Topological properties of the mesoscopic graphene plaquette: Quantum spin Hall effect due to spin imbalance

We study the electronic properties of the confined honeycomb lattice in the presence of the intrinsic spin-orbit (ISO) interaction and perpendicular magnetic field, and report on uncommon aspects of the quantum spin Hall conductance corroborated by peculiar properties of the edge states. The ISO interaction induces two specific gaps in the Hofstadter spectrum, namely the 'weak' topological gap defined by Beugeling et al [Phys. Rev. B 86, 075118 (2012)], and spin-imbalanced gaps in the relativistic range of the energy spectrum. We analyze the evolution of the helical states with the magnetic field and with increasing Anderson disorder. The 'edge' localization of the spin-dependent states and its dependence on the disorder strength is shown. The quantum transport, treated in the Landauer-B\"{u}ttiker formalism, reveals interesting new plateaus of the quantum spin Hall effect (QSHE), and also of the integer quantum Hall effect (IQHE), in the energy ranges corresponding to the spin-imbalanced gaps. The properties of the spin-dependent transmittance matrix that determine the symmetries with respect to the spin, energy and magnetic field of the longitudinal and transverse resistance are shown.