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The instanton vacuum of generalized models

The instanton vacuum of generalized models. L.D. Landau Institute for Theoretical Physics. and A.M.M. Pruisken Institute for Theoretical Physics, University of Amsterdam. I.S. Burmistrov. cond-mat/0407776 accepted in Annals of Physics.

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The instanton vacuum of generalized models

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  1. The instanton vacuum of generalized models L.D. Landau Institute for Theoretical Physics and A.M.M. Pruisken Institute for Theoretical Physics, University of Amsterdam I.S. Burmistrov cond-mat/0407776 accepted in Annals of Physics

  2. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Pruisken ‘84 Wegner ‘79 Efetov, Larkin, Khmelnitzkii ‘80 mean-field longitudinal DC conductivity mean-field Hall DC conductivity Introduction Landau ITP Nonlinear sigma model with topological term is defined on coset Dynamical variable – unitary matrix field Introduction-1

  3. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken n=m=0 n=m=1 n=1, m=N-1 2D disordered electron gas in magnetic field O(3) model model Introduction Landau ITP It contains Introduction-2

  4. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Introduction Landau ITP Dependent on (m, n) Independently on (m, n) • Order of plateau-plateau transitions • Critical exponents for plateau-plateau transitions • Massless chiral edge exictations • Quantum Hall effect, i.e. robust quantization of • Divergent correlation length at = k+1/2 Introduction-3

  5. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Introduction Wegner ’79 Pruisken ’85 Landau ITP Mass terms (linear and bilinear in Q operators) where with Introduction-4

  6. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Landau ITP Non-perturbative renormalization group equations where Perturbative resutls by E. Brezin, S. Hikami and J. Zinn-Justin ‘80 Euler constant Results-1

  7. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Landau ITP Nature of the plateau-plateau transition for different (m,n) FP at zero O(3) FP at nonzero Results-2

  8. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Landau ITP Nature of the plateau-plateau transition for different (m,n) Large m, n Small m, n Results-3

  9. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Landau ITP Quantum Hall Effect for m,n < 1 Renormalization group flow diagram Khmelnitzkii ’83 Pruisken ’83 Results-4

  10. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results QHE in 2DEG (n=m=0) Landau ITP Non-perturbative renormalization group equations Pruisken ’87 (4 times larger coefficient) Fixed point at Results-5

  11. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Plateau-plateau transition Landau ITP Linear environment of FP Divergent localization length Pruisken ‘88 Critical exponents relevant irrelevant Results-6

  12. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Multifractality Landau ITP Inverse participation ratio (IPR) Extended -- zero Localized – finite Wegner ‘79 It can be related with antisymmetric operator as Critical exponent for IPR Results-7 =2

  13. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Multifractality Landau ITP Generalized inverse participation ratio It can be related with higher order antisymmetric operators and written as Critical exponent All exponents are different! Results-8

  14. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Multifractality Landau ITP Legendre transform! New variable Singularity function The result (from NPRGEqs) is parabola Maximum at Results-9

  15. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Comparison with numerics Landau ITP Results-10

  16. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Quantum Hall effect for n=m >0 Landau ITP Localization length exponent Results-11

  17. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Quantum Hall effect for n=m >0 Results-12 Landau ITP Irrelevant exponent

  18. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Results Quantum Hall effect for n=m >0 Results-13 Landau ITP Anomalous dimensions

  19. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Action Landau ITP NL M action Mass terms where Derivation-1

  20. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Boundary conditions Derivation-2 Landau ITP Topological charge If at the boundary then C[Q] is integer valued Why should it be?

  21. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Boundary conditions Derivation-3 Landau ITP Change of variables where at the boundary

  22. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Boundary conditions Derivation-4 Landau ITP split

  23. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Effective action for the edge Derivation-5 Landau ITP where Background fields Effective action for the edge we can write where physical observables

  24. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Effective action for the edge Derivation-6 Landau ITP In the case of finite localization length then No renormalization of k! Skoric, Pruisken, Baranov ‘98 Robust quantization of Hall conductance

  25. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Bulk action Derivation-7 Landau ITP

  26. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Physical observables Derivation-8 Landau ITP conductances Pruisken ’87 masses

  27. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Physical observables Derivation-9 Generators of U(m+n) Landau ITP Specific choice of t Effective action

  28. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Physical observables Derivation-10 Landau ITP Generators Fiertz identity

  29. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Instantons Derivation-11 Landau ITP Instanton solution O(3) instanton Belavin Polyakov ‘75 Action on the instanton solution Finite

  30. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Instantons Derivation-12 Landau ITP then For where Logarithmic divergences in mass terms on the instanton solutions!?

  31. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Quantum fluctuations Derivation-13 Landau ITP where Stereographic projection

  32. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Quantum fluctuations Derivation-14 Landau ITP Spectrum Zero modes size position rotations Eigenfunctions Jacobi polynomials

  33. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Mass terms Derivation-15 Landau ITP Spatially varying masses Transformation preserves logarithms!

  34. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Mass terms Derivation-16 Landau ITP Linear terms

  35. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Pauli-Villars regularization Derivation-17 Landau ITP Set of parameters ‘t Hooft ‘76 Such that Replacement

  36. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Thermodynamic potential Derivation-18 Landau ITP where Quantities are exactly the same as one can obtain in perturbativerenormalization !!

  37. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Transformation Derivation-19 Landau ITP Transformation from curved space to flat space

  38. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Physical observables How is related with ? Derivation-20 Landau ITP where

  39. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Transformation Derivation-21 Landau ITP Local counterterms (‘t Hooft) The action becomes

  40. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Transformations Derivation-22 Landau ITP Renormalization by fluctuations Local counterterms (‘t Hooft) }

  41. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Transformation Derivation-23 Landau ITP where Prescription Similarly

  42. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Conductivities Derivation-24 Landau ITP Non-perturbative results for conductivities Hence (there is no dependence on !)

  43. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Transformation Derivation-25 Landau ITP Similarly where

  44. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Masses Derivation-26 Landau ITP Non-perturbative results for masses ( <0) where “Magnetization”

  45. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Masses Derivation-27 Landau ITP Perturbative results only are needed Hence ( <0) and

  46. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Masses Derivation-28 Landau ITP Non-perturbative results for masses ( >0) then

  47. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Derivation Masses Derivation-29 Landau ITP Non-perturbative results for masses Hence

  48. The instanton vacuum of generalized models I. Burmistrov and A.M.M. Pruisken Conclusions Conclusions-1 Landau ITP Non-perturbative (one instanton) results for beta and gamma (anomalous dimension) functions in generalized models Instanton analysis produces the main features of the QHE QHE in free electron gas (m=n=0) is not the special case of replica limit Instanton analysis provides estimation for critical exponents for plateau-plateau transitions The method lays the foundation for a non-perturbative analysis of the electron gas that includes the effects of electron-electron interaction

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