The Halperin $(m,m^{\prime },n)$ bilayer quantum Hall states are studied on thin cylinders. In this limit, charge-density-wave patterns emerge that are characteristic of the underlying quantum Hall state. The general patterns are worked out from a variant of the plasma analogy. Torus degeneracies are recovered, and for some important special cases a connection to well-known spin chain physics is made. By including interlayer tunneling, we also work out the critical behavior of a possible phase transition between the (331) state and the non-Abelian Moore-Read state in the thin cylinder limit.

• Received 23 January 2008

DOI:https://doi.org/10.1103/PhysRevLett.101.036804