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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 $ u=1$ Moore-Read state and the $ u=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.
In this review the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ($p$-wave) Cooper pairing and are character
We propose ways to create and detect fractionally charged excitations in emph{integer} quantum Hall edge states. The charge fractionalization occurs due to the Coulomb interaction between electrons propagating on different edge channels. The fraction
We investigate the ground state properties of a bosonic Harper-Hofstadter model with local interactions on a finite cylindrical lattice with filling fraction $ u=1/2$. We find that our system supports topologically ordered states by calculating the t
The interplay between interaction and disorder-induced localization is of fundamental interest. This article addresses localization physics in the fractional quantum Hall state, where both interaction and disorder have nonperturbative consequences. W
This paper has been superseded by a new preprint: Kun Yang and Bertrand I. Halperin, arXiv:0901.1429.