Recent experiments with suspended graphene have indicated the crucial role of carrier mobility in the competition be- tween Laughlin collective state and insulating state, probably of Wigner-crystal-type. Moreover, the **fractional** **quantum** **Hall** **effect** (FQHE) in graphene has been observed at a low carrier density where the interaction is reduced as a result of particles dilution. This suggests that the interaction may not be a sole factor in the triggering of FQHE as it was ex- pected to be based on the standard formulation of the composite fermion model. Here, the topological arguments are presented to explain the observed features of the FQHE in graphene and the triggering role of carrier mobility in forma- tion of the Laughlin state.

At moderate Zeeman energies, the distortion of Skyrmions in the **fractional** **quantum** **Hall** **effect** is small and one may describe spin textures using the Belavin-Polyakov solutions. We show that there is a range of parameters where one may self-consistently assume that the charge density of the spin distribution occurs at isolated points. This range is spanned by that realised experimentally. The spin distribution of many Skyrmions depends in a very non-local way upon the Skyrmion coordinates. However, the interplay of Coulomb and Zeeman energies is such that near the classical minimum the behaviour accords with the intuitive picture of equal sized Skyrmions interacting via a residual point-like Coulomb interaction. At low temperatures these form a Wigner crystal with hexagonal symetry. When one takes note of the azimuthal components of the spn texture, the full hexagonal symetry is no longer apparent; the Skyrmions form a face centred rectangular lattice with sides close to the ratio 1 : √ 3. The analytic expression for this dstribution is given by w = hcn(z, e 2πi/3 ). In fact, accounting for the higher moments in the Coulomb interaction, the

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The exact n-body distribution functions are calculated for a two-dimensional, noninteracting **quantum** electron gas in an external magnetic field for any temperature and density. At low temperature and filled lowest-Landau-level, these functions are identical to the exact distribution functions obtained by Jancovici [Phys. Rev. Lett. 46 (1981) 386] for the classical two- dimensional one-component plasma at the special plasma parameter Γ = 2, establishing that the **quantum** state with filling factor ν = 1, associated with the integral **quantum** **Hall** **effect**, is precisely described by an effective classical potential φ (r) = −2 lnr . Further, this Boltzmann factor exactly matches that constructed by Laughlin [Phys. Rev. Lett. 50 (1983) 1395] to account for the **fractional** **quantum** **Hall** **effect**.

**Fractional** **quantum** **Hall** **effect** (FQHE) is investigated by employing normal electrons and the fundamental Hamiltonian without any quasi particle. There are various kinds of electron configu- rations in the Landau orbitals. Therein only one configuration has the minimum energy for the sum of the Landau energy, classical Coulomb energy and Zeeman energy at any **fractional** filling factor. When the strong magnetic field is applied to be upward, the Zeeman energy of down-spin is lower than that of up-spin for electrons. So, all the Landau orbitals in the lowest level are occupied by the electrons with down-spin in a strong magnetic field at 1 < < ν 2 . On the other hand, the Landau orbitals are partially occupied by up-spins. Two electrons with up-spin placed in the nearest orbitals can transfer to all the empty orbitals of up-spin at the specific filling factors

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While several experimental works attempting to demonstrate non-Abelian quasiparticles in the **fractional** **quantum** **Hall** **effect** have recently been performed [37, 38], the interpretation of these experiments is very controversial, and this remains a forefront of research. Several papers in the current issue were aimed at either providing other methods to make this demonstration, for example by measuring tunnelling spectra [39] or examining the additional physics of these experiments (in this case, the Zeno **effect** [22], or disorder in quasiparticle lattices [40]) not previously considered in the predictions. In our entire ’ focus on ’ collection, disappointingly, only a single submitted manuscript was actually a real experiment [41], and this, although a nice experiment demonstrating Aharonov – Bohm oscillations in **quantum** **Hall** samples, remains quite a distance from the desired result that would convincingly show non-Abelian physics.

