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Anderson localization: from theoretical aspects to applications. Antonio M. Garc í a-Garc í a [email protected] 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. - PowerPoint PPT Presentation
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Anderson localization: from theoretical Anderson localization: from theoretical aspects to applicationsaspects to applications
Antonio M. García-Garcí[email protected]
http://phy-ag3.princeton.edu Princeton and ICTP
Theoretical aspects
ApplicationsLocalization in Quantum
Chromodynamics
Existence of a band of metallic states in 1d
Analytical approach to the 3d Anderson transition
Collaborators: Emilio Cuevas, Wang Jiao, James Osborn
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
Scaling Scaling
Perturbative locator expansion
Field theory
Computers
50’
70’
70’
80’
90’
00’Experiments
Anderson localization
Self consistent conditions
Thouless, Wegner, Gang of four, Frolich, Spencer, Molchanov, Aizenman
Abou Chakra, Anderson, Thouless, Vollhardt, Woelfle
Anderson
Efetov, Wegner
Aoki, Schreiber, Kramer, Shapiro
Aspect, Fallani, Segev
Dynamical localization Fishman, Grempel, Prange, Casati
Cayley tree and rbm Efetov, Fyodorov,Mirlin, Klein, Zirnbauer,Kravtsov
1d Kotani, Pastur, Sinai, Jitomirskaya, Mott.
Weak Localization Lee
But my recollection is that, on the whole, But my recollection is that, on the whole, the attitude was one of humoring me.the attitude was one of humoring me.
Tight binding model
Vij nearest neighbors, I random potential
What if I place a particle in a random potential and wait?
Technique: Looking for inestabilities in a locator expansion
Interactions?
Disbelief?, against the spirit of band theory
Correctly predicts a metal-insulator transition in 3d and localization in 1d
Not rigorous! Small denominators
4202 citations!
No control on the approximation.No control on the approximation. It should be a good approx for It should be a good approx for d>>2. d>>2. It predicts correctly localization It predicts correctly localization in 1d and a transition in 3din 1d and a transition in 3d
= 0metal
insulator> 0
metalinsulator
The distribution of the self The distribution of the self energy Senergy Sii (E) is sensitive to (E) is sensitive to localization.localization.
)(Im iESi
Perturbation theory around Perturbation theory around the insulator limit (locator the insulator limit (locator expansion). expansion).
Energy ScalesEnergy Scales1. Mean level spacing:1. Mean level spacing:2. Thouless energy: 2. Thouless energy: ttTT(L) (L) is the travel time to cross a box of size L is the travel time to cross a box of size L
1
TEg Dimensionless Dimensionless
Thouless conductanceThouless conductance
22 ddT LgLLDE Diffusive motion Diffusive motion
without localization without localization correctionscorrections
11
gEgE
T
T
TT thE /
MetalMetal
InsulatorInsulator
Scaling theory of localizationScaling theory of localization Phys. Rev. Lett. 42, 673 (1979), Gang of four. Based on Thouless,Wegner, scaling ideas
Scaling theory of Scaling theory of localizationlocalization
)(lnlog g
Ldgd
0log)(1/)2()(1
/
2
ggegggdgLgg
L
d
The change in the conductance with the system size only depends on the conductance itself)(g
gWeak localizationWeak localization
Predictions of the Predictions of the scaling theory at the scaling theory at the transitiontransition
dttr /22 )(
dd LLDqqD 22 )()(
1. Diffusion becomes anomalous1. Diffusion becomes anomalous
2. Diffusion coefficient become size 2. Diffusion coefficient become size and momentum dependentand momentum dependent
3. g=g3. g=gcc is scale invariant therefore level is scale invariant therefore level statistics are scale invariant as wellstatistics are scale invariant as well
Imry, Slevin
Chalker
1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization) and Diffusons (no localization, semiclassical) can be combined.3. Accurate in d ~2.
Weak localization
Self consistent condition (Wolfle-Volhardt)
No control on the approximation!
Positive correction to the resistivity of a metal at low T
Predictions of the self Predictions of the self consistent theory at the consistent theory at the
transitiontransition
|||)(| /c
r EEer 42/142
1
ddd
1. Critical 1. Critical exponents:exponents:
2. Transition for d>2
Vollhardt, Wolfle,1982
3. Correct for d ~ 2
Disagreement with numerical simulations!!
Why?
