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
Isaac Newton Institute for Mathematical Sciences Mathematics and Physics of Anderson localization: 50 Years After 14 July - 19 December 2008
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
THE ANDERSON MODEL d=1,2: p.p. spectrum, exponential localization d=3: a.c. to p.p. spectrum, metal-insulator transition
Phase diagram 3D Anderson model localized states extended states
UCF MIT dynamical localization QHE QUANTUM CHAOS: - sound - light - matter waves BEC
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) 1469-1564 MIT in 2D heterojunctions, Si-MOS? (PRL, 2008) Charge localization & Polaron formation in Na_xWO_3 (MIT with x) (PRL, 2006) OPTICS!
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)
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)
J. Phys. I France 4 (1994) 1469
THE HAMILTONIAN MATRIX Block Tridiagonal Matrix A block is Hamiltonian matrix of a section
THE TRANSFER MATRIX Eigenvalues of T(E) grow (decay) exponentially in the number of blocks. The rates are the exponents ξ_a(e)
Anderson D=1 tridiagonal random matrices Hatano and Nelson (1996) (Herbert-Jones-Thouless formula)
SPECTRAL DUALITY z^n is an eigenvalue of T(E) iff E is eigenvalue of H(z^n)
determinants of block tridiagonal matrices L.G.M, Linear Algebra and its Applications 429 (2008) 2221
Anderson model: duality Exponents describe decay lenghts of Anderson model. They are obtained from nonherm. energy spectrum via Jensen's identity
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)
the exponents ξ m=3, n=50, w=7
non-hermitian energy spectra (Anderson 2D) m=5 m=10 n=100, w=7, xi=1.5
Anderson 2D (m=3,n=8) (xi fixed, change phase) (change xi and phase)
Non-Hermitian tridiagonal complex matrices I (with G. Lacagnina)
Non-Hermitian tridiagonal complex matrices II
BAND RANDOM MATRICES complex, no symmetry
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?
determinants of tridiagonal matrices
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)