Role of Anderson-Mott localization in Role of Anderson-Mott 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. Based on a recent relation fundamental role in the chiral phase transition. Based on a recent relation

between the Polyakov loop and the spectral properties of the Dirac operator we between the Polyakov loop and the spectral properties of the Dirac operator we discuss how the confinement-deconfinement transition may be related to a discuss how the confinement-deconfinement transition may be related to a metal-insulator transition in the bulk of the spectrum of the Dirac operator. metal-insulator transition in the bulk of the spectrum of the Dirac operator.

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

OutlineOutline1. A few words about localization.1. A few words about localization.

2. Disorder in QCD, Dyakonov – Petrov ideas. 2. Disorder in QCD, Dyakonov – Petrov ideas.

3. A few words about QCD phase transitions.3. A few words about QCD phase transitions.

4. Role of localization in the QCD phase transitions. Results 4. Role of localization in the QCD phase transitions. Results from ILM and lattice.from ILM and lattice.

4.1 The chiral phase transition.4.1 The chiral phase transition.

4.2 The deconfinement transition. In progress. 4.2 The deconfinement transition. In progress.

5. What’s next. Quark diffusion in LHC5. What’s next. Quark diffusion in LHC.

A few words on disordered systemsA few words on disordered systems Quantum particle in a random potential

Anderson localizationQuantum destructive interferencecan induce a transition to an insulatingstate. InsulatorFor d < 3 or, at strong disorder,in d > 3 all eigenstates are localized in space. Metald > 2, Weak disorderEigenstates delocalized Mott localizationInteraction can induce atransition from metal (classical)to insulator. MetalInsulator

Eigenfunction characterizationEigenfunction characterization

1. Eigenfunctions moments:

2. Decay of the eigenfunctions:

MetalV

InsulatorrdrIPR d

n 1

4 1~)(

?/1

/1~)(

/

r

MetalV

Insulatore

r

r

n

Metald

Criticald

Insulatord

Spectral characterization ?

Spectral characterizationSpectral characterization

RMT correlations: Weak disorder (d > 2). Up to Thouless. Poisson correlations: Any disorder d < 2, strong disorder d>2

"In the context of QCD the metallic region corresponds with the infrared limit (constant fields) of the Dirac operator" (Verbaarschot,Shuryak)

2

222 log~

)( Asβes~sP

nnn=nΣ

RMT

Metal

sesP

nnΣ

Poisson

Insulator

)()(

2

2/2 ~)( dLL

Dirac operator has a Dirac operator has a zero modezero mode in the field of an instanton in the field of an instanton

QCD vacuum saturated by weakly interacting (anti) instantons (Shuryak) Density and size of instantons are fixed phenomenologically

Long range hopping in the instanton liquid model (ILM)Long range hopping in the instanton liquid model (ILM) Diakonov - PetrovDiakonov - Petrov

As a consequence of the long range hopping the QCD vacuum is a metal: Zero modes initially bounded As a consequence of the long range hopping the QCD vacuum is a metal: Zero modes initially bounded to an instanton get delocalized due to the overlapping with the rest of zero modes. By increasing to an instanton get delocalized due to the overlapping with the rest of zero modes. By increasing temperature (or other parameters) the QCD vacuum will eventually undergo a metal insulator temperature (or other parameters) the QCD vacuum will eventually undergo a metal insulator transition.transition. What means a metal?What means a metal?

Conductivity Chiral Symmetry breaking Conductivity Chiral Symmetry breaking

Impurities Impurities Instantons Instantons ElectronElectron QuarksQuarks

QCD vacuum, disorder and instantons QCD vacuum, disorder and instantons Diakonov, Petrov, later Verbaarschot, Osborn, Zahed, Osborn & AGG

300 /10 rrψrDψgA+=D ins

μμ

4

3

4 )()ˆ(~)()(

RRuizxiDzxxdT AI

AAIIIA

""Spectral properties of the smallest eigenvalues of the Dirac Spectral properties of the smallest eigenvalues of the Dirac

operator are controled by instantons" operator are controled by instantons" Is that important? Is that important? Yes. Yes.

