Three-fold Weyl points in the Schr\"odinger operator with periodic potentials

TL;DR

This paper proves the existence of three-fold Weyl points in the spectrum of 3D Schrödinger operators with periodic potentials, revealing complex conical intersections involving multiple energy bands and new symmetry conditions.

Contribution

It introduces general conditions and new symmetry techniques to establish the existence of 3-fold Weyl points in 3D periodic Schrödinger operators, extending previous spectral analysis.

Findings

Existence of 3-fold Weyl points under broad potential conditions

Identification of symmetries necessary for conical intersections

Numerical simulations confirming theoretical results

Abstract

Weyl points are degenerate points on the spectral bands at which energy bands intersect conically. They are the origins of many novel physical phenomena and have attracted much attention recently. In this paper, we investigate the existence of such points in the spectrum of the 3-dimensional Schr\"{o}dinger operator $H = - \Delta +V(\textbf{x})$ with $V(\textbf{x})$ being in a large class of periodic potentials. Specifically, we give very general conditions on the potentials which ensure the existence of 3-fold Weyl points on the associated energy bands. Different from 2-dimensional honeycomb structures which possess Dirac points where two adjacent band surfaces touch each other conically, the 3-fold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between. To ensure the 3-fold and 3-dimensional conical structures, more delicate, new symmetries are required. As a consequence, new techniques combining more symmetries are used to justify the existence of such conical points under the conditions proposed. This paper provides comprehensive proof of such 3-fold Weyl points. In particular, the role of each symmetry endowed to the potential is carefully analyzed. Our proof extends the analysis on the conical spectral points to a higher dimension and higher multiplicities. We also provide some numerical simulations on typical potentials to demonstrate our analysis.