Landau levels and snake states of pseudo-spin-1 Dirac-like electrons in gapped Lieb lattices

TL;DR

This paper investigates the electronic structure of a Lieb lattice with spin-1 Dirac-like quasiparticles, revealing flat bands, Landau levels, and snake states under magnetic fields, advancing understanding of topological and flat-band phenomena.

Contribution

It introduces a detailed analysis of Landau levels and snake states in a Lieb lattice with spin-1 Dirac quasiparticles, including exact solutions and field-induced interface states.

Findings

Identification of flat bands between dispersion bands.

Exact determination of Landau levels via polynomial equations.

Existence of snake states at magnetic field interfaces.

Abstract

This work reports the three-band structure associated with a Lieb lattice with arbitrary nearest and next-nearest neighbors hopping interactions. For specific configurations, the system admits a flat band located between two dispersion bands. Three inequivalent Dirac valleys are identified so that the quasi-particles are effectively described by the spin-1 Dirac-type equation. Under external homogeneous magnetic fields, the Landau levels are exactly determined as the third-order polynomial equation for the energy can be solved using Cardano's formula. It is also shown that an external anti-symmetric field promotes the existence of current-carrying states, so-called snake states, confined at the interface where the external field changes its sign.