Quantum Hall liquid, Josephson effect, and hierarchy in a double-layer electron system

Abstract

Based on a bosonic Chern-Simons gauge theory, we present a microscopic theory of a double-layer electron system in which even-denominator fractional quantum Hall (FQH) states have been observed in a strong external magnetic field. In our approach electrons belonging to different layers are interpreted as two distinguishable anyons with appropriate statistics. Neglecting the Zeeman energy, we find in semiclassical approximation that Hall states are realized as condensed states of the bosonized electrons in both of the layers. We calculate the Coulomb energy of the Hall states as well as some excitation spectrum as a function of the distance between the two layers. We also derive the ground-state wave function, which is found to coincide with that of Halperin for a nonvanishing interlayer distance. We find a superfluidity mode in FQH states at the filling ν=1,1/3, 1) / 5 ,..., in which the interlayer tunneling acts as a Josephson junction in the states. It is manifest in our formalism that the Josephson current is given by J=${\mathrm{\Delta }}_{\mathrm{SAS}}$ ${\mathrm{\rho }}_0$ sin(eVt+${\mathrm{\delta }}_0$) for a constant voltage V applied between the two layers, where ${\mathrm{\Delta }}_{\mathrm{SAS}}$ is the gap energy between the symmetric and antisymmetric single-particle states and ${\mathrm{\rho }}_0$ is the average electron density in one layer. The observation of the Josephson effect constitutes a direct experimental test of the existence of condensation of bosonized electrons in FQH states. A hierarchy of the FQH states is also analyzed by using an effective-field theory of vortex solitons (quasiparticles).