Anderson localization: from single particle to many body problems.

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Anderson localizationfrom 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
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Lecture 1-2 Single particle
localization Lecture 2-3 Many-body
localization
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Transport in solids
I
V
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Transport in solids
I
V
Focus of The course
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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 )

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Band metals and insulators
Metals
Insulators
Gapped spectrum
Gapless spectrum
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Metals
Insulators
Gapless spectrum
Gapped spectrum
But clean systems are in fact perfect conductors
Electric field
Current
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Gapless spectrum
Gapped spectrum
But clean systems are in fact perfect conductors
(quasi-momentum is conserved, translational
invariance)
Metals
Insulators
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Finite conductivity by impurity scattering
One impurity
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Finite conductivity by impurity scattering
Finite impurity density
Elastic relaxation time
Elastic mean free path
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Finite conductivity by impurity scattering
Finite impurity density
CLASSICAL
Quantum (single impurity)
Drude conductivity
Quantum (band structure)
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Conductivity and Diffusion
Finite impurity density
Diffusion coefficient
Einstein relation
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Conductivity, Diffusion, Density of States (DoS)
Einstein relation
Density of States (DoS)
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Density of States (DoS)
Clean systems
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Density of States (DoS)
Clean systems
Metals, gapless
Phase transition!!!
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But only disorder makes conductivity finite!!!
Disordered systems
Disorder included
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Spectrum always gapless!!!
Lifshitz tail
No phase transition??? Only crossovers???
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Anderson localization (1957)
Only phase transition possible!!!
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Anderson localization (1957)
Strong disorder
Anderson insulator
Weaker disorder
d3
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Anderson Transition
Coexistence of the localized and extended states
is not possible!!!
extended
Rules out first order phase transition
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Temperature dependence of the conductivity (no
interactions)
DoS
DoS
DoS
Metal
Insulator
Perfect one particle Insulator
No singularities in any thermodynamic
properties!!!
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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

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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
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One could think that diffusion occurs even for

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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
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Resonant pair
Perturbative
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Resonant pair
INFINITE RESONANT PATH ALWAYS EXISTS
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Resonant pair
Decoupled resonant pairs
INFINITE RESONANT PATH ALWAYS EXISTS
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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)
?? ?? (??) ?? ?? ?? ?? 2 ??(??- ?? ?? )

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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)
?? ?? (??) ?? ?? ?? ?? 2 ??(??- ?? ?? )

Insulator
Metal
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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)
Insulator
Metal
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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)
?? ?? (??) ?? ?? ?? ?? 2 ??(??- ?? ?? )

Insulator
Metal
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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
insulator
h?0
metal
h
behavior for a given realization
probability distribution for a fixed energy
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Probability Distribution
Note
metal
insulator
Can not be crossover, thus, transition!!!
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But the Andersons argument is not complete
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On the real lattice, there are multiple
paths connecting two points
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Amplitude associated with the paths interfere
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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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

Metal
Insulator
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• Distribution function of the local densities of
  states is the order parameter for Anderson
  transition

metal
insulator
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Resonant pair
Perturbation theory in (I/W) is convergent!
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Perturbation theory in (I/W) is divergent!
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To establish the metal insulator transition we
have to show the convergence of (W/I)
expansion!!!
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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 many-body
  metal insulator transition

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Why does classical consideration of multiple
scattering events work?
1
Vanish after averaging
2
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Back to Drude formula
Finite impurity density
CLASSICAL
Quantum (single impurity)
Drude conductivity
Quantum (band structure)
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Look for interference contributions that survive
the averaging
Phase coherence
2
1
2
1
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Additional impurities do not break coherence!!!
2
1
2
1
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Sum over all possible returning trajectories
Return probability for classical random work
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Sometimes you may see this
MISLEADING DOES NOT EXIST FOR GAUSSIAN DISORDER
AT ALL
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Quantum corrections (weak localization)
(Gorkov, Larkin, Khmelnitskii, 1979)
Finite but singular
3D
2D
1D
E. Abrahams, P. W. Anderson, D. C. Licciardello,
and T.V. Ramakrishnan, (1979)
Thouless scaling ansatz
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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-dimension
x
U(x)
Nch
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Exact solutions for one-dimension
x
U(x)
Nch
Efetov, Larkin (1983) Dorokhov (1983)
Nch gtgt1
Weak localization
Strong localization
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Other way to analyze the stability of metal
Explicit calculation yields
Metal ???
Metal is unstable
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To take home so far

• Interference corrections due to closed loops are
  singular
• For d1,2 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.

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Regularization of the weak localization
byinelastic scatterings (dephasing)
Does not interfere with
e-h pair
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Regularization of the weak localization
byinelastic scatterings (dephasing)
But interferes with
e-h pair
e-h pair
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Phase difference
e-h pair
e-h pair
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Phase difference
- length of the longest trajectory
e-h pair
e-h pair
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Inelastic rates with energy transfer
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Electron-electron interaction
Altshuler, Aronov, Khmelnitskii (1982)
Significantly exceeds clean Fermi-liquid result
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Almost forward scattering
Ballistic
diffusive
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To take home so far

• Interference corrections due to closed loops are
  singular
• For d1,2 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 ??????

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Transport in deeply localized regime
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Inelastic processes transitions between
localized states
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Phonon-induced hopping
energy difference can be matched by a phonon
Mechanism-dependent prefactor
Optimized phase volume
Any bath with a continuous spectrum of
delocalized excitations down to w 0 will give
the same exponential
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?? ??-??h ? 0 ?????
Drude
Electron phonon Interaction does not enter
metal
insulator
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Q Can we replace phonons with e-h pairs and
obtain phonon-less VRH?
Drude
Electron phonon Interaction does not enter
metal
insulator
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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