1. 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)

2. 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

3. 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)

4. 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

5. 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)

6. 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 /

7. 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

8. 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)

9. 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)

10. 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

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

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

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

14. 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

15. 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

16. Dipole two band model and disorder disorder

17. … and disorder and magnetic field disorder

18. Approach from metallic side Only important new parameter:

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

20. 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)

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

22. 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)

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

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

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

26. 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

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

28. 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)

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

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

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

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

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

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

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

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

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

38. 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.

39. 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.

40. 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.

41. 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.