Quantum Geometry and Landau Levels of Quadratic Band Crossings

TL;DR

This paper explores how the quantum geometry of wave functions influences Landau level spectra in two-band systems with quadratic band crossings, revealing geometric parameters that determine LL shifts and energies.

Contribution

It introduces a geometric characterization of wave functions at quadratic band crossings and links these parameters to Landau level spectra and interband coupling effects.

Findings

Interband coupling parameters are determined by the projected elliptic image of wave functions.

The product of ellipse diameters sets the LL energy shift.

The ratio of diameters influences initial LL energies near QBCP.

Abstract

We study the relation between the quantum geometry of wave functions and the Landau level (LL) spectrum of two-band Hamiltonians with a quadratic band crossing point (QBCP) in two-dimensions. By investigating the influence of interband coupling parameters on the wave function geometry of general QBCPs, we demonstrate that the interband coupling parameters can be entirely determined by the projected elliptic image of the wave functions on the Bloch sphere, which can be characterized by three parameters, i.e., the major $d_1$ and minor $d_2$ diameters of the ellipse, and one angular parameter $\phi$ describing the orientation of the ellipse. These parameters govern the geometric properties of the system such as the Berry phase and modified LL spectra. Explicitly, by comparing the LL spectra of two quadratic band models with and without interband couplings, we show that the product of $d_1$ and $d_2$ determines the constant shift in LL energy while their ratio governs the initial LL energies near a QBCP. Also, by examining the influence of the rotation and time-reversal symmetries on the wave function geometry, we construct a minimal continuum model which exhibits various wave function geometries. We calculate the LL spectra of this model and discuss how interband couplings give LL structure for dispersive bands as well as nearly flat bands.