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

TL;DR

The paper develops a Darboux-transformation framework for spin-1 Dirac-type operators and leverages flat-band states at $E=0$ to construct Hermitian, physically meaningful partner Hamiltonians. By applying the method to the Lieb lattice's pseudospin-1 Dirac equation, it yields four explicit, reflectionless, inhomogeneous systems, each with distinct factorization-energy choices that generate bound states and specific spectral structures. The work clarifies how flat bands can stabilize Darboux-induced potentials and demonstrates a bridge between spin-1 lattice models and graphene-like effective theories, including reduced two-band limits. The results have potential implications for designing tunable, reflectionless transport and for connecting solvable Dirac models across lattice realizations.

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.

Figures (2)

• Figure 1: Lieb lattice with structure composed of nearest neighbor asymmetric shopping $\tau_{1}$ (solid line) and $\tau_{3}$ (double-line) between the atoms $A$ and $B$ in the $\hat{x}$-direction, as well as the hopping $\tau_{2}$ (dashed) and $\tau_{4}$ (double-dashed) for the atoms A and C along the $\hat{y}$-direction. Atoms $B$ and $C$ have a Haldane-like next-nearest neighbor interaction (dotted) denoted by $\pm i\lambda$. The sign is positive if the hooping happens counter-clockwise and negative otherwise. The shaded area depicts a unit cell.
• Figure 2: Components of the new matrix potential $\widetilde{V}(x)$ for the case $\Lambda=diag(\epsilon,0,-\epsilon)$ (a) and $\Lambda=diag(m,0,\epsilon)$ (b). Here, the set of parameters have been fixed as $\{\epsilon=0.75,\hbar=v_{f}=m=1\}$ (a) and $\{\epsilon=-0.25,\hbar=v_{f}=m=1\}$ (b).