We have numerically studied a non-adiabatic charge transport in the quantum Hall system pumped by a magnetic flux, as one of the simplest theoretical realizations of non-adiabatic Thouless pumping. In the adiabatic limit, a pumped charge is quantized, known as Laughlin's argument in a cylindrical lattice. In a uniform electric field, we obtained a formula connecting quantized pumping in the adiabatic limit and no-pumping in the sudden limit. The intermediate region between the two limits is determined by the Landau gap. A randomness or impurity effect is also discussed.
A novel hierarchy of fractional quantum Hall (FQH) states in the lowest Landau level (LL) is proposed to explain recently observed FQH fractions such as nu=5/13, 3/8, or 4/11. Based on the analysis of their interaction pseudopotentials, it is argued that the Laughlin quasiparticles (particles/holes in a partially filled composite fermion LL) form pairs. These pairs are proposed to have Laughlin correlations with one another and to form condensed states at a sequence of fractions which includes all new fractions observed in experiment.
From the analysis of their interaction pseudopotentials, it is argued that (at certain filling factors) Laughlin quasiparticles can form pairs. It is further proposed that such pairs could have Laughlin correlations with one another and form condensed states of a new type. The sequence of fractions corresponding to these states includes all new fractions observed recently in experiment (e.g., ν=5/13, 3/8, or 4/11).
We consider the quantum Hall effect induced by magnetic field and rotation, which can drive the Hall samples into the quantum Hall regime and induce fractional excitations. Both the mass and the charge of the Laughlin quasiparticles are predicted to be fractionally quantized. The observable effects induced by rotation are discussed. Based on the usual Hall samples under rotation, we propose an experimental setup for detecting the macroscopic quantization phenomena and the fractional mass of the Laughlin quasiparticles.
We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry of Laughlin's integer quantum Hall effect argument, at filling factor 1/3. We show that the plasma analogy leads to a periodic density, and that the wave function admits a representation as a ``quantum polymer'', reminiscent of the quantum dimer model by Rokhsar and Kivelson. We explain how the representation can be exploited to compute the normalization and one-particle density in the limit of infinitely many particles.
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…
Laughlin has found "exactly" the wave function which is ascribed to an excitation of fractional charge, such as e/3. We find that the exactness of the wave function is not destroyed by changing the charge to some other quantity such as the magnetic field. Thus e/3 and H can be replaced by e and H/3. Therefore, the wave function need not belong to quasiparticles of charge e/3.
We extend Laughlin's magnetic-flux-threading argument to the quantized thermal Hall effect. A proper analogue of Laughlin's adiabatic magnetic-flux threading process for the case of the thermal Hall effect is given in terms of an external gravitational field. From the perspective of the edge theories of quantum Hall systems, the quantized thermal Hall effect is closely tied to the breakdown of large diffeomorphism invariance, that is, a global gravitational anomaly. In addition, we also give an argument from the bulk perspective in which a free energy, decomposed into its Fourier modes, is adi…
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…
We construct many particle Hamiltonians for which the Laughlin and Jain wavefunctions are exact ground states. The Hamiltonians involve fermions in a magnetic field and with inter-particle interactions. For the Laughlin wave-functions,the interactions involve two- and three-body correlations similar to the Chern-Simons interactions, whereas for the projected Jain wave-functions, N-body interactions (which cannot be explicitly written down in general) are involved.
