On Landau-Zener Transitions in Systems with QuadraticBand-Touching

Harmonic wavepacket motion observed in interband optical experiments with superlattices has confirmed Zener tunneling between coupled bands. Theoretical studies of irregular Bloch-Zener oscillations in various one-dimensional superlattice systems reveal that tunneling effects interweave oscillation frequencies around half the fundamental frequency. Building on the geometric transformation and topological equivalence between pairs of linear bands and quadratic band-touching points, as seen in the Lieb and Kagome lattices, we propose that Landau-Zener transitions (LZT) could also apply to quadratic band-touching points, provided a small bandgap exists between the coupled branches.



This study applies a coherent transport framework to examine Bloch oscillations in 2D systems. Through time evolution of the Hamiltonian, this framework enables precise numerical evaluations of current density under static fields, allowing us to dynamically explore the transformation of Bloch oscillations between Kagome and Lieb systems under applied strain. For strained Kagome systems, a linear expansion of a periodically modulated tight-binding model at two touching points demonstrates that the LZT scattering matrix maintains its characteristic structure even when strain effects are applied, suggesting that the scattering matrix used for linear crossing points is valid for the quadratic case as well.



Additionally, we examine the adiabatic transition mechanism in graphene—a linear band-crossing system—using both theoretical analysis based on the scattering matrix and numerical evaluations within the coherent transport framework. In both quadratic and linear band-touching cases, a zero bandgap prevents Zener tunneling between branches. However, introducing a small bandgap allows strong tunneling between electron and hole branches. Our numerical and theoretical models confirm LZT at quadratic band-touching points in Kagome system, producing two irregular Bloch oscillations in a wave-like pattern that is symmetric around half the regular Bloch oscillation frequency.