Anderson Localization on the Sierpinski Gasket

Anderson Localization on the Sierpinski Gasket G. Mograby 1 M. Zhang 2 1 Department of Physics Technical University of Berlin, Germany 2 Department of Mathematics Jacobs University, Germany 5th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals

Outline 1 Introduction Spectral Aspects 2 3

Outline Introduction Spectral Aspects 1 Introduction Spectral Aspects 2 3

Solid State Physics Spectral Aspects Study of crystals as 3D lattice, atoms on lattice points. Electron diffusion in a crystal induces electrical conduction. One electron approximation.

Periodic Potential Spectral Aspects Electron feels a potential energy from each atom on each lattice point. Periodic arrangement of atoms resulting in periodic potential.

Mathematical Formulation Spectral Aspects Hilbert space l 2 (Z d ) = {u : Z d C n u(n) 2 } with the inner product u, v = n u(n)v(n). Lattice compatible norm n := sup ν=1,...,d n ν. Cube of side length 2L + 1 centered at n 0 Λ L (n 0 ) := {n Z d : n n 0 L}.

Kinetic Energy Spectral Aspects Graph Laplacian (H 0 u)(n) = m n(u(m) u(n)). m n : m are the nearest neighbors of the vertex n. H 0 symmetric and bounded, as H 0 : l 2 (Z d ) l 2 (Z d ), with H 0 u l 2 4d u l 2.

Potential Energy Spectral Aspects Potential due to the atom at the origin of the lattice f : R d R, supp(f ) Λ L (0) for some L. f is called the single site potential. periodic lattice potential of the atom located at i Z d ist f i (x) = f (x i).

Potential Energy Spectral Aspects An electron at the point x feels a potential q f i (x) = q f (x i) due to the atom at the vertex i (q some coupling constant). Total potential V (x) = i Zd q f (x i).

Potential Energy Spectral Aspects Potential energy (alloy type) V ω (x) = i Z d q i (ω) f (x i), ω [0, 1] Zd q i : [0, 1] Zd R, q i (ω) = V max ω i.

Hamiltonian Spectral Aspects Summery Crystal as lattice, Z d. The atomic structure of the crystal is encoded in ω [0, 1] Zd Potential is determined by ω. In the case of crystals, ω is periodic. Electronic states are modeled by the Hamiltonian (H ω u)(n) = (u(m) u(n)) + q i (ω)f (n i)u(n). m n i Z d m n : m are the nearest neighbors of the vertex n.

Outline Introduction Spectral Aspects 1 Introduction Spectral Aspects 2 3

Spectral Aspects Spectrum and its physical Interpretation Hamiltonian with periodic potentials has an absolutely continuous spectrum with band structure. Spectrum of periodic Hamiltonian σ(h) = n=0[a n, b n ], a n < b n a n+1.

Spectral Aspects Spectrum and its physical Interpretation σ(h) = σ ac (H) Physical meaning of this fact? spectrum decomposition physical classification (long time behavior). RAGE theorem formalize the physical language explains the meaning of low or high mobility states.

Spectral Measure Spectral Aspects For a Borel set M R, the characteristic function of M is defined by { 1 if λ M, χ M (λ) = 0 otherwise. Spectral theorem spectral measure of the Hamiltonian H ψ, χ M (H)ψ = χ M (λ) dµ ψ (λ) = µ ψ (M)

Spectral Aspects Decomposition of the Hilbertspace We define the following subsets of H = l 2 (Z d ) Definition H pp = {φ H µ φ is pure point } H sc = {φ H µ φ is singular continuous } H ac = {φ H µ φ is absolutely continuous } These are closed subspaces and mutually orthogonal with H = H pp H sc H ac.

Spectral Aspects σ ac (H) = σ(h Hac ). Periodic Hamiltonian has purely absolute continuous spectrum restrict consideration to states ψ H ac Next theorem shows that these states are of high mobility.

Quantum Dynamics Spectral Aspects Theorem Let H be a self adjoint operator on H = l 2 (Z d ), take ψ H ac and let Λ denote a finite subset of Z d. Then ( e ith ψ(x) 2) = 0 lim t x Λ or equivalently lim t ( e ith ψ(x) 2) = ψ 2. x / Λ

Physical Interpretation Spectral Aspects Explaining the notion low or high mobility of a state through looking at its time evolution. Electron in such a state ψ H ac runs out to infinity as time evolves high mobility. Such electrons are the carrier of electrical conductivity.

Physical Interpretation Spectral Aspects Electron in ψ H pp stay inside a compact set with high probability for arbitrary long time (analog Theorem ). ψ H pp low mobility no contribution to electrical conductivity.

Outline Introduction 1 Introduction Spectral Aspects 2 3

Imperfections in the crystallization process. Mixture of various materials due to the presence of impurities. Periodicity of the potential is disordered.

