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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