We demonstrate numerically that non-Abelian quasihole excitations of the $ν= 5/2$ fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the quasihole spacing is increased, the unitary transformation which describes winding two quasiholes around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four quasiholes become exponentially degenerate. We calculate the length scales for these two decays to be $ξ_{U} \approx 2.7 \ell_0$ and $ξ_{E} \approx 2.3 \ell_0$ respectively. Additionally we determine which fusion channel is lower in energy when two quasiholes are brought close together.
Two- and three-body correlations of incompressible quantum liquids are studied numerically. Pairing of composite fermions (CFs) in the 1/3-filled second CF Landau level is found at nu_e=4/11. It is explained by reduced short-range repulsion due to ring-like single-particle charge distribution. Although Moore-Read state of CFs is unstable in the 1/2-filled second CF level, condensation of its quasiholes is a possible origin of incompressibility at nu_e=4/11. Electron pairing occurs at nu_e=7/3 and 13/3, but with different pair-pair correlations. Signatures of triplets are found at higher fillin…
We construct the wave functions for the Moore-Read $ν= 5/2$ quantum Hall state on a torus in the presence of two quasiholes. 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. The result agrees with the recent discussion by Oshikawa {\it et al.} 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, an…
We present a general method to obtain the central charge and quasihole scaling dimension directly from groundstate and quasihole wavefunctions. Our method applies to wavefunctions satisfying specific clustering properties. We then use our method to examine the relation between Jack symmetric functions and certain W-algebras. We add substantially to the evidence that the (k,r) admissible Jack functions correspond to correlators of the conformal field theory W_k(k+1,k+r), by calculating the central charge and scaling dimensions of some of the fields in both cases and showing that they match. For…
Bilayer quantum Hall system (BLQH) differ from its single layer counterparts (SLQH) by its symmetry breaking ground state and associated neutral gapless mode in the pseudo-spin sector. Due to the gapless mode, qualitatively good groundstate and low energy excited state wavefunctions at any finite distance is still unknown. We investigate this important open problem by the Composite Boson (CB) theory developed by one of the authors to study BLQH systematically. We derive the ground state, quasi-hole and a pair of quasihole wavefunctions from the CB theory and its dual action. We find that the g…
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…
We study the ground state and low-energy excitations of fractional quantum Hall systems on a disk at filling fraction $ν= 5/2$, with Coulomb interaction and background confining potential. We find the Moore-Read ground state is stable within a finite but narrow window in parameter space. The corresponding low-energy excitations contain a fermionic branch and a bosonic branch, with widely different velocities. A short-range repulsive potential can stabilize a charge $+e/4$ quasihole at the center, leading to a different edge excitation spectrum due to the change of boundary conditions for Major…
Using exact diagonalization in the spherical geometry, we investigate systems of localized quasiholes at nu=5/2 for interactions interpolating between the pure Coulomb and the three-body interaction for which the Moore-Read state is the exact ground state. We show that the charge e/4 quasihole can be easily localized by means of a delta-function pinning potential. Using a tuned smooth pinning potential, the quasihole radius can be limited to approximately three magnetic length units. For systems of two quasiholes, adiabatic continuity between the Moore-Read and the Coulomb limit holds for the …
The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree k of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a k+1-body interaction, and so are symmetric polynomials that vanish when any k+1 particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each an…
We construct a wavefunction, generalizing the well known Moore-Read Pfaffian, that describes spinless electrons at filling fraction nu=2/5 (or bosons at filling fraction nu=2/3) as the ground state of a very simple three body potential. We find, analogous to the Pfaffian, that when quasiholes are added there is a ground state degeneracy which can be identified as zero-modes of the quasiholes. The zero-modes are identified as having semionic statistics. We write this wavefunction as a correlator of the Virasoro minimal model conformal field theory M(5,3). Since this model is non-unitary, we con…
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 $ν=1/2$ 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.
