Helicity -- from Clifford to Graphene

TL;DR

This paper explores the relationship between two definitions of helicity from physics and mathematics, proving their connection and applying this insight to the Hamiltonian of graphene, revealing a novel link between abstract algebra and material science.

Contribution

It establishes a formal link between the mathematical and physical definitions of helicity and applies this to analyze the graphene Hamiltonian.

Findings

Mathematical helicity implies physical helicity.

The graphene Hamiltonian is proportional to a trace of an operator related to helicity.

Uncovers a novel connection between Clifford algebra and condensed matter physics.

Abstract

We investigate two seemingly disjoint definitions of helicity, one commonly used in particle physics, the other one used when studying bilinear covariants of Clifford algebras. We can prove that the `mathematical' definition of helicity implies its `physical' counterpart. As an unexpected application of our result we show that the Hamiltonian describing the one-layer superconductor Graphene is proportional to the trace of an operator that is used in the `mathematical' definition of helicity.