Some Remarks on the Quantum Hall Effect

Pasquier, Vincent

doi:10.1007/978-1-4471-4863-0_21

Cited by 3 publications

(3 citation statements)

References 28 publications

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“…Third, the aforementioned duality between quasi-holes and quasi-particles is also visible and better studied for the quantum Calogero-Sutherland N-body system [68,124,127], which degenerates to our quantum periodic Benjamin-Ono in the chiral sector in which the density field is approximately uniform [2,126,144,145]. In light of this hydrodynamic degeneration to quantum periodic Benjamin-Ono, along with the intricacies of the bulk-boundary correspondence, in [154] Wiegmann proposes (1.2.1) as an effective description of edge excitations in the fractional quantum Hall effect.…”

Section: Quantum Quasi-holes and Quasi-particlesmentioning

confidence: 99%

Random Partitions and the Quantum Benjamin-Ono Hierarchy

Moll¹

2015

Preprint

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We derive exact and asymptotic results for random partitions from general results in the semi-classical analysis of coherent states. In geometric quantization of Hermitian affine spaces (M , J, g, ω), a coherent state Υ v (•| ) around a classical state v ∈ M is the reproducing kernel in the Fock space L 2 J-hol (M , dρ ,g ) of J-holomorphic functions on M square-integrable against the Segal-Bargmann Gaussian weight dρ ,g . Under certain regularity assumptions, in any canonical quantization defined by an ordering η, we prove that in the semi-classical limit → 01/2 with mean 0 and variance ||(∇O)| v || 2 g independent of η. These results do not assume integrability of the Hamiltonian flow generated by O but follow directly from the fact that at fixed > 0 the Segal-Bargmann weight on M is already Gaussian with covariance kernel g −1 given by the inverse metric.The classical periodic Benjamin-Ono equation for real 2π-periodic v of mean a ∈ R is Hamiltonian in the leaf (M (a), J, g −1/2 , ω −1/2 ) of the real L 2 -Sobolev space on the circle T at critical regularity s = −1/2 with J the spatial periodic Hilbert transform. We find a classical conserved density dF |v (c|ε) on c ∈ R for this system with dispersion coefficient ε, extending Nazarov-Sklyanin (2013). The authors also give an ordering η N S for an integrable canonical quantization, which we use to construct a quantum conserved density d F η N S (c| , ε)| Ψ . For quantum stationary states, we identify this conserved density with dF λ (c| ε 2 , ε 1 ) the Rayleigh measure of the profile of a partition λ of anisotropy (εAs Jack polynomials are the quantum stationary states and Stanley's Cauchy kernel (1989) is the reproducing kernel, the random values of the quantum periodic Benjamin-Ono hierarchy in a coherent state Υ v (•| ) are a "Jack measure" on partitions, a dispersive generalization of Okounkov's Schur measures (1999). By the above,we have concentration on a limit shape as → 0, the classical conserved density at v, and quantum fluctuations are an explicit Gaussian field. Our results follow from an enumerative asymptotic expansion in and ε of joint cumulants over new combinatorial objects we call "ribbon paths". As above, our results reflect the fact that at fixed > 0 the weight defining Fock space is already a fractional Brownian motion of variance and Hurst index (−s) − 1 2 dim T = + 1 2 − 1 2 = 0.

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Section: Quantum Quasi-holes and Quasi-particlesmentioning

confidence: 99%

Random Partitions and the Quantum Benjamin-Ono Hierarchy

Moll¹

2015

Preprint

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“…In the thermodynamical limit the partition function Z can be obtained by using the saddle-point technique, where the particles are driven into configuration which has the minimum energy [43,44]. For N → ∞, the sum over particles becomes a continuous distribution, which equals the electron density.…”

Section: Appendix: Calculation Of the Statistical Phasementioning

confidence: 99%

Exact solutions of a model for synthetic anyons in a noninteracting system

et al. 2020

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Synthetic anyons can be implemented in a noninteracting many-body system, by using specially tailored localized (physical) probes, which supply the demanded nontrivial topology in the system. We consider the Hamiltonian for noninteracting electrons in two-dimensions (2D), in a uniform magnetic field, where the probes are external solenoids with a magnetic flux that is a fraction of the flux quantum. The Hamiltonian could also be implemented in an ultracold (fermionic) atomic gas in 2D, in a uniform synthetic magnetic field, where the probes are lasers giving rise to synthetic solenoid gauge potentials. We find analytically and numerically the ground state of this system when only the lowest Landau level states are occupied. It is shown that the ground state is anyonic in the coordinates of the probes. We show that these synthetic anyons cannot be considered as emergent quasiparticles. The fusion rules of synthetic anyons are discussed for different microscopic realizations of the fusion process.

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“…The calculation is performed analytically in the thermodynamic limit N →∞ by using the plasma analogy, first introduced by Laughlin [11] (see Refs. [29,30] for details),…”

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confidence: 99%

Quantum Hall Effect with Composites of Magnetic Flux Tubes and Charged Particles

et al. 2018

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Composites formed from charged particles and magnetic flux tubes, proposed by Wilczek, are one model for anyons-particles obeying fractional statistics. Here we propose a scheme for realizing charged flux tubes, in which a charged object with an intrinsic magnetic dipole moment is placed between two semi-infinite blocks of a high-permeability (μ_{r}) material, and the images of the magnetic moment create an effective flux tube. We show that the scheme can lead to a realization of Wilczek's anyons, when a two-dimensional electron system, which exhibits the integer quantum Hall effect, is sandwiched between two blocks of the high-μ_{r} material with a temporally fast response (in the cyclotron and Larmor frequency range). The signature of Wilczek's anyons is a slight shift of the resistivity at the plateau of the IQHE. Thus, the quest for high-μ_{r} materials at high frequencies, which is underway in the field of metamaterials, and the quest for anyons, are here found to be on the same avenue.

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Some Remarks on the Quantum Hall Effect

Cited by 3 publications

References 28 publications

Random Partitions and the Quantum Benjamin-Ono Hierarchy

Random Partitions and the Quantum Benjamin-Ono Hierarchy

Exact solutions of a model for synthetic anyons in a noninteracting system

Quantum Hall Effect with Composites of Magnetic Flux Tubes and Charged Particles

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