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2162-5
Advanced Workshop on Anderson Localization, Nonlinearity and
Turbulence: a Cross-Fertilization
23 August - 3 September, 2010
INTRODUCTORY
Anderson Localization - Introduction
Boris ALTSHULER
Columbia University, Dept. of Physics, New York
NY
U.S.A.
Anderson localization - introduction
Boris Altshuler
Physics Department, Columbia University
Advanced Workshop on Anderson Localization,
Nonlinearity and Turbulence: a Cross-Fertilization
August 23 – September 03, 2010
One--particle Localization
>50 years of Anderson Localization
ƒOne quantum
particle
ƒRandom potential
(e.g., impurities)
Elastic scattering
q.p.
Einstein (1905):
r2
Dt
Random walk
diffusion constant
always diffusion
as long as the system has no memory
Anderson(1958):
It might be that
For quantum
particles
r 2 t
o const
of
D
not always!
Quantum interference
memory
0
Einstein Relation (1905)
V
e DQ
2
Conductivity
dn
Q{
dP
Density of states
Diffusion Constant
U
I
V
G
§I·
¨ ¸ ;
© V ¹V 0
V
L
G
A
wU
D’ 2 U
wt
0
t
t!
x
Einstein Relation (1905)
V
e DQ
2
Conductivity
dn
Q{
dP
Density of states
Diffusion Constant
No diffusion – no conductivity
Localized states – insulator
Extended states - metal
Metal – insulator transition
Localization of single-electron wave-functions:
z
Disorder
extended
I
V
extended
Conductance
localized
localized
Lx Ly
­
½
extended
° V L
°
z
°
°
®
¾
°v exp § Lz · localized
°
¨
¸
°
°
9
© loc ¹
¯
¿
Spin Diffusion
Experiment
Feher, G., Phys. Rev. 114, 1219 (1959); Feher, G. & Gere, E. A., Phys. Rev. 114, 1245 (1959).
Light
Wiersma, D.S., Bartolini, P., Lagendijk, A. & Righini R. “Localization of light in a disordered
medium”, Nature 390, 671-673 (1997).
Scheffold, F., Lenke, R., Tweer, R. & Maret, G. “Localization or classical diffusion of light”,
Nature 398,206-270 (1999).
Schwartz, T., Bartal, G., Fishman, S. & Segev, M. “Transport and Anderson localization in
disordered two dimensional photonic lattices”. Nature 446, 52-55 (2007).
C.M. Aegerter, M.Störzer, S.Fiebig, W. Bührer, and G. Maret : JOSA A, 24, #10, A23, (2007)
Microwave
Dalichaouch, R., Armstrong, J.P., Schultz, S.,Platzman, P.M. & McCall, S.L. “Microwave
localization by 2-dimensional random scattering”. Nature 354, 53, (1991).
Chabanov, A.A., Stoytchev, M. & Genack, A.Z. Statistical signatures of photon localization.
Nature 404, 850, (2000).
Pradhan, P., Sridar, S, “Correlations due to localization in quantum eigenfunctions od
disordered microwave cavities”, PRL 85, (2000)
Sound
Weaver, R.L. Anderson localization of ultrasound. Wave Motion 12, 129-142 (1990).
f = 3.04 GHz
Localized State
Anderson Insulator
f = 7.33 GHz
Extended State
Anderson Metal
Localization of cold atoms
Billy et al. “Direct observation of Anderson localization
of matter waves in a controlled disorder”. Nature 453,
891- 894 (2008).
87Rb
Roati et al. “Anderson localization of a non-interacting
Bose-Einstein condensate“. Nature 453, 895-898 (2008).
• Lattice - tight binding model
Anderson
Model
• Onsite energies
j
Iij
-W < Hi <W
uniformly distributed
i
Hi - random
• Hopping matrix elements Iij
I i and j are nearest
{
Iij =
0
neighbors
otherwise
Anderson Transition
I < Ic
Insulator
All eigenstates are localized
Localization length [
I > Ic
Metal
There appear states extended
all over the whole system
Q:
Why arbitrary
weak hopping I is
not sufficient for
the existence of
the diffusion
?
j
i
Iij
Einstein (1905): Marcovian (no memory)
process J diffusion
!
