Fermionic current from topology and boundaries with applications to higher-dimensional models and nanophysics

We investigate combined effects of topology and boundaries on the vacuum expectation value (VEV) of the fermionic current in the space with an arbitrary number of toroidally compactified dimensions. As a geometry of boundaries we consider two parallel plates on which the fermion field obeys bag boundary conditions. Along the compact dimensions, periodicity conditions are imposed with arbitrary phases. In addition, the presence of a constant gauge field is assumed. The nontrivial topology gives rise to an Aharonov-Bohm effect for the fermionic current induced by the gauge field. It is shown that the VEV of the charge density vanishes and the current density has nonzero expectation values for the components along compact dimensions only. The latter are periodic odd functions of the magnetic flux with the period equal to the flux quantum. In the region between the plates, the VEV of the fermionic current is decomposed into pure topological, single plate, and interference parts. For a massless field the single plate part vanishes and the interference part is distributed uniformly. The corresponding results are generalized for conformally flat spacetimes. Applications of the general formulas to finite-length carbon nanotubes are given within the framework of the Dirac model for quasiparticles in graphene. In the absence of the magnetic flux, two sublattices of the honeycomb graphene lattice yield opposite contributions and the fermionic current vanishes. A magnetic flux through the cross section of the nanotube breaks the symmetry, allowing the current to flow along the compact dimension.

Figure 1

VEV of the current density induced by a single plate in the Kaluza-Klein model with spatial topology $R^3\times S^1$, as a function of the parameter ${\tilde{\alpha }}_D$ (left plot) and of the parameter $m{L}_D$ (right plot), for a fixed distance from the plate corresponding to $mz=0.5$. The numbers near the curves are the values of $m{L}_D$ (left plot) and ${\tilde{\alpha }}_D$ (right plot).

Figure 2

Boundary-induced part of the fermionic current versus ${\tilde{\alpha }}_D$ (left plot) and $L_D/a$ (right plot) in the region between two plates for the model with spatial topology $R^3\times S^1$ and for a massless field. The numbers near the curves are the values of $a/L_D$ (left plot) and of ${\tilde{\alpha }}_D$ (right plot).

Figure 3

Edge-induced part in the fermionic current for semi-infinite metallic (left panel) and semiconducting (right panel) carbon nanotubes as a function of the magnetic flux. The graphs are plotted for $mz=0.5$ and for different values of the parameter $mL$ (numbers near the curves).

Figure 4

Boundary-induced part in the VEV of the fermionic current for a massless field in finite-length metallic (left panel) and semiconducting (right panel) nanotubes, as a function of the magnetic flux. Numbers near the curves correspond to the values of the parameter $a/L$.

Figure 5

VEV of the fermionic current for a massless field in metallic (left panel) and semiconducting (right panel) nanotubes, as a function of the tube length. The graphs are plotted for $\Phi =0.8{\Phi }_0$, and the numbers near the curves correspond to the values of the parameter $L/a_0$.