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Role of Anderson-Mott localization in the QCD phase transitions Antonio M. García-García [email protected] 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. James Osborn In collaboration with PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002 Outline 1. What is localization? 2. QCD vacuum as a conductor. QCD vacuum as a disordered medium. Dyakonov - Petrov ideas. 3. Localization in lattice QCD. 4. A few words about QCD phase transitions. 5. Role of localization in the QCD phase transitions. Results from instanton liquid models and lattice. 5.1 The chiral phase transition. 5.2 The deconfinement transition. In progress. 6. What’s next. Relation confinement and chiral symmetry breaking. Quark diffusion in LHC. 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. Insulator Metal How do we know that a metal is a metal? Texbook answer: Look at the conductivity or other transport properties Other options: Look at eigenvalue and eigenvectors D QCD n in n 1. Eigenvector statistics: Dq = d Metal 2. Eigenvalue statistics: ( q 1) d Pq L H n En n Dq = 0 Insulator 2q n d ( q 1)( d Dq ) (r ) d r ~ L Dq = f(d,q) M-I transition P( s) s i 1 i / Level Spacing distribution: i Number variance: ( L i j / ) = n( L) n( L) 2 Insulator 2 ( L) L ( Poisson ) P( s) e s 2 Metal 2 ( L) ~ log L β As2 ( RMT ) Ps ~ s e 2 QCD : The Theory of the strong interactions High Energy g << 1 Perturbative 1. Asymptotic freedom Quark+gluons, Well understood Low Energy g ~ 1 Lattice simulations The world around us 2. Chiral symmetry breaking ~ (240MeV ) 3 Massive constituent quark 3. Confinement Colorless hadrons V (r ) a / r r How to extract analytical information? Instantons , Monopoles, Vortices QCD vacuum and instantons tHooft, Polyakov, Shuryak, Diakonov, Petrov Instantons (Polyakov,t'Hooft) : Non pertubative solutions of the classical Yang Mills equation. Tunneling between classical vacua. Help explain the chiral anomaly. D = μ + gAμins Dψ0 r 0 ψ0 r 1/ r 3 Dirac operator has a zero mode in the field of an instanton Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons Is that important? Yes. (m) 1 ( ) 1 Tr ( D m) d lim mm0 V m i V Banks-Casher (Kubo) Spectral properties related to SBCS Problem: Non linear equations No superposition Solution: Variational principles(Dyakonov,Petrov), Instanton liquid (Shuryak) Fair agreement with lattice, phenomenology, no confinement 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: Z ILM Nf 0 iD T de det(D m ) SYM k 1 200MeV , T 0 k N 1 fm V 4 IA AI (I A ) ˆ TIA d x ( x z I )iD A ( x z A ) ~ i(u R) R3 4 I 1. ILM explains the chiSB 1 3N N (240MeV ) 2 V 3 1/ 2 c 3 2. Describe non perturbative effects in hadronic correlation functions(Shuryak,Schaefer,Dyakonov,Petrov,Verbaarchot) QCD vacuum as a conductor (T =0) Metal An electron initially bounded to a single atom gets delocalized due to the overlapping with nearest neighbors. QCD Vacuum 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 Differences Dis.Sys: Exponential decay QCD vacuum Power law decay Quarks Nearest neighbors Long range hopping! QCD vacuum as a disordered conductor Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik Instanton positions and color orientations vary Impurities T=0 Instantons Electron Quarks long range hopping 1/Ra, a = 3<4 QCD vacuum is a conductor for any density of instantons AGG and Osborn, PRL, 94 (2005) 244102 Localization in QCD 1. RMT and Thouless energy in QCD. Metallic limit of QCD, Verbaarschot,Osborn, PRL Spectral correlations of the QCD Dirac operator in the infrared limit are universal (Verbaarschot, Shuryak NPA 560 306 ,1993). and can be obtained from a RMT with the symmetries of QCD 81 (1998) 268, Verbaarschot, Shuryak NPA 560 306 ,1993. 2 1 1 F L π E ~ 0 L c m 2 L 2. 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), hep-lat 0705.0018, I. Horvath, N. Isgur, J. McCune, and H. B. Thacker, Phys. Rev. D65, 014502 (2002), I. Horvath et al., Phys. Rev. D66, 034501 (2002), I. Horvath et al., Phys. Rev. D68, 114505 (2003)J. Greensite, S. Olejnik, M. Polikarpov, S. Syritsyn, and V. Zakharov, Phys. Rev. D71, 114507 (2005). q( x) TrF F 1 / 2 p ( x) 5 p 5 ( x) 5 ( x) cut “The general motivation behind the interest issues like localization and (fractal) dimensionality was the hope that the lowest modes are bound to certain singularities of the gauge field that in turn would explain confinement. Here, “singular” is understood in a sense opposite to semiclassical lumps, just in the spirit of Witten’s criticism of instantons.” hep-lat 0705.0018 1. Mobility edge at T = 0 in the infrared region of the spectrum (?). 2. Gauge configurations are multifractal with structures in three and lower dimensions. 