Effect of magnetic field on the electronic properties of an $\alpha$-$T_3$ ring

We consider a quantum ring of a certain radius R built from a sheet of the $\alpha$-$T_3$ lattice and solve for its spectral properties in presence of an external magnetic field. The energy spectrum consists of a conduction band, a valence band and a zero energy flat band, all having a number of discrete levels therein which can be characterized by the angular momentum quantum number, m. The energy levels in the flat band are infinitely degenerate irrespective of the value of $\alpha$. We reveal a two-fold degeneracy of the levels in the conduction band as well as in the valence band for $\alpha$ = 0 and $\alpha$ = 1. However, the m = 0 level for $\alpha$ = 1 is an exception. Corresponding to an intermediate value of $\alpha$, namely, 0 <$\alpha$< 1, the energy levels become nondegenerate. The scenario remains unaltered when the ring is threaded by a magnetic flux which is an integer multiple of the flux quantum. We also calculate the persistent current which exhibits quantum oscillations as a function of the magnetic field with a period of one flux quantum at a particular Dirac point, which is often referred to as a valley. The total current oscillates with a periodicity of one flux quantum for any intermediate value of $\alpha$. We have also explored the effect of a mass term (that breaks the sublattice symmetry) in the Hamiltonian. In the absence of a magnetic field, the energy levels in the flat band become dispersive, except for the m = 0 level in the case of $\alpha$ = 1. In presence of the field, each of the flat band levels becomes dispersive for any $\alpha \neq$ 0. Finally, we also see the effect of the mass term on the behaviour of the persistent current, which shows periodicity of one flux quantum, but the total current remains finite for all values of $\alpha$.