Delocalization for Schrödinger operators with random Dirac masses CEMPI Scientific Day Lille, 10 February 2017
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)
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
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 )
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,...
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.
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)
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)
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 + 1 2 ) 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
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)
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 )
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
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
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)
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) = 1 1 + 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.
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)
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