Disorder and chaos in quantum system: Anderson localization and its generalization
description
Transcript of Disorder and chaos in quantum system: Anderson localization and its generalization
Disorder and chaos in quantum system:
Anderson localization and its generalization
Boris Altshuler (Columbia)Igor Aleiner (Columbia)
(6 lectures)
Lecture # 2• Stability of insulators and Anderson transition• Stability of metals and weak localization
Anderson localization (1957)
extended
localized
Only phase transition possible!!!
Anderson localization (1957)
extended
localized
Strong disorder
Anderson insulator
Weaker disorder
Localized
Localized
Localized
Extended
Extended
d=3
Any disorder, d=1,2
d=3
Anderson Model
• Lattice - tight binding model
• Onsite energies ei - random
• Hopping matrix elements Iij j iIij
-W < ei <W uniformly distributed
Iij =I i and j are nearest neighbors
0 otherwise{ Critical hopping:
Resonant pair
Bethe lattice:
INFINITE RESONANT PATH ALWAYS EXISTS
Resonant pair
Bethe lattice:
INFINITE RESONANT PATH ALWAYS EXISTS
Decoupled resonant pairs
Long hops?
Resonant tunneling requires:
“All states are localized “
means
Probability to find an extended state:
System size
Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
Metal Insulator
Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
InsulatorMetal
Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
InsulatorMetal
Metal Insulator
Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
Order parameter for Anderson transition?
h!0metal
insulator
behavior for agiven realization
metal
insulator
~ h
probability distributionfor a fixed energy
Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
Probability Distribution
metal
insulator
Note:
Can not be crossover, thus, transition!!!
On the real lattice, there are multiple pathsconnecting two points:
Amplitude associated with the pathsinterfere with each other:
To complete proof of metal insulator transition one has to show the stability of the metal
Back to Drude formulaFinite impurity density
Drude conductivity
CLASSICAL
Quantum (band structure)
Quantum (single impurity)
Why does classical consideration of multiple scattering events work?
1
2
Classical Interference
Vanish after averaging
Look for interference contributions that survive the averaging
1
2
12
unitarity
Correction toscattering crossection
Phase coherence
Additional impurities do not break coherence!!!
1
2
12
unitarity
Correction toscattering crossection
Sum over all possible returning trajectories
unitarity1
2
12
Return probability forclassical random
work
Quantum corrections (weak localization)(Gorkov, Larkin, Khmelnitskii, 1979)
3D
2D
1D
Finite but singular
2D
1D
Metals are NOT stable in one- and two dimensions
Localization length:
Drude + corrections
Anderson model,
Exact solutions for one-dimensionx U(x)
Nch
Gertsenshtein, Vasil’ev (1959)
Nch =1
Exact solutions for one-dimensionx U(x)
NchEfetov, Larkin (1983)Dorokhov (1983) Nch >>1
Strong localizationWeak localization
Universal conductancefluctuations
Altshuler (1985); Stone; Lee, Stone
(1985)
We learned today:• How to investigate stability of insulators (locator
expansion).• How to investigate stability of metals (quantum
corrections)• For d=3 stability of both phases implies metal
insulator transition; The order parameter for the transition is the distribution function
• For d=1,2 metal is unstable and all states are localized
Next time:
• Inelastic transport in insulators