Spin-polarized neutron matter: Critical unpairing and BCS-BEC precursor

We obtain the critical magnetic field required for complete destruction of $S$-wave pairing in neutron matter, thereby setting limits on the pairing and superfluidity of neutrons in the crust and outer core of magnetars. We find that for fields $B\ge {10}^{17}$ G the neutron fluid is nonsuperfluid—if weaker spin 1 superfluidity does not intervene—a result with profound consequences for the thermal, rotational, and oscillatory behavior of magnetars. Because the dineutron is not bound in vacuum, cold dilute neutron matter cannot exhibit a proper BCS-BEC crossover. Nevertheless, owing to the strongly resonant behavior of the $nn$ interaction at low densities, neutron matter shows a precursor of the BEC state, as manifested in Cooper-pair correlation lengths being comparable to the interparticle distance. We make a systematic quantitative study of this type of BCS-BEC crossover in the presence of neutron fluid spin polarization induced by an ultrastrong magnetic field. We evaluate the Cooper-pair wave function, quasiparticle occupation numbers, and quasiparticle spectra for densities and temperatures spanning the BCS-BEC crossover region. The phase diagram of spin-polarized neutron matter is constructed and explored at different polarizations.

Figure 1

Phase diagram of neutron matter in the temperature-density plane for several spin polarizations $\alpha$ induced by magnetic fields. The BCS phase is naturally favored over the unpaired phase at lower temperatures and smaller polarizations. The red diamonds locate different points in the phase diagram at which some intrinsic features of low-density neutron matter have been evaluated.

Figure 2

Pairing gap as a function of temperature at constant density index ${log}_{10}(\rho /{\rho }_0)=-1.5$ for three choices of polarization parameter $\alpha$.

Figure 3

Pairing gap as a function of polarization at constant density index ${log}_{10}(\rho /{\rho }_0)=-1.5$ for selected reference temperatures.

Figure 4

Dependence of the kernel $K\left(k\right)$ on momentum (in units of Fermi momentum) for fixed ${log}_{10}(\rho /{\rho }_0)=-1,T=0.25$ MeV, and polarization values color coded in the inset.

Figure 5

Same as Fig. 4, but for ${log}_{10}(\rho /{\rho }_0)=-1.5$ and three polarization values color coded in the inset.

Figure 6

Same as Fig. 4, but for ${log}_{10}(\rho /{\rho }_0)=-2$ and more polarization values.

Figure 7

Dependence of the kernel $K\left(k\right)$ on momentum (in units of Fermi momentum) for fixed ${log}_{10}(\rho /{\rho }_0)=-1,\alpha =0.2$, and temperature values color coded in the inset.

Figure 8

Plots of $\mathrm{\Psi }\left(r\right)$ versus $r$ at fixed temperature $T=0.25$ MeV for three reference densities ${\log }_{10}(\rho /{\rho }_0)=-1$ (a), ${\log }_{10}(\rho /{\rho }_0)=-1.5$ (b), and ${\log }_{10}(\rho /{\rho }_0)=-2$ (c) and polarization values $\alpha =0$ (solid line), $0.1$ (dashed line), $0.2$ (dash-dotted), and 0.3 (dashed-double-dotted).

Figure 9

Same as Fig. 8, except $r^2{\left|\mathrm{\Psi }\left(r\right)\right|}^2$ is plotted versus $r$.

Figure 10

Dependence of the up-spin and down-spin neutron occupation numbers on momentum $k$ (in units of Fermi momentum) for fixed ${log}_{10}(\rho /{\rho }_0)=-1,T=0.25\mathrm{MeV}$, and polarization values color coded in the inset.

Figure 11

Same as Fig. 10, but for ${log}_{10}(\rho /{\rho }_0)=-1.5$ and an additional polarization value.

Figure 12

Same as Fig. 11, but for ${log}_{10}(\rho /{\rho }_0)=-2$ and an additional polarization value.

Figure 13

Dependence of the up-spin and down-spin neutron occupation numbers on momentum $k$ (in units of Fermi momentum) for fixed ${log}_{10}(\rho /{\rho }_0)=-1.5,\alpha =0.2$, and temperatures color coded in the inset.

Figure 14

Dispersion relations for quasiparticle spectra in the BCS condensate, as functions of momentum (in units of Fermi momentum) for three densities ${\log }_{10}(\rho /{\rho }_0)=-1$ (a), ${\log }_{10}(\rho /{\rho }_0)=-1.5$ (b), and ${\log }_{10}(\rho /{\rho }_0)=-2$ (c). The polarization values are $\alpha =0$ (solid line), $0.1$ (dashed line), $0.2$ (dash-dotted), 0.3 (dashed-double-dotted). For each polarization, the upper branch corresponds to the $E_{+}$ solution, the lower branch to the $E_{-}$ solution.

Figure 15

Magnetic field required to create a specified spin polarization as a function of the density for two polarization values $\alpha =0.1$ (a) and 0.2 (b) and temperatures $T=0.25$ (solid line), 0.5 (dashed line), and 0.75 (dash-dotted line).

Figure 16

Same as Fig. 15 for two temperatures $T=0.25\mathrm{MeV}$ (a) and 0.5 MeV (b) and for several polarizations $\alpha =0$ (solid line), 0.2 (dashed line), 0.3 (dash-dotted line).

Figure 17

Ratio of magnetic energy to temperature as a function of density for two polarization values $\alpha =0.1$ (a) and 0.2 (b) and temperatures $T=0.25$ (solid line), 0.5 (dashed line), and 0.75 (dash-dotted line).

Figure 18

Ratio of magnetic energy to temperature as a function of density for two temperatures $T=0.25\mathrm{MeV}$ (a) and 0.5 MeV (b) and for several polarizations $\alpha =0$ (solid line), 0.2 (dashed line), 0.3 (dash-dotted line).

Figure 19

Unpairing magnetic field as a function of density (in units of ${\rho }_0$) for $T=0.05$ (solid line) and $T=0.5\mathrm{MeV}$ (dashed line).