## 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.

### MSC:

 65-XX Numerical analysis 81-XX Quantum theory

### Software:

ScaLAPACK; Pselinv; SelInv; ELPA; SuperLU-DIST
Full Text:

### References:

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