We present a quantitative study of most prominent incompressible quantum Hall states in the partially filled first excited Landau level (LL1) which have been recently studied experimentally by Choi et al. The pseudopotential describing the electron - electron interaction in LL1 is harmonic at short range. It produces a series of incompressible states which is different from its LL0 counterpart. The numerical data indicate that the most prominent states $ν={5/2}$, 7/3, and 8/3 are not produced by Laughlin correlated electrons, but result from a tendency of electrons to form pairs or larger clus…
The quantum Hall effect occuring in two-dimensional electron gases was first explained by Laughlin, who envisioned a thought experiment that laid the groundwork for our understanding of topological quantum matter. His proposal is based on a quantum Hall cylinder periodically driven by an axial magnetic field, resulting in the quantized motion of electrons. We realize this milestone experiment with an ultracold gas of dysprosium atoms, the cyclic dimension being encoded in the electronic spin and the axial field controlled by the phases of laser-induced spin-orbit couplings. Our experiment prov…
We analyze Susskind's proposal of applying the non-commutative Chern-Simons theory to the quantum Hall effect. We study the corresponding regularized matrix Chern-Simons theory introduced by Polychronakos. We use holomorphic quantization and perform a change of matrix variables that solves the Gauss law constraint. The remaining physical degrees of freedom are the complex eigenvalues that can be interpreted as the coordinates of electrons in the lowest Landau level with Laughlin's wave function. At the same time, a statistical interaction is generated among the electrons that is necessary to s…
We have examined the experiments performed by Goldman and Su, de-Picciotto et al, Samanadayar et al and Conforti et al in which it is claimed that a fractional charge of e/3 is found. In all of the measurements, the quantity measured is the product of the charge and the magnetic field but not the charge. It is possible to interpret that charge per unit area has been measured, where the area is the square of the magnetic length. This type of correction to Laughlin's result does not affect the exactness of the calculation. Anderson has suggested the extension of Laughlin's state to particles of …
We report experiments in a large, 2.5 micron diameter Fabry-Perot quantum Hall interferometer with two tunneling constrictions. Interference fringes are observed as conductance oscillations as a function of applied magnetic field (the Aharonov-Bohm flux through the electron island) or a global backgate voltage (electronic charge in the island). Depletion is such that in the fractional quantum Hall regime, filling 1/3 current-carrying chiral edge channels pass through constrictions when the island filling is 2/5. The interferometer device is calibrated with fermionic electrons in the integer qu…
The energy spectra and wavefunctions of up to 14 interacting quasielectrons (QE's) in the Laughlin nu=1/3 fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors nu_QE=1/2, 1/3, 2/3, corresponding to the electron fillings nu=…
We address the question of the stability of the (fractional) quantum Hall effect (QHE) in presence of pseudomagnetic disorder generated by mechanical deformations of a graphene sheet. Neglecting the potential disorder and taking into account only strain-induced random pseudomagnetic fields, it is possible to write down a Laughlin-like trial ground-state wave function explicitly. Exploiting the Laughlin plasma analogy, we demonstrate that in the case of fluctuating pseudomagnetic fluxes of relatively small amplitude both the integer and fractional quantum Hall effects are always stable upon the…
Through exact numerical diagonalization, the von Neumann entropy is calculated for the Laughlin and Pfaffian quantum Hall states in rotating interacting Bose gases at zero temperature in the lowest Landau level limit. The particles comprising the states are indistinguishable, so the required spatial bipartitioning is effected by tracing over a subset of single-particle orbitals. The topological entropy is then extracted through a finite-size scaling analysis. The results for the Laughlin and the Pfaffian states agree with the expected values of $\ln\sqrt{2}$ and $\ln\sqrt{4}$, respectively.
We study current-current correlation in an electronic analog of a beam splitter realized with edge channels of a fractional quantum Hall liquid at Laughlin filling fractions. In analogy with the known result for chiral electrons, if the currents are measured at points located after the beam splitter, we find that the zero frequency equilibrium correlation vanishes due to the chiral propagation along the edge channels. Furthermore, we show that the current-current correlation, normalized to the tunneling current, exhibits clear signatures of the Laughlin quasi-particles' fractional statistics.
Chiral gapless boundary modes are characteristic of quantum Hall (QH) states. For hole-conjugate fractional QH phases counterpropagating edge modes (upstream and downstream) are expected. In the presence of electrostatic interactions and disorder these modes may renormalize into charge and upstream neutral modes. Orthodox models of Laughlin phases anticipate only a downstream charge mode. Here we show that in the latter case, in the presence of a smooth confining potential, edge reconstruction leads to the emergence of pairs of counterpropagating modes, which, by way of mode renormalization, m…
We report experiments on temperature and Hall voltage bias dependence of the superperiodic conductance oscillations in the novel Laughlin quasiparticle interferometer, where quasiparticles of the 1/3 fractional quantum Hall fluid execute a closed path around an island of the 2/5 fluid. The amplitude of the oscillations fits well the quantum-coherent thermal dephasing dependence predicted for a two point-contact chiral edge channel interferometer in the full experimental temperature range 10.2<T<141 mK. The temperature dependence observed in the interferometer is clearly distinct from the behav…
We show that cotunneling in the 5/2 fractional quantum Hall regime allows us to test the Moore-Read wave function, proposed for this regime, and to probe the nature of the fractional charge carriers. We calculate the cotunneling current for electrons that tunnel between two quantum Hall edge states via a quantum dot and for quasiparticles with fractional charges e/4 and e/2 that tunnel via an antidot. While electron cotunneling is strongly suppressed, the quasiparticle tunneling shows signatures characteristic of the Moore-Read state. For comparison, we also consider cotunneling between Laughl…
