We realize a nanoscale-area Mach-Zehnder interferometer with co-propagating quantum Hall spin-resolved edge states and demonstrate the persistence of gate-controlled quantum interference oscillations, as a function of an applied magnetic field, at relatively large temperatures. Arrays of top-gate magnetic nanofingers are used to induce a resonant charge transfer between the pair of spin-resolved edge states. To account for the pattern of oscillations measured as a function of magnetic field and gate voltage, we have developed a simple theoretical model which satisfactorily reproduces the data.
We use a novel scanning electron Mach-Zehnder interferometer constructed in a conventional transmission electron microscope to perform inelastic interferometric imaging with free electrons. An electron wave function is prepared in two paths that pass on opposite sides of a gold nanoparticle, where plasmons are excited before the paths are recombined to produce electron interference. We show that the measured spectra are consistent with theoretical predictions, specifically that the interference signal formed by inelastically scattered electrons is pi out of phase with respect to that formed by…
We study a single-electron pulse injected into the chiral edge-state of a quantum Hall device and subject to a capacitive Coulomb interaction. We find that the scattered multi-particle state remains unentangled and hence can be created itself by a suitable classical voltage-pulse $V(t)$. The application of the inverse pulse $-V(-t)$ corrects for the shake-up due to the interaction and resurrects the original injected wave packet. We suggest an experiment with an asymmetric Mach-Zehnder interferometer where the application of such pulses manifests itself in an improved visibility.
We study Coulomb interactions in the finite bias response of Mach-Zehnder interferometers, which exploit copropagating edge states in the integer quantum Hall effect. Here, interactions are particularly important since the coherent coupling of edge channels is due to a resonant mechanism that is spoiled by inelastic processes. We find that interactions yield a saturation, as a function of bias voltage, of the period-averaged interferometer current, which gives rise to unusual features, such as negative differential conductance, enhancement of the visibility of the current, and nonbounded or ev…
A new type of electron sources has emerged which permits to inject particles in a controllable manner, one at a time, into an electronic circuit. Such single electron sources make it possible to fully exploit the particles' quantum nature. We determine the single-particle coherence length from the decay of the Aharonov-Bohm oscillations as a function of the imbalance of a Mach-Zehnder interferometer connected to a single electron source. The single-particle coherence length is of particular importance as it is an intrinsic property of the source in contrast to the dephasing length.
We propose a new single-step scheme for the generation of a GHZ entangled state of three single-electron excitations (flying qubits). We also present a method to get a generalized GHZ-state. Our idea relies upon the most recent progress in the field of on-demand electron sources and mesoscopic Mach-Zehnder interferometry. We also provide the recipe for the unambiguous detection of this GHZ state via correlations measurements at the output, which imply the violation of a Bell-type inequality which is generalized to the case of three particles. We explain how such measurements can be achieved in…
We use a Mach-Zehnder quantum Hall interferometer of a novel design to investigate the interference effects at fractional filling factors. Our device brings together the advantages of usual Mach-Zehnder and Fabry-Perot quantum Hall interferometers. It realizes the simplest-for-analysis Mach-Zehnder interference scheme, free from Coulomb blockade effects. By contrast to the standard Mach-Zehnder realization, our device does not contain an etched region inside the interference loop. For the first time for Mach-Zehnder interference scheme, the device demonstrates interference oscillations with $Φ…
We propose an electronic quantum eraser in which the electrons are injected into a mesoscopic conductor at the quantum Hall regime. The conductor is composed of a two-path interferometer which is an electronic analogue of the optical Mach-Zehnder interferometer, and a quantum point contact detector capacitively coupled to the interferometer. While the interference of the output current at the interferometer is shown to be suppressed by the which-path information, we show that the which-path information is erased by the zero-frequency cross correlation measurement between the interferometer and…
The recent development of dynamic single-electron sources makes it possible to observe and manipulate the quantum properties of individual charge carriers in mesoscopic circuits. Here, we investigate multi-particle effects in an electronic Mach-Zehnder interferometer driven by dynamic voltage pulses. To this end, we employ a Floquet scattering formalism to evaluate the interference current and the visibility in the outputs of the interferometer. An injected multi-particle state can be described by its first-order correlation function, which we decompose into a sum of elementary correlation fun…
We propose a possible setup of testing the Bell's inequality in mesoscopic conductors. The particular implementation uses two coupled electronic Mach-Zehnder interferometers in which electrons are injected into the conductors in the quantum Hall regime. It is shown that the Bell's inequality is violated for an arbitrary coupling strength between the two interferometers.
