S-duality constraints on 1D patterns associated with fractional quantum Hall states

TL;DR

This paper uses S-duality and modular invariance to derive constraints on 1D patterns related to fractional quantum Hall states, revealing the odd-denominator rule and implications for non-Abelian states.

Contribution

It introduces a framework applying S-duality constraints to 1D patterns, elucidating their relation to fractional quantum Hall states and identifying limitations for certain states.

Findings

Enforces the odd-denominator rule for all 1D patterns with minimal torus degeneracy.

Specifies unique patterns for certain non-Abelian states like Moore-Read and Read-Rezayi.

Shows some states, such as the strong p-wave pairing state, cannot be described by these patterns.

Abstract

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 patterns.