Role of Anderson localization in Role of Anderson localization in the QCD phase transitionsthe QCD phase transitions

Antonio M. García-García

ag3@princeton.eduPrinceton University

ICTP, Trieste

We investigate in what situations Anderson localization may be relevant in the We investigate in what situations Anderson localization may be relevant in the context of QCD. At the chiral phase transition we provide compelling evidence context of QCD. At the chiral phase transition we provide compelling evidence from lattice and phenomenological instanton liquid models that the QCD Dirac from lattice and phenomenological instanton liquid models that the QCD Dirac operator undergoes a metal - insulator transition similar to the one observed in operator undergoes a metal - insulator transition similar to the one observed in

a disordered conductor. This suggests that Anderson localization plays a a disordered conductor. This suggests that Anderson localization plays a fundamental role in the chiral phase transition. fundamental role in the chiral phase transition.

In collaboration with In collaboration with James OsbornJames Osborn PRD,75 (2007) 034503 ,PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002NPA, 770, 141 (2006) PRL 93 (2004) 132002

Conclusions:Conclusions:

nnnQCD iD

0At the same T that the Chiral Phase transition

"A metal-insulator transition in the Dirac operator "A metal-insulator transition in the Dirac operator induces the QCD chiral phase transition"induces the QCD chiral phase transition"

n

n

undergo a metal - insulatormetal - insulator transition

Outline:Outline:

1. Introduction to disordered systems and Anderson localization. 1. Introduction to disordered systems and Anderson localization.

2. QCD vacuum as a conductor. QCD vacuum as a disordered 2. QCD vacuum as a conductor. QCD vacuum as a disordered medium. Dyakonov - Petrov ideas. medium. Dyakonov - Petrov ideas.

3. QCD phase transitions.3. QCD phase transitions.

4. Role of localization in the QCD phase transitions. Results from 4. Role of localization in the QCD phase transitions. Results from instanton liquid models and lattice.instanton liquid models and lattice.

V(x)

X

Ea

Eb

Ec

Anderson (1957):Anderson (1957):

1. 1. How does the quantum dynamics depend on How does the quantum dynamics depend on disorder?disorder?

2. How does the quantum dynamics depend on 2. How does the quantum dynamics depend on energy? energy?

0

A five minutes course A five minutes course on disordered systemson disordered systems

The study of the quantum The study of the quantum motion in a random potentialmotion in a random potential

Insulator:Insulator: For d < 3 or, in d > 3, for strong disorder. Classical diffusion For d < 3 or, in d > 3, for strong disorder. Classical diffusion eventually stops due to destructive interference (Anderson localization). eventually stops due to destructive interference (Anderson localization).

Metal:Metal: For For d > 2 and weak disorder quantum effects do not alter d > 2 and weak disorder quantum effects do not alter significantly the classical diffusion. Eigenstates are delocalized.significantly the classical diffusion. Eigenstates are delocalized.

Metal-Insulator transition: Metal-Insulator transition: For d > 2 in a certain window of For d > 2 in a certain window of energies and disorder. Eigenstates are multifractal.energies and disorder. Eigenstates are multifractal.

Quantum dynamics according to Quantum dynamics according to the one the one

parameter scaling theoryparameter scaling theory

<r2

>

a = ?

Dquan=f(d,W)?

t

Dclast

Dquan

t

Dquanta

Sridhar,et.al

Insulator Metal

How are these different regimes characterized?How are these different regimes characterized?

sesP

D

Poisson

Insulator

)(

0~

)(2

22 ~

)( Asse~sP

dD

GOE

Metal

1. Eigenvector statistics:

2. Eigenvalue statistics:

2~)(4 Ddd

nd LrdrLIPR

i

iissP /)( 1

nnnnnnQCD EHiD

Altshuler, Altshuler, Boulder lecturesBoulder lectures

QCD : The Theory of the strong interactionsQCD : The Theory of the strong interactions

HighHigh EnergyEnergy g << 1 Perturbativeg << 1 Perturbative

1. Asymptotic freedom Quark+gluons, Well understoodQuark+gluons, Well understood

Low Energy Low Energy g ~ 1 Lattice simulationsg ~ 1 Lattice simulations The world around usThe world around us

2. Chiral symmetry breaking2. Chiral symmetry breaking

Massive constituent quark Massive constituent quark

3. Confinement3. Confinement Colorless hadronsColorless hadrons

How to extract analytical information?How to extract analytical information? Instantons , Monopoles, Instantons , Monopoles, VorticesVortices

rrarV /)(

3)240(~ MeV

Instantons:Instantons: Non perturbative solutions of the classical Yang Mills Non perturbative solutions of the classical Yang Mills equation. Tunneling between classical vacua. equation. Tunneling between classical vacua.

