Anderson localization: fifty Anderson localization: fifty years old and still growing years old and still growing Antonio M. García-García [email protected]http://phy-ag3.princeton.edu Princeton University Experiments with cold atoms Theories of localizati on Metal Insulator transition Quantum Chaos & (dynamical) localization MIT in quantum chaos Localization beyond condensed matter What is localizatio n? What is new and why is still a hot topic? What is next? What do we know about localization?
Transcript
Anderson localization: fifty years old Anderson localization: fifty years old and still growingand still growing
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
Anderson Localization (1957)Anderson Localization (1957)
<r2
>
a = ?
Dquan= f(d,dis)?t
DclastDquan
t
Dquanta
Quantum diffusion in a random potential stops due to interference effects.
d = 1 An insulator for any disorderd =2 An insulator for any disorderd > 2 Localization only for disorder strong enough
Still not well understood
Scaling theory Scaling theory
Anderson paper and earlier theories of localization
Weak localization
Computers!
Dynamical localization
50’
60’
70’
80’
90’
00’ Experiments!
Quantum chaos
Mesoscopic physics
Cold atoms
Multifractality
Perturbation theory and band theory
History
Renormalization group
Theory of phase transitions
Theories of localization
Locator expansions
Weak localization expansions
One parameter scaling theory
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!
4002 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.
Predicts correctly localization in Predicts correctly localization in 1d and a MIT in 3d1d and a MIT in 3d
= 0metal
insulator
> 0
metal
insulator
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 Scales
1. 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
TE
g Dimensionless Dimensionless
Thouless conductanceThouless conductance22 dd
T LgLLDE Diffusive motion Diffusive motion without quantum without quantum
correctionscorrections
1
1
gE
gE
T
T
TT thE /
MetalMetal
InsulatorInsulator
Scaling theory of localizationScaling theory of localization Phys. Rev. Lett. 42, 673 (1979), Abraham, P. W. Anderson , Licciardello, , Ramakrishnan.
Scaling theory of Scaling theory of localizationlocalization
)(ln
logg
Ld
gd 0log)(1
/)2()(1/
2
ggegg
gdgLggL
d
The change in the conductance with the system size only depends on the conductance itself)(g
g
Weak localizationWeak localization
Predictions of the Predictions of the scaling theory at the scaling theory at the transitiontransition
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
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
Experimental verification 80’
Predictions of the self Predictions of the self consistent theory at the consistent theory at the
transitiontransition
|||)(| /c
r EEer 42/1
421
d
dd
1. Critical 1. Critical exponents:exponents:
2. Transition for d>2
Vollhardt, Wolfle,1982
3. Exact 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
qqD
LLD
Why do self Why do self consistent methods consistent methods fail for d fail for d 3? 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.
Solution:Solution:
Analytical results combining the scaling theory and the self consistent condition.Critical exponents, critical disorder, 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.
2
1
2
1
d
2
|| cEE
Comparison with numerical results
06.078.075.0
06.084.083.0
07.003.11
06.052.15.1
66
55
44
33
NT
NT
NT
NT
2
1
2
1
d
1. Critical exponents: Excellent
2. Level statistics: Good (problem with gc)
|||)(| /c
r EEer
Localization and metal insulator Localization and metal insulator transition in deterministic (quantum transition in deterministic (quantum chaos) pseudo random systems chaos) pseudo random systems
3. Systems with interactions for which the exact 3. Systems with interactions for which the exact Schrödinger equation cannot be solved.Schrödinger equation cannot be solved.
Quantum Quantum ChaosChaos
Disordered Disordered systemssystems
Random versus Random versus chaoticchaotic
Impact of classical Impact of classical chaos in quantum chaos in quantum
mechanics mechanics
Quantum Quantum mechanics in a mechanics in a
random potentialrandom potential
1. Scaling theory.1. Scaling theory.
2. Ensemble average.2. Ensemble average.
3. Anderson 3. Anderson localization.localization.
Characterization of a metal/insulatorCharacterization of a metal/insulator nnn EH
Random Matrix Random Matrix
22 ~
Asse~sP
dD
sesP
D
)(
0~2Uncorrelated Spectrum
Wigner Dyson statistics
Poisson statistics
Signatures of a metal-insulator transitionSignatures of a metal-insulator transition
)1(2
~)( qDdq
n
qLrdr
Skolovski, Shapiro, Altshuler
varvar
dssPssss nn )(var22
I have created a monster!I have created a monster!19971997
1. Scale invariance of the spectral correlations. Finite size scaling analysis.
3. Eigenstates are multifractals
2.
