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arXiv:0711.3204 · Physical Review B

Fractional quantum Hall hierarchy and the second Landau level

Parsa Bonderson, Johannes K. Slingerland

Published Sep 26, 2008• 134 citations
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Abstract

We generalize the fractional quantum Hall hierarchy picture to apply to arbitrary, possibly non-Abelian, fractional quantum Hall states. Applying this to the $\ensuremath{\nu}=5/2$ Moore-Read state, we construct explicit trial wave functions to describe the fractional quantum Hall effect in the second Landau level. The resulting hierarchy of states, which reproduces the filling fractions of all observed Hall conductance plateaus in the second Landau level, is characterized by electron pairing in the ground state and an excitation spectrum that includes non-Abelian anyons of the Ising type. We propose this as a unifying picture in which $p$-wave pairing characterizes the fractional quantum Hall effect in the second Landau level.

DOI
10.1103/physrevb.78.125323

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arXiv:0908.3481•Physical Review B•Apr 2, 2010•49 citations
We generalize the fractional quantum Hall hierarchy picture to apply to arbitrary, possibly non-Abelian, fractional quantum Hall states. Applying this to the \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=5/2 Moore-Read state, we construct explicit trial wave functions to describe the fractional quantum Hall effect in the second Landau level. The resulting hierarchy of states, which reproduces the filling fractions of all observed Hall conductance plateaus in the second Landau level, is characterized by electron pairing in the ground state and an excitation spectrum that includes non-Abelian anyons of the Ising type. We propose this as a unifying picture in which ppp-wave pairing characterizes the fractional quantum Hall effect in the second Landau level.
We review some theoretical progress towards understanding the origin of the fractional quantum Hall effect at filling factor 5/2 using numerical techniques. The leading candidate thought to describe the physics is the Moore-Read Pfaffian state which supports non-abelian anyonic excitations. Interestingly, these excitations have possible applications in topologically protected quantum information processing. Specifically, we discuss how the finite-thickness of realistic quasi-two-dimensional systems stabilizes the Moore-Read Pfaffian in the fractional quantum Hall effect at 5/2 and the use of bilayer systems in the second Landau level to investigate the possible quantum phase transition between abelian and non-abelian fractional quantum Hall 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 {\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.
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 theoretically examine entanglement in fractional quantum hall states, explicitly taking into account and emphasizing the quasi-two-dimensional nature of experimental quantum Hall systems. In particular, we study the entanglement entropy and the entanglement spectrum as a function of the finite layer thickness, ddd, of the quasi-two-dimensional system for a number of filling fractions $\ensuremath{\nu}$ in the lowest and the second Landau levels: $\ensuremath{\nu}$ = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy, and find that filling fractions 1/3 and 7/3, which are considered to be Laughlin states, weaken with ddd in the lowest Landau level ($\ensuremath{\nu}=1/3$) and strengthen with ddd in the second Landau level ($\ensuremath{\nu}=7/3$). For the enigmatic even-denominator $\ensuremath{\nu}=5/2$ state, we find that entanglement in the ground state is consistent with that of the non-Abelian Moore-Read Pfaffian state at an optimal thickness, ddd. We also find that the single-layer $\ensuremath{\nu}=1/2$ system is not a fractional quantum Hall state consistent with the experimental observation. In general, our theoretical findings based on entanglement considerations are completely consistent with the results based on wave-function overlap calculations.
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.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.