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

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. [Somà, 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 $N_{\ell }$ 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 $N_{\ell }\approx 50$ 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.

Figure 1

Dimension scheme for the Gorkov matrix $\Xi$.

Figure 2

CPU time spent performing specific operations during a typical sc0 calculation as a function of $N_{\max }$ and for different values of $N_{\ell }$. The contribution of the various partial waves $\alpha$ are added. Left panel: time needed to calculate the self-energies and project them to Krylov's subspace (points 1 and 2 of Sec. 3d). Right panel: time required to diagonalize Gorkov's matrix over 100 sc0 iterations (point 3 of Sec. 3d). Dashed lines show scalings of the type ${\left(N_{\max }\right)}^{\gamma }$, with $\gamma =$6 (left panel) and 3 (right panel).

Figure 3

CPU time requirements (in minutes) to perform 100 iterations within typical sc and sc0 self-consistency calculations. Results are shown for different numbers $N_{\ell }$ of Lanczos iterations per pivot as a function of the model-space size $N_{\max }$.

Figure 4

Relative error for the contribution of a given partial wave $\alpha$ to the Koltun sum rule in ${}^{12}$C as a function of $K^{\alpha }$ (see text). Results refer to a single diagonalization and different model-space sizes. The Coulomb interaction has been neglected in this figure.

Figure 5

Relative error in the total binding energy of ${}^{44}$Ca after one second-order iteration as a function of $K^{\prime }$ (see text) for two different model-space sizes. The Coulomb interaction has been neglected in this figure.

Figure 6

Convergence of the (sc0) binding energy of ${}^{44}$Ca as a function of $N_{\ell }$, for different model spaces. The Coulomb interaction has been neglected in this figure.

Figure 7

Density of $J^{\Pi }=1/2^{+}$ states in ${}^{41}$Ti as a function of their excitation energy with respect to the Fermi level ${\mu }_n$ of ${}^{40}$Ti, for increasing $N_{\ell }$. The distribution, discretized in the calculation, is convoluted with Lorentzian curves of 5 MeV width for display purposes.

Figure 8

One-neutron addition and removal spectral strength distribution in ${}^{40}$Ti limited to $J^{\Pi }=1/2^{+}$ final states in ${}^{39,41}$Ti. The distribution, discretized in the calculation, is convoluted with Lorentzian curves of 5 MeV width for display purposes.

Figure 9

Selected neutron and proton effective single-particle energies in ${}^{40}$Ti as a function of the number of Lanczos iterations per pivot $N_{\ell }$. Results are displayed relative to the values obtained for $N_{\ell }=100$. Calculations are performed in an $N_{\max }=13$ model space.

Figure 10

Binding energy of ${}^4$He (left) and ${}^{20}$O (right) as a function of the number $N_{\ell }$ of Lanczos iterations per pivot, for different model-space sizes. Dashed (solid) lines correspond to the sc (sc0) self-consistent scheme.

Figure 11

Binding energies of ${}^{44}$Ca from first-order (upper panel) and second-order (lower panel) Gorkov calculations as a function of the harmonic oscillator spacing $\hslash \Omega$ and for increasing size $N_{\max }$ of the single-particle model space. The inset shows a zoom on the most converged results.

Figure 12

${}^{40}$Ti one-neutron removal spectral strength distribution associated with $J^{\Pi }=1/2^{+}$ final states as a function of the model space dimension $N_{\max }$. The distribution, discretized in the calculation, is convoluted with Lorentzian curves of 5 MeV width for display purposes.

Figure 13

Fishbone-like structure of the Lanczos reduced matrix $E^{\prime }$.