1. Always perturbative around the metallic 1. Always perturbative around the metallic (Vollhardt & Wolfle) or the insulator state (Vollhardt & Wolfle) or the insulator state (Anderson, Abou Chacra, Thouless) .(Anderson, Abou Chacra, Thouless) .
A new basis for localization is neededA new basis for localization is needed
2
2
)()(
d
d
qqDLLD
Why do self Why do self consistent methods consistent methods fail for d = 3?fail for d = 3?
2. Anomalous diffusion at the 2. Anomalous diffusion at the transition (predicted by the scaling transition (predicted by the scaling theory) is not taken into account.theory) is not taken into account.
Proposal:Proposal:
Analytical results combining the scaling theory and the self consistent condition. and level statistics.
2. Right at the transition the quantum dynamics is 2. Right at the transition the quantum dynamics is well described by a process of anomalous diffusion well described by a process of anomalous diffusion with no further localization corrections. with no further localization corrections.
dttr /22 )(
Idea Idea Solve the self consistent equation assuming that Solve the self consistent equation assuming that the diffusion coefficient is renormalized as predicted by the the diffusion coefficient is renormalized as predicted by the scaling theoryscaling theory
AssumptionAssumptions:s:1. All the quantum corrections missing in 1. All the quantum corrections missing in the self consistent treatment are included the self consistent treatment are included by just renormalizing the coefficient of by just renormalizing the coefficient of diffusion following the scaling theory. diffusion following the scaling theory.
Technical details: Critical exponents
The critical exponent ν, can be obtained by The critical exponent ν, can be obtained by solving the above equation for with D solving the above equation for with D (ω) = 0.(ω) = 0.
21
21
d
2
|| cEE
Level Statistics:
Starting point:Starting point: Anomalous diffusion Anomalous diffusion predicted by the scaling predicted by the scaling theorytheory
Semiclassically, only “diffusons”
Two levels correlation function
'EEs
1 d
Cayley tree Aizenman, Warzel
dDd
22
Chalker Kravtsov,Lerner
A linear number variance in the 3d case was obtained by Altshuler et al.’88
2/ dcg dc
Shapiro, Abrahams
Comparison with numerical
results
06.078.075.006.084.083.007.003.1106.052.15.1
66
55
44
33
NT
NT
NT
NT
77.066.07.06.048.05.027.033.0
66
55
44
33
NT
NT
NT
NT
21
21
d
1. Critical exponents: Excellent2, Level statistics: OK? (problem with gc)3. Critical disorder: Not better than before
|||)(| /c
r EEer
Problem: Conditions for the absence of localization in 1d
Motivation
Quasiperiodic potentials
Nonquasiperiodic potentials
Work in progress in collaboration with E Cuevas
Your intuition about localization
V(x)
X
Ea
Eb
Ec
For any of the energies above: For any of the energies above: Will the classical motion be Will the classical motion be strongly affected by quantum effects?strongly affected by quantum effects?
0
Random
The effective 1d random potential is correlated
Speckle potentials
tttt
Localization/Delocalization in 1d:
Random uncorrelated potential
Exponential localization for every energy and disorder
Periodic potentialBloch theorem. Absence
of localization. Band theory
In between?
Quasiperiodic potentials
)cos()( xxV Jitomirskaya, Sinai,Harper,Aubry Critical
InsulatorMetal
111
)exp( AkBak
Jitomirskaya, Bourgain
What it is the least smooth potential that can lead to a band of metallic states?
Similar results
k
k kxaxV )2cos()(
Conjecture 2/3)( CnV
No metallic band if V(x) is discontinuous Jitomirskaya, Aubry, Damanik
CnkanVk
k ))(2cos()( 1||
kAak0
Fourier space:
Long range hopping Localization for >0
Levitov FyodorovMirlin
Delocalization in real space
A metallic band can exist for 0)( CnV
Non quasiperiodic potentials
Physics literature
2/1)(
kkS
Neither of them is accurate
1. Izrailev & Krokhin Metallic band if:
)0()()exp()( VxVikxdxkS
cc kkkkS ,0)( )()( kSk
Born approximation
2. Lyra & Moura 1a. A vanishing Lyapunov exponent does not mean metallic behavior.
1b. Higher order corrections make the Lyapunov exponent > 0
2. Not generic
Localization in correlated potentials: Luck, Shomerus, Efetov, Mirlin,TitovDecaying and sparse potentials (Kunz,Simon, Soudrillard): transition but non ergodic
Mathematical literature:
Kotani’s theory of ergodic operators
Non deterministic potentials
No a.c. spectrum
Deterministic potentials More difficult to tell
Discontinuous potentials
No a.c. spectrum
xVxVxB /1)0()()(
No a.c. spectrum
Kotani, Simon, Kirsch, Minami, Damanik.