We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $ν= 5/2$, based on the disc geometry. Our model includes Coulomb interaction and a semi-realistic confining potential. We also mix in some three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read--like state. Fr…
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 correlatio…
We consider a quaternately generalized Pfaffian QGPf$(\frac{1}{J(z_i,z_j,z_k,z_l)})[J(z_1,...,z_N)]^2$ in which the square of Vandermonde determinant, $[J(z_1,...,z_N)]^2$, implies the upmost Landau level is half filled. This wave function is the unique highest density zero energy state of a special short range interacting Hamiltonian. One can think this quaternate composite fermion liquid as a competing ground state of Moore-Read (MR) Pfaffian state at $ν=5/2$. The degeneracy of the quasihole excitations above the QGPf is higher than that of Moore-Read even Read-Rezayi quasiholes. The QGPf is…
Quasiparticle spin (in the spacetime sense) couples to curvature of space. Here this fact is used to calculate the spin of quasiholes in trial quantum Hall states by adiabatically dragging them around on a sphere, for trial states given by conformal blocks in some conformal field theory. The spin is found to agree with the conformal weight of the corresponding field. The result completes a recent argument that constructions using blocks from non-unitary theories that contain negative quantum dimensions produce contradictions that prevent them from describing topological (gapped) phases of matt…
We study charged spin textures (CST) over the Moore-Read quantum Hall state at filling factor 5/2. We develop an algebraic framework and show that the pairing condition that is inherent in the Moore-Read state naturally leads to a class of charged spin textures, labeled by winding numbers [w_I, w_II]. The fundamental CST, with labels [1,0] and electric charge e/4, is identified with the polar core vortex known in the spin-1 BEC literature. The spin texture carried by the fusion product of fundamental CSTs is correlated with the fusion channel of the underlying non-Abelian quasiholes.
We study charged excitations of the non-abelian Moore-Read liquid at filling factor nu=5/2, allowing for spin depolarization. Using a combination of numerical studies, and taking account of non-zero well widths, we find that at sufficiently low Zeeman energy it is energetically favourable for charge e/4 quasiholes to bind into "skyrmions" of charge e/2. We show that skyrmion formation is further promoted by disorder, and argue that this can lead to a depolarized nu=5/2 ground state in realistic experimental situations. We comment on the consequences for the activated transport.
We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction nu=5/2. We complete the program started in Nucl. Phys. B 506, 685 (1997) and show that the degenerate four-quasihole and six-quasihole wavefunctions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c=1+1/2 conformal field theory. As a consequence, this proves that the non-Abelian statistics…
We use conformal field theory to construct model wavefunctions for a gapless interface between lattice versions of a bosonic Laughlin state and a fermionic Moore-Read state, both at $ν=1/2$. The properties of the resulting model state, such as particle density, correlation function and Rényi entanglement entropy are then studied using the Monte Carlo approach. Moreover, we construct the wavefunctions also for anyonic excitations (quasiparticles and quasiholes). We study their density profile, charge and statistics. We show that, similarly to the Laughlin-Laughlin case studied earlier, some any…
We study fractional quantum Hall states in the cylinder geometry with open boundaries. By truncating the Coulomb interactions between electrons we show that it is possible to construct infinitely many exact eigenstates including the ground state, quasiholes, quasielectrons and the magnetoroton branch of excited states.
We present model wavefunctions for quasielectron (as opposed to quasihole) excitations of the unitary $Z_k$ parafermion sequence (Laughlin/Moore-Read/Read-Rezayi) of Fractional Quantum Hall states. We uniquely define these states through two generalized clustering conditions: they vanish when either a cluster of $k+2$ electrons is put together, or when two clusters of $k+1$ electrons are formed at different positions. For Abelian Fractional Quantum Hall states ($k=1$), our construction reproduces the Jain quasielectron wavefunction, and elucidates the difference between the Jain and Laughlin q…
From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than the one of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including the…
Based on the recently proposed SUSY quantum Hall effect, we show that Laughlin and Moore-Read states are related by a hidden SUSY transformation. Regarding the SUSY Laughlin wavefunction as a master wavefunction, Laughlin and Moore-Read states appear as two extreme limits of component wavefunctions. Realizations of topological excitations on Laughlin and Moore-Read states are also discussed in the SUSY formalism. We develop a streographically projected formulation of the SUSY quantum Hall effect. With appropriate interpretation of Grassmann odd coordinates, we illustrate striking analogies bet…
We provide a robust and generic method to assess the screening properties and extract the scaling exponents of quasiparticle edge excitations of quantum Hall states from model wavefunctions. We numerically implement this method for the fundamental quasihole and hole excitations of several model states. For the Laughlin, Moore-Read, and Z_3-Read-Rezayi states, we find agreement with the predicted edge theory, verifying the bulk-edge correspondence. We also use this to obtain the first clear microscopic demonstration of the pathologies of the Gaffnian wavefunctions.