Motivated by the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor $ν= 1/m$, 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. 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 o…
Possible phase transitions between incompressible quantum Hall states and compressible three-dimensional states are discussed for infinite-layer electron systems in strong magnetic field. By variational Monte Carlo calculation, relative stability of some trial states is studied. If the inter-layer distance is large enough, the Laughlin state is stabilized for the Landau-level filling ν=1/3 of each layer. When the inter-layer distance is comparable to the magnetic length, the Laughlin state becomes unstable against a different fractional quantum Hall state with inter-layer correlation, and/or a…
The fractional quantum Hall effect in 2D electron gases submitted to large magnetic fields remains one of the most striking phenomena in condensed matter physics. Historically, the first observed signature is a Hall resistance quantized to the value (2m+1) when the filling factor (electron density divided by magnetic flux quantum density) of a 2D electron gas is in the vicinity of an inverse odd integer 1/(2m +1). This was one of the first observation of fractional quantum numbers. A large part of our basic theoretical understanding of this effect (and descendants) originates from Laughlin's t…
We propose a quantum matrix oscillator as a model that provides the construction of the quantum Hall states in a direct way. A connection of this model to the regularized matrix model introduced by Polychronakos is established . By transferring the consideration to the Bargmann representation with the help of a particular similarity transformation, we show that the quantum matrix oscillator describes the quantum mechanics of electrons in the lowest Landau level with the ground state described by the Laughlin-type wave function. The equivalence with the Calogero model in one dimension is emphas…
The mean field composite Fermion (CF) picture successfully predicts angular momenta of multiplets forming the lowest energy band in fractional quantum Hall (FQH) systems. This success cannot be attributed to a cancellation between Coulomb and Chern-Simons interactions beyond the mean field, because these interactions have totally different energy scales. Rather, it results from the behavior of the Coulomb pseudopotential V(L) (pair energy as a function of pair angular momentum) in the lowest Landau level (LL). The class of short range repulsive pseudopotentials is defined that lead to short ra…
It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states -- the filling $ν=1/3$ Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize t…
Fractional quantum Hall systems are among the most exciting strongly correlated systems. Accessing them microscopically via quantum simulations with ultracold atoms would be an important achievement toward a better understanding of this strongly correlated state of matter. A promising approach is to confine a small number of bosonic atoms in a quasi-two-dimensional rotating trap, which mimics the magnetic field. For rotation frequencies close to the in-plane trapping frequency, the ground state is predicted to be a bosonic analog of the Laughlin state. Here, we study the problem of the adiabat…
Using the parton construction, we build a three-dimensional (3D) multilayer fractional quantum Hall state with average filling ν= 1/3 per layer that is qualitatively distinct from a stacking of weakly coupled Laughlin states. The state supports gapped charge e/3 fermionic quasiparticles that can propagate both within and between the layers, in contrast to the quasiparticles in a multilayer Laughlin state which are confined within each layer. Moreover, the state has gapless neutral collective modes, a manifestation of an emergent "photon", which is minimally coupled to the fermionic quasipartic…
The tunneling current and shot noise of the current between two Fractional Quantum Hall (FQH) edges in the $ ν=12/5 $ FQH state in electronic Mach-Zehnder interferometer are studied. It is shown that the tunneling current and shot noise can be used to probe the existence of $k=3 $ parafermion statistics in the $ ν=12/5 $ FQH state. More specifically, the dependence of the current on the Aharonov-Bohm flux in the Read-Rezayi state is asymmetric under the change of the sign of the applied voltage. This property is absent in the Abelian Laughlin states. Moreover the Fano factor can exceed 12.7 el…
We present Monte Carlo studies of charge expectation values and charge fluctuations for quasi-particles in the quantum Hall system. We have studied the Laughlin wave functions for quasi-hole and quasi-electron, and also Jain's definition of the quasi-electron wave function. The considered systems consist of from 50 to 200 electrons, and the filling fraction is 1/3. For all quasi-particles our calculations reproduce well the expected values of charge; -1/3 times the electron charge for the quasi-hole, and 1/3 for the quasi-electron. Regarding fluctuations in the charge, our results for the quas…
We show the low-lying excitations at filling factor $ν=n+1/3$ with realistic interactions are contained completely within the well-defined Hilbert space of "Gaffnian quasiholes". Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons". Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bounded and fractionalise with rich dynamical properties. By studying the effective interactions b…
We give a microscopic derivation of the chiral Luttinger liquid theory for the Laughlin states. Starting from the wave function describing an arbitrary incompressibly deformed Laughlin state (IDLS) we quantize these deformations. In this way we obtain the low-energy projections of local microscopic operators and derive the quantum field theory of edge excitations directly from quantum mechanics of electrons. This shows that to describe experimental and numeric deviations from chiral Luttinger liquid theory one needs to go beyond Laughlin's approximation. We show that in the large N limit the I…
We discuss how the braiding properties of Laughlin quasi-particles in quantum Hall states can be understood within a one-dimensional formalism we proposed earlier. In this formalism the two-dimensional space of the Hall liquid is identified with the phase space of a one-dimensional lattice system, and localized Laughlin quasi-holes can be understood as coherent states of lattice solitons. The formalism makes comparatively little use of the detailed structure of Laughlin wavefunctions, and may offer ways to be generalized to non-abelian states.
