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Physical review. B, Condensed matter

Aharonov-Anandan phase and persistent currents in a mesoscopic ring

Xiao-Chun Gao, Tiezheng Qian

Published Mar 15, 1993• 35 citations
Publisher

Abstract

The exact solution for a mesoscopic ring in the presence of a classical, static, inhomogeneous magnetic field is found and used to derive the formulas for persistent currents. From the derivation, we can see the close relation of the Aharonov-Anandan phase to the persistent currents. In the adiabatic limit, our results are shown to be identical with those of Loss et al., who explored the connection between the persistent currents and the Berry phase.

DOI
10.1103/physrevb.47.7128

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The quantum orbital motion of electrons in mesoscopic normal-metal rings threaded by a magnetic flux produces striking interference phenomena such as persistent currents due to the Aharonov-Bohm effect. Similarly, when a quantum spin adiabatically follows a magnetic field that rotates slowly in time, the phase of its state vector acquires an additional contribution known as the Berry phase. We explore the combination of these two quantum phenomena by examining the interplay between orbital and spin degrees of freedom for a charged spin-1/2 particle moving in a mesoscopic ring embedded in a classical, static inhomogeneous magnetic field, i.e, a texture. As a consequence of its orbital motion through the texture, the spin experiences, via the Zeeman interaction, a varying magnetic field. This results in a Berry---or geometric---phase, leading to persistent (i.e., equilibrium) currents of charge and spin. These mesoscopic phenomena are related to (but should be distinguished from) the conventional persistent currents that result from magnetic flux through a ring. We develop a path-integral approach to decouple the orbital and spin motion and, by using an adiabatic approximation, we compute the equilibrium expectation values of the persistent charge and spin currents and the magnetization. We find that the persistent currents depend on the texture in a striking manner through a geometric phase (related to a surface area characterizing the texture) and a geometric vector (related to the projections of this area). In the special case of a cylindrically symmetric texture we use a spectrum obtained by Kuratsuji and Iida to obtain exact results that confirm, independently, the validity of the path-integral approach in the adiabatic limit. We discuss the connection between the geometric vector and quantum-mechanical correlations, and examine quantum fluctuations and the zero-point energy.

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The exact solution for a mesoscopic ring in the presence of a classical, static, inhomogeneous magnetic field is found and used to derive the formulas for persistent currents. From the derivation, we can see the close relation of the Aharonov-Anandan phase to the persistent currents. In the adiabatic limit, our results are shown to be identical with those of Loss et al., who explored the connection between the persistent currents and the Berry phase.
The quantum orbital motion of electrons in mesoscopic normal-metal rings threaded by a magnetic flux produces striking interference phenomena such as persistent currents due to the Aharonov-Bohm effect. Similarly, when a quantum spin adiabatically follows a magnetic field that rotates slowly in time, the phase of its state vector acquires an additional contribution known as the Berry phase. We explore the combination of these two quantum phenomena by examining the interplay between orbital and spin degrees of freedom for a charged spin-1/2 particle moving in a mesoscopic ring embedded in a classical, static inhomogeneous magnetic field, i.e, a texture. As a consequence of its orbital motion through the texture, the spin experiences, via the Zeeman interaction, a varying magnetic field. This results in a Berry---or geometric---phase, leading to persistent (i.e., equilibrium) currents of charge and spin. These mesoscopic phenomena are related to (but should be distinguished from) the conventional persistent currents that result from magnetic flux through a ring. We develop a path-integral approach to decouple the orbital and spin motion and, by using an adiabatic approximation, we compute the equilibrium expectation values of the persistent charge and spin currents and the magnetization. We find that the persistent currents depend on the texture in a striking manner through a geometric phase (related to a surface area characterizing the texture) and a geometric vector (related to the projections of this area). In the special case of a cylindrically symmetric texture we use a spectrum obtained by Kuratsuji and Iida to obtain exact results that confirm, independently, the validity of the path-integral approach in the adiabatic limit. We discuss the connection between the geometric vector and quantum-mechanical correlations, and examine quantum fluctuations and the zero-point energy.
The total geometric phase is composed of the nonadiabatic noncyclic Pancharatnam phase, the usual Aharonov-Bohm (AB) phase, and the effective AB phase. It is found that the persistent current in one-dimensional rings is determined from this phase. As applications, we address first the geometric phase and the persistent current in a ring subject to a cylindrically symmetric electromagnetic field. We show that the Pancharatnam phase recovers the Aharanov-Anandan phase in the case of cyclic evolution, as well as the Berry phase in the adiabatic evolution. Moreover, we discuss the persistent current induced by the spin-orbit-induced geometric phase in the presence of a local magnetic field. Generalization to many-body cases is also addressed.
We present a comprehensive view and details of calculations on Aharonov–Anandan phase for the charged particles in the external electric and magnetic fields for a nonadiabatic process. We derive, with consideration of a spin-orbit interaction and Zeemann Splitting, the persistent currents as a response to an Aharonov–Casher topological interference effect in one-dimensional mesoscopic ring. We also establish a connection to Berry adiabatic phase with deduced dynamical-nature dependence in the nonadiabatic process. The second quantization representation has also been employed in exhibition of persistent currents in the many-body case.
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.
We present a comprehensive view and details of calculations on Aharonov-Anandan phase for the charged particles in the external electric and magnetic fields for a nonadiabatic process. We derive, with consideration of a spin-orbit interaction and Zeemann Splitting, the persistent currents as a response to an Aharonov-Casher topological interference effect in one-dimensional mesoscopic ring. We also establish a connection to Berry adiabatic phase with dednced dynamical-nature dependence in the nonadiabatic process. The second quantization representation has also been employed in exhibition of persistent currents in the many-body case.