1. Role of Anderson-Mott localization in the QCD phase transitions Antonio M. García-García ag3@princeton.edu 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. In collaboration with James Osborn In collaboration with James Osborn PRD,75 (2007) 034503,NPA, 770, 141 (2006) PRL 93 (2004) 132002

2. 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 How to extract analytical information? Instantons, Monopoles, Vortices

3. Instantons (Polyakov,t'Hooft) : Non pertubative solutions of the classical Yang Mills equation. Tunneling between classical vacua. 1. Dirac operator has a zero mode in the field of an instanton 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons 3. Spectral properties related to chiSB. Banks-Casher relation QCD vacuum models based on instantons: 1. Density N/V = 1fm -4. Hopping amplitude 2. Describe chiSB and non perturbative effects in hadronic correlation functions. 3 No confinement. QCD at T=0, instantons and chiSB tHooft, Polyakov, Shuryak, Diakonov, Petrov Dyakonov,Petrov, Shuryak

4. Conductor 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. (Diakonov and Petrov) Zero modes initially bounded to an instanton get delocalized due to the overlapping with the rest of zero modes. (Diakonov and Petrov) Impurities Instantons Electron Quarks Impurities Instantons Electron Quarks Instanton positions and color orientations vary Instanton positions and color orientations vary QCD vacuum as a disordered conductor T = 0 long range hopping T IA ~a IA /R ,   = 3<4 QCD vacuum is a ‘disordered’ conductor for any density of instantons AGG and Osborn, PRL, 94 (2005) 244102

5. QCD at finite T: Phase transitions QCD at finite T: Phase transitions 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: Confining potential vanishes. Chiral Restoration:Matter becomes light.

6. Deconfinement 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). 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

7. What is Anderson localization? A particle in a disordered potential. Classical diffusion stops due to destructive interference. Insulator: For d 3, for strong disorder. Classical diffusion eventually stops. Eigenstates are delocalized. Metal: For d > 2 and weak disorder quantum effects do not alter significantly the classical diffusion. Eigenstates are delocalized. Metal-Insulator transition: For d > 2 in a certain window of energies and disorder. Eigenstates are multifractal. How are these different regimes characterized? 1. Eigenvector statistics: 2. Eigenvalue statistics:

8. 1. Zero modes are localized in space but oscillatory in time. 2. Hopping amplitude restricted to neighboring instantons. 3. Since T IA is short range there must exist a T = T L such that a metal insulator transition takes place. (Dyakonov,Petrov) 4. The chiral phase transition occurs at T=T c. Localization and chiral transition are related if: 1. T L = T c. 2. The localization transition occurs at the origin (Banks-Casher) “This is valid beyond the instanton picutre provided that T IA is short range and the vacuum is disordered enough” Localization and chiral transition

9. At T c 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

10. 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

11. ILM Nf=2 massless. Eigenfunction statistics AGG and J. Osborn, 2006

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

13. Instanton liquid model Nf=2, masless Localization versus chiral transition Localization versus chiral transition Chiral and localizzation transition occurs at the same temperature

14. Lattice QCD AGG, J. Osborn, PRD, 2007 Lattice QCD AGG, J. Osborn, PRD, 2007 1. Simulations around the chiral phase transition T 2. Lowest 64 eigenvalues Quenched Quenched 1. Improved gauge action 2. Fixed Polyakov loop in the “real” Z 3 phase Unquenched Unquenched 1. MILC colaboration 2+1 flavor improved 2. m u = m d = m s /10 3. Lattice sizes L 3 X 4

15. RESULTS ARE THE SAME AGG, Osborn PRD,75 (2007) 034503

16. Localization and order of the chiral phase transition For massless fermions: Localization predicts a (first) order phase transition. Why? 1. Metal insulator transition always occur close to the origin and the chiral condensate is determined by the same eigenvalues. 2. In chiral systems the spectral density is sensitive to localization. For nonzero mass: Eigenvalues up to m contribute to the condensate but the metal insulator transition occurs close to the origin only. Larger eigenvalue are delocalized so we expect a crossover. For nonzero mass: Eigenvalues up to m contribute to the condensate but the metal insulator transition occurs close to the origin only. Larger eigenvalue are delocalized so we expect a crossover. Number of flavors: Disorder effects diminish with the number of flavours. Vacuum with dynamical fermions is more correlated.

17. 1. Eigenvectors of the QCD Dirac operator becomes more localized as the temperature is increased. 2. For a specific temperature we have observed a metal- insulator transition in the QCD Dirac operator in lattice QCD and instanton liquid model. 3. "The Anderson transition occurs at the same T than the chiral phase transition and in the same spectral region“ What’s next? What’s next? 1. How relevant is localization for confinement? 2. How are transport coefficients in the quark gluon plasma affected by localization? 3 Localization and finite density. Color superconductivity. Conclusions THANKS! ag3@princeton.edu

18. Finite size scaling analysis: Finite size scaling analysis: Quenched 2+1 dynamical fermions

19. 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 =

20. 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

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

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

24. Quenched Lattice QCD IPR versus eigenvalue

25. Quenched ILM, Bulk, T=200

26. 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 10 -6 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

27. 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

28. "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

29. 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

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

31. 3. Spectral characterization: Spectral correlations in a metal are given by random matrix theory up to the Thouless energy Ec. 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)

32. 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 560 306,1993). 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.

33. 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