2162-5 advanced workshop on anderson localization,...
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2162-5
Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization
Boris ALTSHULER
23 August - 3 September, 2010
Columbia University, Dept. of Physics, New York NY
U.S.A.
INTRODUCTORY Anderson Localization - Introduction
Anderson localization - introduction
Boris AltshulerPhysics Department, Columbia University
Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization
August 23 – September 03, 2010
OOne--particle Localization
>50 years of Anderson Localization
q.p.
�One quantum particle
�Random potential (e.g., impurities) Elastic scattering
Einstein (1905):Random walk
always diffusion
Dtr �2
diffusion constant
Anderson(1958):For quantum
particles
not always!
constr t���� ��2
It might be that
0�D
as long as the system has no memory
Quantum interference memory
2 0Dt� ��� �
�
�
x
t
t�I V
2 dne Dd
� �
� �
Density of statesConductivity
Einstein Relation (1905)
Diffusion Constant
ALG
VIG
V
�
���
����
�
�
;0
2 dne Dd
� �
� �
Density of statesConductivity
Einstein Relation (1905)
Diffusion Constant
No diffusion – no conductivity
Metal – insulator transition
Localized states – insulatorExtended states - metal
Localization of single-electron wave-functions:
extended
localized
Disorder
IV
Conductanceextended
localizedexp
loc
x y
z
z
L LL
L
�
�
� �� �� �� � �
� �� �� � �� �
���� �
extended
localized
z
Spin DiffusionFeher, G., Phys. Rev. 114, 1219 (1959); Feher, G. & Gere, E. A., Phys. Rev. 114, 1245 (1959).LightWiersma, 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)MicrowaveDalichaouch, 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)
SoundWeaver, R.L. Anderson localization of ultrasound. Wave Motion 12, 129-142 (1990).
Experiment
Localized StateAnderson Insulator
Extended StateAnderson Metal
f = 3.04 GHz f = 7.33 GHz
Billy et al. “Direct observation of Anderson localizationof matter waves in a controlled disorder”. Nature 453,891- 894 (2008).
Localization of cold atoms
87Rb
Roati et al. “Anderson localization of a non-interacting Bose-Einstein condensate“. Nature 453, 895-898 (2008).
Anderson Model
• Lattice - tight binding model
• Onsite energies �i - random
• Hopping matrix elements Iijj iIij
Iij ={-W < �i <Wuniformly distributed
I < Ic I > IcInsulator All eigenstates are localized
Localization length Metal
There appear states extendedall over the whole system
Anderson Transition
I i and j are nearest neighbors
0 otherwise
Q:Why arbitrary weak hopping I is not sufficient for the existence of the diffusion
?Einstein (1905): Marcovian (no memory)
process �� diffusion
j iIij
Quantum mechanics is not marcovian There is memory in quantum propagation!Why?
1�2�
I
���
����
��
2
1ˆ�
�I
IH
Hamiltonian
diagonalize ���
����
��
2
1
00ˆE
EH
! " 221212 IEE #��� ��
���
����
��
2
1ˆ�
�I
IH ! "
III
IEE�����
$#���12
1212221212 ��
������
What about the eigenfunctions ?
1 1 2 2 1 1 2 2, ; , , ; ,E E% � % � & &'
1,212
2,12,1
12
(��
(&
��
���
����
��
#�
���
IO
I
1,22,12,1
12
((&
��
)$
� I
Off-resonanceEigenfunctions are
close to the original on-site wave functions
ResonanceIn both eigenstates the probability is equally
shared between the sites
Anderson insulatorFew isolated resonances
Anderson metalThere are many resonances
and they overlap
Transition: Typically each site is in resonance with some other one
localized and extended never coexist!
DoS DoS
all states arelocalized
I < IcI > Ic
Anderson Transition
- mobility edges (one particle)
extended
Condition for Localization:
i j typW� �� � �
energy mismatch# of n.neighborsI<
energy mismatch
2d�# of nearest neighbors
A bit more precise:
Logarithm is due to the resonances, which are not nearest neighbors
Condition for Localization:
Is it correct?Q:A1:Is exact on the Cayley tree
,lncWI KK K
�is the branching number
2K �
Anderson Model on a Cayley tree
Condition for Localization:
Is it correct?Q:A1:Is exact on the Cayley tree
,lncWI KK K
�is the branching number
2K �
����Is a good approximation at high dimensions.Is qualitatively correct for 3d *
2. Add an infinitesimal Im part i++ to E�
1 2
4 1)2) 0N+
���limits
insulator
metal
1. take descrete spectrum E� of H0
3. Evaluate Im, �
Anderson’s recipe:
4. take limit but only after ��N5. “What we really need to know is the
probability distribution of Im,, not its average…” !
