Theory of fractional quantum Hall interferometers

TL;DR

This paper develops a microscopic theory of fractional quantum Hall interferometers, explaining observed Aharonov-Bohm oscillations with quasi-particle charge and reconciling them with the Byers-Yang theorem.

Contribution

It introduces a low-energy effective theory based on incompressible deformations of Laughlin states, generalizing chiral conformal field theory without ad-hoc assumptions.

Findings

Quasi-particle tunneling operator is single-valued and topology-dependent.

Coherent quasi-particle current oscillates with electronic period, consistent with the Byers-Yang theorem.

Current oscillations with quasi-particle periodicity enable spectroscopy of quantum Hall edge states.

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. 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 quasi-particle 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 quasi-particle current through the interferometer with the help of a master equation. 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 FP interferometer is varied with a modulation gate, current oscillations have the quasi-particle periodicity, thus allowing for spectroscopy of quantum Hall edge states.