Coordinate-space solver for finite-temperature Hartree-Fock-Bogoliubov calculations using the shifted Krylov method

Abstract

Background: In order to study structure of proto-neutron stars and those in subsequent cooling stages, it is of great interest to calculate inhomogeneous hot and cold nuclear matter in a variety of phases. The finite-temperature Hartree-Fock-Bogoliubov (FT-HFB) theory is a primary choice for this purpose, however, its numerical calculation for superfluid (superconducting) many-fermion systems in three dimensions requires enormous computational costs. Purpose: To study a variety of phases in the crust of hot and cold neutron stars, we propose an efficient method to perform the FT-HFB calculation with the three-dimensional (3D) coordinatespace representation. Methods: Recently, an efficient method based on the contour integral of Green's function with the shifted conjugate-orthogonal conjugate-gradient method has been proposed [Phys. Rev. C 95, 044302 (2017)]. We extend the method to the finite temperature, using the shifted conjugateorthogonal conjugate-residual method. Results: We benchmark the 3D coordinate-space solver of the FT-HFB calculation for hot isolated nuclei and fcc phase in the inner crust of neutron stars at finite temperature. The computational performance of the present method is demonstrated. Different critical temperatures of the quadrupole and the octupole deformations are confirmed for 146 Ba. The robustness of the shape coexistence feature in 184 Hg is examined. For the neutron-star crust, the deformed neutron-rich Se nuclei embedded in the sea of superfluid low-density neutrons appear in the fcc phase at the nucleon density of 0.045 fm −3 and the temperature of kBT = 200 keV. Conclusions: The efficiency of the developed solver is demonstrated for nuclei and inhomogeneous nuclear matter at finite temperature. It may provide a standard tool for nuclear physics, especially for the structure of the hot and cold neutron-star matters.

Unified nuclear matter EOSs constrained by the in-medium balance in density-dependent covariant density functionals

“…Adopting spherical and cylindrical approximations for the Wigner-Seitz (WS) cell [73,[88][89][90][91], six types of nuclear matter structures were observed, i.e, droplet, rod, slab, tube, bubble, and uniform. Meanwhile, more complicated structures may emerge if the spherical and cylindrical approximations were not imposed [76,79,[92][93][94][95][96][97][98][99][100][101][102][103]. At densities smaller than neutron drip density (n b < ∼ 0.0003 fm −3 ), the neutron gas vanish and neutron star matter are comprised of finite nuclei in Coulomb lattices, which form the outer crusts of neutron stars as well as white dwarfs.…”

Self-consistent energy density functional approaches to the crust of neutron stars

“…In order to abstain from diagonalizing the large HFB Hamiltonian matrix, which is computationally demanding, the contour integration in the complex energy plane together with iterative solutions of the shifted linear algebraic equations is adopted to construct densities. The method has been extended to the finite-temperature HFB method, by including the Matsubara frequencies as the imaginary shifts [4]. The method was successfully applied to non-uniform nuclear matter with superfluid neutrons at finite temperature.…”

Three-dimensional Skyrme Hartree-Fock-Bogoliubov solver in coordinate-space representation

“…The coordinate-space solvers constitute the second family of HFB codes. Examples of such solvers are: HFBRAD [18] solves spherically symmetric HFB problem using finite differences; HFB-AX [19] is a 2D solver based on B-splines; SkyAx [20] is a highly optimized 2D Hartree-Fock (HF) + BCS code using the fast Fourier transform (FFT) method to compute derivatives; Sky3D [21,22] is a 3D extension of SkyAx; the predecessor of SkyAx and Sky3D is a 1D spherical HF+BCS code using five-point finite differences which was published first in [23] and has meanwhile been developed into a full spherical HFB code Sky1D [24]; the HFB extension of SkyAx is Sky2D [24]; EV8 solves the Skyrme HF+BCS equations using the imaginary time method on a 3D mesh that is limited to one octant by imposing time-reversal and spatial symmetries [25,26]; MOCCa [27,28] is a Skyrme-HFB extension of EV8; MADNESS-HFB [29] is a 3D HFB solver based on multi-resolution analysis and multi-wavelet expansion; LISE is a 3D HFB solver [30] employing the discrete variable representation (or Lagrange-mesh method) and fast Fourier transforms; and there are also 3D HFB solvers based on the contour integral of the Green's function using the shifted Krylov subspace method [31,32].…”