; Wu, Xiaojie. Numerical solution of large scale Hartree–Fock–Bogoliubov equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 763-787. doi: 10.1051/m2an/2020074
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