Damanik,Stolz, Sims
Neighboring values of the potential must be correlated enough in order to avoid destructive interference.
CxV )( > 0 and V(x) and its derivative are bounded.
How to proceed?
A band of metallic states might exist provided
Smoothing uncorrelated random potentials
0)0()(lim)(
VxVxBx
According to the scaling theory in the metallic region motion must be ballistic.
Finite size scaling Finite size scaling analysisanalysis
Thouless, Shklovski, Shapiro 93’
dssPssss nn )(var 22
Spectral correlations are scale invariant at the transition
i
iissP /)( 1
Diffusive Metal
1var)var(0)var(
286.0var)var(
P
WD
ss
s
Clean metal
Insulator
AGG, Cuevas
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
0)0()(lim)(
VxVxBx
A band of metallic states does not exist
Fourier filtering
Resulting potential is analytic
1. Fourier transform of the uncorrelated noise.
2. Remove k > kcut
3. Fourier transform back to real space
0)0()(lim)(
VxVxBx
A band of metallic states do exist
Gruntwald Letnikov operator
2/12/1 /1,/1 NNi
0)0()(lim)(
VxVxBx
Resulting potential is C-+1/2
2/1,)( CnVA band of metallic states exists provided
Is this generic?
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
QCD : The Theory of the strong interactionsQCD : The Theory of the strong interactions HighHigh EnergyEnergy g << 1 Perturbativeg << 1 Perturbative 1. Asymptotic freedom Quark+gluons, Well understoodQuark+gluons, Well understood Low Energy Low Energy g ~ 1 Lattice simulationsg ~ 1 Lattice simulations The world around usThe world around us 2. Chiral symmetry breaking2. Chiral symmetry breaking
Massive constituent quark Massive constituent quark 3. Confinement3. Confinement Colorless hadronsColorless hadrons
How to extract analytical information?How to extract analytical information? Instantons , Monopoles, Instantons , Monopoles, VorticesVortices
rrarV /)(
3)240(~ MeV
Deconfinement and chiral restorationDeconfinement and chiral restoration Deconfinement: Confining potential vanishes:
Chiral Restoration: Matter becomes light:
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.
How to explain these transitions?
Instantons:Instantons: Non perturbative solutions of the classical Yang Mills Non perturbative solutions of the classical Yang Mills
equation. Tunneling between classical vacuaequation. Tunneling between classical vacua..
1. Dirac operator has a zero mode in the field of an instanton1. Dirac operator has a zero mode in the field of an instanton
2. Spectral properties of the smallest eigenvalues of the Dirac operator are 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons controled by instantons
3. 3. Spectral properties related to chiSB. Banks-Casher relation:Spectral properties related to chiSB. Banks-Casher relation:
QCD at T=0, instantons and chiral symmetry breakingQCD at T=0, instantons and chiral symmetry breaking tHooft, Polyakov, Callan, Gross, Shuryak, tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaalDiakonov, Petrov,VanBaal
300 /10 rrψrDψgA+=D ins
μμ
Vm
imdmDTr
V mm
)(lim)()(1
0
1
3
34 )()ˆ(~)()(
RRuizxiDzxxdT AI
AAIIIA
Instanton liquid Instanton liquid models T = 0models T = 0
Multiinstanton Multiinstanton vacuum?vacuum?
NoNo superpositionsuperposition
00
AI
IA
TT
iD
Variational principles(Dyakonov), Variational principles(Dyakonov), Instanton liquid model (Shuryak). Instanton liquid model (Shuryak).
