Numerical solution of large scale Hartree–Fock–Bogoliubov equations

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 $\Or(N_b^2)$ 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 $17280$ CPU cores. This enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.

Unconventional Self-Similar Hofstadter Superconductivity from Repulsive Interactions

“…Additionally, the square HH model can more easily be realized in cold atom systems [76][77][78][79][80][81][82][83], though the focus in that field has been on bosonic [84][85][86][87][88][89] and time-reversal invariant fermionic [90][91][92][93][94][95] HH models (note that the latter coincides with the regular fermionic Hofstadter-Hubbard model at q = 2, i.e. at π-flux).…”

Unconventional self-similar Hofstadter superconductivity from repulsive interactions

“…Additionally, the square HH model can more easily be realized in cold atom systems 77 – 86 , although the focus in that field has been on bosonic 87 – 92 and time-reversal invariant fermionic 93 – 98 HH models (note that the latter coincides with the regular fermionic Hofstadter-Hubbard model at q = 2, i.e., at π -flux). In addition, more recently single layer cuprates exhibiting critical temperatures close to their bulk values have been fabricated 99 , opening an avenue for realizing twisted cuprate moiré systems with square lattices for which our model may be directly applicable.…”

Linear-time generalized Hartree-Fock algorithm for quasi-one-dimensional systems

“…A different approach to improve the performance of gHF based on highly scalable methods for solving the fermionic problem for sparse systems was discussed in Ref. 38, reaching remarkably large systems by parallelizing the computation on several thousand computational cores.…”