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I.L. Aleiner ( Columbia U, NYC, USA ) B.L. Altshuler ( Columbia U, NYC, USA ) K.B. Efetov ( Ruhr-Universitaet,Bochum, Germany) Localization and Critical Diffusion of Quantum Dipoles in Two Dimensions Windsor Summer School August 25, 2012 Phys. Rev. Lett. 107, 076401 (2011)

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2 Outline Outline: 1) Introduction: a) dirty – Localization in two dimensions b) clean – Dipole excitations in clean system 2) Qualitative discussion and results for localization of dipoles: Fixed points accessible by perturbative renormalization group. 3) Modified non-linear model for localization 4) Conclusions

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1. Localization of single-electron wave-functions: extended localized d=1; All states are localized M.E. Gertsenshtein, V.B. Vasilev, (1959) Exact solution for one channel: D.J. Thouless, (1977) Exact solutions for multi-channel: Scaling argument for multi-channel : K.B.Efetov, A.I. Larkin (1983) O.N. Dorokhov (1983) Conjecture for one channel: Sir N.F. Mott and W.D. Twose (1961) Exact solution for for one channel: V.L. Berezinskii, (1973)

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1. Localization of single-electron wave-functions: extended localized d=1; All states are localized d=3; Anderson transition Anderson (1958); Proof of the stability of the insulator

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1. Localization of single-electron wave-functions: extended localized d=1; All states are localized d=3; Anderson transition d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz: If no spin-orbit interaction Instability of metal with respect to quantum (weak localization) corrections: L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979) First numerical evidence: A Maccinnon, B. Kramer, (1981)

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d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz: If no spin-orbit interaction Conductivity Density of state per unit area Diffusion coefficient Dimensionless conductance Thouless energy Level spacing /

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d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz: If no spin-orbit interaction First numerical evidence: A Maccinnon, B. Kramer, (1981) 1 ansatz Locator expansion

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d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz: If no spin-orbit interaction Instability of metal with respect to quantum (weak localization) corrections: L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979) 1 ansatz No magnetic field (GOE)

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d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz: If no spin-orbit interaction Instability of metal with respect to quantum (weak localization) corrections: Wegner (1979) 1 ansatz In magnetic field (GUE)

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2. Quantum dipoles in clean 2-dimensional systems Simplest example: Each site can be in four excited states, + - + - + - + - Short-range part # of dipoles is not conserved Square lattice: z x

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Single dipole spectrum + - + - + - + - + + + + - + - + - + - - - + + - + - - + - + - - Degeneracy protected by the lattice symmetry

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Single dipole spectrum Degeneracy protected by the lattice symmetry Alone does nothing Qualitatively change E-branch

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Single dipole long-range hops + - + - Second order coupling: Fourier transform:

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Single dipole spectrum Degeneracy protected by the lattice symmetry lifted by long-range hops Similar to the transverse-longitudinal splitting in exciton or phonon polaritons

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Single dipole spectrum Goal: To build the scaling theory of localization including long-range hops Similar to the transverse-longitudinal splitting in exciton or phonon polaritons

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Dipole two band model and disorder disorder

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… and disorder and magnetic field disorder

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Approach from metallic side Only important new parameter:

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Scaling results 1 ansatz No magnetic field (GOE) Used to be for A=0

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Scaling results 1 ansatz No magnetic field (GOE) A>0 is not renormalized Instability of insulator, L.S.Levitov, PRL, 64, 547 (1990) Stable critical fixed point Accessible by perturbative RG for Critical diffusion (scale invariant)

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Scaling results In magnetic field (GUE) Used to be for A=0 1 ansatz

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Scaling results 1 ansatz In magnetic field (GUE) A>0 is not renormalized Unstable critical fixed point Accessible by perturbative RG for Metal-Instulator transition (scale invariant)

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23 Orthogonal ensemble: universal conductance (independent of disorder) Unitary ensemble: metal-insulator transition Summary of RG flow:

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Qualitative consideration 1) Long hops (Levy flights) Consider two wave-packets (1) (2)

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Qualitative consideration 1) Long hops (Levy flights) Consider two wave-packets (1) (2)

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Qualitative consideration 1) Long hops (Levy flights) Consider two wave-packets (1) (2)Rate: R Does not depend on the shape of the wave-function Levy flights

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2) Weak localization (first loop) due to the short-range hops [old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)] Constructive interference Destructive interference

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2) Weak localization (first loop) due to the short-range hops [old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)] Constructive interference No magnetic field (GOE) 0 in magnetic field (GUE)

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3) Weak localization (second loop) short hops; In magnetic field; Wegner (1979) 0 no magnetic field (GOE)

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4) New interference term: Second loop: short hops and Levy flight interference: No magnetic field (GOE)

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Scaling results 1 ansatz No magnetic field (GOE) A>0 is not renormalized Stable critical fixed point Accessible by perturbative RG for

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Scaling results 1 ansatz In magnetic field (GUE) A>0 is not renormalized Unstable critical fixed point Accessible by perturbative RG for

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Standard non-linear -model for localization See textbook by K.B. Efetov, Supersymmetry in disorder and chaos, 1997 - supersymmetry Any correlation function

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Free energy functional (form fixed by symmetries) (GOE): Only running constant (one parameter scaling) Standard non-linear -model for localization

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Beyond standard non-linear -model for localization (long range hops) - supersymmetry Any correlation function

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Beyond standard non-linear -model for localization (long range hops)

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37 Orthogonal ensemble: universal conductance (independent of disorder) Unitary ensemble: metal-insulator transition

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38 Conclusions. 1. Dipoles move easier than particles due to long-range hops. 2. Non-linear sigma-model acquires a new term contributing to RG. 3. RG analysis demonstrates criticality for any disorder for the orthogonal ensemble and existence of a metal-insulator transition for the unitary one.

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39 Renormalization group in two dimensions. Integration over fast modes: fast, slow Expansion in and integration over New non-linear -model with renormalized and Gell-Mann-Low equations: A consequence of the supersymmetry Physical meaning: the density of states is constant.

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40 For the orthogonal, unitary and symplectic ensembles Orthogonal: localization Unitary: localization but with a much larger localization length Symplectic: antilocalization Unfortunately, no exact solution for 2D has been obtained. Reason: non-compactness of the symmetry group of Q. Renormalization group (RG) equations.

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41 The explicit structure of Q u,v contain all Grassmann variables All essential structure is in (unitary ensemble) Mixture of both compact and non-compact symmetries rotations: rotations on a sphere and hyperboloid glued by the anticommuting variables.

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