1. Role of Anderson 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. Conclusions: At the same T that the Chiral Phase transition "A metal-insulator transition in the Dirac operator induces the QCD chiral phase transition" metal - insulator undergo a metal - insulator transition

3. Outline: 1. Introduction to disordered systems and Anderson localization. 2. QCD vacuum as a conductor. QCD vacuum as a disordered medium. Dyakonov - Petrov ideas. 3. QCD phase transitions. 4. Role of localization in the QCD phase transitions. Results from instanton liquid models and lattice.

4. V(x) X EaEa EbEb EcEc Anderson (1957): Anderson (1957): 1. How does the quantum dynamics depend on disorder? 2. How does the quantum dynamics depend on energy? 0 A five minutes course on disordered systems The study of the quantum motion in a random potential

5. Insulator: For d 3, for strong disorder. Classical diffusion eventually stops due to destructive interference (Anderson localization). 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. Quantum dynamics according to the one parameter scaling theory a = ? D quan =f(d,W)? t D clas t D quan t D quan t a Sridhar,et.al InsulatorMetal

6. How are these different regimes characterized? 1. Eigenvector statistics: 2. Eigenvalue statistics: Altshuler, Boulder lectures

7. 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

8. Instantons: Non perturbative 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 at T=0, instantons and chiSB tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal

9. Multiinstanton vacuum? Multiinstanton vacuum? Problem: Non linear equations No superposition Sol: Variational principles(Dyakonov,Petrov), Instanton liquid (Shuryak) Typical size and some aspects of the interactions are fixed Typical size and some aspects of the interactions are fixed 1. ILM explains the chiSB 2. Describe non perturbative effects in hadronic correlation functions (Shuryak,Schaefer,Verbaarchot) 3 No confinement. Instanton liquid models T = 0 Instanton liquid models T = 0

10. 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. (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 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! QCD vacuum as a conductor (T =0)

11. QCD vacuum as a disordered conductor Instanton positions and color orientations vary Instanton positions and color orientations vary Impurities Instantons Electron Quarks T = 0 long range hopping 1/R    = 3<4 Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik AGG and Osborn, PRL, 94 (2005) 244102 QCD vacuum is a conductor for any density of instantons

12. 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

13. 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,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). 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) 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

14. 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 picture provided that T IA is short range and the vacuum is disordered enough” Instanton liquid model at finite T

15. 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

16. 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. 2. 3. Eigenstates are multifractals. Skolovski, Shapiro, Altshuler Mobility edge Anderson transition var

17. ILM with 2+1 massless flavors, We have observed a metal-insulator transition at T ~ 125 Mev Spectrum is scale invariant

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

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

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

21. 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

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

23. hiral phase transition and localization Chiral phase transition and localization 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.

24. 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) 032003, hep-lat/0612020 …. but sensitivity to spatial boundary conditions is a criterium (Thouless) for localization! Politely Challenged in: heplat/0703018, Synatschke, Wipf, Wozar

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

26. 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?

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

28. 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

29. 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

30. 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 =