Delocalization for Schrödinger operators with random Dirac masses
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1 Delocalization for Schrödinger operators with random Dirac masses CEMPI Scientific Day Lille, 10 February 2017
2 Disordered Systems E.g. box containing N nuclei Motion of electron in this system High temperature : classical Drude model (1900) Low temperature : quantum effects, interference : Anderson model (1958)
3 Random Schrödinger Operators Simplification : ignore interactions, one body Hamiltonian, Dirac masses We consider the random Schrödinger operator H Ω = + ω Ω δ(x ω), x R d, d = 2, 3 where Ω is a stochastic process on R d Rigorous realization of H Ω : theory of self-adjoint extensions
4 Random displacement model Consider H Ω = + ξ Z d δ(x ξ ω ξ ) where ω ξ are i. i. d. r. v. Probability density P(x) = P( x ), where P C c (R +), P(0) > 0, decreasing. Disorder parameter : δ = E P ( X )
5 Anderson localization Anderson 1958 : strong disorder localized eigenfunctions Definition Let E 2 > E 1 > 0. We say that H Ω is exponentially localized on I = [E 1, E 2 ], if H Ω has a.s. pure point spectrum on I and the eigenfunctions ψ λ satisfy the bound : ψλ (x)ψ λ (y) C ωe x y /L loc where the localization length L loc depends on the choice of I and ω Ω is a sample of the stochastic process. Mathematical results : Goldsheid-Molchanov-Pastur, Fröhlich-Spencer, Simon, Bourgain-Kenig, Klopp, A. Boutet de Monvel, Germinet,...
6 Anderson transition : physical motivation Localization for strong disorder Delocalization for weak disorder? Scaling Theory (Abrahams, Anderson, Licciardello et Ramakrishnan, 1979) : intensity ψ λ 2 dx d = 1 : localization d = 2 : critical case d 3 : transition expected localized Delocalized regime : absolutely continuous spectrum extended eigenstates delocalized Pictures : D. Wiersma et al.
7 Geometry of eigenfunctions Multifractal structure of eigenfunctions near transition point Delocalization : L loc as E E c Want to prove lower bound for L loc test on compact spaces Picture : A. Mirlin (Karlsruhe Institute of Technology)
8 Seba s billiard D rectangular billiard In 1990 Petr Seba studied the singular Hamiltonian H α = +αδ(x x 0 ), α R, x 0 int(d) with Dirichlet boundary conditions on D Consider the restricted Laplacian H 0 = C c (R x 0 ) which admits family of self-adjoint extensions ϕ, ϕ ( π, π)
9 The perturbed spectrum New eigenvalues are solutions to the equation (square torus R 2 /2πZ 2 ) { } 1 ξ 2 λ ξ 2 = C ξ 4 0 tan( ϕ ) ξ Z 2 A plot of the LHS of the spectral equation as a function of λ Eigenfunction G λ (x, x 0 ) = ( λ) 1 δ(x x 0 ) has convenient L 2 -representation G λ (x, x 0 ) = 1 4π 2 i x x e 0,ξ ξ 2 λ ξ Z 2 Singularities at the old eigenvalues
10 Random impurities on the torus Consider the torus T d L = R d /LZ d, d = 2, 3, L 1, and the operator H Ω,L = + N δ(x x j ) where x 1 is fixed and x 2,, x N are i. i. d. uniformly distributed random variables on T d L. Can realize the formal operator above as self-adjoint extensions of the restricted Laplacian j=1 H 0 = C c (T d L O), O = {x 1,, x N } Operator H 0 is positive symmetric with deficiency indices (N, N), extensions parametrized by U(N)
11 Spectrum and eigenfunctions Fix U = e iϕ Id N, ϕ ( π, π), denote corresponding extension by ϕ Green s function : G λ (x, y) = 1 δ(x y), λ λ / σ( ) We have the spectral equation det A ϕ λ = 0 where (A ϕ λ ) kl = G λ (x k, x l ) RG i (x k, x l ) tan( ϕ 2 )IG i(x k, x l )
12 Two types of eigenfunctions : old and new At most N new eigenvalues per Laplacian eigenspace which interlace with distinct Laplacian eigenvalues New eigenfunctions are given by superpositions of Green s functions : N ψ λ (x) = v k G λ (x, x j ), k=1 v ker A ϕ λ Note : the v k are functions of the random variables x 2,, x N T d L
13 Random displacement models Let B L = [ L, L] d and L 1. Consider H Ω,L = + δ(x ξ ω ξ ) ξ Z d B L with Dirichlet BCs on B L Fix small ɛ > 0. Let χ C (B 1 ), χ 0, with supp χ B(0, ɛ) and χ 1 = 1. Denote χ L = L d χ( /L). Let H Ω,L ψ L λ = λψ L λ, ψ L λ 2 = 1. We define the smoothed L 2 -density for any x B (1 ɛ)l. Ψ L λ(x) := ( B L χ L (x x) ψ L λ(x ) 2 dx ) 1/2
14 Definition Let F C 0 (R +) be strictly decreasing and L 1. We say that H Ω,L is F -localized on an interval I = [E 1, E 2 ] if we have for any a I and λ = min(σ(h Ω,L ) [a, E 2 ]) ( ) x, y B (1 ɛ)l, x y L : E Ψ L λ(x)ψ L λ(y) F( x y ). Set F(τ) = e τ/l loc for exponential localization, where L loc depends on the choice of interval I For a given spectral window I consider boxes of size L L loc (I)
15 Delocalization for random displacement models If δ 1, we can show that for E, L large enough H Ω,L fails to be exponentially localized In fact we can rule out any decay but a certain polynomial one Theorem (H.U. 2016) Let d = 2, 3 and F(τ) = τ α d +ɛ. There exist α d > 0, E 0 > 0 and L 0 = L 0 (E 0 ) such that for any interval I = [E 1, E 2 ], E 1 > E 0 and any L L 0, H Ω,L fails to be F -localized.
16 Outlook Delocalization for random Schrödinger operators with generic potentials Explicit construction of extended states Level statistics near transition point Multifractal wave function near Anderson transition. A. Mirlin (Karlsruhe)
17 Thank you for your attention!
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