Multifractal Properties of the Fibonacci Chain
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1 Multifractal Properties of the Fibonacci Chain Frédéric Piéchon Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France Nicolas Macé Anu Jagannathan 1 December 2015 Geo-Dyn,
2 Waves in random media : P.W. Anderson 1958 «gang of four», 1981 scaling theory of metal-insulator transition Weak disorder and d>2: -extended states -normal diffusive motion of wave packet -gaussian distribution of conductance Strong disorder or d<2: -exponentially localized states -no diffusion of wave packet -lognormal distribution of conductance Metallic regime At the transition : -critical states with multifractal properties -anomalous diffusion of wave packet -anomalous power law tail of conductance distribution Insulating regime Critical regime at the MIT
3 Waves in quasiperiodic chains: Fractal spectrum Period of approximant Fractal dimensions:
4 Waves in quasiperiodic chains: Fractal spectrum Self similar Wave functions Period of approximant Fractal dimensions: Wave functions: Local Density of states :
5 Waves in quasiperiodic chains: Fractal spectrum Self similar Wave functions Anomalous diffusion of wavepacket Period of approximant Fractal dimensions: Wave functions: Diffusion exponents : Local Density of states : Fractal properties are generic and tunable
6 Waves in quasiperiodic chains: Fractal spectrum Self similar Wave functions Anomalous diffusion of wavepacket Period of approximant Fractal dimensions: Wave functions: Diffusion exponents : Local Density of states : How to relate spectrum fractal dimensions with wave functions fractal dimensions and diffusion exponents?
7 PART 1 Tight-binding model on the Fibonacci chain Perturbative renormalization group : atoms & molecules Energy spectrum fractal properties 7
8 Tight-binding model on the Fibonacci chain «golden mean» Atoms sites: weak modulation Molecules sites : strong modulation
9 Atoms and molecules : Niu & Nori (1986) Kalugin,Kitaev,Levitov (1986) Energy level Degeneracy Atoms : isolated sites (central) Molecules : isolated dimer Bonding/antibonding (lateral)
10 Perturbative renormalization group Atomic RG step : (decimation of molecules) Molecular RG step: (decimation of atoms) (bonding/antibonding) Niu & Nori (1986) Kalugin,Kitaev,Levitov (1986)
11 Perturbative renormalization group Atomic RG step : (decimation of molecules) Molecular RG step: (decimation of atoms) (bonding/antibonding) Niu & Nori (1986) Zheng (1987) Piéchon, Benakli, Jagannathan (1995)
12 RG reconstruction of energy band spectrum bonding levels atomic levels Niu & Nori (1986) Zheng (1987) Piéchon et al (1995) antibonding levels
13 Density of states : Recursive reconstruction of the IDOS :
14 Fractal dimensions: Nonperturbative formula : (trace map) Kohmoto,Sutherland,Tang (1987) (6 cycle) (2 cycle) (4 cycle) (Van-hove singularity) Rüdinger,Piéchon (1998)
15 RG path of energy bands bonding RG step (left ) atomic RG step (central) antibonding RG step (right) Each energy band is labelled by a sequence of three indices : central band : bottom band: generic band of chain
16 Bands fractal properties Bands anomalous scaling : Lines
17 PART 2 What about the wave functions?
18 Fractal properties of wave functions The aim of the game : Local spectral measure What is known : -Exact wave function fractal properties for central level Kohmoto,Sutherland,Tang (1987) -RG and fractal properties to zero order in Piéchon (1996), Thiem, Schreiber (2013) Lets go beyond
19 RG step for Wave functions Atomic RG step : For E in the central cluster : atom site : molecule site :
20 RG step for Wave functions Atomic RG step : if i is an atom site and E in the central cluster Molecular RG step : if i is a molecule site and E in the lateral clusters
21 Fractal Dimensions RG step for sites with largest probability : For a an energy Fractal dimensions : with renormalzation path
22 Comparison with numerics
23 Comparison with numerics Non perturbative expressions: (obtained from sum rules)
24 Comparison with numerics wave function spatial fractalilty Non perturbative expressions: (obtained from sum rules) energy band fractality
25 Preferential RG path of sites atom 0 mol. left mol. right Each site can be labelled by a sequence of three indices : Reordering the sites according to 0-0 is equivalent to conumerotation
26 Wave functions probability normal position conumeroration position
27 Geometric reconstruction of Wave functions probability
28 Geometric reconstruction of Wave functions probability
29 Geometric reconstruction of Wave functions probability
30 Silver mean quasieriodic chain normal position conumeroration position
31 Harper model normal position «conumeroration» position
32 Sites average fractal dimensions average fractal dimension of wavefunctions: average fractal dimension of the LDOS:
33 Relation between fractal dimensions average fractal dimension of wave functions: average fractal dimension of the LDOS: fractal dimension of the DOS: needs for further numerical check!
34 Summary & perspectives Perturbative RG on the Fibonacci chain : -multifractal properties of the energy spectrum -multifractal properties of the wave functions For a fixed energy the scaling exponent of the band and of the fractal dimensions of wave function depend only on RG path Symmetry between energy RG path for and site RG path (conumerotation) Relation between site average fractal dimensions : «LDOS=Wave function x Spectrum» Work in progress: relation with wave packet anomalous diffusion exponents Waves in 2D/3D tilings : multifractality/anomalous diffusion? Phason disorder
35
36 Local density of states atom site : molecule site :
37 Niu & Nori (1986) Kalugin,Kitaev,Levitov (1986) Atoms and molecules : Energy level Degeneracy Atoms : isolated sites (central) Molecules : isolated dimer Bonding/antibonding (lateral) We can build 3 the effective models : -chain of formed by atom sites : -chains formed by bonding states : -chains formed by antibonding states : is splitted into into into levels levels levels
38 Conumerotation and RG path of sites
39 Quasiperiodic chains: diffusion of wavepacket Wavefunctions Energy spectrum Period of approximant Fractal dimensions: Density of states: Wavefunctions (spatial) : Diffusion exponents : Local Density of states : For a given wavepacket : Guarneri, 1989 Ketzmerick et al, 1997 Initial site average wavepacket : Piéchon, 1996
40 Waves in quasiperiodic chains or tiling: Quasiperiodic order or geometry induces : -multifractal wavefunctions -anomalous diffusion of wavepacket Supplementary difficulty : non trivial energy spectrum (global density of states) - «topological gaps» : number (infinite/finite)? Closed/open? Labelling? -multifractal properties? Generic and tunable! Geometric phason disoder : 2D/3D : suppress multifractal properties leads to delocalization (diffusive regime) 1D : leads to localization?
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