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Department of Applied Mathematics HAIT – Holon Academic Institute of Technology Holon, ISRAEL

H A I T. Towards exact integrability. of replica field theories. Eugene Kanzieper. Department of Applied Mathematics HAIT – Holon Academic Institute of Technology Holon, ISRAEL. Workshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics

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Department of Applied Mathematics HAIT – Holon Academic Institute of Technology Holon, ISRAEL

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  1. H A I T Towards exact integrability of replica field theories Eugene Kanzieper Department of Applied Mathematics HAIT – Holon Academic Institute of Technology Holon, ISRAEL Workshop onRandom Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004

  2. Based on: Replica field theories, Painlevé transcendents, and exact correlation functions, Phys. Rev. Lett. 89, 250201 (2002) Thanks to: Craig Tracy(UCal-Davis),Peter Forrester (UMelb) (for guiding through Painlevé literature) Discussions with: (SUNY) Alex Kamenev(UMin), Ady Stern (WIS),Jac Verbaarschot Supported by: Albert Einstein Minerva Centre for Theoretical Physics (Weizmann Institute of Science, Rehovot, Israel)

  3. Outline  Nonperturbative Methods in Physics of Disorder  Replica sigma models  Supersymmetric sigma model  Keldysh sigma model

  4. Outline  Nonperturbative Methods in Physics of Disorder  Why Replicas? What Are Replicas and The Replica Limit? Two Pitfalls: Analytic Continuation and (Un) controlled Approximations  Message: exact approach to replicas needed 

  5. Outline  Nonperturbative Methods in Physics of Disorder  Why Replicas? What Are Replicas and The Replica Limit? Two Pitfalls: Analytic Continuation and (Un) controlled Approximations  Towards Exact Integrability of Replica Field Theories in 0 Dimensions   Conclusions

  6. Outline Nonperturbative Methods in Physics of Disorder 

  7. Disordered grain Quantum dot Fermi energy mean scattering time  Statistical description: Ensemble of grains  Ensemble averaged observable ?..  Disorder as a perturbation

  8. Disordered grain Quantum dot IMPORTANT PHYSICS LOST  Whole issue of strong localisation

  9. Disordered grain Quantum dot F a i l u r e !  Whole issue of strong localisation  Weak disorder limit: long time particle evolution (times larger than the Heisenberg time)

  10. Disordered grain Quantum dot IMPORTANT PHYSICS LOST NO to perturbative treatment of disorder

  11. Field Theoretic Approaches  Replica sigma models(bosonic and fermionic) Wegner 1979 Larkin, Efetov, Khmelnitskii 1980 Finkelstein 1982  Supersymmetric sigma model Efetov 1982 finite dimensional matrix field  Keldysh sigma model Horbach, Schön 1990 Kamenev, Andreev 1999 symmetry

  12. A Typical Nonlinear Model Nonlinearly constrained matrix field of certain symmetries Generating functional Action

  13. Outline  Nonperturbative Methods in Physics of Disorder  Perturbative approach may become quite useless even in the weak disorder limit Nonperturbative approaches: the three formulations  Replica sigma models Supersymmetric sigma model  Keldysh sigma model 

  14. Outline  Nonperturbative Methods in Physics of Disorder  Why Replicas? What Are Replicas and the Replica Limit?

  15. Field Theoretic Approaches  Replica sigma models(bosonic and fermionic) disorder Wegner 1979 Larkin, Efetov, Khmelnitskii 1980 Finkelstein 1982 interaction disorder  Supersymmetric sigma model Efetov 1982 disorder  Keldysh sigma model interaction Horbach, Schön 1990 Kamenev, Andreev 1999 out-of-equilibrium

  16. Field Theoretic Approaches  Replica sigma models(bosonic and fermionic) disorder Wegner 1979 Larkin, Efetov, Khmelnitskii 1980 Finkelstein 1982 interaction A viable tool to treat an interplay between disorder and interaction!

  17. interaction Field Theoretic Approaches  Replica sigma models(bosonic and fermionic) disorder Wegner 1979 Larkin, Efetov, Khmelnitskii 1980 Finkelstein 1982 A viable tool to treat an interplay between disorder and interaction!

  18. Outline  Nonperturbative Methods in Physics of Disorder  Why Replicas? What Are Replicas and the Replica Limit?

  19. Replica trick  Mean level density out of  Looks easier (replica partition function)  Reconstruct through the replica limit commutativity! based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

  20. Replica trick  Density-density correlation function out of  Looks easier (replica partition function)  Reconstruct through the replica limit commutativity! based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

  21. Replica trick Word of caution: for more than two decades no one could rigorously implement it in mesoscopics !  Reconstruct through the replica limit commutativity! based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

  22. Replica Trick: A Bit of Chronology Bosonic Replicas Fermionic Replicas First Critique (RMT) 1985 1979 1980 Verbaarschot, Zirnbauer Larkin, Efetov,Khmelnitskii F. Wegner  Reconstruct through the replica limit commutativity! based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

  23. Replica Trick: A Bit of Chronology “… the replica trick … suffers from a serious drawback: it is mathematically ill founded.” Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985 “… the replica trick for disordered electron systems is limited to those regions of parameter space where the nonlinear sigma model can be evaluated perturbatively.” Another critique of the replica trick, M. Zirnbauer 1999  Reconstruct through the replica limit commutativity! based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