0�+
Probability Distribution of --=Im ,,
metal
insulator
Look for:
V
+ is an infinitesimal width (Impart of the self-energy due to a coupling with a bath) of one-electron eigenstates
Condition for Localization:
Is it correct?Q:A1:Is exact on the Cayley tree
,lncWI KK K
�is the branching number
2K �
A2:For low dimensions – NO. for All states are localized. Reason – loop trajectories
cI � � 1,2d �
����Is a good approximation at high dimensions.Is qualitatively correct for 3d *
Gertsenshtein & Vasil'ev, 1959
1D Localization
Exactly solved: all states are localized
Mott & Twose, 1961Conjectured:. . .
Condition for Localization:
Is it correct?Q:A2:For low dimensions – NO. for
All states are localized. Reason – loop trajectories cI � � 1,2d �
OPhase accumulated when traveling along the loop
The particle can go around the loop in two directions
Memory!
.� rdp ��( 21( (�
Due to the localization effects diffusion description fails at large scales.Quantum interference Memory
Einstein: there is no diffusion at too shortscales – there is memory, i.e., the process is not marcovian.
Large scales are important diffusion constant depends on the system size
O
21( (�
Conductance2dG L� ���
for a cubic sample of the size L
! "! "
2
2d
g L
e DhG Lh L
�����
�
Einstein relation for the conductivity �
2 dne Dd
� �
� �
! "2
1 dhD Lg LL
�Thouless energy
mean level spacing�
DimensionlessThouless conductance
Scaling theory of Localization(Abrahams, Anderson, Licciardello and Ramakrishnan
1979)
L = 2L = 4L = 8L ....
ET�/L-2 01/�/L-d/
without quantum corrections
ET ET ET ET
01 /01 /01 /01
g g g g
d log g! "d log L! "�1 g! "
g = Gh/e2g = ET / 02Dimensionless Thouless
conductance
d log g! "d log L! "�1 g! "
1 – function is
Universal, i.e., material independentButIt depends on the global symmetries, e.g., it is different with and without T-invariance (in orthogonal and unitary ensembles)Limits:
! " ! "2 11 2dg g L g d Og
1� � ��� � � � # � �
� �
! "1 log 0Lg g e g g 1� � $
0 2?? 2
0 2
ddd
� ��
1 - function ! "gLdgd 1�
loglog
1(g)
g
3D
2D
1D-1
1
1$cg
unstablefixed point
Metal – insulator transition in 3DAll states are localized for d=1,2
RG approach
Effective Field Theory of Localization –Nonlinear �� - model
1 - function ! "gLdgd 1�
loglog
1(g)
g
3D
2D
1D-1
1
1$cg
unstablefixed point
! " 2
2
22
log 2
d
dcl
c ddg L L
Lc dl
� �
3�
� �� � �� �� �
! " 2
? ?
d
d
cg d
cg
1 � � #
� )
!! "! "
12
td F
dv dtP tDt4
� 5�
5.� ! "maxg P tg0
$
2 2log logF Fg v L v Lg D D D l0
4� �
�� �$
1Fv 6 ��
g D � �
2 log Lgl
06
��} 2( )gg
16
� � Universal !!!
O
For d=1,2 all states are localized.
Phase accumulated when traveling along the loop
The particle can go around the loop in two directions
Memory!
.� rdp ��( 21( (�
Weak Localization:The localization length can be large
Inelastic processes lead to dephasing, which is characterized by the dephasing length
If , then only small corrections to a conventional metallic behavior
�
L(L(� ��
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).
