Flat-band engineering of quasi-one-dimensional systems via supersymmetric transformations

TL;DR

The paper tackles the problem of spectrally engineering quasi-one-dimensional crystals described by low-energy Dirac dynamics. It introduces a systematic framework based on supersymmetric (Darboux) transformations to extend a known pseudo-spin-1/2 Dirac model to a nontrivial pseudo-spin-1 partner, enabling deliberate insertion of flat bands and discrete energies within a gapped spectrum. By applying the method to a generalized stub lattice—effectively decorating an SSH chain with a parallel, interacting chain—the authors provide explicit constructions for cases with two bound states plus a flat-band and with a single bound state plus a flat-band, with tunable in-gap energies and local control of hopping patterns. This approach yields a versatile, analytic route to spectral design in quasi-1D lattices, with potential applications across photonic, phononic, and cold-atom platforms where flat bands and bound states play a pivotal role.

Abstract

We introduce a systematic method to spectrally design quasi-one-dimensional crystal models described by the Dirac equation in the low-energy regime. The method is based on the supersymmetric transformation applied to an initially known pseudo-spin-1/2 model. This allows extending the corresponding susy partner so that the new model describes a pseudo-spin-1 system. The spectral design allows the introduction of a flat-band and discrete energies at will into the new model. The results are illustrated in two examples where the Su-Schriefer-Heeger chain is locally converted into a stub lattice.

Figures (8)

• Figure 1: (a) Sketch of the periodic pseudo-one-dimensional stub lattice with three atoms $A$, $B$, and $C$ per unitary cell (dashed rectangle). The hopping parameters are denoted by $t_1$, $t_2$, and $t_3$. (b) Dispersion relations \ref{['dispersion']} in terms of the hopping parameters and the on-site interactions $\mu_{A}$ and $\mu_B=\mu_C$, where $\Delta_{\pm}=\sqrt{(t_1 \pm t_2)^2+t_3^2+\frac{(\mu_A-\mu_C)^2}{4}}$.
• Figure 2: (Color online) Matrix potential terms $\widetilde{V}_{12}(x)$ (blue-solid) and $\widetilde{V}_{13}(x)$ (red-dashed)) obtained from (\ref{['model1interactions']}). Here, the parameters have been fixed to $A=m=0.8$, $\rho=1$, $\lambda=\frac{1}{2}\sqrt{A^2+m^2}$, combined with $\omega=2\omega_{crit}$ (a), $\omega=(1+10^{-2})\omega_{crit}$ (b), $\omega=-(1+10^{-6})\omega_{crit}$ (c), and d) $\omega=-(1+10^{-14})\omega_{crit}$ (d).
• Figure 3: Inhomogeneous hopping parameters $t_1=t_2+\widetilde{V}_{12}$ (red), $t_3=\widetilde{V}_{13}$ (orange) and $t_3$ (gray-dotted) for $\omega=\omega_c$ computed from (\ref{['omegacrit']}), (\ref{['vijcritmodel1']}), and (\ref{['equiv2']}). We fixed $t_2=1$, $m=1$, $A=1$, $\lambda=\frac{1}{2}\sqrt{A^2+m^2}$, and $\rho=1$ (left) and $\rho=0.5$ (right). In the insets, we depict the quasi-one-dimensional chain with the corresponding interactions (the thicker the line, the stronger the coupling), with $t_1$ red, $t_2$ black, and $t_3$ the vertical line.
• Figure 4: (color online) Matrix potential elements $\widetilde{V}_{12}(x)$, $\widetilde{V}_{13}(x)$, and the probability densities of the (normalized) bound states $|w^{\pm}|$, see the inset for the color scheme. The parameters have been fixed as left: $m_0=1$, $\lambda_0=0.5$, $\lambda=0.499$, $\kappa=0.04$, $\rho=0.06$, Right: $\kappa$ and $\rho$ are fixed as in (\ref{['speccase']}), $m_0=1$, $\lambda_0=0.7$, $\lambda=0.1$.
• Figure 5: (color online) Inhomogeneous hopping parameters $t_1=t_2+\widetilde{V}_{12}$ (red), $t_3=\widetilde{V}_{13}$ (orange) and $t_3$ (gray-dotted) as given by (\ref{['H2spec']}). The parameters are fixed as $\lambda_0=0.7$, $m_0=1$ and $\lambda=0.1$.