How can we incorporate Disorder in our model? Impurity potential that encodes the disorder, and perturb the periodic hamiltonian. In the case of ideal crystals, ω [0, 1] Zd was used to encode the atomic structure. Remeber the figure of the one dimensional chain of NaCl ω = (... ω Na, ω Cl, ω Na,... ) is periodic.

Impurity potential Incorporate disorder by randomization of ω : Impurity Potential V ω (x) = i Z d q i (ω) f (x i), ω i [0, 1] random numbers, ω [0, 1] Zd, q i : [0, 1] Zd R, q i (ω) = V max ω i.

Probability space We have just statistical knowledge about the impurities model in probabilistic language. Let ω = (ω n ) Z d be a set of independent, identically distributed random variables with single site distribution p. The probability space is given by the product space Ω = [0, 1] Zd with the usual product σ-algebra. The probability measure on Ω is the product measure P = Z d p From now on we are not interested in properties of H ω for a single ω rather in properties that hold for P-almost all ω.

Anderson Model Random Hamiltonian Crystal as lattice, Z d. ω = (ω n ) Z d independent, identically distributed random variables. Random Hamiltonian (H ω u)(n) = (u(m) u(n)) + q i (ω)f (n i)u(n). m n i Z d m n : m are the nearest neighbors of the vertex n.

Anderson Localization One Definition of the Anderson Localization ist: Definition We say that the random Hamiltonian H ω exhibits spectral localization in an energy interval I if for P-almost all ω the spectrum of H ω in this interval is pure point.

The Laplacian on the periodic triangular lattice finitely ramified Sierpinski fractal field consists of absolutely continuous spectrum and pure poit spectrum. Strichartz and Teplyaev, Spectral Analysis on Infinite Sierpinski Fractafolds.

Anderson Model on the Sierpinski Gasket SG laplacian. Same Random Potential to use for different levels. One type of Impurity possible values {0, 1}. Dirichlet boundary conditions.

Random Potential of level 1

Random Potential of level 4, V max = 1000

First Eigenfunction, SG of level 6

10th Eigenfunction, SG of level 6

Observations V max big enough first couple of eigenfunctions localize. V max increases the number of localized eigenfunctions increases. V max increases support of the localized eigenfunction becomes smaller.

Green s function G(x, fixed point), SG of level 7

Theory of Filoche and Mayboroda Let Ω R n be a bounded open set. Let L = diva(x) + V (x) A real symmetric n n matrix with bounded measurable coefficients, A(x) = {a ij (x)} n i,j=1, x Ω, a ij L (Ω) n a ij (x)ξ i ξ j c ξ 2, ξ R n i,j=1 for some c > 0, and a ij = a ji, i, j = 1,..., n and V L (Ω) nonnegative.

Consider the Dirichlet Problem: f H 1 (Ω) Lu = f, u H 1 (Ω). Green function L x G(x, y) = δ y (x), x, yω, G(., y) H 1 (Ω), y Ω, R n L x G(x, y)v(x)dx = v(y), for y Ω, v H 1 (Ω).

Filoche and Mayboroda did the following Observation: Let λ be an eigenvalue of L, φ H 1 (Ω) corresponding eigenfunction. φ(x) = δ x (y)φ(y)dy Ω = L y G(x, y)φ(y)dy Ω = G(x, y)l y φ(y)dy Ω = λ G(x, y)φ(y)dy Ω

Green function positive, λ 0, eigenfunctions bounded. φ(x) λ φ L (Ω) G(x, y)dy define φ(x) φ L (Ω) Ω λ G(x, y)dy Ω u(x) = G(x, y)dy. Ω

φ(x) φ L (Ω) λu(x) This inequality provides a map of localization in the following sense.

Plotting of the function u, V max = 10000

First cut of u, compared with the first eigenfunction

second cut of u, compared with the second eigenfunction

3rd cut of u, compared with the 3rd eigenfunction

3th cut of u, compared with the 4th eigenfunction

φ(x) φ L (Ω) λu(x) u controls the eigenfunctions as long as λu(x) 1. For higher eigenvalues weaker control is achieved.

References Werner Kirsch, An Invitation to Random Schrödinger Operators. Werner Kirsch, Bandstructure for Mathematician. Peter Stollmann, Caught by Disorder: Bound States in Random Media. Marcel Filoche and Svitlana Mayboroda, The landscape of Anderson localization in a disordered medium. Strichartz and Teplyaev, Spectral Analysis on Infinite Sierpinski Fractafolds.

Figures Festkörperphysik, Gross and Marx. http://wsm.wsu.edu/s/index.php?id=261 http://greenevolution.ru/enc/wiki/epitaksiya/