7
Magnetoresistance
NNo magnetic field ///(2/�/(8
With magnetic field H/////(2�/(8�/8986/7:7;
O O
Length Scales
Magnetoresistance measurements allow to study inelasticcollisions of electrons with phonons and other electrons
Magnetic length LH = (hc/eH)1/2
Dephasing length L( = (D 4()1/2
! " HdLg H fL(
0� �
� � �� �� �
Universalfunctions
Negative Magnetoresistance
Weak Localization
Aharonov-Bohm effectTheory B.A., Aronov & Spivak (1981)
Experiment Sharvin & Sharvin (1981)
Chentsov (1949)
Chemicalpotential
Temperature dependence of the conductivity one-electron picture
DoS DoSDoS
! " 00 ��T� ! " TE Fc
eT�
��
�� ! " TT <� 0�
Assume that all the states
are localized;e.g. d = 1,2 DoS
! " TT <� 0�
Temperature dependence of the conductivity one-electron picture
Inelastic processestransitions between localized states
=
1 energymismatch
(any mechanism)00 �>� �T
Phonon-assisted hopping
Any bath with a continuous spectrum of delocalized excitations down to ?/= 0 will give the same exponential
=
1
Variable Range HoppingN.F. Mott (1968)
Optimizedphase volume
Mechanism-dependentprefactor
1= ��? ���?�
! " 00 ��T�
SSpectral statistics and LLocalization
E== - spectrum (set of eigenvalues)- mean level spacing, determines the density of states- ensemble averaging
- spacing between nearest neighbors
- distribution function of nearest neighbors spacing between
Spectral Rigidity
Level repulsion
! "
! " 4211
00
,,��
��
11ssP
sP
! "sP1
1
0== EEs �
� #
==0 EE �� #11
......
RANDOM MATRIX THEORY
N @/N N �/�ensemble of Hermitian matrices with random matrix element
Spectral statistics
Orthogonal 11�2
Poisson – completely uncorrelated levels
Wigner-Dyson; GOEPoisson
GaussianOrthogonalEnsemble
Unitary1�8
Simplectic1�A
P(s)
( )P s s1�
Anderson Model
• Lattice - tight binding model
• Onsite energies �i - random
• Hopping matrix elements Iijj iIij
-W < �i <Wuniformly distributed
Q: What are the spectral statistics of a finite size Anderson model ?
Is there much in common between Random Matrices and Hamiltonians with random potential ?
I < Ic I > IcInsulator
All eigenstates are localizedLocalization length
MetalThere appear states extended
all over the whole system
Anderson Transition
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
Strong disorder Weak disorder
I0Localized states
InsulatorExtended states
MetalPoisson spectral
statisticsWigner-Dyson
spectral statistics
Anderson Localization andSpectral Statistics
Ic
Extended states:
Level repulsion, anticrossings, Wigner-Dyson spectral statistics
Localized states: Poisson spectral statistics
Invariant (basis independent) definition
In general: Localization in the space of quantum numbers.KAM tori localized states.
GlossaryClassical Quantum
Integrable Integrable
KAM LocalizedErgodic – distributed all over the energy shellChaotic
Extended ?
! "IHH�
00 �IEH�
��B ����
� ,ˆ0
MMany--BBody LLocalization
BA, Gefen, Kamenev & Levitov, 1997Basko, Aleiner & BA, 2005. . .
01 1 1
ˆ ˆˆ ˆ ˆ ˆ ˆN N N
z z z x xi i ij i j i i
i i j i iH B J I H I� � � � �
� 3 � �
� # # � #B B B B
1,2,..., ; 1i
i N N�� ��
�
Perpendicular fieldRandom Ising model
in a parallel field
- Pauli matrices,
Example: Random Ising model in the perpendicular field
12
zi� � )
Will not discuss today in detail
Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion
zi�
01 1 1
ˆ ˆˆ ˆ ˆ ˆ ˆN N N
z z z x xi i ij i j i i
i i j i iH B J I H I� � � � �
� 3 � �
� # # � #B B B B
C D! "0 iH �onsite energy
ˆ ˆ ˆx� � �# �� #hoping between nearest neighbors
Anderson Model on N-dimensional cube
1,2,..., ; 1i
i N N�� ��
�
Perpendicular fieldRandom Ising model
in a parallel field
- Pauli matrices
C Dzi� determines a site
Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion
zi�
Anderson Model on N-dimensional cubeUsually:# of dimensions
system linear size
d const�
L��
Here:# of dimensions
system linear size
d N� ��
1L �
01 1 1
ˆ ˆˆ ˆ ˆ ˆ ˆN N N
z z z x xi i ij i j i i
i i j i iH B J I H I� � � � �
� 3 � �
� # # � #B B B B
6-dimensional cube 9-dimensional cube
Anderson Model on N-dimensional cube
01 1 1
ˆ ˆˆ ˆ ˆ ˆ ˆN N N
z z z x xi i ij i j i i
i i j i iH B J I H I� � � � �
� 3 � �
� # # � #B B B B
Localization: •No relaxation•No equipartition•No temperature•No thermodynamics
Glass ??