The AharonovBohm effect with a twist -- Quantum mechanics on a Möbius Ring

The Möbius ring has a universal fascination from its being a one-sided surface. Analytically solving Schrödinger’s equation for a particle on such a nanoscale twisted surface has been an insurmountable challenge. I use the finite element method, which is based on variation principles, to obtain energy levels, level splitting with and without a magnetic field, and the optical oscillator strengths for transitions between energy levels. The high accuracy FEM allows the determination of these level splittings with an excellent level of precision. We have extended the calculations to 2, 3, 4, and 5 twists on the ring as well. 

The absence of rotational invariance for the Möbius ring manifests itself through the orbital angular momentum Lznot commuting with the Hamiltonian. Its expectation values are found to have nearly integral as well as half-integral values of ħ, and its variances are small. 

The Aharonov-Bohm (AB) effect exhibits itself in the interference of electron waves traveling through multiply-connected regions and the flux linked with the paths for the electron. Here, we consider the transmission of electrons in the presence of a magnetic field through a finite-widthMöbius ring structure with current contacts attached to it. The results are compared with the transport through a flat annular ring and a cylindrical ring, with finite-width input and output contacts attached at the periphery in order to highlight the differences in the transmission patterns. A simple model accounts for the main features associated with the interference effects for propagating states on the Möbius ring. The unusual states with half-integer values of  present on Möbius rings display a different characteristic in transmission. Such resonant states are in constructive interference for transmission at magnetic fields where the contribution from ordinary states with integer  are in destructive interference, and vice versa. This leads to the beautiful result that set of half-integer  states and the set of integer states dominate alternately in transport with increasing magnetic fields.

The conductance of the Möbius ring, given by the Landauer-Büttiker formula, again reflects the alternating dominance of the integer and half-integer  states as a function of the magnetic field and the applied bias at contact reservoirs. The differences in the behavior of the Möbius ring and the untwisted ring structures lend hope for possible for magnetic sensor applications.

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