1. QCD Dirac Spectra and Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007 Eugene Kanzieper Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel Applied Mathematics [ ] 30  arXiv: 0704.2968 [cond-mat.dis-nn]  in collaboration with Vladimir Al. Osipov Bosonic Replicas Chiral GUE and Chiral GUE & Bosonic Replicas

2. Applied Mathematics outline  Nonperturbative description of disordered systems, preferably in presence of p-p interaction © A. M. Chang, Duke Univ [ ] 29   What is the problem and available theoretical tools ? Supersymmetry FT  Replica FT  Keldysh FT  Chiral GUE & Bosonic Replicas  [ Janik, Nowak, Papp & Zahed 1998; Osborn & Verbaarshot 1998; Guhr & Wilke 2001 ] © Guhr & Wilke QCD

3. Applied Mathematics [ ] 28   What is the problem and available theoretical tools ?  Why replicas ? What are the replicas ? What does make them so different from other field theories ? Supersymmetry FT  Replica FT  Keldysh FT   Continuous geometry of replica σ – models Discrete geometry of SUSY and Keldysh  Supersymmetry FT  Keldysh FT  outline Chiral GUE & Bosonic Replicas 

4. Applied Mathematics [ ] 27   What is the problem and available theoretical tools ?  On the asymmetry in performance of fermionic and bosonic replicas and the continuous geometry of replica FTs: GUE  Why replicas ? What are the replicas ? What does make them so different from other field theories ?  Integrability of ( bosonic) replica field theories: Microscopic density of states in chGUE  t - deformed replica partition function  Bilinear identity  Virasoro constraints  KP hierarchy  m-KP hierarchy  Toda Lattice hierarchy  Painlevé and Chazy equations  Conclusions outline Chiral GUE & Bosonic Replicas 

5. Applied Mathematics [ ] 26  Field Theoretic Approaches to Disordered Systems with p-p Interaction  Efetov 1982; Schwiete, Efetov 2004 SUSY FT disorder e-e interaction non- equilibrium ♥ ♥ Replica FT non- equilibrium e-e interaction disorder Wegner 1979; Larkin, Efetov, Khmelnitskii 1980; Finkelstein 1982 ♠ ♠ Keldysh FT disorder e-e interaction non- equilibrium Horbach, Schön 1990, and Kamenev, Andreev 1999 ♣ ♣ Interplay between disorder and p-p interaction  what is the problem and the tools ?  What is the problem and available theoretical tools ? Chiral GUE & Bosonic Replicas

6. Applied Mathematics [ ] 25  Replica FT non- equilibrium e-e interaction disorder Wegner 1979; Larkin, Efetov, Khmelnitskii 1980; Finkelstein 1982 ♠ ♠  Replica FT is a viable tool to treat an interplay between disorder and the p-p interaction  why replicas ?  What is the problem and available theoretical tools ?  Why replicas ?  Quite a remote goal  Sorting out controversies surrounding replica field theories in the RMT limit (interaction off) Field Theoretic Approaches to Disordered Systems with p-p Interaction  Chiral GUE & Bosonic Replicas Random Matrices © NBI

7. Applied Mathematics [ ] 24   what are the replicas ?  What is the problem and available theoretical tools ?  Why replicas ? What are the replicas ? What does make them so different from other field theories ?  What are the replicas? No p-p interaction  Single particle picture  Hamiltonian modelled by a random matrix   Mean level density out of one-point Green function based on Edwards and Anderson 1975; Hardy, Littlewood and Pólya 1934  Replica partition function  Reconstruct through the replica limit commutativity !! T: bosonic replicas: fermionic replicas: Word of caution: For more than two decades no one could rigorously implement the replica method (or trick?!) in mesoscopics Chiral GUE & Bosonic Replicas

8. Applied Mathematics [ ] 23   excursion through the time 1979 1980 Bosonic Replicas Fermionic Replicas F. Wegner Larkin, Efetov & Khmelnitskii  general framework & RG in the context of disordered systems 1985 First Critique (Random Matrices) Verbaarschot & Zirnbauer  first attempt to treat replica FT (RMT) nonperturbatively  fermionic and bosonic replicas brought different results for Chiral GUE & Bosonic Replicas

9. Applied Mathematics [ ] 22  1979 1980 Bosonic Replicas Fermionic Replicas F. Wegner Larkin, Efetov & Khmelnitskii 1985 First Critique (Random Matrices) Verbaarschot & Zirnbauer ill founded  first attempt to treat replica FT (RMT) nonperturbatively  fermionic and bosonic replicas brought different results for  excursion through the time Chiral GUE & Bosonic Replicas ?

