Surface energy of magnetized superconducting matter in the neutron star cores. (arXiv:2208.02738v1 [nucl-th])
<a href="http://arxiv.org/find/nucl-th/1/au:+Kobyakov_D/0/1/0/all/0/1">D. N. Kobyakov</a>

In this paper, an effective field theory for proton superconductor (SC)
interacting with neutron superfluid (SF), both with scalar order parameters, is
developed and applied to the surface energy (SE) of a magnetized SC body in
neutron stars (NS). Essentially, the SE studied here differs from the nuclear
SE: here, the proton SF density decays to zero while the total proton density
is constant across the surface. Interactions between the condensates are
parameterized phenomenologically and their effects determined from calculations
of a planar SE as the ranges of parameters are varied. The critical
Ginzburg-Landau (GL) parameter $kappa_c$ which renders the SE equal to zero is
found analytically by noting that in a system with vanishing SE the
thermodynamic critical MF is equivalent to the upper critical MF. In the case
of weak coupling, $kappa_c$ is shown to be a linear function of SF-SF
density-density coupling, in agreement with the earlier results based on
asymptotic intervortex interactions. Numerical simulations corroborate our
analytical predictions. Coupling due to the mixed term arising from a scalar
product of gradients of the SF densities, which had been considered in the
earlier literature, is seen to have practically no effect on the
superconductivity type. However, this coupling does produce a frozen wave
packet of the SF neutron density localized at the surface. It is shown that the
leading contribution from the gradient coupling arises from a novel mixed
quantum pressure term, but still does not affect the planar SE. The present
calculations provide an initial map of superconductivity types in the
phenomenological effective field theory and will serve as a landmark for future
studies, which require microscopic calculations of the coupling parameters
introduced here phenomenologically.

In this paper, an effective field theory for proton superconductor (SC)
interacting with neutron superfluid (SF), both with scalar order parameters, is
developed and applied to the surface energy (SE) of a magnetized SC body in
neutron stars (NS). Essentially, the SE studied here differs from the nuclear
SE: here, the proton SF density decays to zero while the total proton density
is constant across the surface. Interactions between the condensates are
parameterized phenomenologically and their effects determined from calculations
of a planar SE as the ranges of parameters are varied. The critical
Ginzburg-Landau (GL) parameter $kappa_c$ which renders the SE equal to zero is
found analytically by noting that in a system with vanishing SE the
thermodynamic critical MF is equivalent to the upper critical MF. In the case
of weak coupling, $kappa_c$ is shown to be a linear function of SF-SF
density-density coupling, in agreement with the earlier results based on
asymptotic intervortex interactions. Numerical simulations corroborate our
analytical predictions. Coupling due to the mixed term arising from a scalar
product of gradients of the SF densities, which had been considered in the
earlier literature, is seen to have practically no effect on the
superconductivity type. However, this coupling does produce a frozen wave
packet of the SF neutron density localized at the surface. It is shown that the
leading contribution from the gradient coupling arises from a novel mixed
quantum pressure term, but still does not affect the planar SE. The present
calculations provide an initial map of superconductivity types in the
phenomenological effective field theory and will serve as a landmark for future
studies, which require microscopic calculations of the coupling parameters
introduced here phenomenologically.

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