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. 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) 034503,NPA, 770, 141 (2006) PRL 93 (2004) 132002

2. Outline 1. What is localization? 2. QCD vacuum as a conductor. QCD vacuum as a disordered medium. 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.

3. What is 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

4. How do we know that a metal is a metal? Texbook answer: Look at the conductivity or other transport properties 1. Eigenvector statistics: D q = d Metal D q = 0 Insulator D q = f(d,q) M-I transition 2. Eigenvalue statistics: Level Spacing distribution: Number variance: Other options: Look at eigenvalue and eigenvectors

5. 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 interacting (anti) instantons (Shuryak) Size and (density) of instantons are fixed phenomenologically Long range hopping in the instanton liquid model (ILM) Diakonov - Petrov (80’s) Diakonov - Petrov (80’s) 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 tHooft, Polyakov, Shuryak, QCD vacuum and instantons tHooft, Polyakov, Shuryak, Diakonov, Petrov

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

7. QCD 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 Conductor Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik

8. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons Is that important? Yes. 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 560 306,1993 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. 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 560 306,1993 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. 2.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) 244102 2.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) 244102 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, 094504 (2002),M. Golterman and Y. Shamir, Phys. Rev. D 68, 074501 (2003);V. Weinberg, E.-M. Ilgenfritz, et.al, PoS { LAT2005}, 171 (2005) 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, 094504 (2002),M. Golterman and Y. Shamir, Phys. Rev. D 68, 074501 (2003);V. Weinberg, E.-M. Ilgenfritz, et.al, PoS { LAT2005}, 171 (2005) Conductivity versus chiral symmetry breaking Conductivity versus chiral symmetry breaking Banks-Casher(Kubo)

9. 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) 132002 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) 132002 Random matrix approach beyond the Thouless energy:

10. Agreement with chiral perturbation theory.Agreement with chiral perturbation theory. The number variance beyond the Thouless energy as a 4D disordered system.The number variance beyond the Thouless energy as a 4D disordered system.

11. 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

12. 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 chiral restoration (Wilczek,Pisarski): Universality, epsilon expansion.... too simple? 2. QCD but only consider certain classical solutions (t'Hooft): Instantons (chiral), Monopoles (confinement). Instanton do not dissapear at the transiton (Shuryak,Schafer). 3. Decreasing interactions? Certainly must play a role but QCD is still non perturbative 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

13. 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

14. 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

15. Statistical analysis of spectra and eigenfunctions Statistical analysis of spectra and eigenfunctions 1. Eigenvector statistics: D q = d Metal D q = 0 Insulator D q = f(d,q) M-I transition 2. Eigenvalue statistics: Level Spacing distribution: Number variance:

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, P(s) of the lowest eigenvalues 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. Unquenched ILM, 2 m = 0 Level repulsion s << 1 Level repulsion s << 1 Exponential decay s > 1

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

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

22. Lattice QCD AGG, J. Osborn 2007 Lattice QCD AGG, J. Osborn 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

23. Quenched Lattice QCD IPR versus eigenvalue

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

25. Quenched lattice, close to the origin METAL Transiton from metal to insulator

26. Dynamical, 2+1 Quenched For zero mass, transition sharper with the volume First order For finite mass, the condensate is volume independent Crossover Inverse participation ratio versus T

27. 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 D 2 should modify susceptibility exponents.

28. 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

29. Localization and confinement The dimensionless conductance, g, a measure of localization, is related to the sensitivity of eigenstates to a change in boundary conditions. 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?

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

31. 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?

32. 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 =

33. ● 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

34. 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.

35. 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

36. 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

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

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

40. Quenched ILM, Bulk, T=200

41. 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

42. 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

43. "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

44. 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

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

46. 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)

47. 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.

48. 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