Motivated by the physics of the Moore-Read ν= 1/2 state away from half-filling, we investigate collective states of non-abelian e/4 quasiparticles in a magnetic field. We consider two types of collective states: incompressible liquids and Wigner crystals. In the incompressible liquid case, we construct a natural series of states which can be thought of as a non-abelian generalization of the Laughlin states. These states are associated with a series of hierarchical states derived from the Moore-Read state - the simplest of which occur at filling fraction 8/17 and 7/13. Interestingly, we find that the hierarchical states are abelian even though their parent state is non-abelian. In the Wigner crystal case, we construct two candidate states. We find that they, too, are abelian - in agreement with previous analysis.
We show in this paper that thermopower is enhanced in non-Abelian quantum Hall liquids under appropriate conditions. This is because thermopower measures entropy per electron in the clean limit, while the degeneracy and entropy associated with non-Abelian quasiparticles enhance entropy when they are present. Thus thermopower can potentially probe non-Abelian nature of the quasiparticles, and measure their quantum dimension.
A most interesting feature of certain fractional quantum Hall states is that their quasiparticles obey non-Abelian fractional statistics. So far, candidate non-Abelian wave functions have been constructed from conformal blocks in cleverly chosen conformal field theories. In this work we present a hierarchy scheme by which we can construct daughter states by condensing non-Abelian quasiparticles (as opposed to quasiholes) in a parent state, and show that the daughters have a non-Abelian statistics that differ from the parent. In particular, we discuss the daughter of the bosonic, spin-polarized…
We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors $7/3 <ν_e<8/3$. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at $ν_e=5/2$ (paired Moore--Read "pfafian" state) and 12/5 (clustered Read--Rezayi "parafermion" state). Outside of this range, the noninteracting…
Observation of non-Abelian statistics for the e/4 quasiparticles in the ν=5/2 fractional quantum Hall state remains an outstanding experimental problem. The non-Abelian statistics are linked to the presence of additional low energy states in a system with localised quasiparticles, and hence an additional low-temperature entropy. Recent experiments, which detect changes in the number of quasiparticles trapped in a local potential well as a function of an applied gate voltage, V_G, provide a possibility for measuring this entropy, if carried out over a suitable range of temperatures, T. We prese…
It is a theoretical conjecture that 5/2 fractional quantum Hall state charge e/4 excitations may obey exotic non-Abelian statistics. In edge state interference these purported non-Abelian quasiparticles should display period e/4 Aharonov-Bohm oscillations if the interfering quasiparticle encircles an even number of localized e/4 charges, but suppression of oscillations if an odd number is encircled. To test this hypothesis, here we perform swept area interference measurements at 5/2. We observe an alternating pattern of e/4 and e/2 period oscillations in resistance. This aperiodic alternation …
The quasiparticle propagator of Haldane-Rezayi(HR) fractional quantum Hall (FQH) state is calculated, based on a chiral fermion model (or a Weyl fermion model) equipped with a hidden spin SU(2) symmetry. The spectrum of the chiral fermion model for each total spin and total momentum is shown to be identical to that of the SU(2) c = -2 model introduced to describe the edge spectrum of the HR state.
One-dimensional chains of non-Abelian quasiparticles described by $SU(2)_k$ Chern-Simons-Witten theory can enter random singlet phases analogous to that of a random chain of ordinary spin-1/2 particles (corresponding to $k \to \infty$). For $k=2$ this phase provides a random singlet description of the infinite randomness fixed point of the critical transverse field Ising model. The entanglement entropy of a region of size $L$ in these phases scales as $S_L \simeq \frac{\ln d}{3} \log_2 L$ for large $L$, where $d$ is the quantum dimension of the particles.
We examine interferometric experiments in systems that exhibit non-Abelian braiding statistics, expressing outcomes in terms of the modular S-matrix. In particular, this result applies to FQH interferometry, and we give a detailed treatment of the Read-Rezayi states, providing explicit predictions for the recently observed nu=12/5 plateau.
Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding q…
Non-Abelian e/4 quasiparticles at 5/2 filling factor in a correlated two-dimensional electron gas have a proposed specific property in an interference measurement of their edge propagation: encircling an even number of localized e/4 quasiparticles allows expression of e/4 Aharonov-Bohm (A-B) oscillations, but suppression of these oscillations will occur with an odd number encircled. This picture is tested explicitly here in multiple interferometers of different areas. The encircled localized e/4 quasiparticle number near 5/2 filling factor is changed by sweeping B-field, and oscillations are o…
The quantum Hall states at filling factors $ν=5/2$ and $7/2$ are expected to have Abelian charge $e/2$ quasiparticles and non-Abelian charge $e/4$ quasiparticles. The non-Abelian statistics of the latter has been predicted to display a striking interferometric signature, the even-odd effect. By measuring resistance oscillations as a function of magnetic field in Fabry-Pérot interferometers using new high purity heterostructures, we for the first time report experimental evidence for the non-Abelian nature of excitations at $ν=7/2$. At both $ν=5/2$ and $7/2$ we also examine, for the first time,…
We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a local potential (e.g., induced by an AFM tip) in a microscopic model of fractional quantum Hall liquids with long-range Coulomb interaction and edge confining potential. We find, in particular, at Laughlin filling fraction $ν= 1/3$, both quasihole and quasiparticle states can emerge as the ground state of the system in the presence of the trapping potential. As expected, we find the presence of an Abelian quasihole has no effect on the edge spectrum of the quantum liquid, unlike in the non-Abelian case [Phys. Rev. Let…
We study the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is cons…
In the pattern-of-zeros approach to quantum Hall states, a set of data {n;m;S_a|a=1,...,n; n,m,S_a in N} (called the pattern of zeros) is introduced to characterize a quantum Hall wave function. In this paper we find sufficient conditions on the pattern of zeros so that the data correspond to a valid wave function. Some times, a set of data {n;m;S_a} corresponds to a unique quantum Hall state, while other times, a set of data corresponds to several different quantum Hall states. So in the latter cases, the patterns of zeros alone does not completely characterize the quantum Hall states. In thi…
Quasiparticle tunneling between two counter-propagating edges through point contacts could provide information on the statistics of the quasiparticles. Previous study on a disk found a scaling behavior by varying the tunneling distance. It was found that in the limit with zero tunneling distance, the Abelian quasiparticles tunneling obey the scaling analysis while the non-Abelian quasiparticles exhibit some non-trivial behaviors on the scaling exponents. Because of the limitation of disk geometry, we put the fractional quantum Hall (FQH) state on the surface of a cylinder which has a larger tu…
We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of abelian and nonabelian quantum Hall states. While path dependent geometric phases can perturb the abelian part of the statistics, we find that the nonabelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.
We study quasiparticle tunneling between the edges of a non-Abelian topological state. The simplest examples are a p+ip superconductor and the Moore-Read Pfaffian non-Abelian fractional quantum Hall state; the latter state may have been observed at Landau-level filling fraction nu=5/2. Formulating the problem is conceptually and technically non-trivial: edge quasiparticle correlation functions are elements of a vector space, and transform into each other as the quasiparticle coordinates are braided. We show in general how to resolve this difficulty and uniquely define the quasiparticle tunneli…
We show how shot noise in an electronic Mach-Zehnder interferometer in the fractional quantum Hall regime probes the charge and statistics of quantum Hall quasiparticles. The dependence of the noise on the magnetic flux through the interferometer allows for a simple way to distinguish Abelian from non-Abelian quasiparticle statistics. In the Abelian case, the Fano factor (in units of the electron charge) is always lower than unity. In the non-Abelian case, the maximal Fano factor as a function of the magnetic flux exceeds one.
