Theory of fractional quantum Hall interferometers

Interference of fractionally charged quasiparticles 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 with the shape of a Corbino disk. In such interferometers, the quantum Hall edge states are utilized in place of optical beams, the quantum point contacts play the role of beam splitters connecting different edge channels, and Ohmic contacts represent a source and drain of quasiparticle currents. Depending on the position of Ohmic contacts, one distinguishes interferometers of Fabry-Perot (FP) and Mach-Zehnder (MZ) type. An approximate ground state of such interferometers is described by a Laughlin-type wave function, and low-energy excitations are incompressible deformations of this state. We construct a low-energy effective theory by restricting the microscopic Hamiltonian of electrons to the space of incompressible deformations and show that the theory of the quantum Hall edge so obtained is a generalization of a chiral conformal field theory. In our theory, a quasiparticle tunneling operator is found to be a single-valued function of tunneling point coordinates, and its phase depends on the topology determined by the positions of Ohmic contacts. We describe strong coupling of the edge states to Ohmic contacts and the resulting quasiparticle current through the interferometer with the help of a master equation. We find that the coherent contribution to the average quasiparticle current through MZ interferometers does not vanish after summation over quasiparticle 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 FP interferometer is varied with a modulation gate, current oscillations have the quasiparticle periodicity, thus allowing for spectroscopy of quantum Hall edge states. DOI: 10.1103/PhysRevB.86.245105