Banks-Casher (Kubo) Banks-Casher (Kubo)

Metallic Metallic behavior means chiSB in the ILMbehavior means chiSB in the ILM

Recent developmentsRecent developments::

- Thouless energy in QCD. If the QCD vacuum at T= 0 is a - Thouless energy in QCD. If the QCD vacuum at T= 0 is a metal, one can predict finite size effects. metal, one can predict finite size effects. Verbaarschot,Osborn, Verbaarschot,Osborn, PRL 81 (1998) 268PRL 81 (1998) 268 and Zahed, Janik and Zahed, Janik et.al., et.al., PRL. 81 (1998) 264PRL. 81 (1998) 264

- The QCD Dirac operator can be described by a random - The QCD Dirac operator can be described by a random matrix with long range hopping even beyond the Thouless matrix with long range hopping even beyond the Thouless energy. energy. AGG and Osborn, AGG and Osborn, PRL, 94 (2005) 244102 PRL, 94 (2005) 244102

Conductivity versus chiral symmetry breaking Conductivity versus chiral symmetry breaking

V

mmm

)(lim

0

3

2/1

)240(2

31MeV

V

NNc

Phase transitions in QCD Phase transitions in QCD

Quark- Gluon Plasmaweakly only for T>>Tc

J. Phys. G30 (2004) S1259

Deconfinement and chiral restorationDeconfinement and chiral restoration

Deconfinement•Linear confining potential vanishes. •Matter becomes light

•QCD still non perturbative

Chiral Restoration

How to explain these transitions?

1. Effective model of QCD close to the chiral restoration (Wilczek,Pisarski):

Universality, epsilon expansion.... too simple?

2. QCD but only consider certain classical solutions (t'Hooft): Instantons (chiral), Monopoles (confinement)

No monopoles found, instantons only after lattice cooling, no from QCD

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)

They must be related but nobody* knows exactly how

0~0L

Instanton liquid picture 1.The effective QCD coupling constant g(T) decreases as temperature increases. The 1.The effective QCD coupling constant g(T) decreases as temperature increases. The

density of instantons also decreasesdensity of instantons also decreases

2. Zero modes are exponentially localized in 2. Zero modes are exponentially localized in space but oscillatory in time.space but oscillatory in time.

3. Amplitude hopping restricted to neighboring 3. Amplitude hopping restricted to neighboring instantons.instantons.

4. 4. Localization will depend strongly on the temperature. There must exist a T = TLocalization will depend strongly on the temperature. There must exist a T = TLLsuch such that a MIT takes place.that a MIT takes place.

5. There must exist a T = T5. There must exist a T = Tcc such that such that6. This general picture is valid beyond the instanton liquid approximation (KvBLL, see 6. This general picture is valid beyond the instanton liquid approximation (KvBLL, see

Ilgenfritz talk) provided that the hopping induced by topological objects is short rangeIlgenfritz talk) provided that the hopping induced by topological objects is short range..

Is TIs TLL = T = Tc c ?...Yes Does the MIT occur at the origin? Yes ?...Yes Does the MIT occur at the origin? Yes

0

Localization and chiral transition Localization and chiral transition

2/)0()( c

T)exp()( TRR

)exp(~ ATRTIA

Dyakonov, Dyakonov, PetrovPetrov

μμ

QCD gA+=D

nn

QCD iD inslat

AAA ,

0At Tc , Chiral phase transition

but also the low lying,

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

n

n

undergo a metal-insulator transition.

