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**Numerical solution of large scale Hartree-Fock-Bogoliubov equations.**
*(English)*
Zbl 07405582

Summary: 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 \(\mathcal{O} (N_b^2)\) 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 17 280 CPU cores. This enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.

### Keywords:

Hartree-Fock-Bogoliubov; pole expansion and selected inversion; superconductivity; Hubbard-Hofstadter
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\textit{L. Lin} and \textit{X. Wu}, ESAIM, Math. Model. Numer. Anal. 55, No. 3, 763--787 (2021; Zbl 07405582)

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