RANDOM MATRICES and ANDERSON LOCALIZATION - the Milan ...

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

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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)
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Charge localization & Polaron formation in Na_xWO_3
(MIT with x) (PRL, 2006)
OPTICS!

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

Some basic old ideas
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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 non-
Herm. 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

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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)