From bare interactions, low-energy constants, and unitary gas to nuclear density functionals without free parameters: Application to neutron matter

We further progress along the line of Ref. [D. Lacroix, Phys. Rev. A 94, 043614 (2016)] where a functional for Fermi systems with anomalously large $s$-wave scattering length $a_s$ was proposed that has no free parameters. The functional is designed to correctly reproduce the unitary limit in Fermi gases together with the leading-order contributions in the $s$- and $p$-wave channels at low density. The functional is shown to be predictive up to densities $\sim 0.01 {\mathrm{fm}}^{-3}$ that is much higher densities compared to the Lee-Yang functional, valid for $\rho <{10}^{-6} {\mathrm{fm}}^{-3}$. The form of the functional retained in this work is further motivated. It is shown that the new functional corresponds to an expansion of the energy in $\left(a_s{k}_F\right)$ and $\left(r_e{k}_F\right)$ to all orders, where $r_e$ is the effective range and $k_F$ is the Fermi momentum. One conclusion from the present work is that, except in the extremely low-density regime, nuclear systems can be treated perturbatively in $-{\left(a_s{k}_F\right)}^{-1}$ with respect to the unitary limit. Starting from the functional, we introduce density-dependent scales and show that scales associated with the bare interaction are strongly renormalized by medium effects. As a consequence, some of the scales at play around saturation are dominated by the unitary gas properties and not directly by low-energy constants. For instance, we show that the scale in the $s$-wave channel around saturation is proportional to the so-called Bertsch parameter ${\xi }_0$ and becomes independent of $a_s$. We also point out that these scales are of the same order of magnitude than those empirically obtained in the Skyrme energy density functional. We finally propose a slight modification of the functional such that it becomes accurate up to the saturation density $\rho \simeq 0.16 {\mathrm{fm}}^{-3}$.

Figure 1

Neutron matter energy as a function either of $-\left(a_s{k}_F\right)$ or $\rho$. (a) and (b) show the results to the results obtained by using Eq. (3) where the $A_i$ coefficients are adjusted by imposing the developments given by Eqs. (5) and (6). (c) and (d) show the results of Eq. (7) where $U_i$ and $R_i$ are obtained using Eq. (5) and unitarity limits as constraints. In all panels, the red solid lines correspond to the equation of state including the effective-range dependence while the blue dashed lines correspond to the result obtained by assuming $r_e=0$. In (a) and (c), the black circles and green squares correspond to the QMC results of Refs. [47, 48], respectively, for the cold atom case and AV4 ($s$-wave only) case. In (b) and (d), the black squares and dark blue circles correspond, respectively, to the ab initio results of Refs. [49, 50]. Note finally that the highest density shown in the upper panels corresponds to $\rho =0.01 {\mathrm{fm}}^{-1}$ and corresponds to a very narrow region of (b) and (d).

Figure 2

Same as Fig. 1. The green squares correspond to QMC results of Ref. [47] and the red solid line to results of Eq. (7). The green short dashed line and blue long dashed line correspond to the expansion of Eq. (7) to first order in ${\left(a_s{k}_F\right)}^{-1}$ or to first order in $\left(r_e{k}_F\right)$, respectively.

Figure 3

$E/E_{\mathrm{FG}}$ as a function of $-\left(a{k}_F\right)$ obtained for a Gaussian interaction. The thick black solid line corresponds to Eq. (12) where $A$ and $B$ are calculated from the low-energy constant (with $a_p^3=0$). Note that the energy $E$ corresponds to the sum of the kinetic and Hartree-Fock energy. The blue dotted line stands for the expansion (5) result, while the green long dashed line is obtained by using Eq. (31). The red short dashed line is obtained by using the alternative Eq. (32). The thin solid gray line is the function $1+5/\left(9\pi \right)\left(a_s{k}_F\right)$ that is shown for reference.

Figure 4

Energy of neutron matter obtained in QMC calculations retaining only the $s$-wave contribution (green squares) or using the full AV4 interaction (blue triangle). The red solid corresponds to the result of the functional (7) without the $p$-wave contribution. The blue long-dashed line is obtained by adding the $p$-wave contribution using $a_p^3=0.25 {\mathrm{fm}}^3$. In the inset, the blue long-dashed line represents the $p$-wave term of Eq. (5), i.e., $E_p/E_{\mathrm{FG}}=(\nu +1){\left(a_p{k}_F\right)}^3/\left(3\pi \right)$ using $a_p^3=0.25 {\mathrm{fm}}^3$. This term is compared to the difference between the energy obtained in QMC with AV4 and the QMC result with $s$-wave only (black solid circles).

Figure 5

The red solid lines give the evolution of the quantities (a) ${\tilde{a}}_s\left(\rho \right)$ and (b) ${\tilde{r}}_e\left(\rho \right)$ calculated through the expressions (39) and (40), respectively. The blue short dashed lines represent the approximate expressions (41) [(a)] and (42) [(b)]. In (b), the asymptotic limit at high density given by Eq. (43) is displayed by the black long-dashed line. In both panels, the insets focus on the density region 0.05 ${\mathrm{fm}}^{-3}\le \rho \le 0.2 {\mathrm{fm}}^{-3}$.

Figure 6

Values of the quantities (a) $a_s^{\mathrm{Sk}}$ and (b) $r_e^{\mathrm{Sk}}$, for different widely used sets of Skyrme interaction parameters. In (a) and (b), the green area corresponds to the two windows of ${\tilde{a}}_s\left(\rho \right)$ and ${\tilde{r}}_e\left(\rho \right)$ values given by (44). For completeness we also give in (c) the equivalent to the $p$-wave scattering volume values ${a_p^{\mathrm{Sk}}}^3$ for the same set of Skyrme parameters.

Figure 7

Same as Fig. 1. The red solid line represents the result of the functional with the extra factor ${\displaystyle e^{-c{\left(r_e{k}_F\right)}^4}}$ in the $r_e$ dependence.