Dirac fermions on a surface with localized strain

TL;DR

The paper studies massless Dirac fermions on a curved two-dimensional surface subjected to a localized Gaussian deformation, incorporating both out-of-plane and in-plane displacements via elasticity theory. It derives a curved-space Dirac equation with a spin connection, revealing a geometric potential and a position-dependent Fermi velocity, with Lamé coefficients modulating curvature and state localization. Analytical and numerical results show bound states near the deformation and identify a geometric Aharonov-Bohm phase from spinor holonomy; under an external magnetic field, the deformation acts as a confining region that hosts localized Landau levels whose spacing is influenced by the elastic parameters. The findings illuminate strain-induced electronic effects in Dirac materials like graphene and point to experimental probes such as STM and interference-based ring geometries to detect strain-generated fictitious flux.

Abstract

We study the influence of a localized Gaussian deformation on massless Dirac fermions confined to a two-dimensional curved surface. Both in-plane and out-of-plane displacements are considered within the framework of elasticity theory. These deformations couple to the Dirac spinors via the spin connection and the vielbeins, leading to a position-dependent Fermi velocity and an effective geometric potential. We show that the spin connection contributes an attractive potential centered on the deformation and explore how this influences the fermionic density of states. Analytical and numerical solutions reveal the emergence of bound states near the deformation and demonstrate how the Lamé coefficients affect curvature and state localization. Upon introducing an external magnetic field, the effective potential becomes confining at large distances, producing localized Landau levels that concentrate near the deformation. A geometric Aharonov-Bohm phase is identified through the spinor holonomy. These results contribute to the understanding of strain-induced electronic effects in Dirac materials, such as graphene.

Figures (19)

• Figure 1: Behavior of variations in $\lambda$ and $\mu$, given $h_0$=5, $b$=1, $\mu$=1.
• Figure 2: Field of vectors defined from $u_r$ on the Gaussian surface.
• Figure 3: Variations of the radial component of the metric with $\lambda$ and $\mu$. Where $\delta\lambda=\delta\mu=1$. Here $h_0=b=1$.
• Figure 4: Variations of the angular component of the metric with $\lambda$ and $\mu$. Where $\delta\lambda=\delta\mu=1$. Here $h_0=b=1$.
• Figure 5: Variations $\delta\lambda=\delta\mu=1$. For the continuous curve $h_0=1$, $b=1$, $\lambda=\mu=1$.