Duality in Multi-layered Quantum Hall Systems

1991

We obtain a hierarchy of effective Hamiltonians which allow for a unified treatment of the fractional quantum Hall effect and a gas of fractional-statistics particles (anyons) in two dimensions. Anyon superconductivity is the analog of the fractional quantum Hall effect. For a rational statistics parameter a, P/Q with PQ even, Q anyons bind forming a charge-Qe superfluid.

2002

The current state of the theory of the Fractional Quantum Hall Effect is critically analyzed, especially the generally accepted concept of composite fermions. It is argued that there is no sound theoretical foundation for this concept. A simple one-dimensional model is proposed, which presumably has an energy spectrum similar to that of the FQHE system.

Springer eBooks, 2003

The current state of the theory of the Fractional Quantum Hall Effect is critically analyzed, especially the generally accepted concept of composite fermions. It is argued that there is no sound theoretical foundation for this concept. A simple one-dimensional model is proposed, which presumably has an energy spectrum similar to that of the FQHE system.

2003

IEEE Journal of Quantum Electronics, 1986

We give a brief introduction to the phenomenon of the Fractional Quantum Hall effect, whose discovery was awarded the Nobel prize in 1998. We also explain the composite fermion picture which describes the fractional quantum Hall effect as the integer quantum Hall effect of composite fermions.

In this paper, we report on the study of Abelian and non-Abelian statistics through Fabry-Perot interferometry of fractional quantum Hall (FQH) systems. Our detection of phase slips in quantum interference experiments demonstrates a powerful, new way of detecting braiding of anyons. We confirm the Abelian anyonic braiding statistics in the $\nu = 7/3$ FQH state through detection of the predicted statistical phase angle of $2\pi/3$, consistent with a change of the anyonic particle number by one. The $\nu = 5/2$ FQH state is theoretically believed to harbor non-Abelian anyons which are Majorana, meaning that each pair of quasiparticles contain a neutral fermion orbital which can be occupied or unoccupied and hence can act as a qubit. In this case our observed statistical phase slips agree with a theoretical model where the Majoranas are strongly coupled to each other, and strongly coupled to the edge modes of the interferometer. In particular, an observed phase slip of approximately $...

2007

In Part VI we consider topological field theory that is beyond the descriptions of Feynman diagrams. We begin with the introduction of topological objects in different dimensions in Part VI.1, followed by Part VI.2 describing the phenomenology of integer quantum Hall (IQH) and fractional quantum Hall (FQH) effects in two-dimensional electron gas (2DEG). Topics related to fractional statistics and anyons in two dimensions, and theory of braid and permutation groups, are covered in Part VI.3. Effective field theory based on U(1) Chern-Simons theory for modeling the abelian FQH states and their corresponding edge excitations is described in Part VI.4. Finally in Part VI.5 we consider basic properties of non-abelian anyons, including their braiding and fusion rules, and the potential applications of non-abelian anyons to quantum computation. Candidates of non-abelian FQH states and possible means of experimentally verifying nonabelian statistics will also be briefly discussed.

Physical review letters, 2017

We generalize the notion of Haldane pseudopotentials to anisotropic fractional quantum Hall (FQH) systems that are physically realized, e.g., in tilted magnetic field experiments or anisotropic band structures. This formalism allows us to expand any translation-invariant interaction over a complete basis, and directly reveals the intrinsic metric of incompressible FQH fluids. We show that purely anisotropic pseudopotentials give rise to new types of bound states for small particle clusters in the infinite plane, and can be used as a diagnostic of FQH nematic order. We also demonstrate that generalized pseudopotentials quantify the anisotropic contribution to the effective interaction potential, which can be particularly large in models of fractional Chern insulators.

Journal of Physics: Condensed Matter

Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor ν = 1/m (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelled electron into a different Hilbert subspace from the original Hilbert space of Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions ν ± n,m = n/[n(m ∓ 1) ± 1], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their "conjugates", formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for ν − n,m sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transitions between two topologically distinct phases each corresponding to a family of states: One consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.

Nuclear Physics B, 1995

The edge states of a sample displaying the quantum Hall effect (QHE) can be described by a 1+1 dimensional (conformal) field theory of d massless scalar fields taking values on a d-dimensional torus. It is known from the work of Naculich, Frohlich et al. and others that the requirement of chirality of currents in this scalar field theory implies the Schwinger anomaly in the presence of an electric field, the anomaly coefficient being related in a specific way to Hall conducvivity. The latter can take only certain restricted values with odd denominators if the theory admits fermionic states. We show that the duality symmetry under the O(d, d; Z) group of the free theory transforms the Hall conductivity in a well-defined way and relates integer and fractional QHE's. This means, in particular, that the edge spectra for dually related Hall conductivities are identical, a prediction which may be experimentally testable. We also show that Haldane's hierarchy as well as certain of Jain's fractions can be reproduced from the Laughlin fractions using the duality transformations. We thus find a framework for a unified description of the QHE's occurring at different fractions. We also give a simple derivation of the wave functions for fractions in Haldane's hierarchy.