Quantum Hall states of Cooper pairs and vortices in Josephson arrays

Abstract

We consider regular two-dimensional arrays of small Josephson junctions ( E C ∼ E J) in a transverse magnetic field. The dynamics of this system can be described in terms of interacting quantum particles-charges (for small E J) or vortices (for large E J). We show that at low concentrations of the particles, n ⪡ Λ -2, i.e. in the limit when the range Λ of the interaction is short, the system is equivalent to a Bose gas with weak contact repulsive interaction. The ground state in a strong magnetic field is described by the Laughlin type wave function with even powers. We present numerical results for the largest plateau in the particle density at v = 1 2 and also a toy model that enables us to develop the picture of all plateaus. The Hall conductance σ xy of the array shows the universal quantization at least at the values (2 e) 2/2 mh for small E J and at the values 2 m(2 e) 2/ h for large E J .