Transcript Slide 1

Disorder and chaos in
quantum system:
Anderson localization and
its generalization
(6 lectures)
Igor Aleiner (Columbia)
Boris Altshuler (Columbia)
Lecture # 2
• Stability of insulators and Anderson transition
• Stability of metals and weak localization
Anderson localization (1957)
extended
Only phase
transition possible!!!
localized
Anderson localization (1957)
Strong disorder
extended
d=3
Any disorder, d=1,2
localized
Localized
Extended
Weaker disorder d=3
Localized
Extended
Localized
Anderson insulator
• Lattice - tight binding model
Anderson
Model
ei - random
• Hopping matrix elements Iij
• Onsite energies
j
i
Iij
I i and j are nearest
{
Iij =
0
neighbors
otherwise
-W < ei <W
uniformly distributed
Critical hopping:
Resonant pair
Bethe lattice:
INFINITE RESONANT PATH ALWAYS
EXISTS
Resonant pair
Bethe lattice:
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)
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)
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)
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)
metal
insulator
h!0
insulator
metal
~h
behavior for a
given realization
probability distribution
for a fixed energy
Probability Distribution
metal
Note:
insulator
Can not be crossover, thus, transition!!!
On the real lattice, there are multiple paths
connecting two points:
Amplitude associated with the paths
interfere with each other:
To complete proof of metal insulator transition
one has to show the stability of the metal
Back to Drude formula
Finite impurity density
CLASSICAL
Quantum (single
impurity)
Drude conductivity
Quantum (band
structure)
Why does classical consideration of
multiple scattering events work?
1
2
Classical
Vanish after
averaging
Interference
Look for interference contributions that
survive the averaging
Phase coherence
2
1
Correction to
scattering
crossection
2
1
unitarity
Additional impurities do not break coherence!!!
2
1
Correction to
scattering
crossection
2
unitarity
1
Sum over all possible returning trajectories
2
1
2
1
unitarity
Return probability for
classical random
work
(Gorkov, Larkin, Khmelnitskii, 1979)
Quantum corrections (weak localization)
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-dimension
x
U(x)
Nch
Nch =1
Gertsenshtein, Vasil’ev (1959)
Exact solutions for one-dimension
x
Efetov, Larkin (1983)
Dorokhov (1983)
Nch >>1
U(x)
Nch
Universal conductance
fluctuations
Altshuler (1985);
Stone; Lee, Stone
(1985)
Strong localization
Weak localization
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