an algebraic duality for determinants and applicationsphd.fisica.unimi.it/assets/molinari.pdf ·...
TRANSCRIPT
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 ?