1. Dirac operator has a zero mode in the field of an instanton1. Dirac operator has a zero mode in the field of an instanton

2. Spectral properties of the smallest eigenvalues of the Dirac operator 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons are controled by instantons

3. 3. Spectral properties related to chiSB. Banks-Casher relationSpectral properties related to chiSB. Banks-Casher relation

QCD at T=0, instantons and chiSB QCD at T=0, instantons and chiSB tHooft, Polyakov, Callan, Gross, Shuryak, tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaalDiakonov, Petrov,VanBaal

300 /10 rrψrDψgA+=D ins

μμ

V

m

imdmDTr

V mm

)(lim

)()(

10

1

Multiinstanton vacuum?Multiinstanton vacuum?

Problem:Problem: Non linear equations Non linear equations No superposition No superposition

Sol:Sol: Variational principles(Dyakonov,Petrov), Instanton liquid (Shuryak) Variational principles(Dyakonov,Petrov), Instanton liquid (Shuryak)

Typical size and some aspects of the interactions are fixedTypical size and some aspects of the interactions are fixed

1. ILM explains the chiSB1. ILM explains the chiSB

2. Describe non perturbative effects in hadronic correlation 2. Describe non perturbative effects in hadronic correlation functions functions (Shuryak,Schaefer,Verbaarchot)(Shuryak,Schaefer,Verbaarchot)

3 No confinement.3 No confinement.

3

34 )(

)ˆ(~)()(R

RuizxiDzxxdT AIAAIIIA

3

2/1

)240(2

31MeV

V

NNc

Instanton liquid models T = 0Instanton liquid models T = 0

Metal Metal An electron initially bounded to a single atom gets delocalized due An electron initially bounded to a single atom gets delocalized due

to the overlapping with nearest neighbors.to the overlapping with nearest neighbors.

QCD VacuumQCD Vacuum Zero modes initially bounded to an instanton get delocalized due Zero modes initially bounded to an instanton get delocalized due

to the overlapping with the rest of zero modes. to the overlapping with the rest of zero modes. (Diakonov and (Diakonov and Petrov)Petrov)

Impurities Impurities Instantons Instantons ElectronElectron QuarksQuarks

Differences Differences Dis.Sys:Dis.Sys: Exponential decay N Exponential decay Nearest earest neighborsneighbors QCD vacuumQCD vacuum Power law decayPower law decay Long range hopping! Long range hopping!

QCD vacuum as a conductor (T =0)QCD vacuum as a conductor (T =0)

QCD vacuum as a disordered QCD vacuum as a disordered conductorconductor

Instanton positions and color orientations varyInstanton positions and color orientations vary

Impurities Impurities Instantons Instantons Electron Electron

QuarksQuarksT = 0 long range hopping 1/RT = 0 long range hopping 1/R = 3<4 = 3<4

Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik

AGG and Osborn, AGG and Osborn, PRL, 94 (2005) 244102PRL, 94 (2005) 244102

QCD vacuum is a conductor for any density of instantonsQCD vacuum is a conductor for any density of instantons

QCD at finite T: Phase transitionsQCD at finite T: Phase transitions

Quark- Gluon Plasma perturbation theory only for T>>Tc

J. Phys. G30 (2004) S1259

At which temperature does the transition occur ? What is the nature of transition ?

Péter Petreczky Péter Petreczky

Deconfinement and chiral restorationDeconfinement and chiral restoration

Deconfinement: Confining potential vanishes.

Chiral Restoration:Matter becomes light.

How to explain these transitions?

1. Effective model of QCD close to the phase transition (Wilczek,Pisarski,Yaffe):

Universality, epsilon expansion.... too simple?

2. QCD but only consider certain classical solutions (t'Hooft): Instantons (chiral), Monopoles and vortices (confinement). Instanton do not dissapear at the transiton (Shuryak,Schafer).

