On a Solution to the Dirac Equation with a Triangular Potential Well

TL;DR

This paper provides analytical solutions to the Dirac equation for massless fermions in a triangular potential well, revealing zero-energy modes and chiral anomaly structures relevant to topological materials and graphene nanoribbons.

Contribution

It introduces exact solutions for the Dirac equation with a triangular potential well, connecting quantum field theory anomalies to condensed matter systems.

Findings

Discovery of zero-energy modes in the system

Identification of gauge field-dependent chiral anomalies

Exact solutions expressed with new special functions

Abstract

Chiral anomalies resulting from the breaking of classical symmetries at the quantum level are fundamental to quantum field theory and gaining ever-growing importance in the description of topological materials in condensed matter physics. Here we present analytical solutions of the Dirac equation for massless 3+1 fermions confined to an infinite stripe and placed into a background gauge field forming a triangular potential well across the width of the stripe. Such an effective 1+1 system hosts zero-energy modes resulting in the gauge field-dependent chiral anomaly structure. This problem has a direct relation to a half-bearded graphene nanoribbon placed into an in-plane external electric field and offers it an exact solution in terms of new special functions that are similar but not reducible to Airy functions.