Luca G. MolinariPhysics Department

Universita' degli Studi di Milano

Abstract: The characteristic polynomials of block tridiagonal matrices and their transfer matrices are linked by an algebraic duality. I discuss applications to randomtridiagonal matrices, and the Anderson localization problem.

AN ALGEBRAIC DUALITY FOR DETERMINANTS

and applications

Pisa, may 2011

Some questions:1) Det of block tridiag matrix ?2) Localization of eigenvector and eigenvalue sensitivity to b.c. ?3) Localization of eigenvectors in BRM and Anderson model ?

block tridiagonal matrices

L.G.M, Linear Algebra and its Applications 429 (2008) 2221

THE BLOCK TRIDIAG MATRIX

In general: chain of n ”m-level atoms” with n.n. interactions

THE TRANSFER MATRIX

Eigenvalues of T(E) grow (decay) exponentially in the number of blocks.The rates are the exponents ξ_a(E)

SPECTRAL DUALITY

z^n is an eigenvalue of T(E) iff

E is eigenvalue of H(z^n)

A useful similarity

summary● Det of block tridiagonal matrices and

spectral duality● Hatano Nelson model, hole & halo in

complex tridiag. matrices● Jensen's theorem and spectrum of exps.● Counting exps.● Localization and Non Hermitian Anderson

matrices● Complex BRM

Deformed Anderson D=1 tridiagonal random matricesHatano and Nelson (1996)

(Herbert-Jones-Thouless formula)

Non-Hermitian tridiagonal complex matrices

(with G. Lacagnina)

J.Phys.A: Math.Theor. 42 (2009)

N=100, xi=(.3->.6), (.6->.9)

THE ANDERSON MODEL

● d=1,2: p.p. spectrum, exponential localization

● d=3: a.c. to p.p. spectrum, metal-insulator transition

● QUANTUM CHAOS: dynamical localization

● - sound - light - matter waves

QHE BEC

UCF MIT

Anderson Localization

● 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)

J. Phys. I France 4 (1994) 1469

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)

Phase diagram 3D Anderson model

extended states

localized states

Anderson model: duality

Exponents describe decay lenghts of Anderson model. They are obtained from non-Herm. energy spectrum via Jensen's identity

A formula for the exponents(a deterministic variant of Thouless formula)

m=3

no formula of Thouless type is known in D>1 (only for sum of exps, xi=0)

ξ

non-hermitian energy spectra(Anderson 2D)

m=5 m=10n=100, w=7, xi=1.5

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

(change xi and phase)

BAND RANDOM MATRICEScomplex, no symmetry

Conclusions & big problems

● Spectral duality + Jensen's identity --> exponents of single transfer matrix in terms of eigenvalues of Hamiltonan matrix with non-hermitian b.c.

● ? Distribution of exponents ?● ? Smallest exponent ?● ? Band Random Matrices ?