Ab initioself-consistent Gorkov-Green's function calculations of semi-magic nuclei: Numerical implementation at second order with a two-nucleon interaction

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Background: The newly developed self-consistent Gorkov-Green's function approach represents a promising path to the ab initio description of mid-mass open-shell nuclei. The formalism based on a two-nucleon interaction and the second-order truncation of Gorkov's self-energy has been described in detail in Ref. [Som\`a, Duguet, and Barbieri, Phys. Rev. C 84, 064317 (2011)].Purpose: The objective is to discuss the methodology used to solve Gorkov's equation numerically and to gauge its performance in view of carrying out systematic calculations of medium-mass nuclei in the future. In doing so, different sources of theoretical error and degrees of self-consistency are investigated.Methods: We employ Krylov projection techniques with a multi-pivot Lanczos algorithm to efficiently handle the growth of poles in the one-body Green's function that arises as a result of solving Gorkov's equation self-consistently. We first characterize the numerical scaling of Gorkov's calculations based on full self-consistency and on a partially self-consistent scheme coined as ``sc0''. Using small model spaces, the Krylov projection technique is then benchmarked against exact diagonalization of the original Gorkov matrix. Next, the convergence of the results as a function of the number ${N}_{\ensuremath{\ell}}$ of Lanczos iterations per pivot is investigated in large model spaces. Eventually, the convergence of the calculations with the size of the harmonic oscillator model space is examined.Results: Gorkov self-consistent Green's function (SCGF) calculations performed on the basis of Krylov projection techniques display a favorable numerical scaling that authorizes systematic calculations of mid-mass nuclei. The Krylov projection selects efficiently the appropriate degrees of freedom while spanning a very small fraction of the original space. For typical large-scale calculations of mid-mass nuclei, a Krylov projection making use of ${N}_{\ensuremath{\ell}}\ensuremath{\approx}50$ yields a sufficient degree of accuracy on the observables of interest. The partially self-consistent sc0 scheme is shown to reproduce fully self-consistent solutions in small model spaces at the 1$%$ level. Eventually, Gorkov-Green's function calculations performed on the basis of SRG-evolved interactions show a fast convergence as a function of the model-space size.Conclusions: The end result is a tractable, accurate and gently scaling ab initio scheme applicable to complete isotopic and isotonic chains in the medium-mass region. The partially self-consistent sc0 scheme provides an excellent compromise between accuracy and computational feasibility and will be the workhorse of systematic Gorkov-Green's function calculations in the future. The numerical scaling and performances of the algorithm employed offers the possibility (i) to apply the method to even heavier systems than those (e.g., ${}^{74}$Ni) already studied so far and (ii) to perform converged Gorkov SCGF calculations based on harder, e.g. original chiral interactions.