Anderson Localization (1957)

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Anderson localization from theoretical aspects
to applications
Antonio M. García-García ag3_at_princeton.edu http//
phy-ag3.princeton.edu
Princeton and ICTP
Analytical approach to the 3d Anderson transition
Theoretical aspects
Existence of a band of metallic states in 1d
Localization in Quantum Chromodynamics
Applications
Emilio Cuevas, Wang Jiao, James Osborn
Collaborators
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Problem Get analytical expressions for different
quantities characterizing the metal-insulator
transition in d ? 3 such as ?, level statistics.
Locator expansions One parameter scaling
theory Selfconsistent condition
Quasiclassical approach to the Anderson transition
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Anderson localization
50 70 70 80 90 00
Perturbative locator expansion
Anderson
Self consistent conditions
Abou Chakra, Anderson, Thouless, Vollhardt,
Woelfle
1d
Kotani, Pastur, Sinai, Jitomirskaya, Mott.
Thouless, Wegner, Gang of four, Frolich, Spencer,
Molchanov, Aizenman
Scaling
Dynamical localization
Fishman, Grempel, Prange, Casati
Weak Localization
Lee
Efetov, Wegner
Field theory
Efetov, Fyodorov,Mirlin, Klein,
Zirnbauer,Kravtsov
Cayley tree and rbm
Computers
Aoki, Schreiber, Kramer, Shapiro
Experiments
Aspect, Fallani, Segev
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4202 citations!
What if I place a particle in a random potential
and wait?
Tight binding model
Vij nearest neighbors, ?I random potential
Not rigorous! Small denominators
Technique Looking for inestabilities in a
locator expansion
Correctly predicts a metal-insulator transition
in 3d and localization in 1d
Interactions? Disbelief?, against the spirit of
band theory
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Perturbation theory around the insulator limit
(locator expansion).
No control on the approximation. It should be a
good approx for dgtgt2. It predicts correctly
localization in 1d and a transition in 3d
The distribution of the self energy Si (E) is
sensitive to localization.
metal
insulator
gt 0
metal
0
insulator
h
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Energy Scales
Scaling theory of localization
Phys. Rev. Lett. 42, 673 (1979), Gang of four.
Based on Thouless,Wegner, scaling ideas

• 1. Mean level spacing
• 2. Thouless energy
• tT(L) is the travel time to cross a box of size
  L

Dimensionless
Thouless conductance
Diffusive motion without
localization corrections
Metal
Insulator
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Scaling theory of localization
The change in the conductance with the system
size only depends on the conductance itself

• 
  
  

g
Weak localization
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Predictions of the scaling theory at the
transition

1. Diffusion becomes anomalous
Imry, Slevin
2. Diffusion coefficient become size and momentum
dependent
Chalker
3. ggc is scale invariant therefore level
statistics are scale invariant as well
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Weak localization
Positive correction to the resistivity of a metal
at low T
1.Cooperons (Langer-Neal, maximally crossed,
responsible for weak localization) and Diffusons
(no localization, semiclassical) can be
combined. 3. Accurate in d 2.
Self consistent condition (Wolfle-Volhardt)
No control on the approximation!
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Predictions of the self consistent theory at the
transition
1. Critical exponents
Vollhardt, Wolfle,1982
2. Transition for dgt2
Disagreement with numerical simulations!!
Why?
3. Correct for d 2
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Why do self consistent methods fail for d 3?

• 1. Always perturbative around the metallic
  (Vollhardt Wolfle) or the insulator state
  (Anderson, Abou Chacra, Thouless) .
• A new basis for localization is needed
  

2. Anomalous diffusion at the transition
(predicted by the scaling theory) is not taken
into account.
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Proposal
Analytical results combining the scaling theory
and the self consistent condition. ? and level
statistics.
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Idea Solve the self consistent equation assuming
that the diffusion coefficient is renormalized as
predicted by the scaling theory
Assumptions
1. All the quantum corrections missing in the
self consistent treatment are included by just
renormalizing the coefficient of diffusion
following the scaling theory.

