On the Sector Counting Lemma

TL;DR

This paper proves a sector counting lemma for a class of smooth, strictly convex Fermi surfaces on the plane, extending previous results and enabling the construction of interacting many-fermion models for doped graphene.

Contribution

It generalizes existing sector counting lemmas to a broader class of Fermi surfaces, including quasi-symmetric ones, facilitating new models in condensed matter physics.

Findings

Proved a sector counting lemma for $C^2$-differentiable, strictly convex Fermi surfaces.

Extended previous lemmas to include quasi-symmetric Fermi surfaces.

Enables construction of many-fermion models for doped graphene.

Abstract

In this short note we prove a sector counting lemma for a class of Fermi surface on the plane which are $C^2$-differentiable and strictly convex. This result generalizes the one proved in \cite{FKT} for the class of $C^{2+r}$-differentiable, $r\ge3$, strictly convex and strongly asymmetric Fermi surfaces, and the one proved in \cite{FMRT} and \cite{BGM1}, for the class of $C^2$-differentiable, strictly convex and central symmetric Fermi surfaces. This new sector counting lemma can be used to construct interacting many-fermion models for the doped graphene, in which the Fermi surface is extended and quasi-symmetric.