Role of Anderson-Mott localization in the QCD phase transitions Antonio M. García-García Princeton University ICTP, Trieste We investigate in what situations Anderson localization may be relevant in the context of QCD. At the chiral phase transition we provide compelling evidence from lattice and phenomenological instanton liquid models that the QCD Dirac operator undergoes a metal - insulator transition similar to the one observed in a disordered conductor. This suggests that Anderson localization plays a 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 discuss how the confinement-deconfinement transition may be related to a metal-insulator transition in the bulk of the spectrum of the Dirac operator. In collaboration with James Osborn In collaboration with James Osborn PRD,75 (2007) ,NPA, 770, 141 (2006) PRL 93 (2004)

Outline 1. A few words about localization and spectral analysis. 2. Disorder in QCD, random matrix theory. Dyakonov - Petrov ideas. Dyakonov - Petrov ideas. 3. A few words about QCD phase transitions. 4. Role of localization in the QCD phase transitions. Results from instanton liquid models 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. Relation confinement and chiral symmetry breaking. Quark diffusion in LHC.

A few words on localization Quantum particle in a random potential Anderson-Mott localization Quantum destructive interference, tunneling or interactions can induce a transition to an insulating state. Insulator For d < 3 or, at strong disorder, in d > 3 all eigenstates are localized in space. Classical diffusion is eventually stopped Metal d > 2, Weak disorder Eigenstates delocalized. Quantum effects do not alter significantly the classical diffusion. Metal Insulator

How do we know that a metal is a metal? Texbook answer: Look at the conductivity or other transport properties 1. Eigenfunctions moments: 2. Decay of the eigenfunctions: Other options: Look at eigenvalue and eigenvectors

3. Spectral characterization: Spectral correlations in a metal are given by random matrix theory up to the Thouless energy. Matrix elements are only constrained by symmetry Eigenvalues in an insulator are not correlated. In units of the mean level spacing, the Thouless energy, In units of the mean level spacing, the Thouless energy, In the context of QCD the metallic region corresponds with the infrared limit (constant fields) of the Dirac operator" (Verbaarschot,Shuryak)

Random Matrix Theory: Basic Facts Idea: Spectrum of a RM is highly correlated... not random H ij Random (real/complex Gaussian number) H ij Random (real/complex Gaussian number) Spectral correlators Spectral correlators Signatures of RM spectrum Signatures of RM spectrum Spectral rigidity Level Repulsion Spectral rigidity Level Repulsion

Dirac operator has a zero mode in the field of an instanton Dirac operator has a zero mode 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) Diakonov - Petrov Diakonov - Petrov 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 temperature (or other parameters) the QCD vacuum will eventually undergo a metal insulator transition. Conductivity Chiral Symmetry breaking Conductivity Chiral Symmetry breaking Impurities Instantons Electron Quarks Impurities Instantons Electron Quarks QCD vacuum and instantons QCD vacuum and instantons Diakonov, Petrov, later Verbaarschot, Shuryak, Osborn, Zahed, Osborn & AGG

METAL An electron initially bounded to a single atom gets delocalized due to the overlapping with nearest neighbors. QCD Vacuum QCD Vacuum Zero modes initially bounded to an instanton get delocalized due to the overlapping with the rest of zero modes. Zero modes initially bounded to an instanton get delocalized due to the overlapping with the rest of zero modes. Impurities Instantons Electron Quarks Impurities Instantons Electron Quarks Differences Dis.Sys: Exponential decay Nearest neighbors QCD vacuum Power law decay Long range hopping! Differences Dis.Sys: Exponential decay Nearest neighbors QCD vacuum Power law decay Long range hopping! Diakonov, Petrov, later Verbaarschot Osborn, Shuryak, Zahed QCD vacuum as a disordered medium

Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons" Is that important? Yes. Banks-Casher (Kubo) Banks-Casher (Kubo) Metallic behavior means chiSB in the ILM Metallic behavior means chiSB in the ILM Recent developments: Recent developments: 1. RMT and Thouless energy in QCD. Metallic limit of QCD. Verbaarschot,Osborn, PRL 81 (1998) 268, Zahed, Janik et.al., PRL. 81 (1998) 264, Verbaarschot, Shuryak Nuclear Physics A , RMT and Thouless energy in QCD. Metallic limit of QCD. Verbaarschot,Osborn, PRL 81 (1998) 268, Zahed, Janik et.al., PRL. 81 (1998) 264, Verbaarschot, Shuryak Nuclear Physics A , The QCD Dirac operator can be described by a random matrix with long range hopping even beyond the Thouless energy. AGG and Osborn, PRL, 94 (2005) The QCD Dirac operator can be described by a random matrix with long range hopping even beyond the Thouless energy. AGG and Osborn, PRL, 94 (2005) Localization in the QCD Dirac operator: C. Gattringer, M. Gockeler, et.al. Nucl. Phys. B618, 205 (2001),R.V. Gavai, S. Gupta and R. Lacaze, Phys. Rev. D 65, (2002),M. Golterman and Y. Shamir, Phys. Rev. D 68, (2003);V. Weinberg, E.-M. Ilgenfritz, et.al, PoS { LAT2005}, 171 (2005);F. Farchioni, Ph. de Forcrand, I. Hip, C. B. Lang and K. Splittorff,Phys. Rev. D 62, (2000);P.H. Damgaard, U.M. Heller, et.al.,Nucl. Phys. B583, 347 (2000);M. Gockeler, P.E.L. Rakow,et.al,Phys. Rev. Lett. 87, (2001);J. Greensite, A.V. Kovalenko, et.al Phys. Rev. D 74, (2006), 3. Localization in the QCD Dirac operator: C. Gattringer, M. Gockeler, et.al. Nucl. Phys. B618, 205 (2001),R.V. Gavai, S. Gupta and R. Lacaze, Phys. Rev. D 65, (2002),M. Golterman and Y. Shamir, Phys. Rev. D 68, (2003);V. Weinberg, E.-M. Ilgenfritz, et.al, PoS { LAT2005}, 171 (2005);F. Farchioni, Ph. de Forcrand, I. Hip, C. B. Lang and K. Splittorff,Phys. Rev. D 62, (2000);P.H. Damgaard, U.M. Heller, et.al.,Nucl. Phys. B583, 347 (2000);M. Gockeler, P.E.L. Rakow,et.al,Phys. Rev. Lett. 87, (2001);J. Greensite, A.V. Kovalenko, et.al Phys. Rev. D 74, (2006), Conductivity versus chiral symmetry breaking Conductivity versus chiral symmetry breaking

1. QCD, random matrix theory, Thouless energy: Spectral correlations of the QCD Dirac operator in the infrared limit are universal (Verbaarschot, Shuryak Nuclear Physics A ,1993). They can be obtained from a RMT with the symmetries of QCD. 1. The microscopic spectral density is universal, it depends only on the global symmetries of QCD, and can be computed from random matrix theory. 2. RMT describes the eigenvalue correlations of the full QCD Dirac operator up to E c. This is a finite size effect. In the thermodynamic limit the spectral window in which RMT applies vanishes but at the same time the number of eigenvalues, g, described by RMT diverges.

The spectral correlations of the QCD Dirac operator agree with those of a (chiral) random matrix with the appropiate symmetry. (Verbaarschot,Shuryak) Phys.Rev.Lett. 93 (2004) We describe the spectral correlations of the QCD Dirac operator analytically well beyond this limit by combining the random matrix approach with the phenomenological power-law decay of matrix elements from ILM. Agreement with chiral perturbation theory, Phys.Rev.Lett. 93 (2004) Random matrix approach beyond the Thouless energy:

Phase transitions in QCD Quark- Gluon Plasma perturbation theory only for T>>T c J. Phys. G30 (2004) S1259 At which temperature does the transition occur ? What is the nature of transition ? Péter Petreczky

Deconfinement 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 Anderson localization plays an important role. Nuclear Physics A, 770, 141 (2006) We propose that quantum interference and tunneling, namely, Anderson localization plays an important role. Nuclear Physics A, 770, 141 (2006) They must be related but nobody* knows exactly how

1.The effective QCD coupling constant g(T) decreases as temperature increases. The density of instantons also decreases 2. Zero modes are exponentially localized in space but oscillatory in time. 3. Amplitude hopping restricted to neighboring instantons. 4. Localization will depend strongly on the temperature. There must exist a T = T L such that a MIT takes place. 5. There must exist a T = T c such that 6. This general picture is valid beyond the instanton liquid approximation (KvBLL solutions) provided that the hopping induced by topological objects is short range. Is T L = T c ?...Yes Does the MIT occur at the origin? Yes Localization and chiral transition: Why do we expect a metal insulator transition close to the origin at finite temperature? Dyakonov, Petrov

At T c, Chiral phase transition but also the low lying, "A metal-insulator transition in the Dirac operator induces the chiral phase transition " undergo a metal-insulator transition. Main Result

Signatures of a metal-insulator transition 1. Scale invariance of the spectral correlations. A finite size scaling analysis is then carried out to determine the transition point Eigenstates are multifractals. Skolovski, Shapiro, Altshuler Mobility edge Anderson transition

1. Finite size scaling analysis: Dynamical 2+1 Quenched Unquenched

Quenched Lattice IPR versus eigenvalue

Unquenched ILM, 2 m = 0 2. Level repulsion s << 1 Exponential decay s > 1

Unquenched lattice, close to the origin, 2+1 flavors, 16 3 X6 METAL INSULATOR 3. Transiton from metal to insulator

