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Geometric phase related to point-interaction transport on a magnetic Lobachevsky plane

Sergio A Albeverio, Pavel Exner, Vladimir A. Geyler

Published Jan 1, 2000• 7 citations
OpenAlex

Abstract

We consider a charged quantum particle living in the Lobachevsky plane and interacting with a homogeneous magnetic field perpendicular to the plane and a point interaction which is transported adiabatically along a closed loop C in the plane. We show that the bound-state eigenfunction acquires at that the Berry phase equal to 2 times the number of the flux quanta through the area encircled by C. 1 Introduction The phenomena arising from a geometric phase called Berry phase [Ber] have been put in evidence in many quantum mechanical systems. Recently such a "Berry phase effect" has been observed in some magnetic systems [LSG, MHK]. Moreover, it was shown that the geometric phase can emerge even in a time-independent homogeneous magnetic field when a potential 1 well trapping a two-dimensional particle is transported along a closed loop. An example in which the potential is of zero range is worked out in [EG], where a formula was proved showing that in the absence of an additional conf...

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We consider a charged quantum particle living in the Lobachevsky plane and interacting with a homogeneous magnetic field perpendicular to the plane and a point interaction which is transported adiabatically along a closed loop C in the plane. We show that the bound-state eigenfunction acquires at that the Berry phase equal to 2 times the number of the flux quanta through the area encircled by C. 1 Introduction The phenomena arising from a geometric phase called Berry phase [Ber] have been put in evidence in many quantum mechanical systems. Recently such a "Berry phase effect" has been observed in some magnetic systems [LSG, MHK]. Moreover, it was shown that the geometric phase can emerge even in a time-independent homogeneous magnetic field when a potential 1 well trapping a two-dimensional particle is transported along a closed loop. An example in which the potential is of zero range is worked out in [EG], where a formula was proved showing that in the absence of an additional conf...
The Berry phase [3] is traditionally understood as a geometric phase acquired during cyclic adiabatic evolution, often interpreted as the flux of an effective “magnetic field” through a solid angle in parameter space. In this thesis, we investigate situations in which a geometric phase arises even when no spatial solid angle is enclosed, revealing the limitations of the standard purely spatial interpretation. We first review the Berry phase for a spin-½ particle in a slowly varying magnetic field and analyze scenarios in which the system’s eigenstate is changed during the adiabatic evolution. We show that the resulting phase can be understood through a spacetime formulation analogous to the electrodynamic Aharonov–Bohm effect [1], where both “analogous magnetic” and “analogous electric” fields contribute through a spacetime surface integral. This framework clarifies how nonzero geometric phases can occur even when the projected spatial solid angle vanishes. We then extend this perspective to the Jahn–Teller 𝐸 ⊗ 𝑒 system. While a closed loop around the conical intersection yields the familiar π phase, we show that nontrivial geometric phases can also emerge in paths with zero net enclosed solid angle. Advisors: Herman Batelaan and Ilya Kravchenko
Berry's phase often appears in quantum two-level systems with a degeneracy. An example of such a system is a spin-1/2 particle in a magnetic field. As the magnetic field is slowly evolved through a closed path, the particle has been shown to acquire an additional phase called Berry's phase, in addition to the usual dynamical phase. This phase has been found in two-level quantum systems intrinsic to many superconducting qubits and has particularly been calculated for the charge, flux and Josephson phase qubit. Here, we present an alternative derivation of the Berry's phase in a current-biased Josephson junction qubit. We also calculate the complete Berry's phase evaluated over a closed loop in a frequency parameter space and compare this with the geometric phase. From this comparison, we examine the possibility of using a single phase qubit for topological quantum computing.
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
The authors consider a charged particle confined to a ring encircling a magnetic flux. Supposing that the particle is trapped by a delta-function potential located on the ring, they examine the flux dependence of the energy and the wavefunction of the bound state. This offers a simple explicit model for the Aharonov-Bohm effect on a system localized, but not completely confined, to a simply connected region off a magnetic flux. The result is used to show that Berry's phase for the bound state, when the delta-function potential is slowly transported around the ring, consists not only of the Aharonov-Bohm phase proportional to the flux, but also of an extra term which reflects the incomplete localization of the state; in the limit of strong binding the extra term depends sinusoidally on the flux.
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.