Massless Dirac Fermions in curved surfaces with localized curvature

TL;DR

This paper explores how localized curvature on curved surfaces influences the behavior of massless Dirac fermions, revealing that curvature causes wavefunction localization near bumps while states are free elsewhere.

Contribution

It introduces a geometric model for Dirac fermions on curved surfaces with localized curvature and numerically analyzes their energy spectrum and wavefunctions.

Findings

Energy spectrum is linear and discrete for Dirac fermions on curved surfaces.

Wavefunctions show increased probability density around curved bumps.

States are free waves far from the localized curvature.

Abstract

We investigate how a localized curvature affects the dynamics of massless Dirac fermions in a curved surface. We consider a smooth bump with axial symmetry, adopting two specific geometric models, namely a Gaussian and a volcano-like bumps. By considering a minimal coupling between the spinor and the surface geometry, described by the vielbeins and the spin connection, we study the behavior of the wave function over the surface. By using appropriate numerical methods, we find a linear discrete energy spectrum for the Dirac fermions and its corresponding wavefunctions when the Fermi velocity is considered. It turns out that, since the curvature vanishes asymptotically, the electron states are free waves far from the bumps, but around the curved points, the wave function increases its probability density.