Quantum oscillations in mesoscopic rings and anomalous diffusion

Abstract

We consider the weak localization correction to the conductance of a ring connected to a network. We analyze the harmonics content of the Al'tshuler-Aronov-Spivak (AAS) oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by AAS. The physical origin of this behaviour is the anomalous diffusion of Brownian trajectories around the ring, due to the diffusion in the wires. We show that this problem is related to the anomalous diffusion along the skeleton of a comb. We study in detail the winding properties of Brownian curves around a ring connected to an arbitrary network. Our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales. A general expression of this length is derived for arbitrary networks. More specifically we consider the case of a ring connected to wires, to a square network, and to a Bethe lattice.

17 pages, 7 eps figures

References in corpus (6)

• Spectral determinant on quantum graphs
• Persistent currents on graphs
• Interaction-induced dephasing of Aharonov-Bohm oscillations
• Random walk on Bethe lattice and hyperbolic geometry
• Direct measurement of the phase coherence length in a GaAs/GaAlAs square network
• Weak localization in multiterminal networks of diffusive wires

Cited by in corpus (14)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Functionals of the Brownian motion, localization and metric graphs
• Dephasing due to electron-electron interaction in a diffusive ring
• Green's function approach for quantum graphs: an overview
• Quantum oscillations and decoherence due to electron-electron interaction in metallic networks and hollow cylinders
• Non-linear conductance in mesoscopic weakly disordered wires -- Interaction and magnetic field asymmetry
• Exactly solvable interacting two-particle quantum graphs
• Spectral determinants and zeta functions of Schrödinger operators on metric graphs
• Effect of connecting wires on the decoherence due to electron-electron interaction in a metallic ring
• Winding statistics of a Brownian particle on a ring
• On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach
• Four-terminal resistances in mesoscopic networks of metallic wires: Weak localisation and correlations
• $\zeta$-regularised spectral determinants on metric graphs
• Quantum transport in randomized quantum graphs