We propose that quantum interference and tunneling, namely, Anderson Anderson localization plays an important role. localization plays an important role. Nuclear Physics A, 770, 141 (2006)Nuclear Physics A, 770, 141 (2006)

C. Gattringer, M. Gockeler, et.al. Nucl. Phys. B618, 205 (2001),R.V. Gavai, S. Gupta et.al, PRD 65, 094504 (2002), M.

Golterman and Y. Shamir, Phys. Rev. D 68, 074501 (2003), V. Weinberg, E.-M. Ilgenfritz, et.al, PoS { LAT2005}, 171 (2005), hep-lat 0705.0018, I. Horvath, N. Isgur, J. McCune, and H. B. Thacker, Phys. Rev. D65, 014502 (2002), J. Greensite, S. Olejnik et.al., Phys. Rev. D71, 114507 (2005). V. G. Bornyakov, E.-M. Ilgenfritz, 07064206

They must be related but nobody* knows exactly how

0~0L

1. Zero modes are localized in space but oscillatory in time.1. Zero modes are localized in space but oscillatory in time.

2. Hopping amplitude restricted to neighboring instantons.2. Hopping amplitude restricted to neighboring instantons.

3. Since T3. Since TIAIA is short range there must exist a T = T is short range there must exist a T = TLLsuch that a metal insulator transition takes such that a metal insulator transition takes place. place. (Dyakonov,Petrov)(Dyakonov,Petrov)

4. The chiral phase transition occurs at T=T4. The chiral phase transition occurs at T=Tc.c.

Localization and chiral transition are related if:Localization and chiral transition are related if:

1. T1. TLL = T = Tc . c .

2. The localization transition occurs at the origin 2. The localization transition occurs at the origin (Banks-Casher)(Banks-Casher)

““This is valid beyond the instanton picture provided that TThis is valid beyond the instanton picture provided that TIAIA is short range and the vacuum is is short range and the vacuum is disordered enough”disordered enough”

0

Instanton liquid model at finite T Instanton liquid model at finite T

)exp()( TRR

)exp(~ ATRTIA

nnnQCD iD

At Tc

but also the low lying,

"A metal-insulator transition in the Dirac operator "A metal-insulator transition in the Dirac operator induces the chiral phase transition "induces the chiral phase transition "

n

n

undergo a metal-insulator transition.

Main ResultMain Result

0)(

lim0

V

mmm

Signatures of a metal-insulator transitionSignatures of a metal-insulator transition1. Scale invariance of the spectral correlations.

A finite size scaling analysis is then carried out to determine the transition point.

2.

3. Eigenstates are multifractals.

)1(2

~)( qDdq

n

qLrdr

Skolovski, Shapiro, Altshuler

1~)(

1~)(

sesP

sssPAs

Mobility edge Anderson transition

varvar

dssPssss nn )(var22

ILM with 2+1 massless flavors,

We have observed a metal-insulator transition at T ~ 125 Mev

Spectrum is scale invariant

ILM, close to the origin, 2+1 ILM, close to the origin, 2+1 flavors, N = 200flavors, N = 200

Metal Metal insulator insulator transitiontransition

ILM Nf=2 massless. Eigenfunction ILM Nf=2 massless. Eigenfunction statisticsstatistics

AGG and J. Osborn, 2006 AGG and J. Osborn, 2006

Instanton liquid model Nf=2, maslessInstanton liquid model Nf=2, masless Localization versus Localization versus

chiral transitionchiral transition

Chiral and localizzation transition occurs at the same temperatureChiral and localizzation transition occurs at the same temperature

Lattice QCD Lattice QCD AGG, J. Osborn, PRD, AGG, J. Osborn, PRD,

20072007

1. Simulations around the chiral phase transition 1. Simulations around the chiral phase transition T T

2. Lowest 64 eigenvalues 2. Lowest 64 eigenvalues

QuenchedQuenched

1. Improved gauge action1. Improved gauge action

2. Fixed Polyakov loop in the “real” Z2. Fixed Polyakov loop in the “real” Z33 phase phase

UnquenchedUnquenched

1. MILC colaboration 2+1 flavor improved1. MILC colaboration 2+1 flavor improved

2. m2. muu= m= md d = m= mss/10/10

3. Lattice sizes L3. Lattice sizes L33 X 4 X 4

RESULTS ARE RESULTS ARE THE SAME THE SAME AGG, Osborn AGG, Osborn PRD,75 (2007) PRD,75 (2007) 034503034503

Chiral phase transition and hiral phase transition and localizationlocalization

For massless fermions: For massless fermions: Localization predicts a (first) Localization predicts a (first) order phase transition. Why?order phase transition. Why?

1. Metal insulator transition always occur close to the origin and 1. Metal insulator transition always occur close to the origin and the chiral condensate is determined by the same eigenvalues.the chiral condensate is determined by the same eigenvalues.