The exact n-body distribution functions are calculated for a two-dimensional, noninteracting **quantum** elec- tron gas in an external magnetic field for any temperature and density. At low temperature and filled lowest Landau level, these functions are identical to the exact distribution functions obtained by Jancovici 关 Phys. Rev. Lett. 46, 386 共 1981 兲兴 for the classical two-dimensional one-component plasma 共 2DOCP 兲 at the special plasma parameter ⌫ ⫽ 2. This establishes that the **quantum** state with filling factor ⫽ 1, associated with the integral **quantum** **Hall** **effect**, is precisely described by an effective classical potential (r) ⫽⫺ 2 ln r, so that a classical Boltzmann factor of 2DOCP form can replace the **quantum** Slater sum. Further, this Boltzmann factor exactly matches that constructed by Laughlin 关 Phys. Rev. Lett. 50, 1395 共 1983 兲兴 to account for the **fractional** **quantum** **Hall** **effect**. Additional effective potentials for higher filling factors ⫽ 2,3, . . . are obtained semianalytically from the exact Yvon-Born-Green integral equation and numerically from the approximate hypernetted-chain integral equation. They have the asymptotic form (r) ⬃⫺ (2/ )ln r.

it was named, “**fractional** **quantum** **Hall** **effect**” [2]. In 1983, there is no effort to find the orgin of the quantization particularly at the **fractional** values but Laughlin [3] used incompressibility in the flux quantization to find a wave function which gives quasiparticles of **fractional** charges. It was found that the charge density wave state has higher energy so the fractionally charged state becomes the ground state. For theoretical physics it is sufficient to prove that there is a ground state the excitations of which are fractionally charged. However, the wave functions are not useful for calculating physical properties of solids such as resistivity. By fixing the area in the flux quantization, it is possible to make the **fractional** charge but this is not what happens in the real experiment. There is no mechanism to hold the area to maintain the **fractional** charge and if area is relaxed the **fractional** charge disappears. The spin of the electron also does not play any particular role in holding the area incompressible. In 1985, Shrivastava[4] predicted the **fractional** charges correctly by inventing certain spin symmetries. In this theory, fixing the spin, fixes the charge so that there is spin-charge locking. The symmetries of the predicted charges as well as their magnitudes are in agreement with the experiments [5,6]. When the spin reverses, the charge also changes so that there is an **effect** of Kramer’s time reversal on the change in the charge of the quasiparticles. Therefore, there is a particle-hole symmetry [5]. Usually, the spin of the electron is a positive number but we use also the negative spin [7] so that in going from one charge to another, momentum is conserved. In this way we are able to understand all of the data on the **quantum** **Hall** **effect** [8-14].

The spin polarization of a **fractional** **quantum** **Hall** state shows very interesting properties. The curve of polarization versus magnetic field has wide plateaus. The **fractional** **quantum** **Hall** **effect** is caused by the Coulomb interaction because the 2D electron system without the Coulomb interac- tion yields no energy gap at the **fractional** filling factor. Therefore, the wide plateau in the polari- zation curve is also caused by the Coulomb interaction. When the magnetic field is weak, some electrons have up-spins and the others down-spins. Therein the spin-exchange transition occurs between two electrons with up and down spins via the Coulomb interaction. Then the charge dis- tribution before the transition is the same as one after the transition. So these two states have the same classical Coulomb energy. Accordingly, the partial Hamiltonian composed of the spin ex- change interaction should be treated exactly. We have succeeded in diagonalizing the spin ex- change interaction for the first and second nearest electron pairs. The theoretical results repro- duce the wide plateaus very well. If the interval modulations between Landau orbitals are taken into the Hamiltonian, the total energy has the Peierls instability. We can diagonalize the Hamilto- nian with the interval modulation. The results reproduce wide plateaus and small shoulders which are in good agreement with the experimental data.

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We now proceed to a considerably more difficult case: the low E -field regime. In this regime, as the electric field becomes weaker, the energy gap gets smaller. As a result, there will be an unavoidable point where some L.L.s will overlap (Figure 1(b)), and occupational patterns turn out to be more complex. Inequality (1.7) is no longer true for all n ’s (it will indeed be true only for L.L.s with **quantum** num- ber ≤ i F ). If, for simplicity reasons, we suppose for a moment that the energy

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make them (twigs) to remain oscillatory. The oscillatory **effect** of twigs with their corresponding characteristic frequencies on an electron quanta string follows an adiabatic perturbation and is a manifestation of **fractional** charge quantization. The evidence of this phenomena is supported by the discovery of GMR by Albert Peter and Paul Gruebber and this led them to winning the Nobel prize in the year 2007 [8]. Similarly, the single electron tunneling across the interface state of a transistor will follow a helicon profile with each turn of the helix corresponding to **fractional** **quantum** states (spintronics). The **fractional** **Hall** electric fields are like pearls beaded in an electron quanta string and each string is connected to another electron string with twisting effects as a manifestation of QED behaviour. Thus, we have a **quantum** Garland beaded twigs (sub-quanta) on a single or many electron quanta strings.

