Electronic Effects of a Twisted Graphene Catenoid Bridge
The interplay between non-trivial background geometries and quantum dynamics has emerged as a powerful tool to tailor the electronic properties of 2D materials. In this work, we investigate the effective quantum dynamics of massless Dirac fermions confined to a twisted graphene structure called a catenoid bridge, which connects two single-layer sheets. By adopting a continuum approach, where the electron dynamics is governed by a purely covariant curved Dirac equation, we obtain the effective Hamiltonian containing both curvature and twist interactions. We found that the torsion modifies the electronic states by producing a geometric phase on the wave function. In addition, the twist also deforms the surface geometry, which leads to a new geometric term in the effective Hamiltonian. This twist potential enhances the barrier around the catenoid throat, which increases the suppression of the inter-layer transmission coefficient. Like the spin-curvature interaction, the spin-twist term has a chiral dependence which is invariant under a combined parity and spin flip transformation. As a result, the electronic states can be restricted to the upper or lower layer. These findings provide valuable insights into how mechanical deformations can be harnessed to control quantum transport in graphene-based wormhole architectures.