Magic fermions: Carroll and flat bands

The Carroll algebra is constructed as the c -> 0 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.

Automated summary

This paper investigates Carrollian fermions defined on generic null surfaces. Due to the existence of two different (degenerate) metrics in Carroll spacetimes, there are two possible versions of Carroll Clifford algebras. Both possibilities are considered and explicit representations of Carrollian gamma matrices are constructed. Actions for Carroll fermions are built using these gamma matrices and their properties are explored. The appearance of Carroll symmetry is shown in condensed matter systems where the dispersion relation becomes trivial and bands flatten out, such as twisted bi-layer graphene.