Reference [44] has introduced the closed-form expres- sions for all two-body Haldane pseudopotentials on the torus and cylinder. In this work, inspired by Refs. [42 – 44] as the starting point of our analysis, we provide a complete framework for constructing general **quantum** **Hall** parent Hamiltonians involving N -body pseudopotentials, for fer- mions as well as bosons, in cylindric and toroidal geom- etries. This advance proves particularly important for the non-Abelian states, most of which necessitate many-body pseudopotentials in their parent Hamiltonian class. From the construction scheme laid out in this work, all topo- logical properties of the non-Abelian states such as their modular matrices and topological ground-state degeneracy can now be conveniently studied from their associated toroidal parent Hamiltonian. Complementing previous results for the sphere and infinite plane, our formalism furthermore directly generalizes to multicomponent sys- tems with an arbitrary number of “spin types” or “colors.” Therefore, our construction of many-body clustered Hamiltonians not only applies to arbitrary geometries but also crucially simplifies previous approaches. We illustrate this by numerical examples, including non-Abelian multicomponent states whose parent Hamiltonians were previously unknown.

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The **Hall** **Effect** was discovered by Sir Edwin **Hall** [1] in 1879 while he was under graduate student in Johns Hop- kins University. But at that time, even the electron was not experimentally discovered. Clear understanding had to wait until **quantum** mechanics came in to appearance. A hundred years later, the **Hall** **effect** was revived as a source of wonderful physics. In 1980, Klaus von Klitzing discovered [2] that two dimensional electron gas, at very low temperatures and strong magnetic fields, displays a remarkable quantization of the **Hall** conductance. Namely, the graph of the **Hall** conductance as the function of the magnetic field, is a staircase function, where the value of the **Hall** conductance at the plateaus is, to great accuracy, an integer multiple of e h 2 = 1 25812.807572 . This discovery led to superior standards of resistance and von-Klitzing was awarded the Nobel Prize in 1985 for his discovery. In a 1981 Robert Laughlin put forward [3] an argument for the quantization of the **Hall** conductance. This argument played a seminal role in the development of the theory of the Integer **Hall** **effect**. On the other hand, the lack of symmetry between electric and magnetic fields is one of the oldest puzzles in physics. One of the biggest unresolved questions in theoretical physics is that associated with the quantization of electric charge, i.e. why the observed electric charges in all the electrically charged matter is an integer multiple of a “fundamental charge” ‘e’, the electron charge.Why is it possible to iso-

In the previous paper [4], we proposed a new **quantum** **Hall** experiment where we observe the tunneling **effect** between two **quantum** **Hall** states in a device with the narrow potential barrier in the electron channel. The de- vice proposed in the article [4] is shown in Figure 1. This old type of the device has some difficulties as will be examined below. So we improve it and propose a new type of the device which is illustrated in Figure 2.

density is the result of the existence of the zero-energy LL, which is the fingerprint of a chiral nature of two- dimensional quasiparticles. This contrasts with a grad- ual freeze-out of both **Hall** and dissipative conductivities in semicondutor structures upon their depletion. Hav- ing compared various types of density dependent **Hall** conductivity, we suggest that two kinds of chiral (Berry phase Jπ) quasiparticles specific to monolayer (J = 1) and bilayer (J = 2) systems can be distinguished on the basis of QHE measurements. It is interesting to note that the recent **Hall** **effect** study of ultra-thin films by Novoselov et al. [10] featured both types of σ xy (N ) de- pendence shown in Fig. 4.