Quantum mechanics is not marcovian
There is memory in quantum propagation
Why
?
H2
I
H1
Hamiltonian
Hˆ
§ H1 I ·
¸¸
¨¨
© I H2 ¹
E2 E1
diagonalize
§ E1
¨¨
©0
Hˆ
H 2 H1 2
I
2
0·
¸¸
E2 ¹
Hˆ
§ H1 I ·
¸¸
¨¨
© I H2 ¹
E2 E1
H 2 H1 2
H 2 H 1 H 2 H 1 !! I
I |
I
H 2 H 1 I
2
What about the eigenfunctions ?
I1 , H1 ; I2 , H 2

H 2 H 1 !! I
\ 1, 2
\ 1 , E1 ; \ 2 , E2
H 2 H 1 I
§ I ·
¸¸M 2,1
M1, 2 O¨¨
\
|
M
r
M
H
H
1
,
2
1
,
2
2 ,1
2
1
¹
©
Off-resonance
Eigenfunctions are
close to the original onsite wave functions
Resonance
In both eigenstates the
probability is equally
shared between the sites
Anderson insulator
Few isolated resonances
Anderson metal
There are many resonances
and they overlap
Typically
each
site
is
in
Transition:
resonance with some other one
Anderson Transition
I > Ic
I < Ic
localized and extended
never coexist!
all states are
localized
extended
DoS
- mobility edges (one particle)
DoS
Condition for
Localization:
I<
energy mismatch
# of n.neighbors
energy
mismatch
H i H j typ W
# of nearest
neighbors
2d
A bit more precise:
Logarithm is due to the resonances, which are not nearest neighbors
Condition for Localization:
Q:Is it correct ?
A1:Is exact on the Cayley tree
Ic
W
,
K ln K
K
is the
branching
number
K
2
Anderson Model on a Cayley tree
Condition for Localization:
Q:Is it correct ?
A1:Is exact on the Cayley tree
Ic
W
,
K ln K
K
is the
branching
number
K
a good approximation at high dimensions.
Is
Is qualitatively correct for d t 3
2
Anderson’s recipe:
1. take descrete spectrum EP of H0
2. Add an infinitesimal Im part iK to EP
insulator
3. Evaluate Im6 P
1
2
4
1) N o f
limits 2) K o0
4. take limit K o 0 but only after N o f
5. “What we really need to know is the
probability distribution of Im6, not
its average…”
!
metal
Probability Distribution of *=Im 6
K is an infinitesimal width (Im
metal
insulator
Look for:
V
part of the self-energy due to
a coupling with a bath) of
one-electron eigenstates
Condition for Localization:
Q:Is it correct ?
A1:Is exact on the Cayley tree
Ic
W
,
K ln K
K
is the
branching
number
K
2
a good approximation at high dimensions.
Is
Is qualitatively correct for d t 3
A2:
For low dimensions – NO. I c
f for d 1, 2
All states are localized. Reason – loop trajectories
1D Localization
Gertsenshtein & Vasil'ev,
Exactly solved:
1959
all states are localized
Conjectured:
Mott & Twose, 1961
...
Condition for Localization:
Q:Is it correct ?
For low dimensions – NO. I
f for d 1, 2
A2:All states are localized. Reason – loop trajectories
c
M
& &
³ pdr
Phase accumulated
when traveling
along the loop
M1 M 2
O
The particle
can go around
the loop in
two directions
Memory!
Einstein: there is no diffusion at too short
scales – there is memory, i.e.,
the process is not marcovian.
Due to the localization effects diffusion
description fails at large scales.