We investigate the suppression of the visibility of Aharonov-Bohm oscillations in a two-electron Mach-Zehnder interferometer that leaves the single-electron current unchanged. In the case when the sources emit either spin-polarized or entangled electrons, partial distinguishability of electrons (coming from two different sources) suppresses the visibility. Two-particle entanglement may produce behavior similar to "dephasing" of two-particle interferometry.
We theoretically investigate the thermoelectric properties of heat engines based on Mach-Zehnder interferometers. The energy dependence of the transmission amplitudes in such setups arises from a difference in the interferometer arm lengths. Any thermoelectric response is thus of purely quantum mechanical origin. In addition to an experimentally established three-terminal setup, we also consider a two-terminal geometry as well as a four-terminal setup consisting of two interferometers. We find that Mach-Zehnder interferometers can be used as powerful and efficient heat engines which perform we…
We present a general analysis of shot noise in an electronic Mach-Zehnder interferometer, of the type investigated experimentally [Yang Ji et al., Nature 422, 415 (2003)], under the influence of dephasing produced by fluctuations of a classical field. We show how the usual partition noise expression T(1-T) is modified by dephasing, depending on the power spectrum of the environmental fluctuations.
We performed the conductance and the shot noise measurements in an electronic Mach-Zehnder interferometer. The visibility of the interference is investigated as a function of the electron temperature that is derived from the thermal noise of the interferometer. The non-equilibrium noise displays both h/e and h/2e oscillations vs. the modulation gate voltage.
We study nonequilibrium dephasing in an electronic Mach-Zehnder interferometer. We demonstrate that the shot noise at the beam splitter of the interferometer generates an ensemble of nonequilibrium electron density configurations and that electron interactions induce configuration-specific phase shifts of an interfering electron. The resulting dephasing exhibits two characteristic features, a lobe pattern in the visibility and phase jumps of $π$, in good agreement with experimental data.
The influences of long range Coulomb interaction (LRCI) on Mach-Zehnder interferometer (MZI) constructed on quantum Hall edge states is studied employing bosonization method. The interaction of interchannel zero-modes is shown to give rise to a characteristic energy scale which is of the order of the period of experimentally observed lobe pattern of visibility. The nonmonotonic behavior of visibility as found Chalker \textit{et al.} is understood analytically using asymptotic analysis.
A theoretical non-pertubative treatment is developed to explain the dephasing of electrons in the electronic Mach-Zehnder interferometer via interaction with a near-by partitioned electronic channel, which acts as a "which path" detector. The resulting formula reproduces the recant experimental behavior of the MZI interference visibility. By fitting the model to the experimental results, it is shown that the visibility is strongly influenced by merely ~3 detecting electrons, hence it reflects the Non-Gaussian properties behavior of the detector shot-noise.
We investigate an electronic Mach-Zehnder interferometer with high visibility in the quantum Hall regime. The superposition of the electrostatic potentials from a quantum point contact (QPC) and the residual disorder potential from doping impurities frequently results in the formation of inadvertent quantum dots (QD) in one arm of the interferometer. This gives rise to resonances in the QPC transmission characteristics. While crossing the QD resonance in energy, the interferometer gains a phase shift of $π$ in the interference pattern.
We investigate the visibility of the current and shot-noise correlations of electrical analogs of the optical Mach-Zehnder interferometer and the Hanbury Brown Twiss interferometer. The electrical analogs are discussed in conductors subject to high magnetic fields where electron motion is along edge states. The transport quantities are modulated with the help of an Aharonov-Bohm flux. We discuss the conductance (current) visibility and shot noise visibility as a function of temperature and applied voltage. Dephasing is introduced with the help of fictitious voltage probes. Comparison of these …
This paper has been retracted. The authors have discovered a possible problem in applications of the dephasing terminal approach to shot noise calculations in interference situations: Eq. (7) correctly re-introduces the anticorrelations between the currents in the two arms, but due to its semiclassical nature probably fails to describe the consequential diminishing of exchange effects at the second beamsplitter.
