Relativistic persistent currents in ideal Aharonov–Bohm rings

Abstract

The exact solutions of the complete Dirac equation for fermions moving in ideal Aharonov-Bohm rings are used for deriving the exact expressions of the relativistic partial currents. It is shown that as in the nonrelativistic case, these currents can be related to the derivative of the fermion energy with respect to the flux parameter. A specific relativistic effect is the saturation of the partial currents for high values of the total angular momentum. Based on this property, the total relativistic persistent current at T = 0 is evaluated giving its analytical expression and showing how this depends on the ring parameters.

Relativistic Persistent Currents in Ideal Aharonov-Bohm Rings and Cylinders

“…that has the same form as in the case of the relativistic [1] or non-relativistic AB rings. Note that, in contrast to the circular current, the longitudinal one,…”

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In this report we revisit the results obtained in [1,2] where the relativistic Aharonov-Bohm was studied for the first time. The method is based on the exact solutions of the complete (1+3)-dimensional Dirac equation of fermions moving in ideal Aharonov-Bohm (AB) rings and cylinders which are used for deriving the exact expressions of the relativistic partial currents.

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“…[1,2]. Herein we have shown first that the solutions of the Dirac equation in AB rings and cylinders may be determined as common eigenspinors of a complete systems of commuting operators including the energy, total angular momentum and a specific operator analogous to the well-known Dirac spherical operator of the relativistic central problems [36].…”

“…The relativistic partial currents we have obtained are related to the derivative of the relativistic energies as in the non-relativistic case but, in contrast with this, there appears a crucial difference, i. e. in the non-relativistic theory the partial currents are proportional to the angular momentum while in our approach the relativistic currents tend to saturation in the limit of high total angular momenta. For this reason we have reconsidered the problem of the relativistic persistent currents at T = 0 proposing an approximative analytic formula that matches the numerical calculations with a satisfactory accuracy [1,2].…”

“…Notice that this report brings together the texts of Refs. [1,2] with some improvements and unitary notations. …”