Universal Periods in Quantum Hall Droplets

Abstract

Using the hierarchy picture of the fractional quantum Hall effect, we study the ground-state periodicity of a finite size quantum Hall droplet in a quantum Hall fluid of a different filling factor. The droplet edge charge is periodically modulated with flux through the droplet and will lead to a periodic variation in the conductance of a nearby point contact, such as occurs in some quantum Hall interferometers. Our model is consistent with experiment and predicts that superperiods can be observed in geometries where no interfering trajectories occur. The model may also provide an experimentally feasible method of detecting elusive neutral modes and otherwise obtaining information about the microscopic edge structure in fractional quantum Hall states. . Here we will focus only on the Abelian quantum Hall states, but we will make use of their topological properties to reveal universal periodicities (as a function of magnetic flux through the droplet) in the ground-state energy and edge properties of a quantum Hall droplet inside a surrounding Hall fluid of a different filling factor. The universal periodicities in the groundstate properties can generically be used to probe the quantum Hall edge states in equilibrium settings.Our work is motivated in part by a series of beautiful experiments done on quantum Hall interferometers where superperiods and fractional statistics have purportedly been observed [4]. Several theoretical studies have already addressed these experiments [5], but a complete picture, particularly in the fractional quantum Hall regime, is still lacking. In this Letter we study the universal properties of a finite size quantum Hall droplet inside a quantum Hall fluid of a different filling factor (Fig. 1). For most geometries and droplet filling factors we find that the ground-state energy of the system has a periodicity with magnetic flux through the inner droplet that is determined only by the two filling fractions in the limit that the charging energy of the surrounding fluid edge tends to zero. However, edges such as that of the 2=3 state (which have counterpropagating modes and disorder-influenced excitations [6]) require a special degree of consideration, as we discuss below.Consider a droplet of filling factor d surrounded by a fluid of filling factor s , which itself may be inside an outer fluid of filling o (Fig. 1). As magnetic flux is adiabatically threaded through the inner droplet, its ground-state energy and its radius oscillate with a universal periodicity. This periodicity reveals important information about the microscopic structure of the droplet edge itself, thus providing a mechanism by which theoretical edge models can be directly tested experimentally. We concentrate on two important special cases:General filling fractions with Abelian statistics follow one of the two cases above. While disorder does not play a central role in the physics of the edge in case (i), in case (ii) disorder determines the nature of the edge excitations [6], by causing counterpropagating modes of th...

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