Non-Abelian excitations are an interesting feature of many fractional quantum Hall phases, including those phases described by the Moore-Read (or Pfaffian) wave function. However, the detection of the non-Abelian quasiparticles is challenging. Here, we consider a system described by the Moore-Read wave function, and assume that impurity particles bind to its quasiholes. Then, the angular momentum of the impurities, reflected also by the impurity density, provides a useful witness of the physics of the non-Abelian excitations. By demanding that the impurities are constrained to the lowest Landa…
We review a recent development in the theoretical understanding of the nu=5/2 quantum Hall plateau and propose a new conformal field theory, slightly different from the Moore-Read one, to describe another universality class relevant for this plateau. The ground state is still given by the Pfaffian and is completely polarized, however, the elementary quasiholes are charge 1/2 anyons with abelian statistics theta=pi/2, which obey complete spin-charge separation. The physical hole is represented by two such quasiholes plus a free neutral Majorana fermion. We also compute the periods and amplitude…
We describe a geometric (or gravitational) analogue of the Laughlin quasiholes in the fractional quantum Hall states. Analogously to the quasiholes these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wavefunction, however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wavefunction in the presence of such defects and explain how to assign an electric charge and a spin to each defect, and calculate the adiabatic, non-abelian statisti…
We study a quantum Hall bilayer system of bosons at total filling factor $ν= 1$, and study the phase that results from short ranged pair-tunneling combined with short ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys.~Rev.~Lett.~{\bf 108}, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase while the other is gapless. However, on introduction of a pair tunneling term, the second system becomes gapped and de…
We find the finite-width, i.e., the layer thickness, of experimental quasi-two dimensional systems produces a physical environment sufficient to stabilize the Moore-Read Pfaffian state thought to describe the fractional quantum Hall effect at filling factor $ν=5/2$. This conclusion is based on exact calculations performed in the spherical and torus geometries, studying wavefunction overlap and ground state degeneracy
We provide details of a shorter letter and cond-mat/9702098 and some new results. We describe a Chern-Simons theory for the fractional quantum Hall states in which magnetoplasmon degrees of freedom enter. We derive correlated wavefunctions, operators for creating quasiholes and composite fermions and bosons (which are electrons bound to zeros). We show how the charge of these particles and mass gets renormalized to the final values and compute the effective mass approximately. By deriving a hamiltonian description of the composite fermions and bosons and their charge and current operators, we …
We show model wavefunctions for neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of "pair excitations" works for the magneto-roton modes of single-component Abelian and non-Abeliean FQH states, as well as neutral fermion mode for the Moore-Read (MR) state. The analytic wavefunctions enable computations in the thermodynamic limit previous inaccessible to numerics, and the long wavelength limit of the neutral energy gap of the magneto-roton mod…
The nu=5/2 fractional quantum Hall state is studied numerically, directly including the effects of electron scattering between neighboring Landau levels. Significant reduction of the excitation gap caused by the LL mixing explains the discrepancy between earlier calculations and experiments. On the other hand, LL mixing also considerably reduces overlaps with the Moore--Read wavefunction, raising a question of the actual realization of nonabelian quasiparticles in present experiments.
Some fractional quantum Hall states observed in experiments may be described by first-quantized wavefunctions with special clustering properties like the Moore-Read Pfaffian for filling factor nu = 5/2. This wavefunction has been constructed by constructing correlation functions of a two-dimensional conformal field theory (CFT) involving a free boson and a Majorana fermion. By considering other CFTs many other clustered states have been proposed as candidate FQH states under appropriate circumstances. It is believed that the underlying CFT should be unitary if one wants to describe an incompre…
Generalized Model Fractional Quantum Hall States on Lattices
International Center for Quantum Materials, Peking University（北京大学量子材料国际中心）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）
Vacua of two dimensional incompressible system, such as Fractional Quantum Hall system, are characterized by rational conformal field theory. We develop a method to express the wavefunctions of those systems in terms of chiral vertices. We formulate quantum mechanics on those expressions, which reveals the underlying simple structure of the conformal field theory discription of FQH systems. Also we argue the recent conjecture of Nayak and Wilczek on the spinor statistics of 2n quasihole state in paired quantum hall system.
