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

International audienceBackground: The newly developed self-consistent Gorkov-Green’s function approach represents a promisingpath to the ab initio description of mid-mass open-shell nuclei. The formalism based on a two-nucleon interactionand the second-order truncation of Gorkov’s self-energy has been described in detail in Ref. [Som`a, Duguet, andBarbieri, Phys. Rev. C 84, 064317 (2011)].Purpose: The objective is to discuss the methodology used to solve Gorkov’s equation numerically and to gaugeits 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 handlethe growth of poles in the one-body Green’s function that arises as a result of solving Gorkov’s equation selfconsistently.We first characterize the numerical scaling ofGorkov’s calculations based on full self-consistency andon a partially self-consistent scheme coined as “sc0”. Using small model spaces, the Krylov projection techniqueis then benchmarked against exact diagonalization of the original Gorkov matrix. Next, the convergence of theresults as a function of the number N 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 Krylovprojection techniques display a favorable numerical scaling that authorizes systematic calculations of mid-massnuclei. The Krylov projection selects efficiently the appropriate degrees of freedom while spanning a verysmall fraction of the original space. For typical large-scale calculations of mid-mass nuclei, a Krylov projectionmaking use of N≈ 50 yields a sufficient degree of accuracy on the observables of interest. The partiallyself-consistent sc0 scheme is shown to reproduce fully self-consistent solutions in small model spaces at the1% level. Eventually, Gorkov-Green’s function calculations performed on the basis of SRG-evolved interactionsshow 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 completeisotopic and isotonic chains in the medium-mass region. The partially self-consistent sc0 scheme provides anexcellent compromise between accuracy and computational feasibility and will be the workhorse of systematicGorkov-Green’s function calculations in the future. The numerical scaling and performances of the algorithm employedoffers the possibility (i) to apply the method to even heavier systems than those (e.g., 74Ni) already studiedso far and (ii) to perform converged Gorkov SCGF calculations based on harder, e.g. original chiral interactions