SUSY transformation as the coupler of non-interacting systems

TL;DR

The paper addresses spectral engineering in quasi-one-dimensional lattices by coupling two initially non-interacting chains described by a pseudo-spin-1 Dirac Hamiltonian using the Darboux (SUSY) transformation. By constructing an intertwining relation $L H = ilde{H} L$, the authors generate a coupled Hamiltonian $ ilde{H}$ with inter-chain hoppings that locally realize a saw-chain structure and inherit a flat band from the initially decoupled component. They present two explicit realizations (Model I and Model II) in which the flat band energy is tunable to $ obreak \lambda$ through transformation parameters $(m, obreak obreak obreak obreak obreak obreak ). The work provides a solvable framework for spectral engineering in low-dimensional lattices and suggests avenues for exploring topological properties and experimental implementations of SUSY-generated couplings.

Abstract

Quasi-one-dimensional chains of atoms can be effectively described by one-dimensional Dirac-type equation. Crystal structure of the chain is reflected by pseudo-spin of the quasi-particles. In the article, we present a simple framework where supersymmetric transformation is utilized to generate an interaction between two, initially non-interacting systems described by pseudo-spin-one Dirac-type equation. In the presented example, the transformation converts two asymptotically non-interacting atomic chains into a saw chain locally. The model possesses a flat band whose energy can be fine-tuned deliberately.

Figures (4)

• Figure 1: The saw chain. The upper line formed by the atoms $C_j$ corresponds to forms the $C$-chain. The lower line composed of the atoms $A_j$ and $B_j$ forms the $AB$-chain, $j\in\mathbb{Z}$
• Figure 2: The energy bands $E_j(k)$, $k=1,2,3$, as the solutions of (\ref{['det']}). We set $a=1$, $t_{AA}=0$, $t_{BB}=0$, $t_{AB}=\tilde{t}_{AB}=1$, $t_{AC}=0.2$, $t_{BC}=0.01$ and left: $t_{CC}=0.2$, right: $t_{CC}=1/500$, $a_2=0$ in this case. There is quasi-momentum $k$ on the horizontal axis whereas the vertical axis corresponds to the energy.
• Figure 3: The Hamiltonian (\ref{['H']}) describes two parallel non-interacting chains of atoms, $C$ chain and $AB$ chain. Susy transformation generates the new potential terms (\ref{['vij']}) that correspond to spatially dependent couplings $t_{AC}(x)$ and $t_{BC}(x)$ between the two atomic chains. It can also make the coupling $t_{AB}$ inhomogeneous.
• Figure 4: The two bands of continuum energies with the orange and green threshold enclose the flat band energy $\lambda$.