In conventional treatments, the N -particle Hamiltonian is diagonalized in a single-particle deﬁned Slater determinant basis, but in our treatment, we address the problem by transforming ﬁrst to a set of collective coordinates and then transforming to a hyperspherical representation [4]. While its use in condensed matter physics has been limited, the adiabatic hyperspherical representation has been used successfully in a wide range of problems in few-body physics, including nuclear structure and reactivity, few-electron atoms, positron-electron systems, and Bose condensates. We assert that the technique can also eﬀectively be applied to the many body **quantum** **Hall** system, and that it introduces an interesting and distinct framework for interpreting the system.

Since effectively the local contact vortex velocity dependent part of the Mag- nus force in a Josephson junction array is zero in the classical limit, we pre- dict zero classical **Hall** **effect**. In the **quantum** limit because of the geometric phases due to the finite superfluid density at superconductor grains, rich and complex **Hall** **effect** is found in this **quantum** regime due to the Thouless- Kohmoto-Nightingale-den-Nijs **effect**. 1

The argument of Thouless and collaborators is that, assuming the magnetic flux through the sample is rational, a principal U (1)-bundle can be constructed over the Brillouin zone (momentum space) of the sample, topologically a torus. The authors then build a particular connection on this U (1)-bundle and, using the Kubo formula for conductance from statistical mechanics, show that (up to a universal constant) the **Hall** conductance can be expressed as the integral of the curvature of this connection. One then consults geometric theory to find that the **Hall** conductance is a pairing of a Chern class and a homology class of the Brillouin zone. Thus it is an integer. Thouless et al.’s result was the first to relate the **Hall** conductance to topological data and subsequent papers by, amongst others, Kohmoto [Koh85] showed that the quantisation was stable under small amounts of disorder. The geometric ‘bundle’-viewpoint was a significant step forward in providing an adequate explanation for the quantisation of **Hall** conductance, but relied on the physically unrealistic assumption of rational magnetic flux.

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Coulomb correlations between the layers. The first explicit prediction for a new **quantum** **Hall** state was made in 1987 by Chakraborty and Pietilainen [9]. They numerically modeled two parallel ν = 1/2 layers in the absence of tunneling. Their simulations with four electrons per-layer and found evidence for a **quantum** **Hall** state. Stimulated by these numerical results, in 1989 Fertig pointed out several key properties of this bilayer state [10]: first, the ground state of the system can be written as a pseudospin ferromagnet; second, an equivalent description of the ground states is of a excitonic condensate (an exciton is a neutrally charged electron-hole pair); and third, associated with these broken symmetry ground states is a linearly dispersing Goldstone mode. Each of these predictions has been born out by experiment 2 . Shortly thereafter MacDonald and Rezayi predicted that electron-hole bilayers should form an excitonic superfluid when each layer is half filled [11].

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Recently the nature of the FQHE in graphene has received intense interest (8–14) since the combined spin and valley degrees of freedom are conjectured to yield novel FQHE states within an approximate SU(4) symmetry space (assuming relatively weak spin Zeeman and short-range interaction energies can be ignored). Furthermore, unlike conventional semiconductor systems, which have shown limited evidence for transitions in **fractional** states (15–19), the wide gate tunability of graphene systems coupled with large cyclotron energies allows for the exploration of multiple different SU(4) order parameters for a large range of filling fractions. In bilayer graphene, the capability to force transitions between different spin and valley orderings by cou- pling to electric fields perpendicular to the basal plane as well as magnetic fields provides a unique opportunity to probe interaction-driven symmetry breaking within this expanded SU(4) basis in a fully controllable way (20–23). The most intriguing consequence of this field tunabil- ity is the possibility to induce transitions between different FQHE phases (24–26). Thus, BLG provides a unique model system to experimentally study phase transitions between different topologically-ordered states.

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Most numerical studies of **quantum** **Hall** systems truncate the Hilbert space to the partially filled Landau level in question. The electron-electron interaction is usually taken as the Coulomb interaction projected onto that Landau level, or a model interaction which captures the essential features of the projected interaction. These simplifications make the size of the Hilbert space of the system tractable and amenable to numerical calculations. The projection or approximation of the Coulomb interaction has had much success in the study of Abelian **quantum** **Hall** states since these states seem to be robust to small changes in the electron-electron interactions. The ground state of electrons in the second Landau level, however, seems to be sensitive to such small changes in the interaction between electrons [40, 63, 41, 42]. Though this sensitivity makes the justification of numerical results more difficult, it might also provide a control knob for engineering ground states of **quantum** **Hall** systems.

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