We study fermions in a Mach-Zehnder interferometer, subject to a quantum-mechanical environment leading to inelastic scattering, decoherence, renormalization effects, and time-dependent conductance fluctuations. Both the loss of interference contrast as well as the shot noise are calculated, using equations of motion and leading order perturbation theory. The full dependence of the shot-noise correction on setup parameters, voltage, temperature and the bath spectrum is presented. We find an interesting contribution due to correlations between the fluctuating renormalized phase shift and the ou…
We propose and study a weak value (WV) protocol in the context of a solid state setup, specifically, an electronic Mach-Zehnder interferometer. This is the first specific proposal to measure both the real and imaginary part (i.e., complete tomography) of a WV. We also analyze the manifestation of many-body physics in the WV to be measured, including finite temperature and shot-noise-like contributions.
We investigate the Full Counting Statistics of an electrical Mach-Zehnder interferometer penetrated by an Aharonov-Bohm flux, and in the presence of a classical fluctuating potential. Of interest is the suppression of the Aharonov-Bohm oscillations in the distribution function of the transmitted charge. For a Gaussian fluctuating field we calculate the first three cumulants. The fluctuating potential causes a modulation of the conductance leading in the third cumulant to a term cubic in voltage and to a contribution correlating modulation of current and noise. In the high voltage regime we pre…
Three-dimensional higher-order topological insulators can have topologically protected chiral modes propagating on their hinges. Hinges with two co-propagating chiral modes can serve as a "beam splitter" between hinges with only a single chiral mode. Here we show how such a crystal, with Ohmic contacts attached to its hinges, can be used to realize a Mach-Zehnder interferometer. We present concrete calculations for a lattice model of a first-order topological insulator in a magnetic field, which, for a suitable choice of parameters, is an extrinsic second-order topological insulator with the r…
The spatial distribution of the incompressible edge states (IES) is obtained for a geometry which is topologically equivalent to an electronic Mach-Zehnder interferometer, taking into account the electron-electron interactions within a Hartree type self-consistent model. The magnetic field dependence of these IES is investigated and it is found that an interference pattern may be observed if two IES merge or come very close, near the quantum point contacts. Our calculations demonstrate that, being in a quantized Hall plateau does not guarantee observing the interference behavior.
We study transport through an electronic Mach-Zehnder interferometer recently devised at the Weizmann Institute. We show that this device can be used to probe statistics of quasiparticles in the fractional quantum Hall regime. We calculate the tunneling current through the interferometer as the function of the Aharonov-Bohm flux, temperature and voltage bias, and demonstrate that its flux-dependent component is strongly sensitive to the statistics of tunneling quasiparticles. More specifically, the flux-dependent and flux-independent contributions to the current are related by a power law, the…
We compute the interference pattern of a Mach-Zehnder interferometer operating in the fractional quantum Hall effect. Our theoretical proposal is inspired by a remarkable experiment on edge states in the Integer Quantum Hall effect (IQHE). The Luttinger liquid model is solved via two independent methods: refermionization at nu=1/2 and the Bethe Ansatz solution available for Laughlin fractions. The current differs strongly from that of single electrons in the strong backscattering regime. The Fano factor is periodic in the flux, and it exhibits a sharp transition from sub-Poissonian (charge e/2…
We investigate the temperature and voltage dependence of the quantum interference in an electronic Mach-Zehnder interferometer using edge channels in the integer quantum-Hall-regime. The amplitude of the interference fringes is significantly smaller than expected from theory; nevertheless the functional dependence of the visibility on temperature and bias voltage agrees very well with theoretical predictions. Superimposed on the Aharonov-Bohm (AB) oscillations, a conductance oscillation with six times smaller period is observed. The latter depends only on gate voltage and not on the AB-phase, …
We study nonequilibrium properties of an electronic Mach-Zehnder interferometer built from integer quantum Hall edge states at filling fraction $ν{=}1$. For a model in which electrons interact only when they are inside the interferometer, we calculate exactly the visibility and phase of Aharonov-Bohm fringes at finite source-drain bias. When interactions are strong, we show that a lobe structure develops in visibility as a function of bias, while the phase of fringes is independent of bias, except near zeros of visibility. Both features match the results of recent experiments [Neder \textit{et…
Quantum Logic Processors can be implemented with Mach Zehnder Interferometer(MZI) configurations for the Quantum logic operations and gates. In this paper, its implementation for both optical and electronic system has been presented. The correspondence between Jones matrices for photon polarizations and Pauli spin matrices for electrons gives a representation of all the unitary matrices for the quantum gate operations. A novel quantum computation system based on a Electronic Mach Zehnder Interferometer(MZI) has also been proposed. It uses the electron spin as the primary qubit. Rashba effect i…
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.