We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally-invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or …
We study a tunneling geometry defined by a single point-contact constriction that brings to close vicinity two points sitting at the same edge of a quantum Hall liquid, shortening the trip between the otherwise spatially separated points along the normal chiral edge path. This ``wormhole''-like geometry allows for entrapping bulk quasiparticles between the edge path and the tunnel junction, possibly realizing a topologically protected qubit if the quasiparticles have non-Abelian statistics. We show how either noise or simpler voltage measurements along the edge can probe the non-Abelian nature…
Recent schemes for experimentally probing non-abelian statistics in the quantum Hall effect are based on geometries where current-carrying quasiparticles flow along edges that encircle bulk quasiparticles, which are localized. Here we consider one such scheme, the Fabry-Perot interferometer, and analyze how its interference patterns are affected by a coupling that allows tunneling of neutral Majorana fermions between the bulk and edge. While at weak coupling this tunneling degrades the interference signal, we find that at strong coupling, the bulk quasiparticle becomes essentially absorbed by …
We calculate the non-linear I-V tunneling curves for a two-point-contact tunneling junction between two edges of the $ν={5/2}$ non-abelian fractional quantum Hall state. The non-linear I-V tunneling curves are calculated for both cases with and without an $e/4$ non-abelian quasiparticle between the two contacts. We confirm that, within a dynamical edge theory, the presence of the $e/4$ quasiparticle between the two contacts destroys the interference between the two tunneling paths. We also calculate how the interference reappears as the $e/4$ quasiparticle is moved closer to an edge.
The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at Landau level filling fraction $ν=5/2$. This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the $ν=5/2$ state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically-protected qubit on …
It has been shown that different Abelian and non-Abelian fraction quantum Hall states can be characterized by patterns of zeros described by sequences of integers {S_a}. In this paper, we will show how to use the data {S_a} to calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types and their quantum numbers, as well as the actions of the quasiparticle tunneling and modular transformations on the degenerate ground states on torus.
A stochastic cumulant GW method is presented, allowing us to map the evolution of photoemission spectra, quasiparticle energies, lifetimes and emergence of collective excitations from molecules to bulk-like systems with up to thousands of valence electrons, including Si nanocrystals and nanoplatelet. The quasiparticle energies rise due to their coupling with collective shake-up (plasmon) excitations, and this coupling leads to significant spectral weight loss (up to 50% for the low energy states), shortening the lifetimes and shifting the spectral features to lower energy by as much as 0.6 eV.…
We show that the resistance of the v=5/2 quantum Hall state, confined to an interferometer, oscillates with magnetic field consistent with an Ising-type non-Abelian state. In three quantum Hall interferometers of different sizes, resistance oscillations at v=7/3 and integer filling factors have the magnetic field period expected if the number of quasiparticles contained within the interferometer changes so as to keep the area and the total charge within the interferometer constant. Under these conditions, an Abelian state such as the (3,3,1) state would show oscillations with the same period a…
Fracton phase of matter host fractionalized topological quasiparticles with restricted mobility. A wide variety of fracton models with abelian excitations had been proposed and extensively studied while the candidates for non-Abelian fracton phases are less explored. In this work, we investigate the effect of twisted defect in abelian fracton models. The twisted defect is launched by introducing a branch cut line hosting anyon condensate. In particular, these twisted defects, which alter different types of quasiparticles, carry projective non-Abelian zero modes. En route, such defects can be e…
Entropy of Non-Abelian Anyons from Slow Quasiparticle Dynamics in Quantum Hall Interferometers
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利·萨克勒物理与天文学学院）；Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Center for Quantum Science and Engineering (QSE Center), École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）
We demonstrate how to directly study non-Abelian statistics for a wide class of exactly solvable many-body quantum systems. By employing exact eigenstates to simulate the adiabatic transport of a model's quasiparticles, the resulting Berry phase provides a direct demonstration of their non-Abelian statistics. We apply this technique to Kitaev's honeycomb lattice model and explicitly demonstrate the existence of non-Abelian Ising anyons confirming the previous conjectures. Finally, we present the manipulations needed to transport and detect the statistics of these quasiparticles in the laborato…
Electrons that are confined to a single Landau level in a two dimensional electron gas realize the effects of strong electron-electron repulsion in its purest form. The kinetic energy of individual electrons is completely quenched and all physical properties are dictated solely by many-body effects. A remarkable consequence is the emergence of new quasiparticles with fractional charge and exotic quantum statistics of which the most exciting ones are non-Abelian quasiparticles. A non-integer quantized thermal Hall conductance $κ_{xy}$ (in units of temperature times the universal constant $π^2 k…
Quasiparticles, which obey non abelian statistics, were predicted to exist in different physical systems, but are yet to be observed directly. Possible candidate states, which are expected to support such quasiparticles, are the ν=8/3, ν=5/2 and ν=7/3 fractional quantum Hall states (in the second Landau level). The non abelian quasiparticles are expected to carry charge and a unique form of a chiral neutral edge mode. Recent measurements in the ν=5/2 state detected quasiparticle charge e/4 and an upstream (opposite to charge transport) chiral neutral mode; both agreeing with a non abelian anti…
Fractionally charged quasiparticles in the quantum Hall state with filling factor $ν=5/2$ are expected to obey non-Abelian statistics. We demonstrate that their statistics can be probed by transport measurements in an electronic Mach-Zehnder interferometer. The tunneling current through the interferometer exhibits a characteristic dependence on the magnetic flux and a non-analytic dependence on the tunneling amplitudes which can be controlled by gate voltages.