• 2. Right at the transition the quantum dynamics
  is well described by a process of anomalous
  diffusion with no further localization
  corrections.

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Technical details Critical exponents
2
The critical exponent ?, can be obtained by
solving the above equation for
with D (?) 0.
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Level Statistics
Starting point Anomalous
diffusion predicted by the scaling theory
Semiclassically, only diffusons
Two levels correlation function
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Shapiro, Abrahams
Aizenman, Warzel
Cayley tree
A linear number variance in the 3d case was
obtained by Altshuler et al.88
Chalker Kravtsov,Lerner
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Comparison with numerical results
1. Critical exponents Excellent 2, Level
statistics OK? (problem with gc) 3. Critical
disorder Not better than before
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Problem Conditions for the absence of
localization in 1d
Motivation Quasiperiodic potentials Nonquasiperiod
ic potentials
Work in progress in collaboration with E Cuevas
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Your intuition about localization
Ea
Random
V(x)
Eb
0
Ec
X
For any of the energies above
Will the
classical motion be strongly affected by quantum
effects?
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tt t t
Speckle potentials
The effective 1d random potential is correlated
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Localization/Delocalization in 1d
Exponential localization for every energy and
disorder
Random uncorrelated potential
Bloch theorem. Absence of localization. Band
theory
Periodic potential
In between?
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Quasiperiodic potentials
Jitomirskaya, Sinai,Harper,Aubry
Similar results
What it is the least smooth potential that can
lead to a band of metallic states?
No metallic band if V(x) is discontinuous
Jitomirskaya, Aubry, Damanik
Jitomirskaya, Bourgain
Conjecture
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Fourier space
Localization for ?gt0
Delocalization in real space
Long range hopping
Levitov FyodorovMirlin
A metallic band can exist for
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Non quasiperiodic potentials
Physics literature
1. Izrailev Krokhin
Metallic band if
Neither of them is accurate
Born approximation
1a. A vanishing Lyapunov exponent does not mean
metallic behavior. 1b. Higher order corrections
make the Lyapunov exponent gt 0 2. Not generic
2. Lyra Moura
Decaying and sparse potentials (Kunz,Simon,
Soudrillard) transition but non ergodic
Localization in correlated potentials Luck,
Shomerus, Efetov, Mirlin,Titov
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Mathematical literature
Kotani, Simon, Kirsch, Minami, Damanik.
Kotanis theory of ergodic operators
Non deterministic potentials
No a.c. spectrum
Deterministic potentials
More difficult to tell
Discontinuous potentials
No a.c. spectrum
Damanik,Stolz, Sims
No a.c. spectrum
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A band of metallic states might exist provided
? gt 0 and V(x) and its ? derivative are bounded.
Neighboring values of the potential must be
correlated enough in order to avoid destructive
interference.
According to the scaling theory in the metallic
region motion must be ballistic.
How to proceed?
Smoothing uncorrelated random potentials
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Finite size scaling analysis
Spectral correlations are scale invariant at
the transition

Thouless, Shklovski, Shapiro 93
AGG, Cuevas
Diffusive Metal
Clean metal
Insulator
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Savitzsky-Golay
1. Take np values of V(n) around a given V(n0) 2.
Replace V(n0) by the best fit of the np values
to a polynomial of M degree 3. Repeat for all n0
Resulting potential is not continuous
A band of metallic states does not exist
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Fourier filtering
1. Fourier transform of the uncorrelated
noise. 2. Remove k gt kcut 3. Fourier transform
back to real space
Resulting potential is analytic
A band of metallic states do exist
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Gruntwald Letnikov operator
Resulting potential is C-?1/2
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A band of metallic states exists provided
Is this generic?
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Localization in systems with chiral symmetry and
applications to QCD
1. Chiral phase transition in lattice QCD as a
metal-insulator transition, Phys.Rev. D75 (2007)
034503, AMG, J. Osborn 2. Chiral phase
transition and Anderson localization in the
Instanton Liquid Model for QCD , Nucl.Phys. A770
(2006) 141-161, AMG. J. Osborn 3. Anderson
transition in 3d systems with chiral symmetry,
Phys. Rev. B 74, 113101 (2006), AMG, E.
Cuevas 4. Long range disorder and Anderson
transition in systems with chiral symmetry , AMG,
K. Takahashi, Nucl.Phys. B700 (2004) 361 5.
Chiral Random Matrix Model for Critical
Statistics, Nucl.Phys. B586 (2000) 668-685, AMG
and J. Verbaarschot
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QCD The Theory of the strong interactions