Main ResultMain Result

Spectral characterizationSpectral characterization

RMT correlations: Weak disorder (d > 2). Up to Thouless. Poisson correlations: Any disorder d < 2, strong disorder d>2"In the context of QCD the metallic region corresponds with the infrared limit (constant fields) of the

Dirac operator" (Verbaarschot,Shuryak)

2

222 log~

)( Asβes~sP

nnn=nΣ

RMT

Metal

sesP

nnΣ

Poisson

Insulator

)()(

2

2/2 ~)( dLL

ANDERSON TRANSITIONANDERSON TRANSITIONMain:Main:Non trivial interplay between tunneling and interference leads Non trivial interplay between tunneling and interference leads

to the metal insulator transition (MIT) to the metal insulator transition (MIT)

Spectral correlations Wavefunctions Scale invarianceScale invariance MultifractalsMultifractals

CRITICAL STATISTICSCRITICAL STATISTICS

""Spectral correlations are universal, they depend only on the

dimensionality of the space."

Kravtsov, Muttalib 97

)1(2

~)( qDdq

n

qLrdr

Skolovski, Shapiro, Altshuler

1~)(

1~)(

~)(2

sesP

sssP

nn

As

Mobility edge Anderson transition

Finite size scaling analysis, Dynamical 2+1Finite size scaling analysis, Dynamical 2+1

dssPssss nn )(var22

MasslessMasslessMassiveMassive

Quenched Lattice Quenched Lattice IPR versus eigenvalue IPR versus eigenvalue

Unquenched ILM, 2 m = 0 The transition is located around T =120

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

METALMETAL

INSULATORINSULATOR

Unquenched ILM, close to the origin,

2+1 flavors, N = 200

Instanton liquid model:,condensate and inverse participation ratio versus T Instanton liquid model:,condensate and inverse participation ratio versus T

Unquenched, massive 2+1Unquenched, massive 2+1 Quenched ( also unquenched masless)Quenched ( also unquenched masless)

For zero mass, transition sharper with the volumeFor zero mass, transition sharper with the volume First orderFirst order

For finite mass, the condensate is volume independentFor finite mass, the condensate is volume independent Crossover Crossover

Lattice: and inverse participation ratio versus T Lattice: and inverse participation ratio versus T

Localization and order of the chiral phase Localization and order of the chiral phase transitiontransition

1. Metal insulator transition always occur close to the origin.1. Metal insulator transition always occur close to the origin.

2. Systems with chiral symmetry the spectral density is 2. Systems with chiral symmetry the spectral density is sensitive to localization.sensitive to localization.

3. For zero mass localization predicts a first order phase 3. For zero mass localization predicts a first order phase transition. transition.

4. For a non zero mass m, eigenvalues up to m contribute 4. For a non zero mass m, eigenvalues up to m contribute to the condensate but the metal insulator transition to the condensate but the metal insulator transition occurs close to the origin only. Larger eigenvalue are occurs close to the origin only. Larger eigenvalue are delocalized se we expect a crossover.delocalized se we expect a crossover.

5. Multifractal dimension m=0 should modify susceptibility 5. Multifractal dimension m=0 should modify susceptibility exponents.exponents.

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

Localization and confinementLocalization and confinementThe dimensionless conductance, g, a measure of localization, is related The dimensionless conductance, g, a measure of localization, is related

to the sensitivity of eigenstates to a change in boundary conditions.to the sensitivity of eigenstates to a change in boundary conditions.

Metal Metal

Insulator Insulator

MI transitionMI transition

1.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 T3. Is this region related to a metal-insulator transition at T cc??

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

LLg d 2

Lg 0

Ldgg c )(

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

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?

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

Quenched ILM, T =200, bulk

Mobility edge in the Dirac operator. For T =200 the transition occurs around the center of the spectrum

D2~1.5 similar to the 3D Anderson model. Not related to chiral symmetry

Unquenched ILM, 2+1 flavors

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

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

● For a specific temperature we have observed a For a specific temperature we have observed a metal-insulator transition in the QCD Dirac operator.metal-insulator transition in the QCD Dirac operator.

