Disorder and chaos in quantum system: Anderson localization and its generalization
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Disorder and chaos in quantum system:
Anderson localization and its generalization
Boris Altshuler (Columbia)Igor Aleiner (Columbia)
(6 lectures)
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Lecture # 2• Stability of insulators and Anderson transition• Stability of metals and weak localization
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Anderson localization (1957)
extended
localized
Only phase transition possible!!!
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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
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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:
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Resonant pair
Bethe lattice:
INFINITE RESONANT PATH ALWAYS EXISTS
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Resonant pair
Bethe lattice:
INFINITE RESONANT PATH ALWAYS EXISTS
Decoupled resonant pairs
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Long hops?
Resonant tunneling requires:
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“All states are localized “
means
Probability to find an extended state:
System size
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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
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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
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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
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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?
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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)
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Probability Distribution
metal
insulator
Note:
Can not be crossover, thus, transition!!!
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On the real lattice, there are multiple pathsconnecting two points:
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Amplitude associated with the pathsinterfere with each other:
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To complete proof of metal insulator transition one has to show the stability of the metal
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Back to Drude formulaFinite impurity density
Drude conductivity
CLASSICAL
Quantum (band structure)
Quantum (single impurity)
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Why does classical consideration of multiple scattering events work?
1
2
Classical Interference
Vanish after averaging
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Look for interference contributions that survive the averaging
1
2
12
unitarity
Correction toscattering crossection
Phase coherence
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Additional impurities do not break coherence!!!
1
2
12
unitarity
Correction toscattering crossection
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Sum over all possible returning trajectories
unitarity1
2
12
Return probability forclassical random
work
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Quantum corrections (weak localization)(Gorkov, Larkin, Khmelnitskii, 1979)
3D
2D
1D
Finite but singular
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2D
1D
Metals are NOT stable in one- and two dimensions
Localization length:
Drude + corrections
Anderson model,
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Exact solutions for one-dimensionx U(x)
Nch
Gertsenshtein, Vasil’ev (1959)
Nch =1
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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)
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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
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Next time:
• Inelastic transport in insulators