In-Medium Similarity Renormalization Group For Nuclei

We present a new ab initio method that uses similarity renormalization group (SRG) techniques to continuously diagonalize nuclear many-body Hamiltonians. In contrast with applications of the SRG to two- and three-nucleon interactions in free space, we perform the SRG evolution “in medium” directly in the $A$-body system of interest. The in-medium approach has the advantage that one can approximately evolve $3,\dots ,A$-body operators using only two-body machinery based on normal-ordering techniques. The method is nonperturbative and can be tailored to problems ranging from the diagonalization of closed-shell nuclei to the construction of effective valence-shell Hamiltonians and operators. We present first results for the energies of ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$, which have accuracies comparable to coupled-cluster calculations.

Figure 1

Convergence of the in-medium SRG results at the normal-ordered two-body level, IM-SRG(2), for ${}^4\mathrm{He}$ using the generators ${\eta }^{\mathrm{I}}$ (left) and ${\eta }^{\mathrm{II}}$ (right panel). The filled (open) symbols correspond to solving Eqs. (9, 10, 11) with the underlined terms omitted (included). The ground-state energy $E_0(\infty )$ is given as a function of the harmonic oscillator parameter $\hslash \omega$ with increasing single-particle space $e_{\max }\equiv \max (2n+l)$. The initial $NN$ interaction is a free-space SRG-evolved potential with $\lambda =2.0{\mathrm{fm}}^{-1}$ from the ${\mathrm{N}}^3\mathrm{LO}$ potential of Ref. 16. For comparison we show the coupled-cluster CCSD and CCSD(T) energies in the $e_{\max }=8$ space (calculated at their $\hslash \omega$ minimum).

Figure 2

In-medium SRG evolution of normal-ordered two-body matrix elements ${\Gamma }_{ijkl}$ connecting hole-hole ($hh$) and particle-particle ($pp$) states for ${}^{16}\mathrm{O}$ starting from a smooth-cutoff $V_{\mathrm{low}k}$ with $\Lambda =1.8{\mathrm{fm}}^{-1}$. The color scale is in MeV, and initial and $s$-evolved results are shown. The axes label two-body $jj$-coupled states $|(n_a,l_a,j_a,t_{z_a}),(n_b,l_b,j_b,t_{z_b});J=0⟩$. The ${\Gamma }_{ijkl}$ where $ijkl=ppph$ or $hhhp$, which are not driven to zero with the current generator, are not shown.

Figure 3

Convergence of the IM-SRG(2) energy $E_0(\infty )$ for ${}^4\mathrm{He}$ using the generator ${\eta }^{\mathrm{II}}$ and starting from the “bare” $\mathrm{N}{}^3\mathrm{LO}$ potential of Ref. 16. The notation is the same as in Fig. 1. The converged IM-SRG(2) energy agrees well with the CCSD result (the coupled-cluster energies are taken from Ref. 3), while second- and third-order many-body perturbation theory, MBPT(2) and MBPT(3), clearly break down.

Figure 4

Convergence of the IM-SRG(2) energy $E_0(\infty )$ for ${}^{16}\mathrm{O}$ (left) and ${}^{40}\mathrm{Ca}$ (right panel) using the generator ${\eta }^{\mathrm{II}}$ (solid symbols) and in comparison to CCSD results (dashed lines). The notation is the same as in Fig. 1. The initial $V_{\mathrm{NN}}$ is a smooth-cutoff $V_{\mathrm{low}k}$ with $\Lambda =1.8{\mathrm{fm}}^{-1}$ for ${}^{16}\mathrm{O}$ and a free-space SRG potential with $\lambda =1.8{\mathrm{fm}}^{-1}$ for ${}^{40}\mathrm{Ca}$, both evolved from the $\mathrm{N}{}^3\mathrm{LO}$ potential of Ref. 16.