Two-dimensional quantum ring in a graphene layer in the presence of a Aharonov–Bohm flux

In this paper we study the relativistic quantum dynamics of a massless fermion confined in a quantum ring. We use a model of confining potential and introduce the interaction via Dirac oscillator coupling, which provides ring confinement for massless Dirac fermions. The energy levels and corresponding eigenfunctions for this model in graphene layer in the presence of Aharonov–Bohm flux in the centre of the ring and the expression for persistent current in this model are derived. We also investigate the model for quantum ring in graphene layer in the presence of a disclination and a magnetic flux. The energy spectrum and wave function are obtained exactly for this case. We see that the persistent current depends on parameters characterizing the topological defect.     

Geometric phase of a classical Aharonov–Bohm Hamiltonian

We present a gauge-invariant approach for associating a geometric phase with the phase space trajectory of a classical dynamical system. As an application, we consider the classical analog of the quantum Aharonov–Bohm (AB) Hamiltonian for a charged particle orbiting around a current carrying long thin solenoid. We compute the classical geometric phase of a closed phase space trajectory, and also determine its dependence on the magnetic flux enclosed by the orbit. We study the similarities and differences between this classical geometric phase and the AB phase acquired by the wave function of the quantum AB Hamiltonian. We suggest an experiment to measure the geometric phase for the classical AB system, by using an appropriate optical fiber ring interferometer.     

Geometry of the Aharonov–Bohm Effect

We show that the connection responsible for any Abelian or non-Abelian Aharonov–Bohm effect with n parallel “magnetic” flux lines in ℝ³, lies in a trivial G-principal bundle P→M, i.e. P is isomorphic to the product M×G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space , where path integrals are computed, and the associated bundle P× _G ℂ ^m →M, where the wave function and its covariant derivative are sections.     

Antiresonances modulated by magnetic flux in an Aharonov–Bohm ring

With or without an inserted quantum dot or an attached side resonator, the transmission behavior of an Aharonov–Bohm (AB) ring is investigated in the framework of quantum waveguide theory in the complex-energy plane. The calculated results show that the AB ring possesses antiresonances in the conductance and several kinds of transmission poles in the complex-energy plane. Piercing the ring, magnetic flux causes a series of crossovers among zero-pole pairs, zero-double-pole pairs, peak-double-pole pairs, and peak-pole pairs. With the inserted quantum dot or attached side resonator in one of its arms, the ring produces complicated behaviors in conductance: some of the poles are split into more than two parts. A kind of deformed Fano resonance is found in the conductance of the AB ring with the attached resonator.     

Aharonov–Bohm effect in the ghost interference

Chaotic systems demonstrate complex behaviour in their state variables and their parameters, which generate some challenges and consequences. This paper presents a new synchronisation scheme based on integral sliding mode control (ISMC) method on a class of complex chaotic systems with complex unknown parameters. Synchronisation between corresponding states of a class of complex chaotic systems and also convergence of the errors of the system parameters to zero point are studied. The designed feedback control vector and complex unknown parameter vector are analytically achieved based on the Lyapunov stability theory. Moreover, the effectiveness of the proposed methodology is verified by synchronisation of the Chen complex system and the Lorenz complex systems as the leader and the follower chaotic systems, respectively. In conclusion, some numerical simulations related to the synchronisation methodology is given to illustrate the effectiveness of the theoretical discussions.     

On the Aharonov–Bohm Hamiltonian

Using the theory of self-adjoint extensions, we construct all the possible Hamiltonians describing the nonrelativistic Aharonov–Bohm effect. In general, the resulting Hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.     

Gauge covariance of the Aharonov–Bohm phase in noncommutative quantum mechanics

The gauge covariance of the wave function phase factor in noncommutative quantum mechanics (NCQM) is discussed. We show that the naive path integral formulation and an approach where one shifts the coordinates of NCQM in the presence of a background vector potential leads to the gauge non-covariance of the phase factor. Due to this fact, the Aharonov–Bohm phase in NCQM which is evaluated through the path-integral or by shifting the coordinates is neither gauge invariant nor gauge covariant. We show that the gauge covariant Aharonov–Bohm effect should be described by using the noncommutative Wilson lines, what is consistent with the noncommutative Schrödinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.     

An investigation of the gate dependence in an electronic Aharonov–Bohm interferometer

In this study we have obtained the dependence of the electronic distribution on the gate shape and the applied gate voltage in a quantum Hall effect based Aharonov–Bohm interferometer using a method presented in our previous studies. We have discussed the relation between the distribution of incompressible strips and observation of Aharonov–Bohm oscillations. We have obtained the distributions of the incompressible strips for various gate voltages and have shown that a gate potential sweep and a magnetic field sweep would be equivalent. Our calculations also predict that for wider gate separations it is possible observe a silent region while sweeping the magnetic field or the gate voltage.     