It was recently suggested that a novel type of phase transition may occur in the visibility of electronic Mach-Zehnder Interferometers. Here, we present experimental evidence for the existence of this transition. The transition is induced by strongly non-Gaussian noise that originates from the strong coupling of a quantum point contact to the interferometer. We provide a transparent physical picture of the effect, by exploiting a close analogy to the neutrino-oscillations of particle physics. In addition, our experiment constitutes a probe of the singularity of the elusive full counting statis…
We present an original statistical method to measure the visibility of interferences in an electronic Mach-Zehnder interferometer in the presence of low frequency fluctuations. The visibility presents a single side lobe structure shown to result from a gaussian phase averaging whose variance is quadratic with the bias. To reinforce our approach and validate our statistical method, the same experiment is also realized with a stable sample. It exhibits the same visibility behavior as the fluctuating one, indicating the intrinsic character of finite bias phase averaging. In both samples, the dilu…
We experimentally realize quantum Hall Mach-Zehnder interferometer which operates far beyond the equilibrium. The operation of the interferometer is based on allowed intra-edge elastic transitions within the same Landau sublevel in the regime of high imbalances between the co-propagating edge states. Since the every edge state is definitely connected with the certain Landau sublevel, the formation of the interference loop can be understood as a splitting and a further reconnection of a single edge state. We observe an Aharonov-Bohm type interference pattern even for low-size interferometers. T…
We develop a theoretical description of interaction-induced phenomena in an electronic Mach-Zehnder interferometer formed by integer quantum Hall edge states (with ν=1 and 2 channels) out of equilibrium. Using the non-equilibrium functional bosonization framework, we derive an effective action which contains all the physics of the problem. We apply the theory to the model of a short-range interaction and to a more realistic case of long-range Coulomb interaction. The theory takes into account interaction-induced effects of dispersion of plasmons, charging, and decoherence. In the case of long-…
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 use an electronic Mach-Zehnder interferometer to explore the non-equilibrium coherence of the electron waves within the edge-states that form in the integral quantum Hall effect. The visibility of the interference as a function of bias-voltage and transmission probabilities of the mirrors, which are realized by quantum point-contacts, reveal an unexpected asymmetry at finite bias when the transmission probability $T$ of the mirror at the input of the interferometer is varied between 0 and 100%, while the transmission probability of the other mirror at the output is kept fixed. This can lead…
We address the recently-observed unexpected behavior of Aharonov-Bohm oscillations in the electronic Mach-Zehnder interferometer that was realized experimentally in a quantum Hall system [1]. We argue that the measured lobe structure in the visibility of oscillations and the phase rigidity result from a strong long-range interaction between two adjacent counter-propagating edge states, which leads to a resonant scattering of plasmons. The visibility and phase shift, which we expressed in terms of the transmission coefficient for plasmons, can be used for the tomography of edge states.
