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Relativistic Persistent Currents in Ideal Aharonov-Bohm Rings and Cylinders

Ion I. Cotǎescu

Published Dec 1, 2016• 0 citations
Publisher

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 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.

DOI
10.1515/awutp-2016-0006

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Relativistic Persistent Currents in Ideal Aharonov-Bohm Rings and Cylinders

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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. 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.
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.
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 study the analogue of the Aharonov–Bohm effect for bound states for a neutral particle with a permanent magnetic dipole moment interacting with an external field. We consider a neutral particle confined to moving between two coaxial cylinders and show the dependence of the energy levels on the Aharonov-Casher quantum flux. Moreover, we show that the same flux dependence of the bound states can be found when the neutral particle is confined to a one-dimensional quantum ring and a quantum dot, and we also calculate the persistent currents in each case.
To study the manifestation of the Aharonov-Bohm effect in many-body systems we consider the statistical mechanics of the Gross-Neveu model on a ring (1+1 dimensions) and on a cylinder (2+1 dimensions) with a thin solenoid coinciding with the axis. For such systems with a non-trivial magnetic flux (θθθ) many thermodynamical observables, such as the order parameter, induced current and virial coefficients, display periodic but non-analytic dependence on θθθ. In the 2+1 dimensional case we further find that there is an interval of θ∈(1/3,2/3)θ\in(1/3,2/3)θ∈(1/3,2/3) (modulo integers) where parity is always spontaneously broken, independent of the circumference of the cylinder. We show that the mean-field character of the phase transitions is preserved to the leading order in 1/N1/N1/N, by verifying the θθθ-independence of all the critical exponents. The precise nature of the quasi-particle, locally fermion-like and globally anyon-like, is illuminated through the calculation of the equal-time commutator and the decomposition of the propagator into a sum over paths classified by winding numbers.