Anderson localization from single particle to many body
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 Superconductor I Insulator Conductance: Conductivity: V
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 (Do. S) Einstein relation Density of States (Do. S)
Density of States (Do. S) Clean systems
Density of States (Do. S) 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 Localized Weaker disorder d=3 Extended Localized Anderson insulator
Anderson Transition Coexistence of the localized and extended states is not possible!!! extended Rules out first order phase transition Do. S - mobility edges (one particle)
Temperature dependence of the conductivity (no interactions) Do. S Metal Insulator No singularities in any thermodynamic properties!!! Do. S “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
Anderson Model • Lattice - tight binding model ei - random • Hopping matrix elements Iij • Onsite energies j i Iij { Iij = I i and j are nearest 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 F A L S E Probability for the level with given energy on NEIGHBORING sites 2 d 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) 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 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
Resonant pair Perturbation theory in (I/W) is convergent!
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 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) 3 D 2 D 1 D E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, (1979) Thouless scaling + ansatz: Finite but singular
2 D 1 D Metals are NOT stable in one- and two dimensions Localization length: Drude + corrections Anderson model,
Exact solutions for one-dimension x U(x) 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
Phase difference: e-h pair
Phase difference: - length of the longest trajectory; 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 energy difference can be matched by a phonon Variable Range Hopping 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
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
- Slides: 72