The fractional quantum Hall effect (FQHE) states at half integer Landau fillings ($ν$) have long been of great interest, since they have correlations that differ from those of the fundamental Laughlin states found at odd denominators. At $ν=1/2$ the FQHE has been observed in wide or double quantum wells, and is ascribed to the two-component Halperin-Laughlin $Ψ_{331}$ state. $Ψ_{331}$ excitations carry charge $\pm e/4$, like the carriers of $ν=5/2$ states which are of interest in quantum computation. Further, such an excitation (quasiparticle or -hole) of $Ψ_{331}$ has unequal, opposite charge…
We study bulk and edge correlations in the compressible half-filled state, using a modified version of the plasma analogy. The corresponding plasma has anomalously weak screening properties, and as a consequence we find that the correlations along the edge do not decay algebraically as in the Laughlin (incompressible) case, while the bulk correlations decay in the same way. The results suggest that due to the strong coupling between charged modes on the edge and the neutral Fermions in the bulk, reflected by the weak screening in the plasma analogue, the (attractive) correlation hole is not we…
We measure the Hall conductivity, $σ_{xy}$, on a Corbino geometry sample of a high-mobility AlGaAs/GaAs heterostructure in a pulsed magnetic field. At a bath temperature about 80 mK, we observe well expressed plateaux in $σ_{xy}$ at integer filling factors. In the pulsed magnetic field, the Laughlin condition of the phase coherence of the electron wave functions is strongly violated and, hence, is not crucial for $σ_{xy}$ quantization.
We generalize Laughlin's flux insertion argument, originally discussed in the context of the quantum Hall effect, to topological phases protected by non-on-site unitary symmetries, in particular by parity symmetry or parity symmetry combined with an on-site unitary symmetry. As a model, we discuss fermionic or bosonic systems in two spatial dimensions with CP symmetry, which are, by the CPT theorem, related to time-reversal symmetric topological insulators (e.g., the quantum spin Hall effect). In particular, we develop the stability/instability (or "gappability"/"ingappablity") criteria for no…
We propose a realistic scheme to create motional entangled states of a few bosonic atoms. It can experimentally be realized with a gas of ultra cold bosonic atoms trapped in a deep optical lattice potential. By simultaneously deforming and rotating the trapping potential on each lattice site it is feasible to adiabatically create a variety of entangled states on each lattice well. We fully address the case of N=2 and N=4 atoms per well and identify a sequence of fractional quantum Hall states: the Pfaffian state, the 1/2-Laughlin quasiparticle and the 1/2-Laughlin state. Exact knowledge of the…
A Quantum Antidot electrometer has been used in the first direct observation of the fractionally quantized electric charge. In this paper we report experiments performed on the integer i = 1, 2 and fractional f = 1/3 quantum Hall plateaus extending over a filling factor range of at least 27%. We find the charge of the Laughlin quasiparticles to be invariantly e/3, with standard deviation of 1.2% and absolute accuracy of 4%, independent of filling, tunneling current, and temperature.