10. Applied Mathematics [ ] 21  1979 1980 Bosonic Replicas Fermionic Replicas F. Wegner Larkin, Efetov & Khmelnitskii 1985 First Critique (Random Matrices) Verbaarschot & Zirnbauer 1999 Kamenev & Mézard Replica Symmetry Breaking (RMT) first nonperturbative results Second Critique 1999 Zirnbauer ? “KM procedure is mathematically questionable …” Efetov’s SUSY FT 1982 1983 Nonperturbative RMT results ! 40 : 1 2002 Exact Replicas EK fermionic replicas 2007 Exact Bosonic Replicas VO, EK 2003 SUSY Replicas Splittorff & Verbaarschot  excursion through the time Chiral GUE & Bosonic Replicas 56 : 1

11. GUE Applied Mathematics [ ] 20   Why replicas ? What are the replicas ? What does make them so different from other field theories ? Fermionic replica FT: Bosonic replica FT: Chiral GUE & Bosonic Replicas  what are the replicas ?

12. GUE Applied Mathematics [ ] 19   Why replicas ? What are the replicas ? What does make them so different from other field theories ? How to reconcile ?  analytic continuation  integration over matrices of noninteger dimensions Chiral GUE & Bosonic Replicas  what are the replicas ?

13. Applied Mathematics [ ] 18   Why replicas ? What are the replicas ? What does make them so different from other field theories ?  integration over matrices of noninteger dimensions Continuous geometry of replica σ – models  Discrete geometry of SUSY/Keldysh σ – models  continuous geometry Chiral GUE & Bosonic Replicas GUE

14. Applied Mathematics [ ] 17   outline reminder  What is the problem and available theoretical tools ?  On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs  Why replicas ? What are the replicas ? What does make them so different from other field theories ? Chiral GUE & Bosonic Replicas

15. Applied Mathematics [ ] 16   asymmetry, continuous geometry  On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs The equivalent saddles bring the same DoS without taking the replica limit The equivalent saddles bring totally different DoS in the replica limit Which saddle is correct ? No a-priori way to answer !! GUE: fermionic replicas a-la KM (DoS) The saddle point approach fails to accommodate the true, continuous geometry of fermionic replica field theories !! vs Chiral GUE & Bosonic Replicas EK, 2002 Exact Replicas !! Approximate Replicas

16. Applied Mathematics [ ] 15   On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs The replica limit brings no oscillations in DoS GUE: bosonic replicas a-la KM (DoS) Bosonic replicas are deficient… (Asymmetry !!) Are Bosonic Replicas Faulty ? Chiral GUE & Bosonic Replicas  asymmetry, continuous geometry Approximate Replicas

17. do not analytically continue from an approximate result !! Applied Mathematics [ ] 14   major fault  On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs Chiral GUE & Bosonic Replicas

18. do not analytically continue from an approximate result !! Applied Mathematics [ ] 13   On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs (the major fault of Kamenev & Mezard treatment in 1999)  major fault Chiral GUE & Bosonic Replicas

19. Applied Mathematics [ ] 12   What is the problem and available theoretical tools ?  On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs  Why replicas ? What are the replicas ? What does make them so different from other field theories ?  Integrability of (bosonic) replica field theories: Microscopic density of states in chGUE Chiral GUE & Bosonic Replicas  outline reminder

20. Applied Mathematics [ ] 11   integrability of replicas: general theory  Goal: Nonperturbative evaluation of this RPF  Object: Replica partition function (RPF) Chiral GUE & Bosonic Replicas Dyson’s β = 2 symmetry multi(band) structure “Confinement” potential (allowed to depend on n ) accommodates physical parameters of the theory  Result: Nonlinear differential equation for RPF containing the replica index as a parameter  Method: “Deform and study !!” Date, Kashiwara, Jimbo & Miwa 1983 Mironov & Morozov 1990 Adler, Shiota & van Moerbeke 1995 Deform !! Study !! Project !! ! Nonlinear differential equation for RPF

