Dirac fermions on a surface with localized strain

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.

💡 Research Summary

The paper investigates how a localized Gaussian deformation influences massless Dirac fermions confined to a two‑dimensional curved surface, with a focus on Dirac materials such as graphene. The authors treat both out‑of‑plane (h(r)=h₀ e⁻ʳ²ᐟᵇ²) and in‑plane (uᵣ) displacements within classical elasticity theory, introducing the Lamé coefficients λ and μ to describe the mechanical response. The strain tensor u_{μν}=½(∂μ u_ν+∂ν u_μ+2∂μ h ∂ν h) modifies the metric g{μν}=δ{μν}+2u{μν}, leading to position‑dependent radial (g{rr}) and angular (g_{θθ}) components. Numerical plots show that increasing λ tends to compress the surface while increasing μ tends to stretch it, producing opposite contributions to the metric.

From the curved metric the authors construct the vielbeins e_a^μ and the spin connection ω_{abμ}. The θ‑component of the spin connection, ω_{12θ}, defines an effective geometric vector S_θ whose curl yields a pseudo‑magnetic field B that is proportional to the scalar curvature (B=R/2 in two dimensions). This pseudo‑field acts as an attractive potential near the deformation centre and becomes repulsive at larger radii, forming a finite barrier that vanishes asymptotically. The associated geometric potential Γ_θ(r)=¼ ω_{12θ}√g_{θθ} enters the Dirac Hamiltonian as an additional term.

The Dirac Hamiltonian in polar coordinates (with θ fixed to zero for simplicity) reads
H = –iħv_F