• High
  Energy g ltlt 1 Perturbative
• 
  1. Asymptotic freedom
• 
  Quarkgluons, Well understood
• 
  Low Energy g 1 Lattice simulations
• 
  The world around us
• 
  2. Chiral symmetry breaking
• 
  
• 
  Massive constituent quark
• 
  3. Confinement
• 
  Colorless hadrons
  
  
• 
  
  
  
• How to extract analytical information?
  Instantons , Monopoles, Vortices

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Deconfinement and chiral restoration
Deconfinement Confining potential vanishes
Chiral Restoration Matter becomes light
How to explain these transitions?
1. Effective, simple, model of QCD close to the
phase transition (Wilczek,Pisarski,Yaffe)
Universality. 2. Classical QCD solutions
(t'Hooft) Instantons (chiral), Monopoles and
vortices (confinement).
We propose that quantum interference/tunneling
plays an important role.


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QCD at T0, instantons and chiral symmetry
breaking tHooft, Polyakov,
Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal

• Instantons Non perturbative solutions of the
  classical Yang Mills equation. Tunneling between
  classical vacua.
• 1. Dirac operator has a zero mode in the field of
  an instanton
• 2. Spectral properties of the smallest
  eigenvalues of the Dirac operator are controled
  by instantons
• 3. Spectral properties related to chiSB.
  Banks-Casher relation
• 
  

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Instanton liquid models T 0
Multiinstanton vacuum?
No superposition
Non linear equations
Variational principles(Dyakonov), Instanton
liquid model (Shuryak).
Solution
ILM T gt 0
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QCD vacuum as a conductor (T 0)
Metal An electron initially bounded to a single
atom gets delocalized due to the overlapping with
nearest neighbors
QCD Vacuum Zero modes initially bounded to an
instanton get delocalized due to the overlapping
with the rest of zero modes. (Diakonov and
Petrov)
Differences
Dis.Sys Exponential decay
QCD
vacuum Power law decay
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QCD vacuum as a disordered conductor
Diakonov, Petrov, Verbaarschot,
Osborn, Shuryak, Zahed,Janik
Instanton positions and color
orientations vary
Electron Quarks
Ion Instantons
T 0 TIA 1/R?, ? 3lt4
Tgt0 TIA e-R/l(T)
QCD vacuum is a conductor
A transition is possible
Shuryak,Verbaarschot, AGG and Osborn
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Phys.Rev. D75 (2007) 034503
QCD Dirac operator
Nucl.Phys. A770 (2006) 141
with J. Osborn
At the same Tc that the Chiral Phase transition
undergo a metal - insulator transition
A metal-insulator transition in the Dirac
operator induces the QCD chiral phase transition
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Signatures of a metal-insulator transition
1. Scale invariance of the spectral
correlations.
A finite size scaling
analysis is then carried

out to determine the transition point. 2.
3. Eigenstates are multifractals.






Skolovski, Shapiro, Altshuler
var
Mobility edge Anderson transition
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ILM, close to the origin, 21 flavors, N 200
Metal insulator transition
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Spectrum is scale invariant
ILM with 21 massless flavors,
We have observed a metal-insulator transition at
T 125 Mev
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Localization versus chiral transition
Instanton liquid model Nf2, masless
Chiral and localizzation transition occurs at the
same temperature
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Problem To determine the importance of Anderson
localization effects in deterministic (quantum
chaos) systems
Scaling theory in quantum chaos Metal insulator
transition in quantum chaos
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Quantum chaos studies the quantum properties of
systems whose classical motion is chaotic (or not)
What is quantum chaos?
Bohigas-Giannoni-Schmit conjecture
Classical chaos
Wigner-Dyson

