Reflectionless pseudospin-1 Dirac systems via Darboux transformation and flat band solutions

TL;DR

This paper develops a method using Darboux transformation to construct exactly solvable pseudospin-1 Dirac systems with flat bands, enabling new models for quasi-particles in Lieb lattices with inhomogeneous hopping.

Contribution

It introduces a novel application of Darboux transformation utilizing flat band states to generate Hermitian pseudospin-1 Dirac Hamiltonians with flat bands.

Findings

Constructed new exactly solvable models with flat bands

Demonstrated applicability to Lieb lattice systems

Ensured Hermiticity of transformed Hamiltonians

Abstract

This manuscript explores the Darboux transformation employed in the construction of exactly solvable models for pseudospin-one particles described by the Dirac-type equation. We focus on the settings where a flat band of zero energy is present in the spectrum of the initial system. Using the flat band state as one of the seed solutions substantially improves the applicability of the Darboux transformation, for it becomes necessary to ensure the Hermiticy of the new Hamiltonians. This is illustrated explicitly in four examples, where we show that the new Hamiltonians can describe quasi-particles in Lieb lattice with inhomogeneous hopping amplitudes.