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Spectral and Wavefunction Statistics (II) V.E.Kravtsov, Abdus Salam ICTP

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Wavefunction statistics Porter-Thomas distribution Wavefunction statistics in one-dimensional Anderson model Weak multifractality in 2D disordered conductors

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Wavefunction statistics and Coulomb peaks heights Contact area Bunching

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Porter-Thomas distribution -follows from random matrix theory -describes local distribution of wavefunction intensity in chaotic systems -fails to describe the “web” pattern

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The Anderson model 1D: all states are localized with localization length W 3D: Anderson localization transition at W=16.5 2D: all states are localized with exponentially large localization radius ~exp(-a/W ) 2 2

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Distribution of eigenfunction amplitude for 1D Anderson model Porter-Thomas: Take =2then the distribution for 1D Anderson model can be considered as limit of the Porter-Thomas distribution.

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Poisson distribution as a of the Wigner surmise No surprise that the limit of Porter-Thomas gives the distribution of localized functions in one dimension.

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Distribution of wavefunction amplitude in 2D conductors (L x Porter- Thomas Porter-Thomas with corrections Log-normal

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Sample dimension from local measurement 2D Quasi 1D Porter- Thomas Porter-Thomas with corrections Log-normal in 2D: Stretch- exponential in quasi-1D Zero- dimensional quantum dot For g>>1 RMT behavior for a typical wavefunction Dimension-specific behavior for large amplitudes

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Where the RMT works Porter- Thomas Porter-Thomas with corrections g>>1 For dynamic phenomena g Diffusion displacement for time 1/

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Weak multifractality in 2D conductors 2=dimensionality of space q-dependent multifractal dimensionality Magnetic field makes fractality weaker

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Fractal dimension of this map decreases with increasing the level=“multi”-fractality Multifractality: qualitative picture Map of the regions where exceeds the chosen level Arbitrary chosen level Weight of the dark blue regions scales like

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