The energy spectrum of a two-dimensional electron gas (2DEG) interacting with a valence-band hole is studied in the high magnetic field limit as a function of the filling factor nu and the separation d between the electron and hole layers. For d smaller than the magnetic length lambda, the hole binds one or more electrons to form neutral (X) or charged (X-) excitons. The low-lying states can be understood in terms of Laughlin-like correlations among the constituent charged fermions (electrons and X-). For d comparable to lambda, the electron-hole interaction is not strong enough to bind a full…
We present analytic and numerical calculations on the bipartite entanglement entropy in fractional quantum Hall states of the fermionic Laughlin sequence. The partitioning of the system is done both by dividing Landau level orbitals and by grouping the fermions themselves. For the case of orbital partitioning, our results can be related to spatial partitioning, enabling us to extract a topological quantity (the `total quantum dimension') characterizing the Laughlin states. For particle partitioning we prove a very close upper bound for the entanglement entropy of a subset of the particles with…
Anyon collision experiments have recently demonstrated the ability to discriminate between fermionic and anyonic statistics. However, only one type of anyons associated with the simple Laughlin state at filling factor $ν=1/3$ has been probed so far. It is now important to establish anyon collisions as quantitative probes of fractional statistics for more complex topological orders, with the ability to distinguish between different species of anyons with different statistics. In this work, we use the anyon collider to compare the Laughlin $ν=1/3$ state, which is used as the reference state, wit…
We investigate Laughlin's fractional quantum Hall effect wave function on a cylinder. We show that it displays translational symmetry breaking in the axial direction for sufficiently thin cylinders. At filling factor 1/p, the period is p times the period of the filled lowest Landau level. The proof uses a connection with one-dimensional polymer systems and discrete renewal equations.
We have performed an exact diagonalization study of up to N=12 interacting electrons on a disk at filling $ν={1/3}$ for both Coulomb and $V_1$ short-range interaction for which Laughlin wave function is the exact solution. For Coulomb interaction and $N\geq 10$ we find persistent radial oscillations in electron density, which are not captured by the Laughlin wave function. Our results srongly suggest formation of a chiral edge striped phase in quantum Hall systems. The amplitude of the charge density oscillations decays slowly, perhaps as a square root of the distance from the edge; thus the s…
In 1976 Jackiw and Rebbi found 1/2 of a fermion number by using Dirac equation in 1+1 dimensions. Schrieffer in several proposals made an effort to suggest that there is a fractional charge. The calculations of Peierls distortion, Berry's phase and classical action were presented to accomodate the fractional charge in non-relativistic theory. Laughlin used the antisymmetry to define the charge density per unit area in a two dimensional system. In order to elliminate the area, Laughlin introduced the incompressibility which fixed the area so that the odd number, which determines the antisymmetr…
I discuss results bearing on a variational problem of a new type, inspired by fractional quantum Hall physics. In the latter context, the main result reviewed herein can be spelled as "the phase of independent quasi-holes generated from Laughlin's wave-function is stable against external potentials and weak long-range interactions". The main ingredient of the proof is a connection between fractional quantum Hall wave-functions and statistical mechanics problems that generalize the 2D one-component plasma (jellium model). Universal bounds on the density of such systems, coined "Incompressibilit…
In this note, we study a matrix-regularized version of non-commutative U(1) Chern-Simons theory proposed recently by Polychronakos. We determine a complete minimal basis of exact wavefunctions for the theory at arbitrary level k and rank N and show that these are in one-to-one correspondence with Laughlin-type wavefunctions describing excitations of a quantum Hall droplet composed of N electrons at filling fraction 1/k. The finite matrix Chern-Simons theory is shown to be precisely equivalent to the theory of composite fermions in the lowest Landau level, believed to provide an accurate descri…
We review the recent developments of the SUSY quantum Hall effect [hep-th/0409230, hep-th/0411137, hep-th/0503162, hep-th/0606007, arXiv:0705.4527]. We introduce a SUSY formulation of the quantum Hall effect on supermanifolds. On each of supersphere and superplane, we investigate SUSY Landau problem and explicitly construct SUSY extensions of Laughlin wavefunction and topological excitations. The non-anti-commutative geometry naturally emerges in the lowest Landau level and brings particular physics to the SUSY quantum Hall effect. It is shown that SUSY provides a unified picture of the origin…
We analytically derived the effective two-body interaction for a finite thickness quantum Hall system with a harmonic perpendicular confinement and an in-plane magnetic field. The anisotropic effective interaction in the lowest Landau level (LLL) and first Landau level (1LL) are expanded in the basis of the generalized pseudopotentials (PPs), and we analyze how the coefficients of some prominent isotropic and anisotropic PPs depend on the thickness of the sample and the strength of the in-plane magnetic field. We also investigate the stability of the topological quantum Hall states, especially…
The elementary excitations of a fractional quantum Hall liquid are quasiparticles or quasiholes which are neither bosons nor fermions, but so-called anyons. Here we study impurity particles immersed in a quantum Hall liquid which bind to the quasiholes via repulsive interactions with the liquid. We show that the angular momentum of an impurity is given by the multiple of a fractional "quantum" of angular momentum, and can directly be observed from the impurity density. In a system with several impurities bound to quasiholes, their total angular momentum interpolates between the values for free…