Quantum interference
Memory
M1 M 2
O
Large scales are important
diffusion
constant depends on the system size
Einstein relation for
the conductivity V
V
Conductance
G
d 2
VL
dn
Q{
dP
e DQ
2
2
G
g L
for a cubic sample
of the size L
g L
hD L2
d
1QL
e
d Dh
QL 2
h L
Thouless energy
mean level spacing
Dimensionless
Thouless
conductance
Scaling theory of Localization
(Abrahams, Anderson, Licciardello and Ramakrishnan
1979)
2
Dimensionless Thouless
g = Gh/e
g = ET / G
conductance
L = 2L = 4L = 8L ....
without quantum corrections
ET vL
-2
ET ET ET ET
G1 G1 G1 G1
g
g
g
g
G1vL
-d
d log g
E g
d log L
d log g
E g
d log L
Universal, i.e., material
independent
E – function is
Limits:
g !! 1 g v Ld 2
g 1 g v e L [
E g
But
It depends on the global
symmetries, e.g., it is
different with and
without T-invariance (in
orthogonal and unitary
ensembles)
§1·
d 2 O ¨ ¸
©g¹
E g | log g 0
!0 d!2
?? d 2
0 d2
1
E(g)
E - function
d log g
d log L
unstable
fixed point
3D
gc | 1
-1
E g g
2D
1D
Metal – insulator transition in 3D
All states are localized for d=1,2
RG approach
Effective Field Theory of Localization –
Nonlinear
V
- model
1
E(g)
E - function
d log g
d log L
unstable
fixed point
3D
gc | 1
-1
cd
E g d 2
g
cd ? r ?
E g g
2D
1D
g L V cl Ld 2 cd
d 2
dz2
§ L·
c2 log ¨ ¸ d
©l¹
2
vF dt c
³W Dtcd 2
t
P t d 1
Gg
v F
L
|
log
g
D
DW
v F
2
1
}
SQ
g Q D!
Gg
E (g)
Gg
g
| P tmax 2 v F
L
log
D
l
L
log
S
l
2
2
Sg
Universal !!!
For d=1,2 all states are localized.
M
& &
³ pdr
Phase accumulated
when traveling
along the loop
M1 M 2
O
The particle
can go around
the loop in
two directions
Memory!
Weak Localization:
The localization length 9 can be large
Inelastic processes lead to dephasing, which is
characterized by the dephasing length L
M
If 9 !! LM , then only small corrections to a
conventional metallic behavior
R.A. Chentsov “On the variation of electrical resistivity of tellurium in
magnetic field at low temperatures”, Zh. Exp. Theor. Fiz. v.18, 375-385, (1948).
Magnetoresistance
)
O
No magnetic field
M M
O
With magnetic field H
MM S))
Length Scales
1/2
Magnetic length
LH = (hc/eH)
1/2
Dephasing length
LM = (D WM)
G g H § LH
fd ¨
¨ LM
©
·
¸¸
¹
Universal
functions
Magnetoresistance measurements allow to study inelastic
collisions of electrons with phonons and other electrons
Weak Localization
Negative
Magnetoresistance
Chentsov
(1949)
Aharonov-Bohm effect
Theory
Experiment
B.A., Aronov & Spivak (1981)
Sharvin & Sharvin (1981)
Temperature dependence of the conductivity
one-electron picture
Chemical
potential
DoS
DoS
V T o 0 ! 0
V T v e
E c H F
T
DoS
V T 0 T
Temperature dependence of the conductivity
one-electron picture
Assume that all the
states
are localized;
e.g. d = 1,2
DoS
V T 0 T
Inelastic processes
transitions between localized states
E
energy
mismatch
D
T
0 Ÿ V
0
(any
mechanism)
Phonon-assisted hopping
!Z
E
D
!Z
HD H E
V T
0 0
Variable Range
Hopping
N.F. Mott (1968)
Mechanism-dependent
prefactor
Optimized
phase volume
Any bath with a continuous spectrum of delocalized
excitations down to Z= 0 will give the same exponential
Spectral statistics and
Localization
Spectral
statistics
RANDOM MATRIX THEORY
ensemble of Hermitian matrices
with random matrix element
N uN
N o’
ED
- spectrum (set of eigenvalues)
G 1 { ED 1 ED
- mean level spacing,
determines the density of states
......
