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 p
eculiar 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.
The specific topology of the line centered square lattice (known also as the Lieb lattice) induces remarkable spectral properties as the macroscopically degenerated zero energy flat band, the Dirac cone in the low energy spectrum, and the peculiar Ho
fstadter-type spectrum in magnetic field. We study here the properties of the finite Lieb lattice with periodic and vanishing boundary conditions. We find out the behavior of the flat band induced by disorder and external magnetic and electric fields. We show that in the confined Lieb plaquette threaded by a perpendicular magnetic flux there are edge states with nontrivial behavior. The specific class of twisted edge states, which have alternating chirality, are sensitive to disorder and do not support IQHE, but contribute to the longitudinal resistance. The symmetry of the transmittance matrix in the energy range where these states are located is revealed. The diamagnetic moments of the bulk and edge states in the Dirac-Landau domain, and also of the flat states in crossed magnetic and electric fields are shown.