Energy is the only integral of motion
Momentum is not a good quantum number
Delocalization


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Gutzwiller-Berry-Tabor conjecture
Poisson statistics (Insulator)
Integrable classical motion
Integrability
P(s)
Canonical momenta are conserved
s
System is localized in momentum space
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Dynamical localization
Fishman, Prange, Casati
Exceptions to the BGS conjecture
1. Kicked systems
Classical
ltp2gt
Quantum
Dynamical localization in momentum space
t
2. Harper model 3. Arithmetic billiards
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Characterization
Random
Deterministic
d gt 2 Weak disorder
Wigner-Dyson Delocalization Normal diffusion
Chaotic motion
Always?
d 1,2 d gt 2
Strong disorder
Poisson Localization Diffusion stops

Integrable motion
Bogomolny Altshuler, Levitov Casati, Shepelansky
Critical statistics Multifractality Anomalous
diffusion
d gt 2 Critical disorder
??????????
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Adapt the one parameter scaling theory in quantum
chaos in order to

• Determine the class of systems in which
  Wigner-Dyson statistics applies.
• Does this analysis coincide with the BGS
  conjecture?

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Scaling theory and anomalous diffusion
de fractal dimension of
the spectrum.
Compute g
Universality
weak localization?
Wigner-Dyson ?(g) gt 0 Poisson
?(g) lt 0
Two routes to the Anderson transition
1. Semiclassical origin 2. Induced by quantum
effects
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Wigner-Dyson statistics in non-random systems
1. Estimate the typical time needed to reach the
boundary (in real or momentum space) of the
system.
In billiards ballistic travel time. In kicked
rotors time needed to explore a fixed basis.
2. Use the Heisenberg relation to estimate
thedimensionless conductance g(L) .
Wigner-Dyson statistics applies if
and
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Anderson transition in quantum chaos
Conditions
1. Classical phase space must be homogeneous.

2. Quantum power-law
localization.
3.
Examples
1D ?1, de1/2, Harper model, interval exchange
maps (Bogomolny) ?2, de1, Kicked rotor
with classical singularities (AGG, WangJiao)



2D ?1, de1, Coulomb billiard (Altshuler,
Levitov). 3D ?2/3, de1, 3D Kicked rotor at
critical coupling.
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3D kicked rotator
Finite size scaling analysis shows there is a
transition at kc 2.3
At k kc 2.3 diffusion is anomalous
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1D kicked rotor with singularities
Classical Motion
Normal diffusion
Anomalous Diffusion
Quantum Evolution
Quantum anomalous diffusion
No dynamical localization for ?lt0
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1. ? gt 0 Localization Poisson
2. ? lt 0
Delocalization Wigner-Dyson 3.
? 0 MIT Critical statistics
Anderson transition for log and step singularities
AGG, Wang Jjiao, PRL 2005
Results are stable under perturbations and
sensitive to the removal of the singularity
Possible to test experimentally
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Analytical approach From the kicked rotor to
the 1D Anderson model with long-range hopping
Fishman,Grempel, Prange
1d Anderson model
Tm pseudo random
Always localization
Insulator for ? ?0
Explicit analytical results are possible,
Fyodorov and Mirlin
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Conclusions
1. Anderson localization depends on the degree
of differentiability of the potential.
2. Critical exponents and level statistics are
acessible to analytical techniques
3. The adaptation of the scaling theory to
quantum chaos provides a powerful tool to predict
localization effects in non random systems
4. Anderson localization plays a role in the
chiral phase transition of QCD
ag3_at_princeton.edu http//phy-ag3.princeton.edu
Thanks!
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NEXT
1. Find a way to compute analytically the
critical disorder and others quantities that
characterize the Anderson transition.
2. Adapt localization theories to the
peculiarities of cold atoms.
3. Mathematicians Prove delocalization
ag3_at_princeton.edu http//phy-ag3.princeton.edu