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arXiv:1502.07563 · International Journal of Modern Physics B

Relativistic persistent currents in ideal Aharonov–Bohm rings

Ion I. Cotăescu, Doru-Marcel Băltăţeanu, Ion Cotăescu

Published Nov 27, 2015• 6 citations
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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.

DOI
10.1142/s0217979215502458

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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.
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. It is shown that these currents can be related to the derivative of the fermion energy with respect to the flux parameter, just as in the non-relativistic case. However, a new and remarkable 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 currents at T = 0 is evaluated for rings and cylinders obtaining approximative simple closed formulas. Notice that this report brings together the texts of Refs. [1, 2] with some improvements and unitary notations.
Abstract 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. It is shown that these currents can be related to the derivative of the fermion energy with respect to the flux parameter, just as in the non-relativistic case. However, a new and remarkable 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 currents atT= 0 is evaluated for rings and cylinders obtaining approximative simple closed formulas. Notice that this report brings together the texts of Refs. [1, 2] with some improvements and unitary notations.
Based upon a tight-binding model Hamiltonian and by applying the Jordan-Wigner transformation, the Aharonov-Bohm (AB) effect for charged hard-core bosons in one-dimensional (1D) mesoscopic rings is investigated for the first time. By introducing a specific unitary transformation, the energy spectrum and the persistent charge current are analytically derived for ID perfect lattices. Similar to electron systems, it has been found that the energy and persistent current are periodic in AB flux with the period Φ0=hc/q. More interestingly, we have exactly shown that, in the absence of the external AB flux, the charge current can be self-sustained via the AB effect, which is in fact the ground state of the system regardless of the number of hard-core bosons being even or odd.
Exact analytic solutions are found to the Dirac equation in 2+12+12+1 dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.
We have calculated the persistent spin current of an open ring induced by the Aharonov–Casher phase. For unpolarized electrons there exist no persistent charge currents, but persistent spin currents. We show that, in general, the magnitude of the persistent spin current in a ring depends on the direction of the direct current flow from one reservoir to another. The persistent spin current is modulated by the cosine function of the spin precession angle. The nonadiabatic Aharonov–Casher phase gives anomalous behaviors. The Aharonov–Anandan phase is determined by the solid angle of spin precession. When the nonadiabatic Aharonov–Anandan phase approaches a constant value with the increase of the electric field, the periodic behavior of the spin persistent current occurs in an adiabatic limit. In this limit the periodic behavior of the persistent spin current could be understood by the effective spin-dependent Aharonov–Bohm flux.