Chiral states induced by symmetry-breaking in $\alpha-T_3$ lattices: Magnetic field effect

The sublattice-symmetry breaking in the $\alpha-T_3$ lattice leads to a bandgap opening. A defect line in the substrate on which the $\alpha-T_3$ lattice is deposited can be viewed as a topological change in the substrate that induces translational in-plane symmetry breaking, resulting in mid-gap states. These topologically protected states are confined along the defect line and exhibit preferential directional motion, with different signs for the different Dirac valleys. Within this context, we investigate how these unidirectional interface chiral states are affected in the presence of a perpendicular magnetic field and how they can be tuned by varying the controlling system parameter $\alpha$. The latter tunes the $\alpha-T_3$ structure from a honeycomb-like lattice ($\alpha=0$) to a dice lattice ($\alpha=1$). Our theoretical framework is based on the continuum approximation described by a $3\times 3$ matrix Hamiltonian with a sublattice symmetry-breaking term given by $\Delta(x) diag(1,\quad -1,\quad 1)$, assuming $\Delta(x)$ as a kink-like mass potential profile. Results for dispersion relations and wavefunction distributions for different $\alpha$ parameters and magnetic field amplitudes are discussed. We demonstrate lifting of Landau levels degeneracy and of valley degeneracy. Our findings pave the way for proposing valley filter devices based on any evolutionary stage between the honeycomb-like and dice lattice structures of the $\alpha-T_3$ phase, controlled by external fields.