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

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

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

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

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

 Outline Nonperturbative Methods in Physics of Disorder

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Replica Trick: A Bit of Chronology Reconstruct through the replica limit  based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934 commutativity! Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985 “… the replica trick … suffers from a serious drawback: it is mathematically ill founded.” “… 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

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

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

0D limit: Disordered grain Quantum dot RMT Two Pitfalls

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

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

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

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

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

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

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

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

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

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

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

Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002)  Let’s do everything exactly !! 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 GUE N from fermionic replicas (EK, 2002)  Let’s do everything exactly !! duality   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, (2002) “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

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

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

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

Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002)

Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002)

Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002) Hankel determinant !! Darboux Theorem !!

Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002) (Positive) Toda Lattice Equation

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

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

Kyoto School’s Formalism Okamoto, Noumi, Yamada discovered a link between Toda Lattices and Painlevé equations No Photo Yet Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002)

Toda Lattice can be reduced to Painlevé equation Painlevé equation contains the replica index as 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  Loosely speaking: Correctness of a such an analytic continuation can independently be proven (details in the papers), but uniqueness …  Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002)

+ boundary conditions How it looks in the very end… Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002)

get exact answer ! How it looks in the very end… Exact Integrability in 0 Dimensions DoS in GUE N from fermionic replicas (EK, 2002) + boundary conditions

How it looks in the very end… Exact Integrability in 0 Dimensions DoS-DoS in GUE N from fermionic replicas (EK, 2002) + boundary conditions

How it looks in the very end… Exact Integrability in 0 Dimensions DoS in chGUE from fermionic replicas (EK, 2002) + boundary conditions

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

Established:  Conclusions Shown:  Derived:  Link between fermionic replica field theories in 0D and the theory of nonlinear integrable hierarchies Toda hierarchy of replica partition functions intimately related to the unitary symmetry of the replica field theory Exact nonperturbative correlation functions in the GUE, Ginibre’s ensemble of complex random matrices, and in chiral GUE (Dustermaat-Heckman theorem is violated) – by reduction of Toda Lattices to Painlevé equations combined with a “simple minded” analytic continuation

 symmetry classes (Pfaff Lattices)  Puzzle of bosonic replicas Questions left unanswered  What is about the uniqueness of analytic continuation? Is there exact integrability of 0D replica field theories with interaction ?

Workshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004 No Photo Yet