QCD Dirac Spectra and Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007...

QCD Dirac Spectra and

Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007

Eugene KanzieperDepartment of Applied

MathematicsH.I.T. - Holon Institute of

TechnologyHolon 58102, Israel

Applied Mathematics

[ ]30

arXiv: 0704.2968 [cond-mat.dis-nn]

in collaboration with

Vladimir Al. Osipov

Bosonic ReplicasChiral GUE

and

Chiral GUE & Bosonic Replicas

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


[ Janik, Nowak, Papp & Zahed 1998; Osborn & Verbaarshot 1998; Guhr & Wilke 2001 ]

© G

uh

r &

Wilke

QCD

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


Applied Mathematics

[ ]27


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


Applied Mathematics

[ ]26

Field Theoretic Approaches to Disordered Systems with p-p

Interaction

Efetov 1982; Schwiete,

Efe

tov 2

004

SUSY FT

disorder

e-e

in

tera

cti

on

non-

equili

brium

♥

♥

Replica FT non

-

equili

brium

e-e

in

tera

cti

on

disorder

Wegner 1979; Larkin, Efetov,

Kh

meln

itskii 1

980;

Fin

kels

tein

1982

♠

♠

Keldysh FT

disorder

e-e

in

tera

cti

on non

-

equili

brium

Horbach, Schön 1990, and

Kam

en

ev,

An

dre

ev

1999

♣

♣

Interplay between disorderand p-p interaction

what is the problem and the tools ?



Applied Mathematics

[ ]25

Replica FT non

-

equili

brium

e-e

in

tera

cti

on

disorder

Wegner 1979; Larkin, Efetov,

Kh

meln

itskii 1

980;

Fin

kels

tein

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


Random

Matrice

s

© NBI

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


Applied Mathematics

[ ]23

excursion through the time

1979 1980

BosonicReplicas

FermionicReplicas

F. Wegner Larkin, Efetov&

Khmelnitskii

general framework &

RG in the context of

disordered systems

1985

First Critique (Random Matrices)

Verbaarschot

& Zirnba

uer

first attempt to treat

replica FT (RMT)

nonperturbatively

fermionic and bosonic

replicas brought different results

for


Applied Mathematics

[ ]22

1979 1980

BosonicReplicas

FermionicReplicas


Khmelnitskii

1985


Verbaarschot

& Zirnba

uer

ill founded

first attempt to treat

replica FT (RMT)

nonperturbatively

fermionic and bosonic

replicas brought different results

for

excursion through the timeChiral GUE & Bosonic Replicas

?

Applied Mathematics

[ ]21

1979 1980

BosonicReplicas

FermionicReplicas


Khmelnitskii

1985


Verbaarschot

& Zirnba

uer

1999

Kamenev

& Mézard

Replica SymmetryBreaking (RMT)

firstnonperturbative

results

Second Critique199

9Zirnbauer

?“KM procedure

ismathematically

questionable …”

Efetov’s SUSY FT

1982 1983Nonperturbativ

eRMT results

!

40:1

2002

Exact Replica

s

EK

ferm

ion

ic

rep

licas

2007

Exact Bosonic Replica

s

VO,EK

2003

SUSY Replicas

Splittorff &Verbaarsch

ot

excursion through the timeChiral GUE & Bosonic Replicas

56:1

GUE

Applied Mathematics

[ ]20


Fermionic replica FT:

Bosonic replica FT:



GUE

Applied Mathematics

[ ]19


How to reconcile ? analytic continuation

integration over matrices of noninteger dimensions



Applied Mathematics

[ ]18


integration over matrices of noninteger dimensions Continuous geometry of replica σ–

models

Discrete geometry of SUSY/Keldysh σ–models

continuous geometryChiral GUE & Bosonic Replicas

GUE

Applied Mathematics

[ ]17

outline reminder


On the asymmetry in performance of fermionic and bosonic replicas and continuous geometry of replica FTs



Applied Mathematics

[ ]16

asymmetry, continuous geometry


• 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 ReplicasEK

, 2002

Exact

Replicas !!

Ap

pro

xim

ate

R

ep

licas

Applied Mathematics

[ ]15


• 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

Ap

pro

xim

ate

R

ep

licas

do not analytically continue from an approximate result !!

Applied Mathematics

[ ]14

major fault



do not analytically continue from an approximate result !!

Applied Mathematics

[ ]13


(the major fault of Kamenev & Mezard treatment in 1999)

major fault


Applied Mathematics

[ ]12




Integrability of (bosonic) replica field theories: Microscopic density of states in chGUE

Chiral GUE & Bosonic Replicas outline reminder

Applied Mathematics

[ ]11

integrability of replicas: general theory

Goal: Nonperturbative evaluation of this RPF

Object: Replica partition function (RPF)


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 1990Adler, Shiota & van Moerbeke 1995

Deform !!

Study !!

Project !!

!

Nonlinear differential

equation for RPF

Applied Mathematics

[ ]10




Method: “Deform and study !!”

Date, Kashiwara, Jimbo & Miwa 1983 Mironov & Morozov 1990Adler, 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

Applied Mathematics

[ ]09







Kadomtsev-Petviashvili hierarchy modified KP hierarchy multicomponent KP hierarchy Toda Lattice hierarchy

First Equation of the KP Hierarchy

in the t –space

Applied Mathematics

[ ]08









more of

an art

Calculated in

terms of

Applied Mathematics

[ ]07









Projection onto and


t–Toda → Toda Lattice in physical parameterst–KP Eq → Painlevé-like in physical parameters

Applied Mathematics

[ ]06

example: microscopic density in chGUE

The chGUE model


def: bosonic partition function

bosonic partition function after replica mapping

Applied Mathematics

[ ]05


The chGUE model


bosonic partition function after replica mapping

general theory applies

Applied Mathematics

[ ]04


The chGUE model


general theory applies







Projection onto and


t–Toda → Toda Lattice in physical parameterst–KP Eq → Painlevé-like in physical parameters

First KP equation

Virasoro constraints

nonlinear differential equation for

nonlinear differential

Applied Mathematics

[ ]03


The chGUE model


nonlinear differential equation for

+ boundary conditions

chGUE bosonic partition function after replica mapping

Bottom

Line Bosonic

replicas !!

Conclusions

Applied Mathematics

[ ]02

conclusions


Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007

Vladimir Al. OsipovEugene Kanzieper

Applied Mathematics

[ ]01

arXiv: 0704.2968 [cond-mat.dis-nn]

Bosonic ReplicasChiral GUE

and


QCD Dirac Spectra and Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007...

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Transcript of QCD Dirac Spectra and Random Matrix Theory: Recent Applications, Niels Bohr Institute, May 8, 2007...