Abstract
The effect of vacuum polarization of charged massive fermions in an Aharonov-Bohm (AB) potential in 2+1 dimensions is investigated. The causal Green's function of the Dirac equation with the AB potential is represented via the regular and irregular solutions of the two-dimensional radial Dirac equation. It is shown that the vacuum current density contains the contribution from free filled states of the negative energy continuum as well as that from a bound unfilled state, which can emerge in the above background due to the interaction of the fermion spin magnetic moment with the AB magnetic field while the induced charge density contains only the contribution from the bound state. The expressions for the vacuum charge and induced current densities are obtained (recovered for massless fermions) for the graphene in the field of infinitesimally thin solenoid perpendicular to the plane of a sample. We also find the bound state energy as a function of magnetic flux, fermion spin and the radius of solenoid as well as discuss the role of the so-called self-adjoint extension parameter and determine it in terms of the physics of the problem.
Highlights
IntroductionImportant physical quantities are the vacuum charge and current densities induced by the background field
Quantum systems of relativistic fermions in external fields in 2 + 1 dimensions attract considerable interest, which is related to the possibility of applying the results obtained for simplified models to the study of fundamental physical phenomena such as the Aharonov–Bohm and quantum Hall effects [1,2], as well as high-temperature superconductivity [3]
When the external field configuration has the cylindrical symmetry, a natural assumption is that the relevant quantummechanical system is invariant along the symmetry (z) axis and the system becomes essentially two-dimensional in the x y plane
Summary
Important physical quantities are the vacuum charge and current densities induced by the background field. The induced vacuum current in graphene in the field of a solenoid was investigated in [20] and a wonderful phenomenon was revealed: the induced current turns out to be a finite periodic function of the magnetic flux. Induced vacuum condensates such as the induced charge density and current (and other ones) for massless fermions in the background of a singular magnetic vortex in (2 + 1)-dimensional space-time were investigated in [21,22].
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