Non linear equationsNon linear equations
Solution
ILM T > 0 ))(/exp(~ TlRTIA
QCD vacuum as a conductor (T =0)QCD vacuum as a conductor (T =0)Metal:Metal: An electron initially bounded to a single atom An electron initially bounded to a single atom gets delocalized due to the overlapping with nearest gets delocalized due to the overlapping with nearest neighborsneighborsQCD Vacuum:QCD Vacuum: Zero modes initially bounded to an Zero modes initially bounded to an instanton get delocalized due to the overlapping with instanton get delocalized due to the overlapping with the rest of zero modes. (Diakonov and Petrov)the rest of zero modes. (Diakonov and Petrov)
Dis.Sys:Dis.Sys: Exponential decay Exponential decay QCD vacuum:QCD vacuum: Power law Power law
decay decay
DifferencesDifferences
QCD vacuum as a disordered QCD vacuum as a disordered conductorconductor
Instanton positions and color Instanton positions and color orientations varyorientations vary
Ion Ion InstantonsInstantonsT = 0 TT = 0 TIAIA~ 1/R~ 1/R = = 3<43<4
Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik
Shuryak,Verbaarschot, AGG and OsbornShuryak,Verbaarschot, AGG and Osborn
QCD vacuum is a conductorQCD vacuum is a conductor
Electron Electron QuarksQuarks
T>0 TT>0 TIAIA~ e~ e-R/l(T)-R/l(T)
A transition is possibleA transition is possible
nnnQCD iD
0At the same Tc that the Chiral Phase transition
A metal-insulator transition in the Dirac operator A metal-insulator transition in the Dirac operator induces the QCD chiral phase transitioninduces the QCD chiral phase transition
n
n
undergo a metal - insulatormetal - insulator transition
with J. Osborn
Phys.Rev. D75 (2007) 034503
Nucl.Phys. A770 (2006) 141
QCD Dirac operator μμ
QCD gA+=D
Signatures of a metal-insulator transitionSignatures of a metal-insulator transition1. 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.
)1(2 ~)( qDdq
n
qLrdr
Skolovski, Shapiro, Altshuler
1~)(1~)(
sesPsssP
As
Mobility edge Anderson transition
varvar
dssPssss nn )(var 22
ILM, close to the origin, 2+1 ILM, close to the origin, 2+1 flavors, N = 200flavors, N = 200
Metal Metal insulator insulator transitiontransition
ILM with 2+1 massless flavors,
We have observed a metal-insulator transition at T ~ 125 Mev
Spectrum is scale invariant
Instanton liquid model Nf=2, maslessInstanton liquid model Nf=2, masless Localization versus Localization versus chiral transitionchiral transition
Chiral and localizzation transition occurs at the same temperatureChiral and localizzation transition occurs at the same temperature
Problem: To determine the Problem: To determine the importance of Anderson localization importance of Anderson localization effects in deterministic (quantum effects in deterministic (quantum chaos) systems chaos) systems
Scaling theory in quantum chaos
Metal insulator transition in quantum chaos
Quantum chaos studies the Quantum chaos studies the quantum properties of systems quantum properties of systems whose classical motion is whose classical motion is chaotic (or not)chaotic (or not)
Bohigas-Giannoni-Schmit Bohigas-Giannoni-Schmit conjectureconjecture
Classical chaos Wigner-Classical chaos Wigner-DysonDyson
Momentum is not a good quantum number Momentum is not a good quantum number DelocalizationDelocalization
What is quantum chaos?
Energy is the only integral of motionEnergy is the only integral of motion
Gutzwiller-Berry-Tabor Gutzwiller-Berry-Tabor conjectureconjecture
Poisson Poisson statisticsstatistics
(Insulator(Insulator
))
s
P(s)
Integrable classical motion
Integrability
Canonical Canonical momenta are momenta are conservedconserved
System is localized in System is localized in momentum spacemomentum space
Dynamical Dynamical localizationlocalization
n
nTtVpH )()(2
cos)( KV Dynamical localization Dynamical localization in momentum spacein momentum space
2. Harper model2. Harper model3. Arithmetic 3. Arithmetic billiardsbilliards
<p2
>
t
Classical
Quantum
Exceptions to the BGS Exceptions to the BGS conjectureconjecture
1. Kicked systems1. Kicked systems
Fishman, Prange, Casati
Random Deterministic
d = 1,2 d > 2 Strong disorder
d > 2Weak disorder
d > 2Critical disorder
Chaotic motion
Integrable Integrable motionmotion
??????????
Wigner-Dyson
Delocalization
Normal diffusion
Poisson
Localization
Diffusion stops
Critical statistics
Multifractality
Anomalous diffusion
CharacterizationCharacterization
cgg
0g
g
Always?
Bogomolny
Altshuler, Levitov
Casati, Shepelansky
Determine the class of systems in which Determine the class of systems in which Wigner-Dyson statistics applies. Wigner-Dyson statistics applies.
Does this analysis coincide with the BGS Does this analysis coincide with the BGS conjecture?conjecture?
