The Casimir effect for fermionic currents in conical rings with applications to graphene ribbons

TL;DR

This paper analyzes how boundaries and topology influence vacuum charge and current densities for fermions in conical geometries, with applications to graphene ribbons, revealing effects of magnetic flux, boundary conditions, and zero modes.

Contribution

It provides a detailed study of vacuum expectation values of charge and current densities in conical rings with magnetic flux, considering various boundary conditions and their implications for graphene ribbons.

Findings

Charge and current densities increase with planar angle deficit.

Edge contributions to densities are explicitly calculated.

Discontinuities in densities occur at half-integer flux ratios due to zero modes.

Abstract

We investigate the combined effects of boundaries and topology on the vacuum expectation values (VEVs) of the charge and current densities for a massive 2D fermionic field confined on a conical ring threaded by a magnetic flux. Different types of boundary conditions on the ring edges are considered for fields realizing two inequivalent irreducible representations of the Clifford algebra. The related bound states and zero energy fermionic modes are discussed. The edge contributions to the VEVs of the charge and azimuthal current densities are explicitly extracted and their behavior in various asymptotic limits is considered. On the ring edges the azimuthal current density is equal to the charge density or has an opposite sign. We show that the absolute values of the charge and current densities increase with increasing planar angle deficit. Depending on the boundary conditions, the VEVs are continuous or discontinuous at half-integer values of the ratio of the effective magnetic flux to the flux quantum. The discontinuity is related to the presence of the zero energy mode. By combining the results for the fields realizing the irreducible representations of the Clifford algebra, the charge and current densities are studied in parity and time-reversal symmetric fermionic models. If the boundary conditions and the phases in quasiperiodicity conditions for separate fields are the same the total charge density vanishes. Applications are given to graphitic cones with edges (conical ribbons).