Disorder and chaos in quantum system: Anderson localization and its generalization

1
Disorder and chaos in quantum systemAnderson
localization and its generalization
(6 lectures)
Igor Aleiner (Columbia)
2
Lecture 2

• Stability of insulators and Anderson transition
• Stability of metals and weak localization

3
Anderson localization (1957)
Only phase transition possible!!!
4
Anderson localization (1957)
Strong disorder
Anderson insulator
Weaker disorder
d3
5
Anderson Model

• Lattice - tight binding model
• Onsite energies ei - random
• Hopping matrix elements Iij

i
j
Iij
Critical hopping
-W lt ei ltW uniformly distributed
6
One could think that diffusion occurs even for

7
is F A L S E
Probability for the level with given energy on
NEIGHBORING sites
Probability for the level with given energy in
the whole system
2d attempts
Infinite number of attempts
8
Resonant pair
Perturbative
9
Resonant pair
INFINITE RESONANT PATH ALWAYS EXISTS
10
Resonant pair
Decoupled resonant pairs
INFINITE RESONANT PATH ALWAYS EXISTS
11
Long hops?
Resonant tunneling requires
12
All states are localized
means
Probability to find an extended state
System size
13
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)
14
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)
Insulator
Metal
15
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)
Insulator
Metal
16
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)
Insulator
Metal
17
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
insulator
h!0
metal
h
behavior for a given realization
probability distribution for a fixed energy
18
Probability Distribution
Note
metal
insulator
Can not be crossover, thus, transition!!!
19
On the real lattice, there are multiple
paths connecting two points
20
Amplitude associated with the paths interfere
with each other
21
To complete proof of metal insulator transition
one has to show the stability of the metal
22
Back to Drude formula
Finite impurity density
CLASSICAL
Quantum (single impurity)
Drude conductivity
Quantum (band structure)
23
Why does classical consideration of multiple
scattering events work?
1
Vanish after averaging
2
24
Look for interference contributions that survive
the averaging
Phase coherence
2
1
2
1
25
Additional impurities do not break coherence!!!
2
1
2
1
26
Sum over all possible returning trajectories
Return probability for classical random work
27
Quantum corrections (weak localization)
(Gorkov, Larkin, Khmelnitskii, 1979)
Finite but singular
3D
2D
1D
28
2D
1D
Metals are NOT stable in one- and two dimensions
Localization length
Drude corrections
Anderson model,
29
Exact solutions for one-dimension
x
U(x)
Nch
30
Exact solutions for one-dimension
x
U(x)
Nch
Efetov, Larkin (1983) Dorokhov (1983)
Nch gtgt1
Weak localization
Strong localization
31
We learned today

• How to investigate stability of insulators
  (locator expansion).
• How to investigate stability of metals (quantum
  corrections)
• For d3 stability of both phases implies metal
  insulator transition The order parameter for the
  transition is the distribution function
• For d1,2 metal is unstable and all states are
  localized

32
Next time

• Inelastic transport in insulators