● For lattice and ILM, and for quenched and For lattice and ILM, and for quenched and unquenched we have found two transitions close to unquenched we have found two transitions close to the origin and in the UV part of the spectrum and. the origin and in the UV part of the spectrum and.

MAINMAIN

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

“ “ Confinement-Deconfinemente transition has to Confinement-Deconfinemente transition has to do with localization-delocalization in time do with localization-delocalization in time direction” direction”

ConclusionsConclusions

What's next?

1. How critical exponents are 1. How critical exponents are affected by localization? affected by localization?

2. Confinement and localization, 2. Confinement and localization, analytical result?analytical result?

3. How are transport coefficients in 3. How are transport coefficients in the quark gluon plasma affected by the quark gluon plasma affected by localization? localization?

4. Localization in finite density. Color 4. Localization in finite density. Color superconductivity. superconductivity.

QCD : The Theory of the strong QCD : The Theory of the strong interactionsinteractions

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

Analytical information?Analytical information? Instantons , Monopoles, VorticesInstantons , Monopoles, Vortices

rrarV /)(

2

41)(

GmgAiLqQCD

3)240(~ MeV

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

IPR, two massless flavors D2 ~ 1.5 (bulk) D2~2.3(origin)

dssPs=A

AA

AA=W

RMTP

RMT

0

2

Spectrum Unfolding Spectral Spectrum Unfolding Spectral CorrelatorsCorrelators

How to get information from a bunch of levelsHow to get information from a bunch of levels

Quenched ILM, Bulk, T=200

Colliding Nuclei HardCollisions QG Plasma ?

Hadron Gas & Freeze-out

1 2 3 4

sNN = 130, 200 GeV(center-of-mass energy per nucleon-nucleon collision)

1.1. Cosmology Cosmology 1010-6-6 sec after Bing Bang, neutron stars (astro) sec after Bing Bang, neutron stars (astro)

2.2. Lattice QCD Lattice QCD finite size effects. finite size effects. Analytical, Analytical, N=4 super YMN=4 super YM ??

3.3. High energy Heavy Ion Collisions. High energy Heavy Ion Collisions. RHIC, LHCRHIC, LHC

Nuclear (quark) matter at finite temperatureNuclear (quark) matter at finite temperature

MultifractalityMultifractalityIntuitive: Intuitive: Points in which the modulus of the Points in which the modulus of the

wave function is bigger than a (small) wave function is bigger than a (small) cutoffcutoff MM.. If If the fractal dimension depends on thethe fractal dimension depends on the cutoff M,cutoff M, the wave function is the wave function is multifractal.multifractal.

Kravtsov, Chalker,Aoki,Schreiber,Castellani

24

2

D

L

d

nLrdrψ=IIPR d

"QCD vacuum saturated by interacting (anti) instantons"

Density and size of (a)instantons are fixed phenomenologically

The Dirac operator D, in a basis of single I,A:

1. ILM explains the chiSB

2. Describe non perturbative effects in hadronic correlation functions (Shuryak,Schaefer,dyakonov,petrov,verbaarchot)

0

0

AI

IA

T

TiD

4

3

4 )()ˆ(~)()(

RRuizxiDzxxdT AI

AAIIIA

41,200 fmV

NMeV

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

Eight light Bosons (), no parity doublets.

)1()3(

)1()1()3()3(

VV

AVVA

USU

UUSUSU

)1( 5, RL

MeVqq 3)250(

QCD Chiral SymmetriesQCD Chiral Symmetries

ClassicalClassical

QuantumQuantum

U(1)U(1)A A explicitly broken by the anomaly.explicitly broken by the anomaly.

SU(3)SU(3)AA spontaneously broken by the QCD vacuum spontaneously broken by the QCD vacuum

Dynamical massDynamical mass

Quenched lattice QCD simulations Symanzik 1-loop glue with asqtad valence