Graphene is a very promising test-bed for the field of electron quantum optics. However, a fully tunable and coherent electronic beam splitter is still missing. We report the demonstration of electronic beam splitters in graphene that couple quantum Hall edge channels having opposite valley polarizations. The electronic transmission of our beam splitters can be tuned from zero to near unity. By independently setting the beam splitters at the two corners of a graphene PN junction to intermediate transmissions, we realize a fully tunable electronic Mach-Zehnder interferometer. This tunability al…
We study dephasing in an electronic Mach-Zehnder (MZ) interferometer based on quantum Hall (QH) edge states by a micromiter-sized Ohmic contact embedded in one of its arms. We find that at the filling factor $ν=1$, as well as in the case where an Ohmic contact is connected to an MZ interfeoremter by a quantum point contact (QPC) that transmits only one electron channel, the phase coherence may not be fully suppressed. Namely, if the voltage bias $Δμ$ and the temperature $T$ are small compared to the charging energy of the Ohmic contact $E_C$, the free fermion picture is manifested, and the vis…
Mach-Zehnder interferometry has been proposed as a probe for detecting the statistics of anyonic quasiparticles in fractional quantum Hall (FQH) states. Here we focus on interferometers made of multimode edge states with upstream modes. We find that the interference visibility is suppressed due to downstream-upstream mode entanglement; the latter serves as a "which path" detector to the downstream interfering trajectories. Our analysis tackles a concrete realization of filling factor 2/3, but its applicability goes beyond that specific case, and encompasses the recent observation of ubiquitous…
We study the evolution of a single-electron packet of Lorentzian shape along an edge of the integer quantum Hall regime or in a Mach-Zehnder interferometer, considering a capacitive Coulomb interaction and using a bosonization approach. When the packet propagates along a chiral quantum Hall edge, we find that its electron density profile becomes more distorted from Lorentzian due to the generation of electron-hole excitations, as the interaction strength increases yet stays in a weak interaction regime. However, as the interaction strength becomes larger and enters a strong interaction regime,…
Spin filtering through quantum spin interference is addressed exactly, in two dimensions, in a medium that has both Rashba and Dresselhaus spin-orbit couplings and an applied external magnetic field. The problem is addressed from the fully non-Abelian Gauge formulation. We propose an experimentally feasible electronic Mach Zehnder Interferometer and solve for the perfect spin filtering conditions. We find two broad solutions, one where filtering is achieved in the original incoming basis, that is purely a non-Abelian solution, and the other where one needs a tilted axis to observe the polarize…
We consider dephasing in the electronic Mach-Zehnder interferometer strongly coupled to current noise created by a voltage biased quantum point contact (QPC). We find the visibility of Aharonov-Bohm oscillations as a function voltage bias and express it via the cumulant generating function of noise. In the large-bias regime, high-order cumulants of current add up to cancel the dilution effect of a QPC. This leads to an abrupt change in the dependence of the visibility on voltage bias which occurs at the QPC's transparency T=1/2. Quantum fluctuations in the vicinity of this point smear out the …
We investigate a Mach-Zehnder interferometer driven by a time-dependent voltage. Motivated by recent experiments, we focus on a train of Lorentzian voltage pulses which we compare to a sinusoidal and a constant voltage. We discuss the visibilities of Aharonov-Bohm oscillations in the current and in the noise. For the current, we find a strikingly different behavior in the driven as compared to the static case for voltage pulses containing multiple charges. For pulses containing fractional charges, we find a universality at path-length differences equal to multiples of the spacing between the v…
We propose a simple physical model which describes dephasing in the electronic Mach-Zehnder interferometer at filling factor 2. This model explains very recent experimental results, such as the unusual lobe-type structure in the visibility of Aharonov-Bohm oscillations, phase rigidity, and the asymmetry of the visibility as a function of transparencies of quantum point contacts. According to our model, dephasing in the interferometer originates from strong Coulomb interaction at the edge of two-dimensional electron gas. The long-range character of the interaction leads to a separation of the s…
We consider a ballistic Mach-Zehnder interferometer for electrons propagating chirally in one dimension (such as in an integer Quantum Hall effect edge channel). In such a system, dephasing occurs when the finite range of the interaction potential is taken into account. Using the tools of bosonization, we discuss the decay of coherence as a function of propagation distance and energy. We supplement the exact solution by a semiclassical approach that is physically transparent and is exact at high energies. In particular, we study in more detail the recently predicted universal power-law decay o…