The low energy physics of fractional quantum Hall (FQH) states -- a paradigm of strongly correlated topological phases of matter -- to a large extent is captured by weakly interacting quasiparticles known as composite fermions (CFs). In this paper, based on numerical simulations and effective field theory, we argue that some \emph{high energy} states in the FQH spectra necessitate a different description based on \emph{parton} quasiparticles. We show that Jain states at filling factor $ν{=}n/(2pn\pm1)$ with integers $n,p{\geq}2$, support two kinds of collective modes: in addition to the well-k…
We provide a comprehensive theoretical investigation of the Fermi liquid quasiparticle description in two-dimensional electron gas interacting via the long-range Coulomb interaction by calculating the electron self-energy within the leading-order approximation, which is exact in the high-density limit. We find that the quasiparticle energy is larger than the imaginary part of the self-energy up to very high energies, implying that the basic Landau quasiparticle picture is robust up to far above the Fermi energy. We find, however, that the quasiparticle picture becomes fragile in a small discre…
We consider the tunneling current through a double point-contact Fabry-Perot interferometer such as used in recent experimental studies of the fractional quantum Hall plateau at filling fraction nu=5/2. We compare the predictions of several different models of the state of the electrons at this plateau: the Moore-Read, anti-Pfaffian, SU(2)_2 NAF, K=8 strong pairing, and (3,3,1) states. All of these predict the existence of charge e/4 quasiparticles, but the first three are non-Abelian while the last two are Abelian. We give explicit formulas for the scaling of charge e/2 and charge e/4 quasipa…
In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport, and provide information on both the charge and statistics of the quasiparticles involved. Here we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charge e/4 (non-Abelian) and e/2 (Abelian) may co-exist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for e/2 and e/4 quasiholes to tunnel through the bulk of the Moore…
Several states were proposed as candidates for the $ν=5/2$ quantum Hall plateau. We suggest an experiment which can determine the physical state. The proposal involves transport measurements in the geometry with three quantum Hall edges connected by two quantum point contacts. In contrast to interference experiments, this approach can distinguish the Pfaffian and anti-Pfaffian states as well as different states with identical Pfaffian or anti-Pfaffian statistics. The transport is not sensitive to the fluctuations of the number of the quasiparticles trapped in the system.
We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex-vortex interactions are shown to lead to the lifting of the topological degeneracy and the energy is discovered to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice…
The fractional quantum Hall effect, where plateaus in the Hall resistance at values of coexist with zeros in the longitudinal resistance, results from electron correlations in two dimensions under a strong magnetic field. Current flows along the edges carried by charged excitations (quasi particles) whose charge is a fraction of the electron charge. While earlier research concentrated on odd denominator fractional values of $ν$, the observation of the even denominator $ν=5/2$ state sparked a vast interest. This state is conjectured to be characterized by quasiparticles of charge e/4, whose sta…
Starting from a collective description of the incompressible fractional quantum Hall liquid as an elastic medium that supports gapped neutral excitations I show that the one-electron spectral function of this system exhibits a sharp peak at the lowest available excitation energy, well separated from the continuum spectrum at higher energy. I interpret this peak as the signature of the integral charge quasiparticle recently predicted by Peterson and Jain\cite{Jain05}, and calculate its spectral weight for different filling factors.
