Transcript Aleiner Lecture 1 and 2

Anderson localization:
from single particle to many
body problems.
(4 lectures)
Igor Aleiner
( Columbia University in the City of New York, USA )
Windsor Summer School, 14-26 August 2012
Lecture # 1-2 Single particle localization
Lecture # 2-3 Many-body localization
Transport in solids
Metal
I
Superconductor
V
Insulator
Conductance:
Conductivity:
Transport in solids
I
Metal
V
Insulator
Focus of
The course
Conductance:
Conductivity:
Lecture # 1
•
•
•
•
Metals and insulators – importance of disorder
Drude theory of metals
First glimpse into Anderson localization
Anderson metal-insulator transition (Bethe lattice
argument; order parameter … )
Band metals and insulators
Metals
Gapless spectrum
Insulators
Gapped spectrum
Metals
Gapless spectrum
Insulators
Gapped spectrum
But clean systems are in fact perfect conductors:
Current
Electric field
Gapless spectrum
Gapped spectrum
But clean systems are in fact perfect conductors:
(quasi-momentum is conserved,
translational invariance)
Metals
Insulators
Finite conductivity by impurity scattering
One impurity
Probability
density
Incoming
flux
Scattering
cross-section
Finite conductivity by impurity scattering
Finite impurity density
Elastic mean
free path
Elastic relaxation
time
Finite conductivity by impurity scattering
Finite impurity density
CLASSICAL
Quantum (single
impurity)
Drude conductivity
Quantum (band
structure)
Conductivity and Diffusion
Finite impurity density
Diffusion coefficient
Einstein relation
Conductivity, Diffusion,
Density of States (DoS)
Einstein relation
Density of States
(DoS)
Density of States (DoS)
Clean systems
Density of States (DoS)
Clean systems
Metals,
gapless
Insulators,
gapped
Phase transition!!!
But only disorder makes conductivity
finite!!!
Disordered
systems
Clean
Disordered
Disorder
included
Disordered
Spectrum always gapless!!!
Lifshitz tail
No phase transition???
Only crossovers???
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
Anderson Transition
Coexistence of the localized and
extended states is not possible!!!
extended
Rules out first order phase transition
DoS
- mobility edges (one particle)
Temperature dependence of the
conductivity (no interactions)
DoS
DoS
Metal
Insulator
No singularities in any
thermodynamic properties!!!
DoS
“Perfect” one particle
Insulator
To take home so far:
• Conductivity is finite only due to broken
translational invariance (disorder)
• Spectrum (averaged) in disordered system is
gapless
• Metal-Insulator transition (Anderson) is
encoded into properties of the wave-functions
• 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:
One could think that diffusion occurs
even for
:
Random walk on the lattice
Golden rule:
?
Pronounce words:
Self-consistency
Mean-field
Self-averaging
Effective medium
…………..
is
FALSE
Probability for the level
with given energy on
NEIGHBORING sites
2d attempts
Probability for the level
with given energy in the
whole system
Infinite number of attempts
Perturbative
Resonant pair
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)
𝜈𝑖 (𝜀) =
Metal
𝛼
𝜓𝛼 𝑖
2
𝛿(𝜀 − 𝜉𝛼 )
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)
𝜈𝑖 (𝜀) =
Metal
𝛼
𝜓𝛼 𝑖
2
𝛿(𝜀 − 𝜉𝛼 )
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)
𝜈𝑖 (𝜀) =
Metal
𝛼
𝜓𝛼 𝑖
2
𝛿(𝜀 − 𝜉𝛼 )
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!!!
But the Anderson’s argument is not
complete:
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
Summary of Lecture # 1
• Conductivity is finite only due to broken
translational invariance (disorder)
• Spectrum (averaged) in disordered system is
gapless (Lifshitz tail)
• Metal-Insulator transition (Anderson) is encoded
into properties of the wave-functions
extended
Metal
Insulator
localized
• Distribution function of the local densities of
states is the order parameter for Anderson
transition
insulator
metal
I < Ic
Perturbation theory in (I/W) is convergent!
Resonant pair
I > Ic
Perturbation theory in (I/W) is divergent!
To establish the metal insulator transition
we have to show the convergence of
(W/I) expansion!!!
Lecture # 2
•
•
•
•
Stability of metals and weak localization
Inelastic e-e interactions in metals
Phonon assisted hopping in insulators
Statement of many-body localization and manybody metal insulator transition
Why does classical consideration of
multiple scattering events work?
1
2
Classical
Vanish after
averaging
Interference
Back to Drude formula
Finite impurity density
CLASSICAL
Quantum (single
impurity)
Drude conductivity
Quantum (band
structure)
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
Sometimes you may see this…
MISLEADING…
DOES NOT EXIST FOR GAUSSIAN
DISORDER AT ALL
(Gorkov, Larkin, Khmelnitskii, 1979)
Quantum corrections (weak localization)
3D
2D
1D
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
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
Other way to analyze the stability of metal
insulator
metal
Explicit calculation yields:
Metal ???
Metal is unstable
To take home so far:
• Interference corrections due to closed loops are
singular;
• For d=1,2 they diverges making the metalic
phase of non-interacting particles unstable;
• Finite size system is described as a good metal,
if
, in other words
• For
, the properties are well described by
Anderson model with
replacing lattice
constant.
Regularization of the weak localization by
inelastic scatterings (dephasing)
Does not interfere with
e-h pair
Regularization of the weak localization by
inelastic scatterings (dephasing)
But interferes with
e-h pair
e-h pair
Phase difference:
e-h pair
e-h pair
Phase difference:
- length
of the longest trajectory;
e-h pair
e-h pair
Inelastic rates with energy transfer
Electron-electron interaction
Altshuler, Aronov, Khmelnitskii (1982)
Significantly exceeds clean
Fermi-liquid result
Almost forward scattering:
Ballistic
diffusive
To take home so far:
• Interference corrections due to closed loops are singular;
• For d=1,2 they diverges making the metalic
phase of non-interacting particles unstable;
• Interactions at finite T lead to finite
• System at finite temperature is described as a good metal,
• if
,
in other words
•
For
, the properties are well described by ??????
Transport in deeply localized regime
Inelastic processes:
transitions between localized states

energy
mismatch

(inelastic lifetime)–1
(any mechanism)
Phonon-induced hopping


energyHopping
difference can be matched by a phonon
Variable Range
Sir N.F. Mott (1968)
Mechanism-dependent
prefactor
Without Coulomb gap
A.L.Efros, B.I.Shklovskii (1975)
Optimized
phase volume
Any bath with a continuous spectrum of delocalized excitations
down to w = 0 will give the same exponential
𝜆𝑒−𝑝ℎ ⟶ 0 ?????
Drude
“metal”
“insulator”
Electron phonon
Interaction does not enter
Q: Can we replace phonons with
e-h pairs and obtain phonon-less VRH?
Drude
“metal”
“insulator”
Electron phonon
Interaction does not enter
Metal-Insulator Transition and many-body
Localization:
[Basko, Aleiner, Altshuler (2005)]
and all one particle state are localized
Drude
metal
insulator
(Perfect Ins)
Interaction strength