Monte Carlo evaluation of non-Abelian statistics

We develop a general framework to (numerically) study adiabatic braiding of quasiholes in fractional quantum Hall systems. Specifically, we investigate the Moore-Read (MR) state at nu=1/2 filling factor, a known candidate for non-Abelian statistics, which appears to actually occur in nature. The non-Abelian statistics of MR quasiholes is demonstrated explicitly for the first time, confirming the results predicted by conformal field theories.     

Nature of excitations of the 5/2 fractional quantum Hall effect

It is shown, with the help of exact diagonalization studies on systems with up to 16 electrons, in the presence of up to two delta function impurities, that the Pfaffian model is not accurate for the actual quasiholes and quasiparticles of the 5/2 fractional quantum Hall effect. Implications for non-Abelian statistics are discussed.     

Quasiparticles in the multicomponent Zhang–Hansson–Kivelson model

We study the vortex solutions in a multicomponent Zhang–Hansson–Kivelson model for the fractional quantum Hall effect, at the self-dual point. Vortices with minimal free energy represent Laughlin quasiholes. We find at least two classes of solutions, distinguished by their global invariance, or by the number of conserved charges.     

Correlation-hole induced paired quantum Hall states in the lowest Landau level

A theory is developed for the paired even-denominator fractional quantum Hall states in the lowest Landau level. We show that electrons bind to quantized vortices to form composite fermions, interacting through an exact instantaneous interaction that favors chiral p-wave pairing. There are two canonically dual pairing gap functions related by the bosonic Laughlin wave function (Jastrow factor) due to the correlation holes. We find that the ground state is the Moore-Read Pfaffian in the long-wavelength limit for weak Coulomb interactions, a new Pfaffian with an oscillatory pairing function for intermediate interactions, and a Read-Rezayi composite Fermi liquid beyond a critical interaction strength. Our findings are consistent with recent experimental observations of the 1/2 and 1/4 fractional quantum Hall effects in asymmetric wide quantum wells.     

Reconstructing the electron in a fractionalized quantum fluid

The low-energy physics of the fractional Hall liquid is described in terms of quasiparticles that are qualitatively distinct from electrons. We show, however, that a long-lived electronlike quasiparticle also exists in the excitation spectrum: the state obtained by the application of an electron creation operator to a fractional quantum Hall ground state has a nonzero overlap with a complex, high energy bound state containing an odd number of composite-fermion quasiparticles. The electron annihilation operator similarly couples to a bound complex of composite-fermion holes. We predict that these bound states can be observed through a conductance resonance in experiments involving a tunneling of an external electron into the fractional quantum Hall liquid. A comment is made on the origin of the breakdown of the Fermi liquid paradigm in the fractional Hall liquid.     

A plasma analogy and Berry matrices for non-abelian quantum Hall states

We present an approach to the computation of the non-abelian statistics of quasiholes in quantum Hall states, such as the Pfaffian state, whose wavefunctions are related to the conformal blocks of minimal model conformal field theories. We use the Coulomb gas construction of these conformal field theories to formulate a plasma analogy for the quantum Hall states. A number of properties of the Pfaffian state follow immediately, including the Berry phases, which demonstrate the quasiholes' fractional charge, the abelian statistics of the two-quasihole state, and equal-time ground state correlation functions. The non-abelian statistics of multi-quasihole states follows from an additional assumption.     

Extracting excitations from model state entanglement

We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.     

Microscopic origin of the next-generation fractional quantum Hall effect

Most of the fractions observed to date belong to the sequences nu=n/(2pn+/-1) and nu=1-n/(2pn+/-1), n and p integers, understood as the familiar integral quantum Hall effect of composite fermions. These sequences fail to accommodate, however, many fractions such as nu=4/11 and 5/13, discovered recently in ultrahigh mobility samples at very low temperatures. We show that these "next generation" fractional quantum Hall states are accurately described as the fractional quantum Hall effect of composite fermions.     