Inspired by recent theoretical discovery of robust fractional topological phases without a magnetic field, we search for the non-Abelian quantum Hall effect (NA-QHE) in lattice models with topological flat bands (TFBs). Through extensive numerical studies on the Haldane model with three-body hard-core bosons loaded into a TFB, we find convincing numerical evidence of a stable $ν=1$ bosonic NA-QHE, with the characteristic three-fold quasi-degeneracy of ground states on a torus, a quantized Chern number, and a robust spectrum gap. Moreover, the spectrum for two-quasihole states also shows a fini…
Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin, Moore-Read and Read-Rezayi wavefunctions, belong to a special class of orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant). This fundamental observation allows to point out two different recurrence relations for the coefficients of the permanent (Slater) decomposition of the bosonic (fermionic) states. Here we provide an explicit Fock space representation for these wavefunctions by introducing a two-body squeezing operator which represents them as a Jastrow operator applied to reference states…
We analyze transitions between quantum Hall ground states at prominent filling factors $ν$ in the spherical geometry by tuning the width parameter of the Zhang-Das Sarma interaction potential. We find that incompressible ground states evolve adiabatically under this tuning, whereas the compressible ones are driven through a first order phase transition. Overlap calculations show that the resulting phase is increasingly well described by appropriate analytic model wavefunctions (Laughlin, Moore-Read, Read-Rezayi). This scenario is shared by both odd ($ν=1/3, 1/5, 3/5, 7/3, 11/5, 13/5$) and even…
We perform numerical studies to determine if the fractional quantum Hall state observed at filling nu=5/2 is the Moore-Read wavefunction or its particle hole conjugate, the so-called AntiPfaffian. Using a truncated Hilbert space approach we find that for realistic interactions, including Landau-level mixing, the ground state remains fully polarized and the AntiPfaffian is strongly favored.
We further develop an approach to identify the braiding statistics associated to a given fractional quantum Hall state through adiabatic transport of quasiparticles. This approach is based on the notion of adiabatic continuity between quantum Hall states on the torus and simple product states---or "patterns"---in the thin torus limit, together with a suitable coherent state Ansatz for localized quasiholes that respects the modular invariance of the torus. We give a refined and unified account of the application of this method to the Laughlin and Moore-Read states, which may serve as a pedagogi…
Many trial wavefunctions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wavefunctions for certain paired states of fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wavefunctions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely-separated quasiholes, which has been predicted to be non-Abelian in a variety of cases.…
We analyze pairing of fermions in two dimensions for fully-gapped cases with broken parity (P) and time-reversal (T), especially cases in which the gap function is an orbital angular momentum ($l$) eigenstate, in particular $l=-1$ (p-wave, spinless or spin-triplet) and $l=-2$ (d-wave, spin-singlet). For $l\neq0$, these fall into two phases, weak and strong pairing, which may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topological invariant. For the spinless p-wave case, the weak-pairing phase has a p…
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. Two canonically dual pairing gap functions are related by the bosonic Laughlin wavefunction (Jastraw 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 of an oscillatory pairing function for intermedi…
We study the edge-mode excitations of a fractional quantum Hall droplet by expressing the edge state wavefunctions as linear combinations of Jack polynomials with a negative parameter. We show that the exact diagonalization within subspace of Jack polynomials can be used to generate the chiral edge-mode excitation spectrum in the Laughlin phase and the Moore-Read phase with realistic Coulomb interaction. The truncation technique for the edge excitations simplifies the procedure to extract reliably the edge-mode velocities, which avoids the otherwise complicated analysis of the full spectrum th…
The fractional quantum Hall effect (FQHE) in the second orbital Landau level at filling factor 5/2 remains enigmatic and motivates our work. We consider the effect of the quasi-2D nature of the experimental FQH system on a number of FQH states (fillings 1/3, 1/5, 1/2) in the lowest, second, and third Landau levels (LLL, SLL, TLL,) by calculating the overlap, as a function of quasi-2D layer thickness, between the exact ground state of a model Hamiltonian and the consensus variational wavefunctions (Laughlin wavefunction for 1/3 and 1/5 and the Moore-Read Pfaffian wavefunction for 1/2). Using la…
