$S$-Duality Constraints on 1D Patterns Associated with Fractional Quantum Hall States

Using the modular invariance of the torus, constraints on the 1D patterns are derived that are associated with various fractional quantum Hall ground states, e.g., through the thin torus limit. In the simplest case, these constraints enforce the well-known odd-denominator rule, which is seen to be a necessary property of all 1D patterns associated to quantum Hall states with minimum torus degeneracy. However, the same constraints also have implications for the non-Abelian states possible within this framework. In simple cases, including the $\nu =1$ Moore-Read state and the $\nu =3/2$ level 3 Read-Rezayi state, the filling factor and the torus degeneracy uniquely specify the possible patterns, and thus all physical properties that are encoded in them. It is also shown that some states, such as the “strong $p$-wave pairing state,” cannot in principle be described through 1D patterns.

Figure 1

A generic periodic 1D pattern as it might appear in the thin torus limit of a fractional quantum Hall state, for a given topological sector. In addition, the individual orbitals may carry a pseudospin label, as in Ref. 6. The complete set of patterns for all topological sectors are subject to the duality constraint (4).