The Hausdorff dimension of fractal sets and fractional quantum Hall effect

Cruz, Wellington da

doi:10.1016/s0960-0779(02)00580-5

Cited by 12 publications

(20 citation statements)

References 9 publications

(21 reference statements)

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“…(23). By direct substitution we can prove the next, vortex generation technique for solutions of this equation.…”

Section: Vortex Solutions Generating Techniquementioning

confidence: 95%

“…It describes the time dependent stationary vortex lattice solution of (23). Positions of vortices in the lattice do not change with time.…”

Section: Sine-hermite Solution and Stationary Vortex Latticementioning

confidence: 99%

“…They can play essential role in the loop gravity and quantum spacetime, leading to formation of the Cantorian Space [21], fractalization of the microspace geometry, which has been discussed recently in the context of the fractional Quantum Hall Effect [22,23] and Cosmic Strings [24].…”

Section: Introductionmentioning

confidence: 99%

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Integrable vortex dynamics in anisotropic planar spin liquid model

Gürkan

Pashaev

2008

Chaos, Solitons & Fractals

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The problem of magnetic vortex dynamics in an anisotropic spin liquid model is considered. For incompressible flow the model admits reduction to saturating Bogomolny inequality analytic projections of spin variables, subject the linear holomorphic Schrö dinger equation. It allows us to construct N vortex configurations in terms of the complex Hermite polynomials. Using complex Galilean boost transformations, the interaction of the vortices and the vortex chain lattices (vortex crystals) is studied. By the complexified Cole-Hopf transformation, integrable N vortex dynamics is described by the holomorphic Burgers equation. Mapping of the point vortex problem to N-particle problem, the complexified Calogero-Moser system, showing its integrability and the Hamiltonian structure, is given.

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“…(23). By direct substitution we can prove the next, vortex generation technique for solutions of this equation.…”

Section: Vortex Solutions Generating Techniquementioning

confidence: 95%

“…It describes the time dependent stationary vortex lattice solution of (23). Positions of vortices in the lattice do not change with time.…”

Section: Sine-hermite Solution and Stationary Vortex Latticementioning

confidence: 99%

See 1 more Smart Citation

Integrable vortex dynamics in anisotropic planar spin liquid model

Gürkan

Pashaev

2008

Chaos, Solitons & Fractals

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“…Among the assumptions that go into the derivation of (4.4) from (4.8) are that the temperature is sufficiently low and that the sample is sufficiently pure, but unfortunately the analyses in [13,14,17,82] are unable to quantify exactly what "sufficiently" means. Basically one must simply assume that (4.8) contains the most relevant terms for obtaining the long wavelength, low frequency response functions, but obviously this is not always true even at the lowest temperatures, for example it is believed that, at zero temperature, a Wigner crystal will form for filling fractions below about 1/7 and (4.8), being rotationally invariant, cannot allow for this 11 .…”

Section: Duality and The Quantum Hall Ef Fectmentioning

confidence: 99%

Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

Dolan¹

2007

SIGMA

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Abstract. The parallel rôles of modular symmetry in N = 2 supersymmetric Yang-Mills and in the quantum Hall effect are reviewed. In supersymmetric Yang-Mills theories modular symmetry emerges as a version of Dirac's electric -magnetic duality. It has significant consequences for the vacuum structure of these theories, leading to a fractal vacuum which has an infinite hierarchy of related phases. In the case of N = 2 supersymmetric Yang-Mills in 3 + 1 dimensions, scaling functions can be defined which are modular forms of a subgroup of the full modular group and which interpolate between vacua. Infra-red fixed points at strong coupling correspond to θ-vacua with θ a rational number that, in the case of pure SUSY Yang-Mills, has odd denominator. There is a mass gap for electrically charged particles which can carry fractional electric charge. A similar structure applies to the 2 + 1 dimensional quantum Hall effect where the hierarchy of Hall plateaux can be understood in terms of an action of the modular group and the stability of Hall plateaux is due to the fact that odd denominator Hall conductivities are attractive infra-red fixed points. There is a mass gap for electrically charged excitations which, in the case of the fractional quantum Hall effect, carry fractional electric charge.

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“…There are now some theoretical and logical indications for the existence of a fractal Hall effect [50], If these indications are correct, then Ε-infinity theory is destined to play a major role in this area [50,51]. notably Goldfain [56] and Ji-Huan He [57], Further fundamental research on the application of Ε-infinity in cosmology is due to Agop, Iovane, Tomashits [58] and Sidharth [16].…”

Section: Fractal Hall Effectmentioning

confidence: 99%