Skyrme-based extrapolation for the static response of neutron matter

The study of inhomogeneous neutron matter can provide insights into the structure of neutron stars as well as their dynamics in neutron-star mergers. In this work we tackle pure neutron matter in the presence of a periodic external field by considering a finite (but potentially large) number of particles placed in periodic boundary conditions. We start with the simpler setting of a noninteracting gas and then switch to a Skyrme-Hartree-Fock approach, showing static-response results for five distinct Skyrme parametrizations. We explain both the technical details of our computational approach, as well as the significance of these results as a general finite-size extrapolation scheme that may be used by ab initio practitioners to approach the static-response problem of neutron matter in the thermodynamic limit.

Figure 1

Relative error of $N$ particle energy compared to TL energies shows magnitude of finite-size effects versus particle number. Inset: Energy FS effects on a linear scale.

Figure 2

Noninteracting Fermi gas linear response functions at a density of $0.1{\mathrm{fm}}^{-3}$. The circles and squares correspond to 66 and 8250 particles respectively. The line is the TL response function.

Figure 3

Relative error of $N$ particle energy compared to TL energies shows magnitude of finite-size effects. SLy4 parametrization at ${\rho }_0=0.1{\mathrm{fm}}^{-3}$.

Figure 4

SLy4 density of 8250 particles with external potential parameters: $2v_q=0.25E_F$, 1 period in the box and average density $0.10{\mathrm{fm}}^{-3}$.

Figure 5

SLy4 density of 66 particles with external potential parameters: $2v_q=0.25E_F$, one period in the box and average density $0.10{\mathrm{fm}}^{-3}$.

Figure 6

SLy4 density of 8250 particles with external potential parameters: $2v_q=0.25E_F$, five periods in the box and average density $0.10{\mathrm{fm}}^{-3}$.

Figure 7

SLy4 density of 66 particles with external potential parameters: $2v_q=0.25E_F$, one period in the box and average density $0.04{\mathrm{fm}}^{-3}$.

Figure 8

SLy4 density of 8250 particles with external potential parameters: $2v_q=0.25E_F$, one period in the box and average density $0.04{\mathrm{fm}}^{-3}$.

Figure 9

SLy4 density of 4224 particles with external potential parameters: $2v_q=0.25E_F$, one period in the box and average density $0.04{\mathrm{fm}}^{-3}$.

Figure 10

SLy4 density of 8250 particles with external potential parameters: $2v_q=0.25E_F$, two periods in the box and average density $0.04{\mathrm{fm}}^{-3}$.

Figure 11

SLy4 change in energy per particle versus the potential strength parameter for 8250 particles at an average density of $0.10{\mathrm{fm}}^{-3}$ with five periods of the potential in the box. The dashed line is the fourth degree fit.

Figure 12

SLy4 change in energy per particle versus the potential strength parameter for 66 particles at an average density of $0.04{\mathrm{fm}}^{-3}$. Here (and below) the straight lines serve to guide the eye.

Figure 13

SLy4 change in energy per particle versus the potential strength parameter for 8250 particles at an average density of $0.04{\mathrm{fm}}^{-3}$.

Figure 14

Linear response functions of the Skyrme parametrizations KDE0v1, SLy4, SKRA, and SkM (left to right) at a density of $0.04{\mathrm{fm}}^{-3}$. The dotted lines are the response in the TL. Solid and hollow squares correspond to 66 and 8250 particle responses, respectively.

Figure 15

Energy per particle versus external potential strength for SLy4 at a density of $0.04{\mathrm{fm}}^{-3}$. The solid (circles) and dashed line (squares) correspond to 66 particles with one and two periods of the potential in the box, respectively. The dotted line (triangles) correspond to 8250 particles with the same periodicity as the solid line system.

Figure 16

$0.04{\mathrm{fm}}^{-3}$ SLy4 response functions for different particle numbers. The circles and squares and line correspond to 38 particles, 114 particles, and the TL respectively.

Figure 17

$0.04{\mathrm{fm}}^{-3}$ energy versus external potential strength for 38 particles for the two smallest periodicities. The circles and squares correspond to one and two periods of the potential traversing the box.

Figure 18

SLy4 change in energy per particle contribution to the FS fix versus the potential strength parameter at an average density of $0.04{\mathrm{fm}}^{-3}$.

Figure 19

Free-gas (circles) and NRAPR (squares) response functions at a density of $0.04{\mathrm{fm}}^{-3}$. Solid and hollow symbols correspond to 66 and 8250 particles, respectively. The solid and dotted lines are the corresponding TL responses of the noninteracting and NRAPR systems.

Figure 20

Averaged Skyrme change in energy per particle contribution to the FS fix versus the potential strength parameter at an average density of $0.04{\mathrm{fm}}^{-3}$.