We analyze the Moore-Read Pfaffian state on a thin torus. The known six-fold degeneracy is realized by two inequivalent crystalline states with a four-and two-fold degeneracy respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension 2 n−1 for 2n − k holes and k particles at fixed positions and assign each a charge ±e/4. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model hamiltonian and a small number of particles supports the presence of a stable phase with degenerate vacua and quarter charged domain walls also away from the thin torus limit. A spin chain hamiltonian encodes the degenerate vacua and the various domain walls.One of the most intriguing aspects of the quantum Hall (QH) system is the possibility of experimentally observing non-abelian statistics. In particular it has been proposed that the fractional filling part of the observed [1] ν = 5/2 state is well described by the non-abelian Moore-Read, or Pfaffian, wave function [2],where Ψ 1/2 is the bosonic Laughlin state at ν = 1/2 and z i are the complex electron coordinates in the plane. By now, a great deal has been learnt about (1) and its quasihole excitations, and we list some of the pertinent results: I. Eq.(1) is the exact ground state of a certain local three-body interaction [3,4]. II. There are six degenerate ground states on a torus, and the electronic wave functions are explicitly known [3]. III. The quasiholes have charge e/4 and can only be created in pairs [2]. The dimensionality of the Hilbert space for 2n holes at fixed positions in the plane is 2 n−1 , and the wave functions in a particular "preferred basis" have been constructed [5,6]. IV. The quasihole wave functions are the conformal blocks of a correlator in a c = 3/2 rational conformal field theory involving a bosonic vertex operator and a majorana fermion. The conformal blocks have been explicitly constructed for four holes, where the Hilbert space is twodimensional. The braiding properties, or monodromies, of the conformal blocks translate into non-abelian statistics for the quasiholes [2,7]. V. The ground state in eq.(1) can be viewed as a triplet pairing state of composite fermions, and the quasiholes as vortex excitations. The pairing picture nicely explains the presence of quarter charged holes [3,8].Finally we should mention that the great recent interest in the non-abelian QH states has to a large extent been spurred by the proposals to use them to build decoherence free quantum computational devices [9].Recently, it was shown that studying the lowest Lan-dau level (LLL) on a thin torus, with circumference L 1 , both allows for a simple understanding of already established phenomena, and for arriving at new results [10,11]. In particular, it was shown how the states in the Jain series ν = p/(2pm + 1) [12] become gapped crystals, with a unit c...

We analyze the non-abelian Read-Rezayi quantum Hall states on the torus, where it is natural to employ a mapping of the many-body problem onto a one-dimensional lattice model. On the thin torus-the Tao-Thouless (TT) limit-the interacting many-body problem is exactly solvable. The Read-Rezayi states at filling ν = k kM +2 are known to be exact ground states of a local repulsive k + 1-body interaction, and in the TT limit this is manifested in that all states in the ground state manifold have exactly k particles on any kM + 2 consecutive sites. For M = 0 the two-body correlations of these states also imply that there is no more than one particle on M adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domain walls between the ground states, and we show that the number of distinct domain wall patterns gives rise to the nontrivial degeneracies, required by the non-abelian statistics of these states. In the second part of the paper we consider the quasihole degeneracies from a conformal field theory (CFT) perspective, and show that the counting of the domain wall patterns maps one to one on the CFT counting via the fusion rules. Moreover we extend the CFT analysis to topologies of higher genus. PACS numbers: 73.43.Cd, 71.10.Pm I. INTRODUCTION Microscopic wave functions have ever since Laughlin's original work [1] back in 1983 been instrumental for the understanding of the fractional quantum Hall effect (FQHE). At Landau level filling fraction ν = 1/q, q odd, Laughlin's construction shows why an incompressible quantum liquid, with fractionally charged excitations, may form to minimize the electron-electron repulsion by optimizing the short-range correlations as two particles approach each-other. The Moore-Read (MR) [2] and Read-Rezayi (RR) states [3] provide a natural extension of this, where the wave functions vanish as clusters of k + 1 particles are formed. The latter has filling fraction ν = k/(kM + 2), describing fermions (bosons) for M odd (even).Conformal field theory (CFT) plays a central role in the theory of the FQHE, as it e.g. describes the edge theory and gives a method for construction of trial wave functions. The understanding of this connection was boosted by Moore and Read with their seminal paper from 1991 [2], where they suggested a general CFT-FQHE connection, and in particular showed that (at least) some FQHE wave functions can be obtained from correlators in certain so called rational CFT's. Not only did they reproduce the Laughlin wave functions, but also put forward the so called Moore-Read (aka pfaffian) state which supports non-abelian excitations. It was first suggested in [4] that this state can describe the enigmatic state observed [5] at filling ν = 5/2. By now, there is ample (numerical) evidence that this is indeed the case [6,7,8]. It is exciting that the first experimental steps towards determining the nature of the ν = 5/2 quantum Hall state have recently been made [9].Read and Rezayi [10] provided support for the suggestions that the k = 3, M = 1 RR state...

We study dipolar bosons in a 1D optical lattice and identify a region in parameter space---strong coupling but relatively weak on-site repulsion---hosting a series of stable charge-density-wave (CDW) states whose low-energy excitations, built from "fractional domain walls," have remarkable similarities to those of non-Abelian fractional quantum Hall states. Here, a conventional domain wall between translated CDW's may be split by inserting strings of degenerate, but inequivalent, CDW states. Outside these insulating regions, we find numerous supersolids as well as a superfluid regime. The mentioned phases should be accessible experimentally and, in particular, the fractional domain walls can be created in the ground state using single-site addressing, i.e., by locally changing the chemical potential.Comment: 6 pages, 3 figures. Extended discussion. As publishe