Department of Applied Mathematics HAIT – Holon Academic Institute of Technology Holon, ISRAEL Workshop on Random Matrix Theory: Condensed Matter, Statistical

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  • Department of Applied MathematicsHAIT Holon Academic Institute of TechnologyHolon, ISRAELWorkshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004Towards exact integrabilityof replica field theoriesEugene Kanzieper

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

  • OutlineNonperturbative Methods in Physics of DisorderReplica sigma modelsSupersymmetric sigma modelKeldysh sigma model

  • Why Replicas? What Are Replicas and The Replica Limit? Two Pitfalls: Analytic Continuation and (Un) controlled ApproximationsMessage: exact approach to replicas neededOutlineNonperturbative Methods in Physics of Disorder

  • Towards Exact Integrability of Replica Field Theories in 0 DimensionsConclusionsWhy Replicas? What Are Replicas and The Replica Limit? Two Pitfalls: Analytic Continuation and (Un) controlled ApproximationsOutlineNonperturbative Methods in Physics of Disorder

  • OutlineNonperturbative Methods in Physics of Disorder

  • Statistical description: Ensemble of grainsEnsemble averaged observableDisorder as a perturbation?..mean scattering timeFermi energy

  • IMPORTANTPHYSICSLOSTWhole issue of strong localisation

  • Weak disorder limit: long time particle evolution (times larger than the Heisenberg time)Whole issue of strong localisation

  • to perturbative treatment of disorderIMPORTANTPHYSICSLOSTNO

  • Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982Supersymmetric sigma modelEfetov 1982Keldysh sigma modelHorbach, Schn 1990 Kamenev, Andreev 1999Field Theoretic Approachesfinite dimensionalmatrix field symmetry

  • A Typical NonlinearNonlinearly constrained matrix field of certain symmetriesGenerating functionalActionModel

  • OutlineNonperturbative Methods in Physics of DisorderReplica sigma modelsSupersymmetric sigma modelKeldysh 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? OutlineNonperturbative Methods in Physics of Disorder

  • Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982Supersymmetric sigma modelEfetov 1982Keldysh sigma modelHorbach, Schn 1990 Kamenev, Andreev 1999Field Theoretic Approachesdisorderinteractioninteractionout-of-equilibriumdisorderdisorder

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

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

  • Why Replicas? What Are Replicas and the Replica Limit? OutlineNonperturbative Methods in Physics of Disorder

  • Replica trickMean level density out of Looks easier (replica partition function)

    Reconstruct through the replica limitbased on Edwards, Anderson 1975; Hardy, Littlewood, Plya 1934commutativity!

  • Replica trickDensity-density correlation function out of Looks easier (replica partition function)

    Reconstruct through the replica limitbased on Edwards, Anderson 1975; Hardy, Littlewood, Plya 1934commutativity!

  • Reconstruct through the replica limitbased on Edwards, Anderson 1975; Hardy, Littlewood, Plya 1934commutativity!Replica trickWord of caution:

    for more than two decades no one could rigorously implement it in mesoscopics !

  • Replica Trick: A Bit of Chronology197919801985Bosonic ReplicasFermionicReplicasF. WegnerLarkin, Efetov, KhmelnitskiiFirst Critique(RMT)Verbaarschot, ZirnbauerReconstruct through the replica limitbased on Edwards, Anderson 1975; Hardy, Littlewood, Plya 1934commutativity!

  • Replica Trick: A Bit of ChronologyReconstruct through the replica limitbased on Edwards, Anderson 1975; Hardy, Littlewood, Plya 1934commutativity!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 Chronology197919801985Bosonic ReplicasFermionicReplicasF. WegnerLarkin, Efetov, KhmelnitskiiFirst Critique(RMT)Verbaarschot, ZirnbauerReconstruct through the replica limitbased on Edwards, Anderson 1975; Hardy, Littlewood, Plya 1934commutativity!1999Kamenev, MzardReplica SymmetryBreaking (RMT)Second Critique(RMT)ZirnbauerAsymptotically Nonperturbative Results: ? 19821983SUSYEfetov

  • Two Pitfalls: Analytic Continuation and (Un) controlled ApproximationsWhy Replicas? What Are Replicas and the Replica Limit? OutlineNonperturbative Methods in Physics of Disorder

  • 0D limit:Disordered grainQuantum dotRMTTwo Pitfalls

  • (a) Analytic ContinuationOriginal recipeField theoretic realisation

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

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsknown explicitlyReplica Symmetry Breaking for causal saddle points

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsknown explicitlybreaks down at Analytic continuation

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsbreaks down at Does NOT existIs NOT unique: Re-enumerate Saddles!!Analytic continuationIn the vicinity n=0:Kamenev, Mzard 1999Zirnbauer 1999Kanzieper 2004 (unpublished)

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsbreaks down at Analytic continuation..?Whats the reason(s) for the failure ?

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsbreaks down at Analytic continuation..?

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsbreaks down at Analytic continuation..?

  • (b) (Un) Controlled Approximationsmade prior to analytic continuationDoS in GUEN from fermionic replicas a-l Kamenev-Mzard (1999)Saddle point evaluation for matrices of large dimensionsbreaks down at Analytic continuation..?Its a bit too dangerous to make analyticcontinuation based on an approximate result !

  • Two Pitfalls: Analytic Continuation and (Un) controlled ApproximationsWhy Replicas? What Are Replicas and the Replica Limit? OutlineNonperturbative Methods in Physics of DisorderTowards Exact Integrability of Replica Field Theories in 0 Dimensions

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)Saddle point evaluation for large matrices (KM, 1999)

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)Lets 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

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)Lets 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)

  • Paul Painlev(1863-1933)Gaston Darboux (1842-1917)No PhotoYetMorikazu Todaborn 1917French Prime MinisterSeptember-November 1917April-November 1925

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)Lets do everything exactly !!

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)Lets do everything exactly !!

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (EK, 2002)

  • Exact Integrability in 0 DimensionsDoS in GUEN from fermionic replicas (