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

Instanton liquid model Nf=2, masless 4. Localization versus chiral transition

Unquenched, 2+1 Quenched For zero mass, transition sharper with the volume First order For finite mass, the condensate is volume independent Crossover Lattice: and inverse participation ratio versus T

Localization and order of the chiral phase transition 1. Metal insulator transition always occur close to the origin. 2. Systems with chiral symmetry the spectral density is sensitive to localization. 3. For zero mass localization predicts a (first) phase transition not crossover. 4. For a non zero mass m, eigenvalues up to m contribute to the condensate but the metal insulator transition occurs close to the origin only. Larger eigenvalue are delocalized se we expect a crossover. 5. Multifractal dimension m=0 should modify susceptibility exponents.

Confinement and spectral properties Idea:Polyakov loop is expressed as the response of the Dirac operator to a change in time boundary conditions Idea: Polyakov loop is expressed as the response of the Dirac operator to a change in time boundary conditions Gattringer,PRL 97 (2006) , hep-lat/ …. but sensitivity to spatial boundary conditions is a criterium (Thouless) for localization! Politely Challenged in: heplat/ , Synatschke, Wipf, Wozar

Localization and confinement The dimensionless conductance, g, a measure of localization, is related to the sensitivity of eigenstates to a change in boundary conditions. MetalInsulator MI transition 1.What part of the spectrum contributes the most to the Polyakov loop?.Does it scale with volume? 2. Does it depend on temperature? 3. Is this region related to a metal-insulator transition at T c ? 4. What is the estimation of the P from localization theory?

Accumulated Polyakov loop versus eigenvalue Confinement is controlled by the ultraviolet part of the spectrum P

IPR (red), Accumulated Polyakov loop (blue) for T>T c as a function of the eigenvalue. Localization and Confinement Localization and Confinement Metalprediction MI 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 Bulk D2~2.3(origin) Multifractal IPR X N =

Quenched ILM, T =200, bulk Mobility edge in the Dirac operator. For T =200 the transition occurs around the center of the spectrum D 2 ~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 more localized as the temperature is increased. ● For a specific temperature we have observed a metal- insulator transition in the QCD Dirac operator. ● For lattice and ILM, and for quenched and unquenched we have found two transitions close to the origin and in the UV part of the spectrum and. MAIN MAIN "The Anderson transition occurs at the same T than the chiral phase transition and in the same spectral region" “ Confinement-Deconfinemente transition has to do with localization-delocalization in time direction” Conclusions

What's next? 1. How critical exponents are affected by localization? 2. Confinement and localization, analytical result? 3. How are transport coefficients in the quark gluon plasma affected by localization? 4. Localization in finite density. Color superconductivity.

QCD : The Theory of the strong interactions QCD : The Theory of the strong interactions High Energy g << 1 Perturbative High Energy g << 1 Perturbative 1. Asymptotic freedom Quark+gluons, Well understood Low Energy g ~ 1 Lattice simulations Low Energy g ~ 1 Lattice simulations The world around us The world around us 2. Chiral symmetry breaking 2. Chiral symmetry breaking Massive constituent quark Massive constituent quark 3. Confinement 3. Confinement Colorless hadrons Colorless hadrons Analytical information? Instantons, Monopoles, Vortices Analytical information? Instantons, Monopoles, Vortices

Quenched ILM, Origin, N = 2000 For T < 100 MeV we expect (finite size scaling) to see a (slow) convergence to RMT results. T = , the metal insulator transition occurs

IPR, two massless flavors D 2 ~ 1.5 (bulk) D 2 ~2.3(origin)

Spectrum Unfolding Spectral Correlators How to get information from a bunch of levels

Quenched ILM, Bulk, T=200

Colliding NucleiHard Collisions QG Plasma ? Hadron Gas & Freeze-out 1234  s NN = 130, 200 GeV (center-of-mass energy per nucleon-nucleon collision) 1.Cosmology sec after Bing Bang, neutron stars (astro) 2.Lattice QCD finite size effects. Analytical, N=4 super YM ? 3.High energy Heavy Ion Collisions. RHIC, LHC Nuclear (quark) matter at finite temperature

Multifractality Intuitive: Points in which the modulus of the wave function is bigger than a (small) cutoff M. If the fractal dimension depends on the cutoff M, the wave function is multifractal. Kravtsov, Chalker,Aoki, Schreiber,Castellani

"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) Instanton liquid models T = 0

Eight light Bosons (  ), no parity doublets. QCD Chiral Symmetries Classical Quantum U(1) A explicitly broken by the anomaly. SU(3) A spontaneously broken by the QCD vacuum Dynamical mass

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