Weyl-Gauge Symmetry of Graphene

TL;DR

This paper investigates the Weyl symmetry in graphene's low-energy Dirac theory, revealing geometric constraints for shape-dependent electronic properties and connecting these to structures like the Virasoro algebra and Liouville equation.

Contribution

It uncovers the local Weyl symmetry in graphene's Dirac theory and links geometric shapes of graphene to algebraic structures and relativistic-like behaviors.

Findings

Weyl symmetry constrains graphene's shape for consistent electronic density

Mathematical structures like Virasoro algebra emerge in the analysis

Connections to three-dimensional gravity and geometric profiles

Abstract

The conformal invariance of the low energy limit theory governing the electronic properties of graphene is explored. In particular, it is noted that the massless Dirac theory in point enjoys local Weyl symmetry, a very large symmetry. Exploiting this symmetry in the two spatial dimensions and in the associated three dimensional spacetime, we find the geometric constraints that correspond to specific shapes of the graphene sheet for which the electronic density of states is the same as that for planar graphene, provided the measurements are made in accordance to the inner reference frame of the electronic system. These results rely on the (surprising) general relativistic-like behavior of the graphene system arising from the combination of its well known special relativistic-like behavior with the less explored Weyl symmetry. Mathematical structures, such as the Virasoro algebra and the Liouville equation, naturally arise in this three-dimensional context and can be related to specific profiles of the graphene sheet. Speculations on possible applications of three-dimensional gravity are also proposed.