We construct the wavefunctions for the Moore–Read ν = 5/2 quantum Hall state on a torus in the presence of two quasiholes. These explicit wavefunctions allow us to compute the monodromy matrix that describes the effect of quasihole motion on the space of degenerate ground states. The result agrees with the recent discussion by Oshikawa et al ( Ann. Phys. at press). Our calculation provides a conformal field theory explanation of why certain transitions between ground states are forbidden. It is because taking a quasihole around a generator of the torus can change the fusion channel of the two quasiholes, and this requires a change of parity of the electron number in some of the ground states.
We construct the wavefunctions for the Moore–Read ν = 5/2 quantum Hall state on a torus in the presence of two quasiholes. These explicit wavefunctions allow us to compute the monodromy matrix that describes the effect of quasihole motion on the space of degenerate ground states. The result agrees with the recent discussion by Oshikawa et al ( Ann. Phys. at press). Our calculation provides a conformal field theory explanation of why certain transitions between ground states are forbidden. It is because taking a quasihole around a generator of the torus can change the fusion channel of the two quasiholes, and this requires a change of parity of the electron number in some of the ground states.
Since fractional statistics of Laughlin states is reflected in its topological degeneracy, it is natural to expect non-Abelian statistics to have some effect on topological degeneracy. To study the topological degeneracy of the Moore-Read nu = 5/2 quantum Hall state on a torus, we construct the wave functions of this state in the presence of two quasiholes [6]. These explicit wave functions allow us to compute the monodromy matrix that describes the effect of quasihole motion on the space of degenerate ground states. Our calculation provides a conformal field theory explanation of why certain transitions between ground states are forbidden. It is because taking a quasihole around a generator of the torus can change the fusion channel of the two quasiholes, and this requires a change of parity of the electron number in sonic of the ground states.
The Abelian hierarchy of quantum Hall states accounts for most of the states in the lowest Landau level, and there is evidence of a similar hierarchy of non-Abelian states emanating from the ν=5/2 Moore–Read state in the second Landau level. Extending a recently developed formalism for hierarchical quasihole condensation, we present a theory that allows for the explicit construction of the ground state wave function, as well as its quasiparticle excitations, for any state based on the Abelian hierarchy. We relate our construction to structures in rational conformal field theory and stress the importance of using coherent state wave functions, which allows us to formulate an extension of the bulk–edge correspondence that was conjectured by Moore and Read. Finally, we study the proposed ground state wave functions in the limiting geometry of a thin torus and argue that they coincide with the known exact results.
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 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.
The abelian hierarchy of quantum Hall states accounts for most of the states in the lowest Landau level, and there is evidence of a similar hierarchy of non-abelian states emanating from the {\nu} = 5/2 Moore-Read state in the second Landau level. Extending a recently developed formalism for hierarchical quasihole condensation, we present a theory that allows for the explicit construction of the ground state wave function, as well as its quasiparticle excitations, for any state based on the abelian hierarchy. We relate our construction to structures in rational conformal field theory and stress the importance of using coherent state wave functions which allows us to formulate an extension of the bulk - edge correspondence that was conjectured by Moore and Read. Finally, we study the proposed ground state wave functions in the limiting geometry of a thin torus and argue that they coincide with the known exact results.
This work concerns Ising quasiholes in Moore-Read type lattice wave functions derived from conformal field theory. We commence with constructing Moore-Read type lattice states and then add quasiholes to them. By use of Metropolis Monte Carlo simulations, we analyze the features of the quasiholes, such as their size, shape, charge, and braiding properties. The braiding properties, which turn out to be the same as in the continuum Moore-Read state, demonstrate the topological attributes of the Moore-Read lattice states in a direct way. We also derive parent Hamiltonians for which the states with quasiholes included are ground states. One advantage of these Hamiltonians lies therein that we can now braid the quasiholes just by changing the coupling strengths in the Hamiltonian since the Hamiltonian is a function of the positions of the quasiholes. The methodology exploited in this article can also be used to construct other kinds of lattice fractional quantum Hall models containing quasiholes, for example, investigation of Fibonacci quasiholes in lattice Read-Rezayi states.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.