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. 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

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 EnergyLow 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 (Polyakov,t'Hooft)(Polyakov,t'Hooft) : : Non pertubative solutions of the classical Non pertubative solutions of the classical

Yang Mills equation. Tunneling between classical vacua. Yang Mills 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 are 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons controled by instantons

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

QCD vacuum models based on instantons:QCD vacuum models based on instantons:1. Density N/V = 1fm1. Density N/V = 1fm-4-4. Hopping amplitude . Hopping amplitude

2. Describe chiSB and non perturbative effects in hadronic correlation functions2. Describe chiSB and non perturbative effects in hadronic correlation functions..

3 No confinement.3 No confinement.

QCD at T=0, instantons and chiSBQCD at T=0, instantons and chiSB tHooft, Polyakov, Shuryak, tHooft, Polyakov, Shuryak, Diakonov, PetrovDiakonov, Petrov

300 /10 rrψrDψgA+=D ins

μμ

V

m

imdmDTr

V mm

)(lim

)()(

10

1

Dyakonov,PetroDyakonov,Petrov,Shuryakv,Shuryak

3

3)()ˆ(~

RRuiT AI

IA

Conductor Conductor An electron initially bounded to a single atom gets delocalized An electron initially bounded to a single atom gets delocalized

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

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

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

Impurities Impurities Instantons Instantons ElectronElectron QuarksQuarks

Instanton positions and color orientations varyInstanton positions and color orientations vary

QCD vacuum as a disordered conductor QCD vacuum as a disordered conductor

T = 0 long range hopping TT = 0 long range hopping TIAIA~a~aIAIA/R/R,, = 3<4 = 3<4

QCD vacuum is a ‘disordered’ conductor for any density of instantonsQCD vacuum is a ‘disordered’ conductor for any density of instantons

AGG and Osborn, AGG and Osborn,

PRL, 94 (2005) PRL, 94 (2005)

244102244102

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: Confining potential vanishes.

Chiral Restoration:Matter becomes light.

0L

0~

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):

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)

They must be related but nobody* knows exactly how

0~0L

What is Anderson localization?What is Anderson localization?A particle in a disordered potential. A particle in a disordered potential. Classical dClassical diffusion stops iffusion stops due to destructive interference.due to destructive interference.

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. Eigenstates are delocalized. eventually stops. Eigenstates are delocalized.

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.

How are these different regimes characterized?How are these different regimes characterized?1. Eigenvector statistics:

2. Eigenvalue statistics:

sesP

D

Poisson

Insulator

)(

0~

)(2

22 ~

)( Asβes~sP

dD

RMT

Metal

2~)(4 Ddd

nd LrdrLIPR

i

iissP /)( 1

1

10 2

se~sP

ss~sPdD

MITAs

β

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

0

Localization and chiral transition Localization and chiral transition

)exp()( TRR

)exp(~ ATRTIA

μμ

QCD gA+=D

nn

QCD iD inslat

AAA ,

At Tc

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

0)(

lim0

V

mmm

ILM with 2+1 massless flavors, P(s) of the lowest eigenvalues

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

Spectrum is scale invariant

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

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

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

transitiontransition

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

transitiontransition

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

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

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

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

Finite size Finite size scaling analysis: scaling analysis: dssPssss nn )(var

22

QuenchedQuenched2+1 dynamical fermions2+1 dynamical fermions

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, 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 Lattice QCD Quenched Lattice QCD IPR IPR

versus eigenvalueversus eigenvalue

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

2

222 log~)(

)( Asβes~sP

Enn=En

RMT

Metal

sesP

EEn

Poisson

Insulator

)(

)(

)(

2

3. Spectral characterization:3. Spectral characterization:

Spectral correlations in a metal are given by random Spectral correlations in a metal are given by random

matrix theory up to the Thouless energy Ec. Matrix matrix theory up to the Thouless energy Ec. Matrix

elements are only constrained by symmetryelements are only constrained by symmetry

Eigenvalues in an insulator are not correlated. Eigenvalues in an insulator are not correlated.

In units of the mean level spacing, the In units of the mean level spacing, the

Thouless energy, Thouless energy,

In the context of QCD the metallic region In the context of QCD the metallic region

corresponds with the infrared limit (constant corresponds with the infrared limit (constant

fields) of the Dirac operator" fields) of the Dirac operator"

(Verbaarschot,Shuryak) (Verbaarschot,Shuryak)

2 dE Lg c

2/1 LEc

1. QCD, random matrix theory, Thouless 1. QCD, random matrix theory, Thouless energy:energy:Spectral correlations of the QCD Dirac operator in the infrared Spectral correlations of the QCD Dirac operator in the infrared

limit are universal limit are universal (Verbaarschot, Shuryak Nuclear Physics A 560 (Verbaarschot, Shuryak Nuclear Physics A 560

306 ,1993).306 ,1993). They They can be obtained from a RMT with the can be obtained from a RMT with the symmetries of QCD.symmetries of QCD.

1. The microscopic spectral density is universal, it depends only 1. The microscopic spectral density is universal, it depends only on the global symmetries of QCD, and can be computed from on the global symmetries of QCD, and can be computed from random matrix theory. random matrix theory.

2. RMT describes the eigenvalue correlations of the full QCD 2. RMT describes the eigenvalue correlations of the full QCD Dirac operator up to EDirac operator up to Ecc. This is a finite size effect. In the . This is a finite size effect. In the thermodynamic limit the spectral window in which RMT thermodynamic limit the spectral window in which RMT applies vanishes but at the same time the number of applies vanishes but at the same time the number of eigenvalues, g, described by RMT diverges. eigenvalues, g, described by RMT diverges.

L0~2

2

L

FE π

c

)()(~)( 21

20 sJsJss

LLFE

g πc 22~

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