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Torsional oscillations of magnetized neutron stars have been well studiedsince they may be excited in magnetar starquakes and relevant to the observedquasiperiodic oscillations in the magnetar giant flares. In the crustal regionof a magnetar, the strong magnetic field can alter the equation of state andcomposition of the crust due to the Landau-Rabi quantization of electronmotion. In this paper, we study this effect on the torsional oscillation modesof neutron stars with mixed poloidal-toroidal magnetic fields in generalrelativity under the Cowling approximation. Furthermore, the inner and outercrusts are treated consistently based on the nuclear-energy density functionaltheory. Depending on the magnetic-field configurations, we find that theLandau-Rabi quantization of electron can change the frequencies of thefundamental torsional oscillation mode of $1.4 M_\odot$ neutron star modelswith a normal fluid core by about $10\%$ when the magnetic field strength atthe pole reaches the order of $10^{16}$ G. The shift can even approach $20\%$at a field strength of $10^{15}$ G for neutron stars with a simple model ofsuperconducting core where the magnetic field is assumed to be expelledcompletely from the core and confined only in the crust.

Torsional oscillations of magnetized neutron stars: Impacts of Landau-Rabi quantization of electron motion