Constraining the Skyrme energy density functional with quantum Monte Carlo calculations

We study the problem of an impurity in fully polarized (spin-up) low-density neutron matter with the help of an accurate quantum Monte Carlo method in conjunction with a realistic nucleon-nucleon interaction derived from chiral effective field theory at next-to-next-to leading order. Our calculations show that the behavior of the proton spin-down impurity is very similar to that of a polaron in a fully polarized unitary Fermi gas. We show that our results can be used to put tight constraints on the time-odd parts of the Skyrme energy density functional, independent of the time-even parts, in the density regime relevant to neutron-rich nuclei and compact astrophysical objects such as neutron stars and supernovae.

Figure 1

The energy of the neutron spin-down impurity in the units of the Fermi energy of the spin-up neutrons. The red filled squares are our QMC results with the ${\mathrm{NNLO}}_{\mathrm{opt}}$ interaction. The green filled circles are the GFMC calculations with an $s$-wave interaction fit to the $nn$ scattering length and effective range [14]. The black dashed lines are predictions from various density functionals (see text).

Figure 2

The energy of the (a) proton-spin-up and (b) proton-spin-down impurities in units of the Fermi energy of the spin-up neutrons. The red filled squares are our QMC results with the ${\mathrm{NNLO}}_{\mathrm{opt}}$. interaction. The green solid lines are the results from second-order perturbation theory. The black dashed lines are predictions from various density functionals (see text).

Figure 3

The difference between the energies of the proton spin-up and spin-down impurities in units of the Fermi energy of the spin-up neutrons. The red filled squares are our QMC results with the chiral ${\mathrm{NNLO}}_{\mathrm{opt}}$ interaction. The green solid line is the prediction from second-order perturbation theory. The blue dot-dashed line is a fit of the form $A-\frac{B}{k_F\left|a_s\right|}-C{k}_F{r}_e$. The black dashed lines are predictions from various density functionals (see text).