Role of Anderson-Mott localization in the QCD phase transitions

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 PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002

2. Outline 1. A few words about localization. 2. Disorder in QCD, Dyakonov – Petrov ideas. 3. A few words about QCD phase transitions. 4. Role of localization in the QCD phase transitions. Results from ILM and lattice. 4.1 The chiral phase transition. 4.2 The deconfinement transition. In progress. 5. What’s next. Quark diffusion in LHC.

3. A few words on disordered systems Quantum particle in a random potential Anderson localization Quantum destructive interference 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. Metal d > 2, Weak disorder Eigenstates delocalized Mott localization Interaction can induce a transition from metal (classical) to insulator. Metal Insulator

4. Eigenfunction characterization 1. Eigenfunctions moments: 2. Decay of the eigenfunctions: Spectral characterization ?

5. Spectral characterization RMT correlations:Weak disorder (d > 2). Up to Thouless. Poisson correlations: Any disorder d < 2, strong disorder d>2 "In the context of QCD the metallic region corresponds with the infrared limit (constant fields) of the Dirac operator" (Verbaarschot,Shuryak)

6. QCD vacuum, disorder and instantons Diakonov, Petrov, later Verbaarschot, Osborn, Zahed, Osborn & AGG 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 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.What means a metal? Conductivity Chiral Symmetry breaking Impurities Instantons ElectronQuarks

7. Conductivity versus chiral symmetry breaking "Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons" Is that important? Yes. Banks-Casher (Kubo) Metallic behavior means chiSB in the ILM Recent developments: - Thouless energy in QCD. If the QCD vacuum at T= 0 is a metal, one can predict finite size effects. Verbaarschot,Osborn, PRL 81 (1998) 268 and Zahed, Janik et.al., PRL. 81 (1998) 264 - 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

8. Phase transitions in QCD J. Phys. G30 (2004) S1259 • Quark- Gluon Plasma • weakly only for T>>Tc

9. Deconfinement and chiral restoration They must be related but nobody* knows exactly how • Deconfinement • Linear confining potential vanishes. Chiral Restoration • Matter becomes light • QCD still non perturbative 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 We propose that quantum interference and tunneling, namely,Anderson localization plays an important role. Nuclear Physics A, 770, 141 (2006)

10. Localization and chiral transition Instanton liquid picture 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 = TLsuch that a MIT takes place. 5. There must exist a T = Tc such that 6. This general picture is valid beyond the instanton liquid approximation (KvBLL, see Ilgenfritz talk) provided that the hopping induced by topological objects is short range. Is TL = Tc ?...Yes Does the MIT occur at the origin? Yes Dyakonov, Petrov

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

12. Spectral characterization RMT correlations:Weak disorder (d > 2). Up to Thouless. Poisson correlations: Any disorder d < 2, strong disorder d>2 "In the context of QCD the metallic region corresponds with the infrared limit (constant fields) of the Dirac operator" (Verbaarschot,Shuryak)

13. ANDERSON TRANSITIONMain:Non trivial interplay between tunneling and interference leads to the metal insulator transition (MIT) Spectral correlationsWavefunctions Scale invarianceMultifractals CRITICAL STATISTICS "Spectral correlations are universal, they depend only on the dimensionality of the space." Kravtsov, Muttalib 97 Skolovski, Shapiro, Altshuler Mobility edge Anderson transition

14. Finite size scaling analysis, Dynamical 2+1 Massless Massive

15. Quenched Lattice IPR versus eigenvalue

16. Unquenched ILM, 2 m = 0 The transition is located around T =120

17. Unquenched lattice, close to the origin, 2+1 flavors, N = 200 INSULATOR METAL

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

19. Instanton liquid model:,condensate and inverse participation ratio versus T

20. Lattice: and inverse participation ratio versus T Unquenched, massive 2+1 Quenched ( also unquenched masless) For zero mass, transition sharper with the volumeFirst order For finite mass, the condensate is volume independentCrossover

21. 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 order phase transition. 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.

22. Confinement and spectral properties Idea:Polyakov loop is expressed as the response of the Dirac operator to a change in time boundary conditionsGattringer,PRL 97 (2006) 032003, hep-lat/0612020 …. but sensitivity to boundary conditions is a criterium (Thouless) for localization!

23. Localization and confinement The dimensionless conductance, g, a measure of localization, is related to the sensitivity of eigenstates to a change in boundary conditions. Metal Insulator 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 Tc? 4. What is the estimation of the P from localization theory?

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

25. Localization and Confinement IPR (red), Accumulated Polyakov loop (blue) for T>Tc as a function of the eigenvalue. MI transition? Metal prediction

26. Quenched ILM, IPR, N = 2000 Metal IPR X N= 1 Insulator IPR X N = N Multifractal IPR X N = Similar to overlap prediction Morozov,Ilgenfritz,Weinberg, et.al. Origin D2~2.3(origin) Bulk

27. 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

28. Unquenched ILM, 2+1 flavors We have observed a metal-insulator transition at T ~ 125 Mev

29. Conclusions • 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 "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”

30. 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.

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

32. 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

33. IPR, two massless flavors D2 ~ 1.5 (bulk) D2~2.3(origin)

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

36. Quenched ILM, Bulk, T=200

37. Nuclear (quark) matter at finite temperature • Cosmology 10-6 sec after Bing Bang, neutron stars (astro) • Lattice QCD finite size effects. Analytical, N=4 super YM? • High energy Heavy Ion Collisions. RHIC, LHC 1 2 3 4 Hadron Gas & Freeze-out Colliding Nuclei HardCollisions QG Plasma ? sNN = 130, 200 GeV (center-of-mass energy per nucleon-nucleon collision)

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

39. Instanton liquid models T = 0 "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)

40. QCD Chiral Symmetries Classical Quantum U(1)A explicitly broken by the anomaly. SU(3)A spontaneously broken by the QCD vacuum Dynamical mass Eight light Bosons (p,K,h), no parity doublets.

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