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Fractal‐to‐Euclidean Crossover in Quantum Percolation
Author(s) -
Hałdaś G.,
Kolek A.,
Stadler A.W.
Publication year - 2002
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/1521-3951(200203)230:1<249::aid-pssb249>3.0.co;2-g
Subject(s) - percolation threshold , percolation critical exponents , percolation (cognitive psychology) , scaling , physics , fractal , critical exponent , lyapunov exponent , statistical physics , condensed matter physics , mathematics , quantum mechanics , mathematical analysis , phase transition , geometry , neuroscience , nonlinear system , biology , electrical resistivity and conductivity
Extensive numerical studies of quantum percolation in 2D show no indications of localization–delocalization transition. At the percolation threshold, i.e. for p = p c , the scaling curve β ≡ ∂ ln g /∂ ln L exhibits a fractal‐like behavior. For ln g ≪ 0 it senses superlocalization: it has the slope d ϕ ≅ 1.14. For ln g ≫ 0 it saturates at — t / ν + d — 2 ≅ — 1, where t and ν are percolation critical exponents. For small size L (∼10) of percolation cluster the distribution of variable λ 1 = arccosh ( $ 1/ \sqrt {T_1} $ ), where T 1 is the first transmission eigenvalue, has the exponential tail P ( λ 1 ) ∼ exp (— λ 1 ), which is characteristic for chaotic cavities with one‐moded leads. For intermediate sizes P ( λ 1 ) changes to Wigner surmise typical for metallic states. For large sizes the shape of P ( λ 1 ) results from the “convolution” of the first Lyapunov exponent γ 1 (which is Gaussian) and chemical length l (which has a tail for large l ). For p > p c we observe a crossover from fractal‐like behavior for L ≪ ξ p , ξ p is the percolation correlation length, to Euclidean‐like behavior, characteristic for homogeneous disorder, for L ≫ ξ p .