Non-abelian anyons are prospective candidates for fault-tolerant topological quantum computation due to their long-range entanglement. Curiously these quasiparticles are charge-neutral, hence elusive to most conventional measurement techniques. A proposed host of such quasiparticles is the $ν$=5/2 quantum Hall state. The gapless edge modes can provide the topological order of the state, which in turn identifies the chirality of the non-abelian mode. Since the $ν$=5/2 state hosts a variety of edge modes (integer, fractional, neutral), a robust technique is needed to isolate the fractional chann…
Two-dimensional electron gases (2DEGs) in SrTiO$_3$ have become model systems for engineering emergent behaviour in complex transition metal oxides. Understanding the collective interactions that enable this, however, has thus far proved elusive. Here we demonstrate that angle-resolved photoemission can directly image the quasiparticle dynamics of the $d$-electron subband ladder of this complex-oxide 2DEG. Combined with realistic tight-binding supercell calculations, we uncover how quantum confinement and inversion symmetry breaking collectively tune the delicate interplay of charge, spin, orb…
Real-time imaging of quasiparticle dynamics at a topological defect in an electronic crystal
Jozef Stefan Institute（约瑟夫·斯蒂芬研究所）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Jülich Supercomputing Centre, Institute for Advanced Simulation, Forschungszentrum Jülich（于利希研究中心先进模拟所超级计算中心）
The quasiparticles in Quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could in principle be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes both in Abelian and non-Abelian QH states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons, and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.
Quasiparticles with fractional charge and fractional statistics are key features of the fractional quantum Hall effect. We discuss in detail the definitions of fractional charge and statistics and the ways in which these properties may be observed. In addition to theoretical foundations, we review the present status of the experiments in the area. We also discuss the notions of non-Abelian statistics and attempts to find experimental evidence for the existence of non-Abelian quasiparticles in certain quantum Hall systems.
We present a microscopic theory of ground-state spectral function of bilayer quantum Hall systems that includes interactions between Hartree-Fock quasiparticles and quantum fluctuations of the order parameter field. The collective modes in these systems are properly described only when fluctuations in direct and exchange particle-hole channels are taken into account. Using an auxiliary field functional integral approach, we present a generalization of the random phase approximation \emph{for quasiparticle self-energy} which captures fluctuations in both channels. We discuss its relationship to…
We analyze tunneling of non-Abelian quasiparticles between the edges of a quantum Hall droplet at Landau level filling fraction nu=5/2, assuming that the electrons in the first excited Landau level organize themselves in the non-Abelian Moore-Read Pfaffian state. We formulate a bosonized theory of the modes at the two edges of a Hall bar; an effective spin-1/2 degree of freedom emerges in the description of a point contact. We show how the crossover from the high-temperature regime of weak quasiparticle tunneling between the edges of the droplet, with 4-terminal R_{xx} scaling as T^{-3/2}, to …
Non-Abelian states of matter, in which the final state depends on the order of the interchanges of two quasiparticles, can encode information immune from environmental noise with the potential to provide a robust platform for topological quantum computation. We demonstrate that phonons can carry non-Abelian frame charges at the band crossing points of their frequency spectrum, and that external stimuli can drive their braiding. We present a general framework to understand the topological configurations of phonons from first principles calculations using a topological invariant called Euler cla…
Collective motions of electrons in solids are often conveniently described as the movements of quasiparticles. Here we show that these quasiparticles can be hierarchical. Examples are valley electrons, which move in hyperorbits within a honeycomb lattice and forms a valley pseudospin, or the self-rotation of the wave-packet. We demonstrate that twist can induce higher level motions of valley electrons around the moire superlattice of bilayer systems. Such larger scale collective movement of the valley electron, can be regarded as the self-rotation (spin) of a higher-level quasiparticle, or wha…
The nu=5/2 fractional quantum Hall effect state has attracted great interest recently, both as an arena to explore the physics of non-Abelian quasiparticle excitations, and as a possible architecture for topological quantum information processing. Here we use the conformal field theoretic description of the Moore-Read state to provide clear tunneling signatures of this state in an Aharonov-Bohm geometry. While not probing statistics directly, the measurements proposed here would provide a first, experimentally tractable step towards a full characterization of the 5/2 state.
