Bloch Oscillations and Landau-Zener Transitions in Flat-Band Lattices with Quadratic and Linear Band Touchings

TL;DR

This work investigates how Landau-Zener transitions shape Bloch oscillations in flat-band lattices by comparing Kagome (quadratic touching) and Lieb (flat plus two linear bands) geometries. It employs a coherent transport framework to compute time-resolved currents and a scattering-matrix approach near band touchings, linking irregular BO frequencies to interband tunneling strength. In Kagome, a finite gap at the quadratic touching yields two irregular BO frequencies around $rac{ ext{BO fundamental}}{2}$, with a breakdown of simple LZT descriptions at stronger coupling. In Lieb, independent LZT processes between each dispersive band and the flat band produce a linearly gap-dependent irregular BO response, with strain gradually morphing the Kagome pattern into the Lieb pattern while preserving the LZT mechanism. The results illuminate nonadiabatic quantum transport in three-level flat-band systems and offer insights for strain-tunable topological photonic/quantum devices, with robust LZT behavior under geometric transformations between lattice types.

Abstract

Bloch oscillations (BOs) describe the coherent oscillatory motion of electrons in a periodic lattice under a constant external electric field. Deviations from pure harmonic wave packet motion or irregular Bloch oscillations can occur due to Zener tunneling (Landau-Zener Transitions or LZTs), with oscillation frequencies closely tied to interband coupling strengths. Motivated by the interplay between flat-band physics and interband coupling in generating irregular BOs, here we investigate these oscillations in Lieb and Kagome lattices using two complementary approaches: coherent transport simulations and scattering matrix analysis. In the presence of unavoidable band touchings, half-fundamental and fundamental BO frequencies are observed in Lieb and Kagome lattices, respectively -- a behavior directly linked to their distinct band structures. When avoided band touchings are introduced, distinct BO frequency responses to coupling parameters in each lattice are observed. Scattering matrix analysis reveals strong coupling and potential LZTs between dispersive bands and the flat band in Kagome lattices, with the quadratic band touching enhancing interband interactions and resulting in BO dynamics that is distinct from systems with linear crossings. In contrast, the Lieb lattice -- a three level system -- shows independent coupling between the flat band and two dispersive bands, without direct LZTs occurring between the two dispersive bands themselves. Finally, to obtain a unifying perspective on these results, we examine BOs during a strain-induced transition from Kagome to Lieb lattices, and link the evolution of irregular BO frequencies to changes in band connectivity and interband coupling.

Figures (10)

• Figure 1: (a) The top view of $3 \times 3 \times 1$ superlattice structure, (b) corresponding band diagram, and (c) numerical current evolution over an extended time range computed using RK45 and symplectic ODE solvers, for the Kagome graphene lattice.
• Figure 2: Bloch oscillation frequencies as a function of the coupling parameter $\delta$ in the graphene system (see Sec. \ref{['subsec:scattering_matrix']}). The frequencies correspond to oscillations in the current density: (a) calculated using the coherent transport framework, (b) obtained from the linearly expanded tight binding model, and (c) comparison between theoretical predictions and numerical results.
• Figure 3: (a) Geometric structure of the Kagome lattice, with the red dashed line marking the unit cell, and (b) its electronic band diagram.
• Figure 4: (a-c) Band structure of the Kagome lattice calculated from Equation \ref{['path']} (black lines) and Equation \ref{['expansion']} (red dashed lines) for coupling parameter $\delta$ = 0, 0.1 and 0.2, respectively.
• Figure 5: Bloch oscillation frequencies as a function of $\delta$ in the Kagome system: (a) extracted from the coherent transport framework, (b) based on the linearly expanded tight-binding model, and (c) comparison between numerical and theoretical results.