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Microscopic Theory of Fractional Quantum Hall Interferometers

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Mesoscopic Quantum Hall Effect

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Abstract

Interference of fractionally charged quasi-particles is expected to lead to Aharonov-Bohm oscillations with periods larger than the flux quantum. However, according to the Byers-Yang theorem, observables of an electronic system are invariant under an adiabatic insertion of a quantum of singular flux. We resolve this seeming paradox by considering a microscopic model of electronic interferometers made from a quantum Hall liquid at filling factor 1/m. We find that the coherent contribution to the average quasi-particle current through Mach-Zehnder interferometers does not vanish after summation over quasi-particle degrees of freedom. However, it acquires oscillations with the electronic period, in agreement with the Byers-Yang theorem. Importantly, our theory does not rely on any ad-hoc constructions, such as Klein factors, etc. When the magnetic flux through an Fabry-Perot interferometer is varied with a modulation gate, current oscillations have the quasi-particle periodicity.

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Notes

  1. 1.

    Very recently, unexpected values of quasi-particle charges, determined via shot noise measurements, have been reported in [8]. These results may indicate that the Fano factor of a weak backscattering current is not determined solely by the quasi-particle charge.

  2. 2.

    The use of the term “charge fractionalization” in this context is somewhat unfortunate, because, in contrast to the quasi-particle fractionalization, the corresponding process is completely classical in nature. In fact, it is very similar to the displacement current in electrical circuits.

  3. 3.

    Note, in particular, that the average values of physical observables are not determined solely by the energy spectrum of a system, but also by the matrix elements of the observables. Thus, one is not able to make a definitive conclusion on the periodicity of observables based on the consideration of the spectrum alone.

  4. 4.

    Note that the functional integration in Eq. (8.11) over \(\rho (z)\) is constrained to the domain \(\rho (z)\ge 0\).

  5. 5.

    It is expected that two-dimensional electrostatics for an ideal Coulomb plasma holds for \(1/\nu =m< 7\), while for larger “inverse temperature” \(m\) there is a tendency to Wigner crystallization [55].

  6. 6.

    To be precise, the operators (8.22) are not unitary if \(N\) and \(M\) are finite. However, in the thermodynamic limit (\(N,M\rightarrow \infty \)) these operators become unitary and satisfy the commutation relations (8.23).

  7. 7.

    This commutation relation follows form the definition of the derivative operator (8.28), which differs from the standard one.

  8. 8.

    On may check that solving the detailed balance equation with tunneling rates calculated with the amplitudes (8.54) and (8.55) leads to the distribution (8.69).

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  1. Department of Theoretical Physics, University of Geneva, 1211, Geneva, Switzerland

    Ivan Levkivskyi

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Levkivskyi, I. (2012). Microscopic Theory of Fractional Quantum Hall Interferometers. In: Mesoscopic Quantum Hall Effect. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30499-6_8

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  • DOI: https://doi.org/10.1007/978-3-642-30499-6_8

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