  24. Replica Trick: A Bit of Chronology ? Second Critique (RMT) Replica Symmetry Breaking (RMT) Bosonic Replicas Fermionic Replicas First Critique (RMT) SUSY Asymptotically Nonperturbative Results: 1985 1999 1979 1980 1982 1983 Efetov Verbaarschot, Zirnbauer Larkin, Efetov,Khmelnitskii Kamenev, Mézard Zirnbauer F. Wegner  Reconstruct through the replica limit commutativity! based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

  25. Outline  Nonperturbative Methods in Physics of Disorder Why Replicas? What Are Replicas and the Replica Limit?  Two Pitfalls: Analytic Continuation and (Un)controlled Approximations 

  26. 5.73 4.28 0.18 9.33 4.58 9.27 7.30 4.03 4.05 1.59 6.49 9.19 4.78 8.45 0.02 9.52 6.97 4.20 1.14 9.93 5.94 6.49 5.03 4.50 6.41 4.02 0.01 5.17 9.32 4.73 3.00 3.19 0.74 8.03 4.38 1.30 1.83 2.47 8.03 6.60 4.34 9.47 9.93 5.94 6.49 4.78 4.85 3.28 4.06 7.37 9.03 8.05 4.51 3.95 4.00 3.05 3.58 7.10 4.48 9.37 4.78 8.45 0.02 9.52 6.97 4.20 8.03 7.94 5.29 1.18 4.38 3.01 0.09 5.32 3.86 8.22 0.36 0.88 0.28 2.40 1.39 6.60 4.34 9.47 1.18 2.87 1.14 9.93 5.94 6.49 4.78 8.45 0.02 9.52 6.97 4.20 3.00 5.29 3.57 5.29 8.83 7.17 2.40 1.39 5.73 4.28 0.18 9.33 0.28 2.40 1.39 5.73 6.41 4.02 0.01 5.17 5.07 7.35 4.78 8.45 8.45 7.30 4.03 4.05 1.59 6.49 9.19 3.02 4.39 4.04 9.03 8.10 9.93 5.94 6.49 4.78 4.85 3.28 7.24 8.04 0.39 1.83 2.47 8.03 Two Pitfalls RMT Disordered grain Quantum dot 0D limit:

  27. (a) Analytic Continuation  Original recipe  Field theoretic realisation

  28. (a) Analytic Continuation van Hemmen and Palmer 1979 Verbaarschot and Zirnbauer 1985 Zirnbauer 1999 U n i q u e n e s s ?..

  29. (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)    Saddle point evaluation for matrices of large dimensions Replica Symmetry Breaking for “causal” saddle points known explicitly

  30. (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)  Analytic continuation…  Saddle point evaluation for matrices of large dimensions known explicitly breaks down at

  31. diverges for (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)  Analytic continuation… Kamenev, Mézard 1999 Zirnbauer 1999 In the vicinity n=0: Kanzieper 2004 (unpublished) Does NOT exist Is NOT unique: Re-enumerate Saddles!!  Saddle point evaluation for matrices of large dimensions breaks down at

  32. diverges for (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999) ..?  Analytic continuation… What’s the reason(s) for the failure ?  Saddle point evaluation for matrices of large dimensions breaks down at

  33. diverges for (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999) ..?  Analytic continuation…  Saddle point evaluation for matrices of large dimensions breaks down at

  34. diverges for (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999) ..?  Analytic continuation…  Saddle point evaluation for matrices of large dimensions breaks down at

  35. diverges for (b) (Un)Controlled Approximations made prior to analytic continuation DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999) ..?  Analytic continuation… It’s a bit too dangerous to make analytic continuation based on an approximate result !  Saddle point evaluation for matrices of large dimensions breaks down at

  36. Outline  Nonperturbative Methods in Physics of Disorder Why Replicas? What Are Replicas and the Replica Limit?   Two Pitfalls: Analytic Continuation and (Un)controlled Approximations  Towards Exact Integrability of Replica Field Theories in 0 Dimensions

  37. duality Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)    Saddle point evaluation for large matrices (KM, 1999)

  38. duality Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)    Let’s do everything exactly !! “There is of course little hope that the multi-dimensional integrals appearing in the replica formalism … can ever be evaluated non-perturbatively … ” Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985

  39. duality “There is of course little hope that the multi-dimensional integrals appearing in the replica formalism … can ever be evaluated non-perturbatively … ” Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985 Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)    Let’s do everything exactly !! Exact evaluation is possible as there exists an exact link between 0D replica field theories and the theory of nonlinear integrable hierarchies. EK: PRL 89, 250201 (2002)

  40. Gaston Darboux(1842-1917) born 1917 No Photo Yet Paul Painlevé (1863-1933) French Prime Minister September-November 1917 April-November 1925 Morikazu Toda

  41. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)    Let’s do everything exactly !!

  42. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)  Let’s do everything exactly !!

  43. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)

  44. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)

  45. Hankel determinant !! Darboux Theorem !! Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)

  46. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002) (Positive) Toda Lattice Equation

  47. S Y M M E T R Y Toda Lattice Hierarchy for replica partition functions is a fingerprint of exact integrability hidden in replica field theories ! Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)

  48. S Y M M E T R Y Does it help us perform analytic continuation ? from to Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002)

  49. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002) Kyoto School’s Formalism Okamoto, Noumi, Yamada discovered a link between Toda LatticesandPainlevé equations No Photo Yet

  50. Exact Integrability in 0 Dimensions DoS in GUEN from fermionic replicas (EK, 2002) Loosely speaking:  Toda Latticecan be reduced toPainlevé equation  Painlevé equation contains thereplica indexas a parameter in its coefficients  A “simple minded” analytic continuation of so- obtained replica partition function away from integers leads to a correct replica limit  Correctness of a such ananalytic continuation can independently be proven (details in the papers), but uniqueness …

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