Multifold degeneracy points of quantum systems and singularities of matrix varieties

TL;DR

This paper investigates the geometric and algebraic structure of multifold degeneracy points in quantum systems, providing an upper bound on resulting Weyl points by analyzing singularities in matrix varieties using algebraic geometry.

Contribution

It introduces a method to compute an upper bound on Weyl points from multifold degeneracies by analyzing matrix variety singularities, bridging physics and mathematics.

Findings

Derived an upper bound for Weyl points from multifold degeneracies.

Analyzed the multiplicity of singular points in matrix varieties.

Connected algebraic geometry tools with quantum degeneracy analysis.

Abstract

Parameter-dependent quantum systems often exhibit energy degeneracy points, whose comprehensive description naturally lead to the application of methods from singularity theory. A prime example is an electronic band structure where two energy levels coincide in a point of momentum space. It may happen, and this case is the focus of our work, that three or more levels coincide at a parameter point, called multifold degeneracy. Upon a generic perturbation, such a multifold degeneracy point is dissolved into a set of Weyl points, that is, generic two-fold degeneracy points. In this work, we provide an upper bound to the number of Weyl points born from the multifold degeneracy point. To compute this upper bound, we describe the geometric degeneracy variety in the space of complex matrices. We compute its multiplicity at certain singular points corresponding to a multifold degeneracy, and the multiplicity of holomorphic map germs with respect to this variety. Our work covers physics and mathematics aspects in detail, and attempts to bridge the two disciplines and communities. For self-containedness, we survey examples of multi-fold degeneracies in quantum systems and condensed-matter physics, as well as the established tools of local algebraic geometry that we use to identify the upper bound.