Incorporating Brueckner-Hartree-Fock correlations in energy density functionals

Recently, a microscopically motivated nuclear energy density functional was derived by applying the density-matrix expansion to the Hartree-Fock (HF) energy obtained from long-range chiral effective-field theory two- and three-nucleon interactions. However, the HF approach cannot account for all many-body correlations. One class of correlations is included by Brueckner-Hartree-Fock (BHF) theory, which gives an improved definition of the one-body HF potential by replacing the interaction by a reaction matrix $G$. In this paper, we find that the difference between the $G$ matrix and the similarity renormalization group evolved nucleon-nucleon potential $V_{\mathrm{SRG}}$ can be well accounted for by a truncated series of contact terms. This is consistent with renormalization-group decoupling generating a series of counterterms as short-distance physics is integrated out. The coefficients $C_n$ of the power series expansion $\sum C_n{q}^n$ for the counterterms are examined for two potentials at different renormalization-group resolutions and at a range of densities. The success of this expansion for $G-V_{\mathrm{SRG}}$ means we can apply the density-matrix expansion at the HF level with low-momentum interactions and density-dependent zero-range interactions to model BHF correlations.

Figure 1

Correlation plots for the matrix $G-V_{\mathrm{SRG}}$ between the AV18 [45] and N3LO [46] potentials in the ${}^{1}S_0$ channel at flow parameter (a) $\lambda =\infty$, (b) $2.0{\mathrm{fm}}^{-1}$ and (c) $1.5{\mathrm{fm}}^{-1}$ [note that the $x, y$ axis scales in panels (a)–(c) are different]. The $G$ matrix is evaluated at the saturation density ($k_{\mathrm{F}}=1.3{\mathrm{fm}}^{-1}$). The potentials are separated into three different regions, the low-momentum region ($k,k^{\prime }<k_{\text{F}}$), the high-momentum region ($k,k^{\prime }>k_{\text{F}}$), and the coupling region ($k>k_{\text{F}}, k^{\prime }<k_{\text{F}}$ or $k<k_{\text{F}}, k^{\prime }>k_{\text{F}}$).

Figure 2

Comparison of $G-P{V}_{\mathrm{SRG}}P$ (solid line) with $V_{\mathrm{CT}}$ (dots) for the ${}^{1}S_0$ and ${}^{3}S_1$ channels. The left panel (a) shows off-diagonal elements and the right panel (b) shows diagonal elements. $V_{\mathrm{SRG}}$ is the N3LO potential evolved by the SRG to $\lambda =1.5{\mathrm{fm}}^{-1}$ at $k_{\text{F}}=1.3{\mathrm{fm}}^{-1}$.

Figure 3

Comparison of $G-P{V}_{\mathrm{SRG}}P$ (solid line) with $V_{\mathrm{CT}}$ (dots) for the ${}^{3}S_1-{}^{3}D_1$ and ${}^{3}D_1$ channels. The left panel (a) shows off-diagonal elements and the right panel (b) shows diagonal elements. $V_{\mathrm{SRG}}$ is the N3LO potential evolved by the SRG to $\lambda =1.5{\mathrm{fm}}^{-1}$ at $k_{\text{F}}=1.3{\mathrm{fm}}^{-1}$.

Figure 4

The coefficients of the counterterms for the AV18 and N3LO potentials as a function of Fermi momentum in the ${}^{1}S_0$ channel with SRG flow parameters $\lambda =10$, 2, and $1.5{\mathrm{fm}}^{-1}$.

Figure 5

Counterterms for different partial waves as a function of Fermi momentum obtained from SRG-evolved N3LO and AV18 potentials at flow parameter $\lambda =3.0{\mathrm{fm}}^{-1}$. Each column of plots is a single partial-wave channel, given at the top, while each row of plots is one of the counterterms ($C_0–C_4^{\prime }$) given on the left.

Figure 6

Counterterms as a function of Fermi momentum for different partial waves obtained from SRG-evolved N3LO and AV18 potentials at flow parameter $\lambda =3.0{\mathrm{fm}}^{-1}$. Each column of plots is a single partial-wave channel, given at the top, while each row of plots is one of the counterterms ($C_0–C_4^{\prime }$) given on the left.

Figure 7

The density-dependent Skyrme-like couplings (a) $x_0$ and (b) $t_0$ from Eq. (11) are plotted as a function of the isoscalar density ${\rho }_0$, using SRG-evolved N3LO and AV18 potential at flow parameter $\lambda =3.0{\mathrm{fm}}^{-1}$.