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 Hofstadter-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.

Spectral and transport properties of electrons in confined phosphorene systems are investigated in a five hopping parameter tight-binding model, using analytical and numerical techniques. The main emphasis is on the properties of the topological edge states accommodated by the quasi-flat band that characterizes the phosphorene energy spectrum.We show, in the particular case of phosphorene, how the breaking of the bipartite lattice structure gives rise to the electron-hole asymmetry of the energy spectrum. The properties of the topological edge states in the zig-zag nanoribbons are analyzed under different aspects: degeneracy, localization, extension in the Brillouin zone, dispersion of the quasi-flat band in magnetic field.The finite-size phosphorene plaquette exhibits a Hofstadter-type spectrum made up of two unequal butterflies separated by a gap, where a quasi-flat band composed of zig-zag edge states is located.The transport properties are investigated by simulating a four-lead Hall device (importantly, all leads are attached on the same zig-zag side), and using the Landauer-Büttiker formalism. We find out that the chiral edge states due to the magnetic field yield quantum Hall plateaus, but the topological edge states in the gap do not support the quantum Hall effect and prove a dissipative behavior. By calculating the complex eigenenergies of the non-Hermitian effective Hamiltonian that describes the open system (plaquette+leads), we prove the superradiance effect in the energy range of the quasi-flat band, with consequences for the density of states and electron transmission properties.

We investigate new properties of the Dirac electrons in the finite graphene sample under perpendicular magnetic field that emerge when an in-plane electric bias is also applied. The numerical analysis of the Hofstadter spectrum and of the edge-type wave functions evidentiate the presence of shortcut edge states that appear under the influence of the electric field. The states are characterized by a specific spatial distribution, which follows only partially the perimeter, and exhibit ridges that connect opposite sides of the graphene plaquette. Two kinds of such states have been found in different regions of the spectrum, their particular spatial localization being shown along with the diamagnetic moments that reveal their chirality. By simulating a four-lead Hall device, we investigate the transport properties and observe new, unconventional plateaus of the integer quantum Hall effect, which are associated with the presence of the shortcut edge states. The contributions of the novel states to the conductance matrix that determine the new transport properties are shown. The shortcut edge states resulting from the splitting of the n=0 Landau level represent a special case, giving rise to non-trivial transverse and longitudinal resistance.