I.L. Aleiner ( Columbia U, NYC, USA ) B.L. Altshuler ( Columbia U, NYC, USA )

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 Phys. Rev. Lett. 107, 076401 (2011) Windsor Summer School August 25, 2012

2. 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 s-model for localization 4) Conclusions

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

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

5. extended localized 1. Localization of single-electron wave-functions: d=1; All states are localized d=3; Anderson transition d=2; All states are localized If no spin-orbit interaction Thouless scaling + ansatz: E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) 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. Density of state per unit area d=2; All states are localized If no spin-orbit interaction Thouless scaling + ansatz: E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless energy Dimensionless conductance / Level spacing Conductivity Diffusion coefficient

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

26. Qualitative consideration 1) Long hops (Levy flights) Consider two wave-packets R Rate: (2) (1) 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 A>0 No magnetic field (GOE) 1 ansatz Stable critical fixed point Accessible by perturbative RG for is not renormalized

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

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

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

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

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

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

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. Renormalization group in two dimensions. Integration over fast modes: fast, slow Expansion inand 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. Renormalization group (RG) equations. For the orthogonal, unitary and symplectic ensembles Orthogonal: localizationUnitary: 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.

41. The explicit structure of Q u,vcontain 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.