Superfluid Band Theory for the Rod Phase in the Magnetized Inner Crust Matter: Entrainment, Spin-orbit, Spin-triplet Pairing

TL;DR

This work develops a two-dimensional superfluid band theory for the magnetized rod phase of neutron-star inner crust matter, combining Skyrme energy density functionals, Bloch-band theory, and Hartree-Fock-Bogoliubov pairing under finite magnetic fields. It shows that strong fields ($B_\star \sim 10^3$) substantially enhance the neutron effective mass via band-structure effects, indicating strong entrainment, and that spin-orbit coupling enables spin polarization even for field directions where it would be absent in simpler geometries. Spin-triplet pairing components emerge coherently: a polarization-driven rank-0 condensate appears irrespective of the pairing channel, while a rank-2 condensate requires explicit spin-triplet interactions; band-structure effects modify magnitudes but leave the qualitative trends intact. These microscopically self-consistent results have implications for neutron-star crust dynamics in magnetars and motivate extensions to time-dependent and fully three-dimensional crystalline structures for more complete astrophysical modeling.

Abstract

The inner crust of neutron stars hosts a rich variety of nuclear phenomena and provides a unique environment for exploring microscopic nuclear properties relevant to diverse astrophysical observations. Particularly magnetars, which possess extremely strong magnetic-fields, have attracted increasing attention in connection with nuclear spin dynamics and unconventional pairing correlations. This work is dedicated to develop a comprehensive theoretical framework to describe the structures and properties of two-dimensional (rod-phase) matter in the neutron star inner crust, incorporating band-structure effects, neutron spin-triplet pairing, and strong magnetic-fields on an equal footing. The main results of this study can be summarized as follows. In the first place, the magnetic-fields of the order of $10^{16}\,$G are found to substantially enhance the neutron effective mass by a factor of approximately $1.5$, indicating a significant modification of entrainment properties in strongly magnetized crustal matter. In the second place, while the overall behavior of pairing phase transitions is qualitatively similar to that observed in one-dimensional systems studied previously, the present two-dimensional calculations reveal a nontrivial role of the spin-orbit interaction in inducing spin-polarization under magnetic fields. In the third place, concerning spin-triplet superfluidity, the rank-0 component is shown to emerge as a consequence of magnetic-field-induced spin-polarization, irrespective of the presence of spin-triplet pairing interactions, whereas the rank-2 component appears only when the corresponding interaction channel is included.

Figures (6)

• Figure 1: Calculated neutron effective masses as a function of magnetic-field strength. The red solid line indicates the results when the magnetic direction is along $x$-axis, while the green dash-dotted line is dedicated to $z$-axis result.
• Figure 2: The energy band diagram along the Brillouin zone path of the square lattice. The left figure shows the results of no magnetic-field case, whereas the right is for $B_z = 1000B_\star$ case. The black dashed line in two figures indicate the chemical potentials of neutrons.
• Figure 3: The averaged pairing gaps of spin-singlet neutron superfluidity for a range of magnetic-field strengths. The left figure shows the result for the case where the magnetic-field is imposed along $x$-axis, while the right one is for $z$-axis case. In both figures, the results both when the spin-orbit EDF is present and absent is demonstrated.
• Figure 4: The same with Fig. \ref{['fig:Delta_singlet']}, but for the spin-polarization profiles.
• Figure 5: The pairing condensation of each component of spin-triplet neutron superfluidity. In the left figure, the result when the magnetic-field is imposed along $x$-axis is shown, while the right figure is for the $z$-axis case. In both figures, the red solid line, green dotted line, violet dash-dotted line correspond to the sequences of rank-0, rank-1 and rank-2 components, respectively.