RANDOM MATRICES and ANDERSON LOCALIZATION

Transcription

1 RANDOM MATRICES and ANDERSON LOCALIZATION Luca G. Molinari Physics Department Universita' degli Studi di Milano Abstract: a particle in a lattice with random potential is subject to Anderson localization, which affects low T transport properties of disordered materials. After 50 years the Anderson model continues to be an active area of research. I present some analytic properties of block tridiagonal matrices, for the study of localization in d>1 Milan, april 2, 2009

2 Isaac Newton Institute for Mathematical Sciences Mathematics and Physics of Anderson localization: 50 Years After 14 July - 19 December 2008

3 summary The Anderson model Determinants of block tridiagonal matrices and spectral duality Jensen's theorem and the spectrum of exponents. Energy spectra of non Hermitian Anderson matrices The Argument Principle, hole & halo in complex spectra of tridiagonal matrices

4 THE ANDERSON MODEL d=1,2: p.p. spectrum, exponential localization d=3: a.c. to p.p. spectrum, metal-insulator transition

5 Phase diagram 3D Anderson model localized states extended states

6 UCF MIT dynamical localization QHE QUANTUM CHAOS: - sound - light - matter waves BEC

7 Low T conductivity of amorphous semicond. σ exp [-(c/t)^⅟4] (Mott, 1979: phononassisted hopping between localized states) Weakly disordered metal films 1/σ -log T Random alloys Ref: B.Kramer and A.MacKinnon, Localization: theory and experiment, Rep. Progr. Phys. 56 (1993) MIT in 2D heterojunctions, Si-MOS? (PRL, 2008) Charge localization & Polaron formation in Na_xWO_3 (MIT with x) (PRL, 2006) OPTICS!

8 THEORY Theorems (Spencer, Ishii, Pastur,...) Kubo formula weak disorder (Stone, Altshuler,...) Energy levels and b.c. (Thouless, Hatano & Nelson, level curvatures,... ) Transfer matrix and Lyap spectrum scaling (Kramer&MacKinnon), DMPK eq., conductance &scattering (Buttiker and Landauer),... Supersymmetry, BRM (Efetov, Fyodorov & Mirlin)

9 Some basic old ideas Adimensional conductance g(l)=h/e² L^(d-2)σ Scattering ( lead-sample-lead) g ~ tr tt* (t=transm. matrix) DMPK Periodic b.c.: Thouless conductance g ~ d²e/dφ² /Δ (Bloch phase) One parameter scaling d(log g) / d(log L) = β(g)

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11 J. Phys. I France 4 (1994) 1469

12 THE HAMILTONIAN MATRIX Block Tridiagonal Matrix A block is Hamiltonian matrix of a section

13 THE TRANSFER MATRIX Eigenvalues of T(E) grow (decay) exponentially in the number of blocks. The rates are the exponents ξ_a(e)

14 Anderson D=1 tridiagonal random matrices Hatano and Nelson (1996) (Herbert-Jones-Thouless formula)

15 SPECTRAL DUALITY z^n is an eigenvalue of T(E) iff E is eigenvalue of H(z^n)

16 determinants of block tridiagonal matrices L.G.M, Linear Algebra and its Applications 429 (2008) 2221

17 Anderson model: duality Exponents describe decay lenghts of Anderson model. They are obtained from nonherm. energy spectrum via Jensen's identity

18 A formula for the exponents (a deterministic variant of Thouless formula) m=3 ξ no formula of Thouless type is known in D>1 (only for sum of exps, xi=0)

19 the exponents ξ m=3, n=50, w=7

20 non-hermitian energy spectra (Anderson 2D) m=5 m=10 n=100, w=7, xi=1.5

21 Anderson 2D (m=3,n=8) (xi fixed, change phase) (change xi and phase)

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23 Non-Hermitian tridiagonal complex matrices I (with G. Lacagnina)

24 Non-Hermitian tridiagonal complex matrices II

25 BAND RANDOM MATRICES complex, no symmetry

26 conclusions Spectral duality + Jensen's identity --> exponents of single transfer matrix in terms of eigenvalues of Hamiltonan matrix with non-hermitian b.c. Spectral duality + Argument principle --> holes in spectrum of Hamiltonian matrix with non Hermitian b.c. Theory can be extended to T*T (Lyapunov exponents)? Metal insulator transition (D=3)?? Band Random Matrices?

27 determinants of tridiagonal matrices

28 A formula for the exponents (a deterministic variant of Thouless formula) m=3 ξ no formula of Thouless type is known in D>1 (only for sum of exps, xi=0)