Adapt the one parameter scaling theory in quantum chaos in order to:
Scaling theory and anomalous diffusionScaling theory and anomalous diffusion
2)(
eclas
T
ddLELg clas
clasquanclas 000 quanclas
0)( g
)()( gfg clas
weak weak localization?localization?L Wigner-DysonWigner-Dyson (g) (g)
> 0> 0Poisson Poisson (g) (g) < 0< 0
eddLtq /2 dde e fractal fractal dimension dimension of the of the spectrum.spectrum.
Two routes to the Anderson transitionTwo routes to the Anderson transition
1. Semiclassical origin 1. Semiclassical origin 2. Induced by quantum effects2. Induced by quantum effects
Compute g
Universality
Wigner-Dyson statistics in non-random Wigner-Dyson statistics in non-random systemssystems
02)(
eclas
T
ddLELg clas
tq 2
1. Estimate the typical time needed to 1. Estimate the typical time needed to reach the “boundary” (in real or reach the “boundary” (in real or momentum space) of the system.momentum space) of the system.
In billiards: ballistic travel time.In billiards: ballistic travel time.In kicked rotors: time needed to explore a fixed In kicked rotors: time needed to explore a fixed basis.basis.2. Use the Heisenberg relation to estimate 2. Use the Heisenberg relation to estimate
thedimensionless conductance g(L) .thedimensionless conductance g(L) .
Wigner-Dyson statistics applies Wigner-Dyson statistics applies ifif
and
0quan
tk
ddLELg
eclas
T clas 202)(
1D 1D =1, d=1, dee=1/2, Harper model, interval exchange maps =1/2, Harper model, interval exchange maps (Bogomolny)(Bogomolny)
=2, d=2, dee=1, Kicked rotor with classical singularities =1, Kicked rotor with classical singularities (AGG, WangJiao)(AGG, WangJiao)
2D 2D =1, d=1, dee=1, Coulomb billiard =1, Coulomb billiard (Altshuler, Levitov).(Altshuler, Levitov).
3D 3D =2/3, d=2/3, dee=1, 3D Kicked rotor at critical coupling.=1, 3D Kicked rotor at critical coupling.
Anderson transition in quantum chaosAnderson transition in quantum chaos Conditions:Conditions:
1. 1. Classical phase space must be homogeneous. Classical phase space must be homogeneous. 2. 2. Quantum power-law localization. Quantum power-law localization. 3. 3.
Examples:Examples:
3D kicked rotator3D kicked rotator
Finite size scaling analysis shows Finite size scaling analysis shows there is a transition at kthere is a transition at kc c ~ 2.3~ 2.3
)cos()cos()cos(),,( 321321 kV
3/22 ~)( ttpquan
At k = kAt k = kc c ~ 2.3 diffusion ~ 2.3 diffusion is anomalous is anomalous
1D kicked rotor with singularities 1D kicked rotor with singularities
||)( V
)4
exp()/)(exp()4
exp(ˆ2
2
2
2
TiVTU
n
nTtVpH )()(2
cos)( KV Classical Motion
Quantum Evolution
Anomalous Diffusion
11||/1),( tkktkP
'' ||/1),( tkktkP Quantum anomalous diffusion
No dynamical localization for <0
Normal diffusion
||log)( V
11||)(
tkLELg clas
T clas
AGG, Wang Jjiao, PRL 2005
1. 1. > 0 Localization Poisson > 0 Localization Poisson 2. 2. < 0 Delocalization Wigner-Dyson < 0 Delocalization Wigner-Dyson 3. 3. = 0 MIT Critical statistics = 0 MIT Critical statistics Anderson transition Anderson transition for for
log and step singularitieslog and step singularities
Results are Results are stablestable under perturbations and under perturbations and sensitive to the removal of the singularitysensitive to the removal of the singularity
Possible to test experimentally
Analytical approach: From the kicked rotor to the 1D Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping Anderson model with long-range hopping
),()()(),(21),( 2
2
tntVttt
in
1
1 r
Wr
Explicit analytical results are possible, Fyodorov and Mirlin
Insulator for 0
0r
mrmrmm EuuWuT
cos)( KV
||)( V
Fishman,Grempel, PrangeFishman,Grempel, Prange
1d Anderson 1d Anderson modelmodel
Tm pseudo random
1,1, rrrW Always localization
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
Thanks! [email protected]://phy-ag3.princeton.edu
[email protected]://phy-ag3.princeton.edu
NEXT1. 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