Neutron polaron as a constraint on nuclear density functionals

We study the energy of an impurity (polaron) that interacts strongly in a sea of fermions when the effective range of the impurity-fermion interaction becomes important, thereby mapping the Fermi polaron of condensed matter physics and ultracold atoms to strongly interacting neutrons. We present quantum Monte Carlo results for this neutron polaron, and compare these with effective field theory calculations that also include contributions beyond the effective range. We show that state-of-the-art nuclear density functionals vary substantially and generally underestimate the neutron polaron energy. Our results thus provide constraints for adjusting the time-odd components of nuclear density functionals to better characterize polarized systems.

Figure 1

Neutron polaron energy $E_{\mathrm{pol}}$ (in units of the Fermi energy $E_F$) as a function of Fermi momentum $k_F$. The arrows mark the values at unitarity $\eta =-0.607$ and $-0.616$ (see text). Results are shown from the dEFT calculation with and without an effective range (the $r_e=0$ result approaches the unitarity value with increasing $k_F$) and from QMC calculations with 33 + 1 particles and S-wave interactions. The two bands are based on chiral $NN$ interactions at N${}^3$LO, including also contributions beyond effective-range effects, at the level of third-order particle-particle (pp) ladder contributions (yellow band; third-order pp) and resumming pp ladders (red band; pp resummed). The width of the bands reflects the variation from using different cutoffs, different single-particle energies, and for the third-order pp band, the difference from second- to third-order ladders.

Figure 2

Dependence of the polaron energy $E_{\mathrm{pol}}$ (solid line, left axis, in units of the Fermi energy $E_F$) on the number of spin-up neutrons $N_{\uparrow }$ for the largest density ($k_F=0.7{\mathrm{fm}}^{-1}$). This follows the dependence of noninteracting neutrons $E_{\mathrm{free}}$ (dashed line, right axis, in units of the Fermi-gas energy $E_{FG}$) except for the smallest system $N_{\uparrow }=7$, where the box is only 4 times larger than the effective range.

Figure 3

Comparison of the microscopic results for the polaron energy of Fig. 1 with predictions of nuclear density functionals [12, 13, 49, 50, 51, 52, 53, 54] (see text). The thick solid (green) curve is the new UNEDF1-pol density functional, which is based on Ref. [55] plus a fit to the QMC $E_{\mathrm{pol}}$ results.