Detection of Majorana fermions in an Aharonov–Bohm interferometer

By using the non-equilibrium Green's function technique, we investigate the electronic transport properties in an Aharonov–Bohm interferometer coupling with Majorana fermions. We find a fixed unit conductance peak which is independent of the other factors when the topological superconductor is grounded. Especially, an additional phase appears when the topological superconductor is in the strong Coulomb regime, which induces a new conductance resonant peak compared with the structure of replacing the topological superconductor by a quantum dot, and the conductance oscillation with the magnetic flux reveals a 2π phase shift by raising(lowering) a charge on the capacitor.     

Does the Aharonov–Bohm Effect Exist?

We draw a distinction between the Aharonov–Bohm phase shift and the Aharonov–Bohm effect. Although the Aharonov–Bohm phase shift occurring when an electron beam passes around a magnetic solenoid is well-verified experimentally, it is not clear whether this phase shift occurs because of classical forces or because of a topological effect occurring in the absence of classical forces as claimed by Aharonov and Bohm. The mathematics of the Schroedinger equation itself does not reveal the physical basis for the effect. However, the experimentally observed Aharonov–Bohm phase shift is of the same form as the shift observed due to electrostatic forces for which the consensus view accepts the role of the classical forces. The Aharonov–Bohm phase shift may well arise from classical electromagnetic forces which are simply more subtle in the magnetic case since they involve relativistic effects of the order v ² /c ² . Here we first review the experimentally observable differences between phenomena arising from classical forces and phenomena arising from the quantum topological effect suggested by Aharonov and Bohm. Second we point out that most discussions of the classical electromagnetic forces involved when a charged particle passes a solenoid are inaccurate because they omit the Faraday induction terms. The subtleties of the relativisitic v ² /c ² classical electromagnetic forces between a point charge and a solenoid have been explored by Coleman and Van Vleck in their analysis of the Shockley–James paradox; indeed, we point out that an analysis exactly parallel to that of Coleman and Van Vleck suggests that the Aharonov–Bohm phase shift is actually due to classical electromagnetic forces. Finally we note that electromagnetic velocity fields penetrate even excellent conductors in a form which is unfamiliar to many physicists. An ohmic conductor surrounding a solenoid does not screen out the magnetic field of the passing charge, but rather the time-integral of the magnetic field is an invariant; this time integral is precisely what is involved in the classical explanation of the Aharonov–Bohm phase shift. Thus the persistence of the Aharonov–Bohm phase shift when the solenoid is surrounded by a conductor does not exclude a classical force-based explanation for the phase shift. At present there is no experimental evidence for the Aharonov–Bohm effect.     

The Aharonov–Bohm effect in scattering theory

The Aharonov–Bohm effect is considered as a scattering event with nonrelativistic charged particles of the wavelength which is less than the transverse size of an impenetrable magnetic vortex. The quasiclassical WKB method is shown to be efficient in solving this scattering problem. We find that the scattering cross section consists of two terms, one describing the classical phenomenon of elastic reflection and another one describing the quantum phenomenon of diffraction; the Aharonov–Bohm effect is manifested as a fringe shift in the diffraction pattern. Both the classical and the quantum phenomena are independent of the choice of a boundary condition at the vortex edge, providing that probability is conserved. We show that a propagation of charged particles can be controlled by altering the flux of a magnetic vortex placed on their way.     

The Aharonov–Bohm effect in graphene rings

This is a review of electronic quantum interference in mesoscopic ring structures based on graphene, with a focus on the interplay between the Aharonov–Bohm effect and the peculiar electronic and transport properties of this material. We first present an overview on recent developments of this topic, both from the experimental as well as the theoretical side. We then review our recent work on signatures of two prominent graphene-specific features in the Aharonov–Bohm conductance oscillations, namely Klein tunneling and specular Andreev reflection. We close with an assessment of experimental and theoretical development in the field and highlight open questions as well as potential directions of the developments in future work.     

Line of magnetic monopoles and an extension of the Aharonov–Bohm effect

In the Landau problem on the two-dimensional plane, physical displacement of a charged particle (i.e., magnetic translation) can be induced by an in-plane electric field. The geometric phase accompanying such magnetic translation around a closed path differs from the topological phase of Aharonov and Bohm in two essential aspects: The particle is in direct contact with the magnetic field and the geometric phase has an opposite sign from the Aharonov–Bohm phase. We show that magnetic translation on the two-dimensional cylinder implemented by the Schrödinger time evolution truly leads to the Aharonov–Bohm effect. The magnetic field normal to the cylinder’s surface corresponds to a line of magnetic monopoles of uniform density whose simulation is currently under investigation in cold atom physics. In order to characterize the quantum problem, one needs to specify the value of the magnetic flux (modulo the flux unit) that threads but not in touch with the cylinder. A general closed path on the cylinder may enclose both the Aharonov–Bohm flux and the local magnetic field that is in direct contact with the charged particle. This suggests an extension of the Aharonov–Bohm experiment that naturally takes into account both the geometric phase due to local interaction with the magnetic field and the topological phase of Aharonov and Bohm.     