- ensemble averaging
s{
ED 1 ED
P s G1
- spacing between nearest
neighbors
- distribution function of nearest
neighbors spacing between
Spectral Rigidity P s 0 0
Level repulsion
P s 1 v s
E
E 1,2,4
P(s)
Wigner-Dyson; GOE
Poisson
Gaussian
Orthogonal
Ensemble
Orthogonal
E P( s ) v s E
Unitary
E Simplectic
E Poisson – completely
uncorrelated
levels
• Lattice - tight binding model
Anderson
Model
• Onsite energies
j
i
Hi - random
• Hopping matrix elements Iij
Iij
-W < Hi <W
uniformly distributed
Is there much in common between Random Matrices
and Hamiltonians with random potential ?
Q:
What are the spectral statistics
of a finite size Anderson model
?
Anderson Transition
Strong disorder
Weak disorder
I < Ic
I > Ic
Insulator
Metal
All eigenstates are localized
Localization length [
There appear states extended
all over the whole system
The eigenstates, which are
localized at different places
will not repel each other
Any two extended
eigenstates repel each other
Poisson spectral statistics
Wigner – Dyson spectral statistics
Anderson Localization and
Spectral Statistics
0
Ic
I
Localized states
Insulator
Extended states
Metal
Poisson spectral
statistics
Wigner-Dyson
spectral statistics
Extended Level repulsion, anticrossings,
Wigner-Dyson spectral statistics
states:
Localized Poisson spectral statistics
states:
Invariant (basis independent) definition
In general:
Localization in the space of quantum numbers.
KAM tori
localized states.
Glossary
Classical
Quantum
Integrable
Integrable
H0
&
H0 I
Hˆ 0
¦P EP P
P,
KAM
Localized
Ergodic – distributed all
over the energy shell
Chaotic
Extended ?
P
&
I
Many--Body
Localization
BA, Gefen, Kamenev & Levitov, 1997
Basko, Aleiner & BA, 2005. . .
Example: Random Ising model in the perpendicular field
Will not discuss today in detail
Hˆ
N
N
i 1
i 1
z
z z
x
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
B
V
J
V
V
I
V
{
H
I
V
¦ i i ¦ ij i j ¦ i 0 ¦ i
i 1
&
N
iz j
Random Ising model
in a parallel field
V i - Pauli matrices, V
z
i
i 1, 2,..., N ; N !! 1
Perpendicular
field
1
r
2
V
z
i
Without perpendicular field all
commute with the Hamiltonian, i.e.
they are integrals of motion
Hˆ
N
N
i 1
i 1
z
z z
x
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
B
V
J
V
V
I
V
{
H
I
V
¦ i i ¦ ij i j ¦ i 0 ¦ i
i 1
iz j
Random Ising model
in a parallel field
&
N
V i - Pauli matrices
i 1, 2,..., N ; N !! 1
Perpendicular
field
Withoutz perpendicular field
all V i commute with the
Hamiltonian, i.e. they are
integrals of motion
Anderson Model on ^V z ` determines
i
N-dimensional cube
x
ˆ
ˆ
ˆ
V
V
V
H ^V `
0
i
onsite energy
hoping between
nearest neighbors
a site
Hˆ
N
N
N
i 1
i 1
z
z z
x
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
{
B
V
J
V
V
I
V
H
I
V
¦ i i ¦ ij i j ¦ i 0 ¦ i
i 1
iz j
Anderson Model on N-dimensional cube
Usually:
Here:
# of dimensions
d o const
# of dimensions
system linear size
Lof
system linear size
d
N of
L 1
6-dimensional cube
9-dimensional cube
Hˆ
N
N
N
i 1
i 1
z
z z
x
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
{
B
V
J
V
V
I
V
H
I
V
¦ i i ¦ ij i j ¦ i 0 ¦ i
i 1
iz j
Anderson Model on N-dimensional cube
Localization: •No relaxation
•No equipartition
•No temperature
•No thermodynamics
Glass ??