1~)(
1~)(
sesP
sssPAs
1+2+3 = Critical statistics
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)
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
d
dL
ELg 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) .
Determine the universality class in quantum chaos Determine the universality class in quantum chaos related to the metal-insulator transition.related to the metal-insulator transition.
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:
1D kicked rotor with singularities 1D kicked rotor with singularities
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
),()()(),(2
1),(
2
2
tntVttt
in
1
1 r
Wr
Explicit analytical results are possible, Fyodorov and Mirlin
Finite size scaling analysis Finite size scaling analysis shows there is a transition shows there is a transition a MIT at ka MIT at kc c ~ 3.3~ 3.3
)cos()cos()cos(),,( 221321 kV
3/22 ~)( ttpquan
ttpclas
~)(2
In 3D, for =2/3
cgg
Localization beyond condensed matter physics
A two minute course on non A two minute course on non perturbative QCDperturbative QCD
State of the art
T = 0 T = 0 low low energyenergy
What?What? How? How?
1. Lattice QCD
2. Effective models:
Instantons….
Chiral Symmetry breaking and Confinement
T = TT = Tcc
Chiral and deconfinement
transition
Universality (Wilczek and Pisarski)
T > TT > Tcc Quark- gluon plasma
QCD non perturbative!
AdS-CFT
N =4 Super Yang Mills
μ large Color superconductivity Bulk BCS
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 neighborsneighbors
QCD 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
Impurities Impurities InstantonsInstantons
T = 0 long range hopping 1/RT = 0 long range hopping 1/R = 3<4 = 3<4
AGG and Osborn, AGG and Osborn, PRL, 94 (2005) 244102PRL, 94 (2005) 244102
QCD vacuum is a conductor for any density of instantonsQCD vacuum is a conductor for any density of instantons
Electron Electron QuarksQuarks
QCD at finite T: Phase transitionsQCD at finite T: Phase transitions
Quark- Gluon Plasma perturbation theory only for T>>Tc
J. Phys. G30 (2004) S1259
At which temperature does the transition occur ? What is the nature of transition ?
Péter Petreczky Péter Petreczky
Deconfinement and chiral restorationDeconfinement and chiral restoration
Deconfinement: Confining potential vanishes:
Chiral Restoration: Matter becomes light:
1. Effective model of QCD close to the phase transition (Wilczek,Pisarski,Yaffe): Universality, epsilon expansion.... too simple?
2. Classical QCD solutions (t'Hooft): Instantons (chiral), Monopoles and vortices (confinement). Instanton do not dissapear at the transiton (Shuryak,Schafer).
We propose that quantum interference and tunneling, namely, Anderson localizationAnderson localization plays an important role.
How to explain these transitions?
Main contributionMain contribution
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
ILM with 2+1 massless flavors,
We have observed a metal-insulator transition at T ~ 125 Mev
Spectrum is scale invariant
ILM, close to the origin, 2+1 ILM, close to the origin, 2+1 flavors, N = 200flavors, N = 200
Metal Metal insulator insulator transitiontransition
Experimental studies of
localization
Experiments: Difficult!!!!
Localization is quite fragile phenomenon: broken by inellastic scaterring, even at low temperatures, unquenched disorder….
No control over interactions or details of the disordered potential.
People see things ….but is it really localization?
No control on absorption by the medium.
Interactions and disorder are controlled with great precision!!!!
Electronic systems:
Light:
Cold atoms:
Weak localization in 2D. OK
The effective random potential is correlated
Speckle potentials
The effective random potential is correlated
Π0(t) population with zero velocity
Π0 (t) = constant
Π0 (t) ~t -1/2
Insulator
arXiv:0709.4320
Metal
Test of quantum mechanics?
Why is the localization problem still interesting?Why is the localization problem still interesting?