Fractionalization via Z(2) gauge fields at a cold-atom quantum Hall transition

We study a single species of fermionic atoms in an "effective" magnetic field at total filling factor ν(f)=1, interacting through a p-wave Feshbach resonance, and show that the system undergoes a quantum phase transition from a ν(f)=1 fermionic integer quantum Hall state to ν(b)=1/4 bosonic fractional quantum Hall state as a function of detuning. The transition is in the (2+1)D Ising universality class. We formulate a dual theory in terms of quasiparticles interacting with a Z(2) gauge field and show that charge fractionalization follows from this topological quantum phase transition. Experimental consequences and possible tests of our theoretical predictions are discussed.     

The stability of the fractional quantum Hall effect in topological insulators

With the recent observation of graphene-like Landau levels at the surface of topological insulators, the possibility of fractional quantum Hall effect, which is a fundamental signature of strong correlations, has become of interest. Some experiments have reported intra-Landau level structure that is suggestive of fractional quantum Hall effect. This paper discusses the feasibility of fractional quantum Hall effect from a theoretical perspective, and argues that while this effect should occur, ideally, in the n=0 and |n|=1 Landau levels, it is ruled out in higher |n| Landau levels. Unlike graphene, the fractional quantum Hall effect in topological insulators is predicted to show an interesting asymmetry between n=1 and n=−1 Landau levels due to spin-orbit coupling.     

Theory of four-dimensional fractional quantum Hall states

We propose a pseudo-potential Hamiltonian for the Zhang-Hu’s generalized fractional quantum Hall states to be the exact and unique ground states. Analogously to Laughlin’s quasi-hole (quasi-particle), the excitations in the generalized fractional quantum Hall states are extended objects. They are vortex-like excitations with fractional charges +(−)1/m³ in the total configuration space CP³. The density correlation function of the Zhang-Hu states indicates that they are incompressible liquid.     

Thermally activated dissipative conductivity in the fractional quantum Hall effect regime

Experimental investigations of thermally activated dissipative conductivity σ_xx in the fractional quantum Hall effect at filling factors v=1/3 and near zero were performed with GaAs/AlGaAs heterojunction based field-effect transistors with an electronic channel. To within our accuracy of 10%, the pre-exponential factor measured for v=1/3 is equal to 2e*2/h (e*=e/3 is the charge of the quasiparticles), the value expected for the case when the quasielectrons and quasiholes make the same contribution to the conductivity. The observed change in the temperature dependence of the conductivity when n deviates from 1/3 is associated with the change in the filling of the energy levels of the quasielectrons and quasiholes and indicates that there is no gap in the quasiparticle density of states averaged over the sample. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 1, 67–72 (10 January 1996)     

Higher-energy composite fermion levels in the fractional quantum Hall effect

Even though composite fermions in the fractional quantum Hall liquid are well established, it is not yet known up to what energies they remain intact. We probe the high-energy spectrum of the 1/3 liquid directly by resonant inelastic light scattering, and report the observation of a large number of new collective modes. Supported by our theoretical calculations, we associate these with transitions across two or more composite fermions levels. The formation of quasiparticle levels up to high energies is direct evidence for the robustness of topological order in the fractional quantum Hall effect.     

Exact results for systems of electrons in the fractional quantum Hall regime

There has been a great deal of interest over the last two decades on the fractional quantum Hall effect, a novel quantum many-body liquid state of strongly correlated two-dimensional electronic systems in a strong perpendicular magnetic field. The most pronounced fractional quantum Hall states occur at odd denominator filling factors of the lowest Landau level and are described by the Laughlin wave function. It is well known that exact closed-form solutions for many-body wave functions, including the Laughlin wave function, are generally very rare and hard to obtain. In this work we present some exact results corresponding to small systems of electrons in the fractional quantum Hall regime at odd denominator filling factors. Use of Jacobi coordinates is the key tool that facilitates the exact calculation of various quantities. Expressions involving integrals over many variables are considerably simplified with the help of Jacobi coordinates allowing us to calculate exactly various quantities corresponding to systems with several electrons.     