Classical Electromagnetism and the Aharonov–Bohm Phase Shift

Although there is good experimental evidence for the Aharonov–Bohm phase shift occurring when a solenoid is placed between the beams forming a double-slit electron interference pattern, there has been very little analysis of the relevant classical electromagnetic forces. These forces between a point charge and a solenoid involve subtle relativistic effects of order v ² /c ² analogous to those discussed by Coleman and Van Vleck in their treatment of the Shockley–James paradox. In this article we show that a treatment exactly analogous to that given by Coleman and Van Vleck predicts classical electromagnetic forces which provide the basis for the Aharonov–Bohm phase shift. The magnetic force on the solenoid due to the passing charge leads to a displacement of the solenoid center of energy which must be balanced by the displacement of the passing charge. This classical displacement of the passing charge is exactly what is required to account for the Aharonov–Bohm phase shift. Also, we discuss a magnetic moment model which appears frequently in the literature and note that although the model provides conservation of linear momentum, it does not satisfy the general requirements for relativistic theories. We give an example suggesting that the new equation of motion for a magnetic moment proposed by Aharonov, Pearle, and Vaidman based upon the hidden momentum of the magnetic moment is completely inappropriate. Finally, we emphasize that the Aharonov–Casher phase shift is also explained by classical electromagnetic forces exactly parallel to those explaining the Aharonov–Bohm phase shift.     

Exotic electron states and tunable magneto-transport in a fractal Aharonov–Bohm interferometer

A Sierpinski gasket fractal network model is studied in respect of its electronic spectrum and magneto-transport when each ‘arm’ of the gasket is replaced by a diamond shaped Aharonov–Bohm interferometer, threaded by a uniform magnetic flux. Within the framework of a tight binding model for non-interacting, spinless electrons and a real space renormalization group method we unravel a class of extended and localized electronic states. In particular, we demonstrate the existence of extreme localization of electronic states at a special finite set of energy eigenvalues, and an infinite set of energy eigenvalues where the localization gets ‘delayed’ in space (staggered localization). These eigenstates exhibit a multitude of localization areas. The two terminal transmission coefficient and its dependence on the magnetic flux threading each basic Aharonov–Bohm interferometer is studied in details. Sharp switch on–switch off effects that can be tuned by controlling the flux from outside, are discussed. Our results are analytically exact.     

Perturbative Study of Bremsstrahlung and Pair-Production by Spin-½ Particles in the Aharonov–Bohm Potential

In the presence of an external Aharonov–Bohm potential, we investigate the two QED processes of the emission of a bremsstrahlung photon by an electron, and the production of an electron–positron pair by a single photon. Calculations are carried out using the Born approximation within the framework of covariant perturbation theory to lowest non-vanishing order in α. The matrix element for each process is derived, and the corresponding differential cross-section is calculated. In the non-relativistic limit, the resulting angular and spectral distributions and some polarization properties are considered, and compared to results of previous works.     

Loop Quantum Gravity à la Aharonov–Bohm

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein–Maxwell–Klein–Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman–Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work (Bi?ák et al. in Class Quantum Gravity 27(17):175011, 2010) and show that, unlike the conformal scalar field, the (Maxwell–)Klein–Gordon field has negatively semi-definite mass-loss formula.     

Aharonov–Bohm oscillations caused by non-topological surface states in Dirac nanowires

One intriguing fingerprint of surface states in topological insulators is the Aharonov–Bohm effect in magnetoconductivity of nanowires. We show that surface states in nanowires of Dirac materials (bismuth, bismuth antimony, and lead tin chalcogenides) being in non-topological phase, exhibit the same effect as amendment to magnetoconductivity of the bulk states. We consider a simple model of a cylindrical nanowire, which is described by the 3D Dirac equation with a general T-invariant boundary condition. The boundary condition is determined by a single phenomenological parameter whose sign defines topological-like and non-topological surface states. The non-topological surface states emerge outside the gap. In a longitudinal magnetic field B, they lead to Aharonov–Bohm amendment for the density of states and correspondingly for the conductivity of the nanowire. The phase of these magnetic oscillations increases with B from π to 2π.