Broken Symmetry and Josephson-like Tunneling in Quantum Hall Bilayers

The various quantum Hall effects are among the most remarkable many-body phenomena discovered in the second half of the twentieth century. 2, 3, 4 The fractional effect has yielded fractional charge, spin and statistics, as well as unprecedented order parameters. There are beautiful connections with a variety of different topological and conformal field theories of interest in nuclear and high energy physics. The quantum Hall effect (QHE) takes place in a two-dimensional electron gas formed in a quantum well in a semiconductor host material and subjected to a very high magnetic field. In essence it is a result of a commensuration between the number of electrons, N , and the number of flux quanta, NΦ, in the applied magnetic field. The electrons condense into distinct and highly non-trivial ground states (‘vacua’) formed at each rational fractional value of the filling factor ν ≡ N/NΦ. The essential feature of (most) of these exotic states is the existence of an excitation gap. The electron fluid is incompressible and flows rigidly past obstacles (impurities in the sample) with no dissipation. A weak external electric field will cause the fluid to move, but the excitation gap prevents the fluid from absorbing any energy from the electric field. Hence the current flow must be exactly at right angles to the field and the conductivity tensor takes the remarkable universal form