Bilayer silicene in a magnetic field: spin localization and its effect on the integer and fractional Hall conductivity*

The Hall conductivity of many condensed matter systems presents step structure when a uniform perpendicular magnetic field is applied. We report the emergence of fractions of the Hall conductivity due to the grouping in pairs of an electron with a bilayer silicene sublattice site due to strong spin-orbit coupling. The collective behavior of this pairing causes the conductivity to exhibit steps of e²/h or 4/5 e²/h high for the low-lying spectrum, depending on the dominating sublattices of Landau levels. This coexistence of integer and fractional Quantum Hall States arises from the interplay of lattice geometry, atomic interaction, spin-orbit coupling, and external magnetic field. We also report the significant effect of an external electric field on the evolution of integer and fractional quantum Hall effects.