Magic Fermions: Carroll and Flat Bands

TL;DR

This paper explores Carrollian fermions on null surfaces, constructs their algebraic representations, and links Carroll symmetry to flat band phenomena in condensed matter systems like twisted bilayer graphene.

Contribution

It introduces Carroll fermions, constructs their gamma matrices, and connects Carroll symmetry to flat band systems, including explicit examples like twisted bilayer graphene.

Findings

Carroll fermions can be constructed with explicit gamma matrices.

Carroll symmetry appears in systems with flat energy bands.

Connection established between Carroll fermions and magic-angle graphene.

Abstract

The Carroll algebra is constructed as the $c\to0$ limit of the Poincare algebra and is associated to symmetries on generic null surfaces. In this paper, we begin investigations of Carrollian fermions or fermions defined on generic null surfaces. Due to the availability of two different (degenerate) metrics on Carroll spacetimes, there is the possibility of two different versions of Carroll Clifford algebras. We consider both possibilities and construct explicit representations of Carrollian gamma matrices and show how to build higher spacetime dimensional representations out of lower ones. Actions for Carroll fermions are constructed with these gamma matrices and the properties of these actions are investigated. We show that in condensed matter systems where the dispersion relation becomes trivial i.e. the energy is not dependent on momentum and bands flatten out, Carroll symmetry generically appears. We give explicit examples of this including that of twisted bi-layer graphene, where superconductivity appears at so called magic angles and connect this to Carroll fermions.