Electron fluid and quasiparticles in the quantum Hall effect

To explain fractional quantum Hall effect, it is necessary to take into account both the interaction between electrons and their interaction with impurities. We propose a simple model, where the Coulomb repulsion is replaced by a short range potential. For this model we are able to find many-body wave functions of the electron system interacting with impurities and calculate the Hall conductivityσ _xy. A simple physical picture, arising in the framework of this model, provides the understanding of a general reason for both fractional and integral quantum Hall effect. In the model, electrons forming a two-dimensional system, is supposed to occupy the first Landau level. The interaction of electrons is regarded as being small compared with the distance between the Landau levels. The radius of interaction is much less than the magnetic length. The following statements have been proved (Pokrovsky and Talapov 1985a,b; Trugman and Kivelson 1985). For the fillingν=1/m of the first Landau level the ground state is nondegenerate and has the wave functionΩ _w, proposed by Laughlin (1983). Forν, which is slightly less than 1/m the ground state is highly degenerate in the absence of impurities. It can be described as a system of noninteracting quasiholes as proposed by Laughlin (1983). These quasiholes float in the uniform incompressible fluid. Each quasihole has the charge |e|/m. The total number of quasiholes isq=S?mN, whereS is a number of states on the Landau level,N is the number of electrons. The impurities capture quasiholes. If the number of quasiholesq is less than the number of impuritiesN _i, then the ground state becomes nondegenerate. This fact permits us to calculateσ _xy (Pokrovsky and Talapov 1985b). Let there be a small electric fieldE in the system. In the absence of impurities the electron fluid is at rest in the frame of reference, moving with velocityν=cE/H. In this frame of reference the impurities move with the velocity ?v, carrying captured quasiholes. Therefore, the quasihole currents isj _q=(?ν)(| e|/m)q. Hence, in the initial frame of reference the total current isj=Nev+j _q=Sev/m. This means thatσ _xy=(1/m)e ²/2π?).     

to -12pt [-12pt]to 16ptRapid Note Wavefunctions for the Luttinger liquid

Standard bosonization techniques lead to phonon-like excitations in a Luttinger liquid (LL), reflecting the absence of Landau quasiparticles in these systems. Yet in addition to the above excitations some LL are known to possess solitonic states carrying fractional quantum numbers (e.g. the spin 1/2 Heisenberg chain). We have reconsidered the zero modes in the low-energy spectrum of the Gaussian boson LL Hamiltonian both for fermionic and bosonic LL: in the spinless case we find that two elementary excitations carrying fractional quantum numbers allow to generate all the charge and current excited states of the LL. We explicitly compute the wavefunctions of these two objects and show that one of them can be identified with the 1D version of the Laughlin quasiparticle introduced in the context of the Fractional Quantum Hall effect. For bosons, the other quasiparticle corresponds to a spinon excitation. The eigenfunctions of Wen's chiral LL Hamiltonian are also derived: they are quite simply the one dimensional restrictions of the 2D bulk Laughlin wavefunctions. Received 26 January 1999 and Received in final form 21 April 1999     

Metamorphosis of the quantum Hall ferromagnet at nu=2/5

We report on the dramatic evolution of the quantum Hall ferromagnet in the fractional quantum Hall regime at nu=2/5 filling. A large enhancement in the characteristic time scale gives rise to a dynamical transition into a novel quantized Hall state. The observed Hall state is determined to be a zero-temperature phase distinct from the spin-polarized and spin-unpolarized nu=2/5 fractional quantum Hall states. It is characterized by a strong temperature dependence and puzzling correlation between temperature and time.     

Quantum hall conductivity in a Landau type model with a realistic geometry II

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractional quantization of the Hall conductivity depending on the value of NB/2π (N is the number of charge carriers and B is the magnetic field). When NB/2π is irrational, we show that monovaluated wave functions can be constructed only on the graph of a free group with two generators. When NB/2π is rational, the relevant space becomes a punctured Riemann surface. We finally discuss our results from a phenomenological viewpoint.     

Disorder and the fractional quantum Hall effect

The fractional quantum Hall effect is studied in the 2D electron gas of four GaAs-Al_xGa_1?xAs heterostructures. Localization due to disorder, known to give rise to the wide integral quantum Hall plateaus, is demonstrated to inhibit the fractional effect, which is observed only in the higher mobility samples.