Screening and antiscreening of the pairing interaction in low-density neutron matter

We study pairing in low-density neutron matter including the screening interaction due to the exchange of particle-hole and RPA excitations. As bare force, we employ the effective low-momentum interaction $V_{\text{low}k}$, while the Fermi-liquid parameters are taken from a phenomenological energy density functional (SLy4) which correctly reproduces the equation of state of neutron matter. At low density, we find screening, i.e., pairing is reduced, while at higher densities, we find antiscreening, i.e., pairing is enhanced. This enhancement is mostly due to the strongly attractive Landau parameter $f_0$. We discuss in detail the critical temperature $T_c$ in the limit of low densities and show that the suppression of $T_c$ predicted by Gor'kov and Melik-Barkhudarov can only be reproduced if the cutoff of the $V_{\text{low}k}$ interaction is scaled with the Fermi momentum. We also discuss the effect of noncondensed pairs on the density dependence of $T_c$ in the framework of the Nozières-Schmitt-Rink theory.

Figure 1

Feynman diagrams representing the induced interaction. The wiggly line in diagram (b) is meant to include the RPA bubble summation.

Figure 2

Elements of Feynman diagrams to clarify the notation. Left, particle-particle interaction; right, particle-hole propagator.

Figure 3

Fermi-liquid parameters $m^{*}/m$, $G_0$, and $F_0$ used in the present work, obtained from different phenomenological effective interactions: Skyrme parametrization SLy4 (solid lines) and Gogny D1N (short dashes) and D1 (long dashes) parametrizations.

Figure 4

Convergence of the induced interaction with respect to variation of the maximum angular momentum $j_{\text{max}}$ used in the partial wave expansion of the bare interaction. The figure shows the diagonal matrix elements for $k_F=0.8{\text{fm}}^{-1}$. Upper thin curves, results for diagram (a); lower thin curves, results for diagram (b); thick curves, sum of (a)+(b). The bare interaction in this example is $V_{\text{low}k}$, and the Fermi-liquid parameters are those of SLy4.

Figure 5

Induced pairing interaction due to the exchange of $S=0$ (dashed lines) and $S=1$ (dotted lines) excitations. The thin lines represent the contributions of diagrams (a) only, and the thick lines are the sums of diagrams (a) and (b). The thick solid line is the sum of $S=0$ and $S=1$ contributions. The parameters are the same as in Fig. 4.

Figure 6

Critical temperature $T_c$ as a function of the Fermi momentum $k_F$, obtained with the $V_{\text{low}k}$ interaction (Fermi-liquid parameters from the Skyrme force SLy4). Dashes, result obtained using only the bare interaction; dots, result obtained including diagram (a); solid line, full result including also diagram (b).

Figure 7

Same as Fig. 6, but here the Gogny D1N and D1 interactions are used in the particle-particle channel and for the Fermi-liquid parameters.

Figure 8

Measure of the contribution of different momenta to the gap equation as defined in Eq. (32), for two different densities $[k_F=0.012{\text{fm}}^{-1}$ (solid line) and $1.45{\text{fm}}^{-1}$ (dashes), indicated by the thin vertical lines]. The integrals were calculated with the $V_{\text{low}k}$ interaction $m^{*}$ from the Skyrme force SLy4) at the respective critical temperatures.

Figure 9

Matrix elements $V(q,k_F)$ of the bare ($V_{\text{low}k}$, dashes) and of the screened (solid line) interaction for $k_F=0.012{\text{fm}}^{-1}$. For comparison, we display also the screened interaction obtained with the analytical approximation Eq. (28) (dots, almost indistinguishable from the solid line). The screening correction is so tiny that it is almost invisible on the big graph; see the inset for an enlargement. The thin vertical line indicates $q=k_F$.

Figure 10

Higher order ladder diagrams in the 3p1h vertices which are not included in the present work.

Figure 11

Same as Fig. 9 but now calculated with an interaction $V_{\text{low}k}$ evolved to a much lower cutoff $\mathrm{\Lambda }=0.03{\text{fm}}^{-1}=2.5k_F$ (and with an exponential instead of Fermi-Dirac regulator; see text).

Figure 12

Reduction of the critical temperature due to the screening correction $V_{\text{ind}}=V_a+V_b$ as a function of $k_F$, obtained with the constant cutoff $\mathrm{\Lambda }=2{\text{fm}}^{-1}$ (dashes) and with the density-dependent cutoff $\mathrm{\Lambda }=2.5k_F$ (solid line), respectively.

Figure 13

Dependence of $V_{\text{ind}}$ on the momentum $K$ of the center of mass. For a density $k_F=0.2{\text{fm}}^{-1}$ (left panels), the $K$ dependence is extremely weak even for momenta $K$ exceeding $2k_F$. For $k_F=0.8{\text{fm}}^{-1}$ (upper right panel), the $K$ dependence is somewhat stronger but still too weak to make a significant contribution.

Figure 14

The unsubtracted correlated density ${\rho }_{\text{corr},1}$ as a function of the Fermi momentum $k_F^0$ with and without the screening correction, calculated at the respective critical temperatures $T_c$. Here, the black solid lines and the red dashed lines show the results for the two cutoffs of $\mathrm{\Lambda }=2.0{\text{fm}}^{-1}$ and $\mathrm{\Lambda }=2.5k_F^0$. The thin lines contain only $V_0$, while the thick lines include the induced interactions. The inset in the figure magnifies the cutoff dependence in ${\rho }_{\text{corr},1}$ at low densities. The Fermi-liquid parameters are calculated using the SLy4 interaction.

Figure 15

Subtracted correlated density ${\rho }_{\text{corr}}$ as a function of $k_F^0$ with and without screening, calculated at the respective critical temperatures. See Fig. 14 for details.

Figure 16

$T_c$ vs $k_F$: (left panel) results with fixed cutoff $\mathrm{\Lambda }=2{\text{fm}}^{-1}$; (right panel) density-dependent cutoff $2.5k_F$. The green dash-dotted lines are the full results including the induced interaction $V_{\text{ind}}$ and the correlated density ${\rho }_{\text{corr}}$ in the NSR framework. For comparison, we also show the BCS result (only $V_0$ and ${\rho }_0$, black solid lines) and the results obtained with the induced interaction $V_{\text{ind}}$ but without the NSR correction (red dashed lines).