We present a numerical study of a multichannel electronic Mach-Zehnder interferometer, based on magnetically-driven non-interacting edge states. The electron path is defined by a full-scale potential landscape on the two-dimensional electron gas at filling factor two, assuming initially only the first Landau level as filled. We tailor the two beam splitters with 50% interchannel mixing and measure Aharonov-Bohm oscillations in the transmission probability of the second channel. We perform time-dependent simulations by solving the electron Schroedinger equation through a parallel implementation…
We have studied the zero-temperature statistics of charge transfer between the two edges of Quantum Hall liquids with filling factors $ν_{0,1}=1/(2 m_{0,1}+1)$ forming Mach-Zehnder interferometer. The known Bethe ansatz solution for symmetric interferometer is used to obtain the cumulant-generating function of charge at constant voltage $V$ between the edges. Its low-$V$ behavior can be interpreted in terms of electron tunneling, while its large-$V$ asymptotics reproduces the $m$-state dynamics ($m\equiv 1+m_{0}+m_{1}$) of quasiparticles with fractional (for $m>1$) charge and statistics. We al…
We develop a theoretical description of a Mach-Zehnder interferometer built from integer quantum Hall edge states, with an emphasis on how electron-electron interactions produce decoherence. We calculate the visibility of interference fringes and noise power, as a function of bias voltage and of temperature. Interactions are treated exactly, by using bosonization and considering edge states that are only weakly coupled via tunneling at the interferometer beam-splitters. In this weak-tunneling limit, we show that the bias-dependence of Aharonov-Bohm oscillations in source-drain conductance and …
We investigate a Mach-Zehnder interferometer fed by two time-dependently driven single-particle sources, one of them placed in front of the interferometer, the other in the center of one of the arms. As long as the two sources are operated independently, the signal at the output of the interferometer shows an interference pattern, which we analyze in the spectral current, in the charge and energy currents, as well as in the charge current noise. The synchronization of the two sources in this specifically designed setup allows for collisions and absorptions of particles at different points of t…
We report on a new mechanism of giant phase modulation. The phenomenon arises when a dispersed photonic mode (slow light) strongly couples to an excitonic resonance. In such a case, even a small amount of optically injected carriers creates a potential barrier for the propagating exciton-polariton which provokes a considerable phase shift. We evidence this effect by fabricating an exciton-polariton Mach-Zehnder interferometer, modulating the output intensity by constructive or destructive interferences controlled by optical pumping of a micrometric size area. The figure of merit for a π phase …
We consider tunneling between two edges of Quantum Hall liquids (QHL) of filling factors $ν_{0,1}=1/(2 m_{0,1}+1)$, with $m_0 \geq m_1\geq 0$, through two point contacts forming Mach-Zehnder interferometer. Quasi-particle description of the interferometer is derived explicitly through the instanton duality transformation of the initial electron model. For $m_{0}+m_{1}+1\equiv m>1$, tunneling of quasiparticles of charge $e/m$ leads to non-trivial $m$-state dynamics of effective flux through the interferometer, which restores the regular ``electron'' periodicity of the current in flux. The exact…
We develop the theory of electronic Mach-Zehnder interferometers built from quantum Hall edge states at Landau level filling factor ν= 2, which have been investigated in a series of recent experiments and theoretical studies. We show that a detailed treatment of dephasing and non-equlibrium transport is made possible by using bosonization combined with refermionization to study a model in which interactions between electrons are short-range. In particular, this approach allows a non-perturbative treatment of electron tunneling at the quantum point contacts that act as beam-splitters. We find a…
It was recently pointed out that Halperin's 113 topological order explains the transport experiments in the quantum Hall liquid at filling factor $ν=5/2$. The 113 order, however, cannot be easily distinguished from other likely topological orders at $ν=5/2$ such as the non-Abelian Pfaffian and anti-Pfaffian states and the Abelian Halperin 331 state in Fabry-Perot interferometry. We show that an electronic Mach-Zehnder interferometer provides a clear identification of these candidate $ν=5/2$ states. Specifically, the $I$-$V$ curve for the tunneling current through the interferometer is more asy…
We propose direct experimental tests of the effective models of fractional quantum Hall edge states. We first recall a classification of effective models based on the requirement of anomaly cancellation and illustrate the general classification with the example of a quantum Hall fluid at filling factor 2/3. We show that, in this example, it is impossible to describe the edge states with only one chiral channel and that there are several inequivalent models of the edge states with two fields. We focus our attention on the four simplest models of the edge states of a fluid with filling factor 2/…