21. Applied Mathematics [ ] 10   integrability of replicas: general theory  Object: Replica partition function (RPF) Chiral GUE & Bosonic Replicas  Method: “Deform and study !!” Date, Kashiwara, Jimbo & Miwa 1983 Mironov & Morozov 1990 Adler, Shiota & van Moerbeke 1995 Study !! Describes response of the quantum system to an infinite dimensional perturbation that preserves the system symmetry – Integrable Hierarchies Reflect invariance of the tau-function under the change of integration variables – Loop Equations  First ingredient: Bilinear identity  Second ingredient: (Linear) Virasoro constraints  Projection onto and Generates nonlinear differential equations and hierarchies for RPF in terms of physical parameters

22. Applied Mathematics [ ] 09   integrability of replicas: general theory  Object: Replica partition function (RPF) Chiral GUE & Bosonic Replicas  Method: “Deform and study !!” Describes response of the quantum system to an infinite dimensional perturbation that preserves the system symmetry – Integrable Hierarchies  First ingredient: Bilinear identity  Kadomtsev-Petviashvili hierarchy  modified KP hierarchy  multicomponent KP hierarchy  Toda Lattice hierarchy First Equation of the KP Hierarchy in the t –space

23. Applied Mathematics [ ] 08   integrability of replicas: general theory  Object: Replica partition function (RPF) Chiral GUE & Bosonic Replicas  Method: “Deform and study !!” Describes response of the quantum system to an infinite dimensional perturbation that preserves the system symmetry – Integrable Hierarchies  First ingredient: Bilinear identity Reflect invariance of the tau-function under the change of integration variables – Loop Equations  Second ingredient: (Linear) Virasoro constraints more of an art Calculated in terms of

24. Applied Mathematics [ ] 07   integrability of replicas: general theory  Object: Replica partition function (RPF) Chiral GUE & Bosonic Replicas  Method: “Deform and study !!” Describes response of the quantum system to an infinite dimensional perturbation that preserves the system symmetry – Integrable Hierarchies  First ingredient: Bilinear identity Reflect invariance of the tau-function under the change of integration variables – Loop Equations  Second ingredient: (Linear) Virasoro constraints  Projection onto and Generates nonlinear differential equations and hierarchies for RPF in terms of physical parameters t – Toda → Toda Lattice in physical parameters t – KP Eq → Painlevé-like in physical parameters

25. Applied Mathematics [ ] 06   example: microscopic density in chGUE  The chGUE model Chiral GUE & Bosonic Replicas def: bosonic partition function bosonic partition function after replica mapping

26. Applied Mathematics [ ] 05   example: microscopic density in chGUE  The chGUE model Chiral GUE & Bosonic Replicas bosonic partition function after replica mapping general theory applies

27. Applied Mathematics [ ] 04   example: microscopic density in chGUE  The chGUE model Chiral GUE & Bosonic Replicas general theory applies  Object: Replica partition function (RPF)  Method: “Deform and study !!” Describes response of the quantum system to an infinite dimensional perturbation that preserves the system symmetry – Integrable Hierarchies  First ingredient: Bilinear identity Reflect invariance of the tau-function under the change of integration variables – Loop Equations  Second ingredient: (Linear) Virasoro constraints  Projection onto and Generates nonlinear differential equations and hierarchies for RPF in terms of physical parameters t – Toda → Toda Lattice in physical parameters t – KP Eq → Painlevé-like in physical parameters First KP equation Virasoro constraints nonlinear differential equation for nonlinear differential

28. Applied Mathematics [ ] 03   example: microscopic density in chGUE  The chGUE model Chiral GUE & Bosonic Replicas nonlinear differential equation for + boundary conditions chGUE bosonic partition function after replica mapping Bottom Line Bosonic replicas !!

29.  Conclusions Applied Mathematics [ ] 02   conclusions Chiral GUE & Bosonic Replicas

30. Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007 Vladimir Al. Osipov Eugene Kanzieper Applied Mathematics [ ] 01  arXiv: 0704.2968 [cond-mat.dis-nn]  Bosonic Replicas Chiral GUE and Chiral GUE & Bosonic Replicas