2. In chiral systems the spectral density is sensitive to localization2. In chiral systems the spectral density is sensitive to localization..

For nonzero mass:For nonzero mass: Eigenvalues up to m contribute to the Eigenvalues up to m contribute to the condensate but the metal insulator transition occurs close to condensate but the metal insulator transition occurs close to the origin only. Larger eigenvalue are delocalized so we expect the origin only. Larger eigenvalue are delocalized so we expect a crossover.a crossover.

Number of flavors:Number of flavors: Disorder effects diminish with the number Disorder effects diminish with the number of flavours. Vacuum with dynamical fermions is more correlated. of flavours. Vacuum with dynamical fermions is more correlated.

V

mmm

)(lim

0

2

22

1

11

)()1(

)()1()(2

8

1)(

2

1

z

z

zNz

zNz

N

xz

xzx

NxL

),(),()( ,1

, txvtxvx R

N

tL

),(),( 44 NxzUNxU

2

2

1

1)1()1(2

8

1)( 21

zz

Nz

Nz

N zzV

xLP

Confinement and spectral propertiesIdea:Idea: Polyakov loop is expressed as the response of the Dirac operator to a Polyakov loop is expressed as the response of the Dirac operator to a change in time boundary conditionschange in time boundary conditions Gattringer,PRL 97 (2006) 032003, hep-lat/0612020

……. . but sensitivity to but sensitivity to spatialspatial boundary conditions boundary conditions is a criterium (Thouless) for localization!is a criterium (Thouless) for localization!

Politely Challenged in:Politely Challenged in:

heplat/0703018, heplat/0703018,

Synatschke, Wipf, WozarSynatschke, Wipf, Wozar

Localization and confinementLocalization and confinement1.What part of the spectrum contributes the most to the 1.What part of the spectrum contributes the most to the

Polyakov loop?.Does it scale with volume?Polyakov loop?.Does it scale with volume?

2. Does it depend on temperature?2. Does it depend on temperature?

3. Is this region related to a metal-insulator transition at 3. Is this region related to a metal-insulator transition at TTcc??

4. What is the estimation of the P from localization theory?4. What is the estimation of the P from localization theory?

5. Can we define an order parameter for the chiral phase 5. Can we define an order parameter for the chiral phase transition in terms of the sensitivity of the Dirac transition in terms of the sensitivity of the Dirac operator to a change in spatial boundary conditions? operator to a change in spatial boundary conditions?

IPR (red), Accumulated Polyakov loop (blue) for T>TIPR (red), Accumulated Polyakov loop (blue) for T>Tcc as a as a

function of the eigenvalue.function of the eigenvalue.

Localization and ConfinementLocalization and Confinement

MetalMetal

predictionprediction

MI MI transition?transition?

Accumulated Polyakov loop versus eigenvalueAccumulated Polyakov loop versus eigenvalue

Confinement is controlled by the ultraviolet part of the spectrum Confinement is controlled by the ultraviolet part of the spectrum

PP

1. Eigenvectors of the QCD Dirac operator becomes 1. Eigenvectors of the QCD Dirac operator becomes more localized as the temperature is increased. more localized as the temperature is increased.

2. For a specific temperature we have observed a 2. For a specific temperature we have observed a metal-insulator transition in the QCD Dirac operator metal-insulator transition in the QCD Dirac operator in lattice QCD and instanton liquid model.in lattice QCD and instanton liquid model.

3. "The Anderson transition occurs at the same 3. "The Anderson transition occurs at the same T than the chiral phase transition and in the T than the chiral phase transition and in the same spectral region“same spectral region“

What’s next?What’s next?

1. How relevant is localization for confinement? 1. How relevant is localization for confinement?

2. How are transport coefficients in the quark gluon plasma 2. How are transport coefficients in the quark gluon plasma affected by localization?affected by localization?

3 Localization and finite density. Color superconductivity3 Localization and finite density. Color superconductivity..

ConclusionsConclusions

THANKS! THANKS! ag3@princeton.eduag3@princeton.edu

Quenched ILM, Origin, N = 2000

For T < 100 MeV we expect (finite size scaling) to see a (slow) convergence to RMT results.

T = 100-140, the metal insulator transition occurs

Quenched ILM, IPR, N = 2000

Similar to overlap prediction

Morozov,Ilgenfritz,Weinberg, et.al.

Metal

IPR X N= 1

Insulator

IPR X N = N

Origin

BulkD2~2.3(origin)

Multifractal

IPR X N = 2DN