Aharonov-Bohm effect in the chiral Luttinger liquid

Edge states of the quantum Hall fluid provide an almost unparalled opportunity to study mesoscopic effects in a highly correlated electron system. In this paper we develop a bosonization formalism for the finite-size edge state, as described by chiral Luttinger liquid theory, and use it to study the Aharonov-Bohm effect. The problem we address may be realized experimentally by measuring the tunneling current between two edge states through a third edge state formed around an antidot in the fractional quantum Hall effect regime. The finite size $L$ of the antidot edge state introduces a temperature scale $T_0\equiv ħv/\pi k_{B}L,$ where $v$ is the edge-state Fermi velocity. A renormalization group analysis reveals the existence of a two-parameter universal scaling function $\tilde{G}\mathrm{}(X,Y)\mathrm{}$ that describes the Aharonov-Bohm conductance resonances. We also show that the strong renormalization of the tunneling amplitudes that couple the antidot to the incident edge states, together with the nature of the Aharonov-Bohm interference process in a chiral system, prevent the occurrence of perfect resonances as the magnetic field is varied, even at zero temperature. In an experimentally realizable strong-antidot-coupling regime, where the source-to-drain transmission is weak, and at bulk filling factor $g=1/q$ with $q$ an odd integer, we predict the low-temperature $(T\ll T_0)$ Aharonov-Bohm amplitude to vanish with temperature as $T^{2q-2},$ in striking contrast to a Fermi liquid $\mathrm{}(q=1\mathrm{}).$ Near $T_0,$ there is a pronounced maximum in the amplitude, also in contrast to a Fermi liquid. At high temperatures $\left(T\gg T_0\right),$ however, we predict a crossover to a $T^{2q-1}{e}^{-{qT/T}_0}$ temperature dependence, which is qualitatively similar to chiral Fermi liquid behavior. Careful measurements in the strong-antidot-coupling regime above $T_0$ should be able to distinguish between a Fermi liquid and our predicted nearly Fermi liquid scaling. In addition, we predict an interesting high-temperature nonlinear response regime, where the voltage satisfies $V>T>\mathrm{}{T}_0,$ which may also be used to distinguish between chiral Fermi liquid and chiral Luttinger liquid behavior. Finally, we predict mesoscopic edge-current oscillations, which are similar to the persistent current oscillations in a mesoscopic ring, except that they are not reduced in amplitude by weak disorder. In the fractional quantum Hall effects regime, these “chiral persistent currents” have a universal non-Fermi-liquid temperature dependence and may be another ideal system to observe a chiral Luttinger liquid.