3. Instantons are irrelevant… is that true? Phase transitions in QCD At which temperature does the transition occur ? What is the nature of transition ? Deconfinement: Confining potential vanishes. L 0 Chiral Restoration:Matter becomes light. Péter Petreczky J. Phys. G30 (2004) S1259 ~ 0 Quark- Gluon Plasma perturbation theory only for T>>Tc Deconfinement and chiral restoration They must be related but nobody* knows exactly how Deconfinement: Confining potential vanishes. L Chiral Restoration:Matter becomes light. 0 ~ 0 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 (confinement). Instanton do not dissapear at the transiton (Shuryak,Schafer). We propose that quantum interference and tunneling, namely, Anderson localization plays an important role. Nuclear Physics A, 770, 141 (2006) Localization and chiral transition: Why do we expect a metal insulator transition close to the origin at finite temperature? 1.The effective QCD coupling constant g(T) decreases as temperature increases. The density of instantons also decreases. 2. Zero modes are exponentially localized ( R) exp(TR ) in space but oscillatory in time. 3. Hopping amplitude restricted to T ~ exp( ATR) IA neighboring instantons. 4. Localization will depend strongly on the temperature. There must exist a T = TLsuch that a metal insulator transition takes place. Dyakonov, 5. There must exist a T = Tc such that 0 Petrov 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 TL = Tc ?...Yes Does the MIT occur at the origin? Yes Main Result D = + gA QCD μ A A , A lat At Tc μ ins D QCD lim m m0 i n n (m) V 0 but also the low lying, n undergo a metal-insulator transition. n "A metal-insulator transition in the Dirac operator induces the chiral phase transition " Statistical analysis of spectra and eigenfunctions D QCD n in n 1. Eigenvector statistics: Dq = d Metal ( q 1) d Pq L H n En n Dq = 0 Insulator 2q n d ( q 1)( d Dq ) (r ) d r ~ L Dq = f(d,q) M-I transition 2. Eigenvalue statistics: P( s) s i 1 i / Level Spacing distribution: i Number variance: ( L i j / ) = n( L) n( L) 2 Insulator 2 ( L) L ( Poisson ) P( s) e s 2 2 Metal 2 ( L) ~ log L β As2 ( RMT ) Ps ~ s e 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. Skolovski, Shapiro, Altshuler 2. P( s) ~ s P(s) ~ e As s 1 2 ( n) ~ s 1 var n 3. Eigenstates are multifractals. (r ) d r ~ L 2q d Dq ( q 1) var s 2 s 2 s n s n P(s)ds n Mobility edge Anderson transition Spectrum is scale invariant ILM with 2+1 massless flavors, P(s) of the lowest eigenvalues We have observed a metal-insulator transition at T ~ 125 Mev Metal insulator transition ILM, close to the origin, 2+1 flavors, N = 200 Level repulsion s << 1 Exponential decay s > 1 Unquenched ILM, 2 m = 0 ILM Nf=2 massless. Eigenfunction statistics AGG and J. Osborn, 2006 Localization versus chiral transition Instanton liquid model Nf=2, masless Chiral and localizzation transition occurs at the same temperature Lattice QCD AGG, J. Osborn 2007 1. Simulations around the chiral phase transition T 2. Lowest 64 eigenvalues Quenched 1. Improved gauge action 2. Fixed Polyakov loop in the “real” Z3 phase Unquenched 1. MILC colaboration 2+1 flavor improved 2. mu= md = ms/10 3. Lattice sizes L3 X 4 Quenched Lattice QCD IPR versus eigenvalue Finite size scaling analysis: var s s 2+1 dynamical fermions 2 2 s s P(s)ds n Quenched n Transiton from metal to insulator Quenched lattice, close to the origin METAL Inverse participation ratio versus T Dynamical, 2+1 Quenched For zero mass, transition sharper with the volume First order For finite mass, the condensate is volume independent Crossover Localization and order of the chiral phase transition lim m m0 ( m) V 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 so we expect a crossover. 5. Multifractal dimension D2 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 Gattringer,PRL 97 (2006) 032003, hep-lat/0612020 U 4 ( x, N ) zU 4 ( x, N ) 2 N ( x ) (1 z1 ) Nz1 ( x ) z1 z1 1 L( x ) N 8 N (1 z 2 ) z ( x ) z 2 2 z 2 Politely Challenged in: heplat/0703018, Synatschke, Wipf, Wozar 1 N N N P L( x ) 2 (1 z1 ) z1 (1 z2 ) z2 8V z1 z2 ( x ) v ( x, t ) v ( x, t ) N t 1 L, R, …. but sensitivity to spatial boundary conditions is a criterium (Thouless) for localization! 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 Tc? 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? Accumulated Polyakov loop versus eigenvalue Confinement is controlled by the ultraviolet part of the spectrum P Localization and Confinement IPR (red), Accumulated Polyakov loop (blue) for T>Tc as a function of the eigenvalue. Metal prediction MI transition? Conclusions ● ● ● Eigenvectors of the QCD Dirac operator becomes more localized as the temperature is increased. For a specific temperature we have observed a metalinsulator 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“ Wish “ Confinement-Deconfinemente transition has to do with localization-delocalization in time direction” 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 and finite density. Color superconductivity. Quenched ILM, IPR, N = 2000 Metal IPR X N= 1 Insulator IPR X N = N Multifractal Similar to overlap prediction Origin Morozov,Ilgenfritz,Weinberg, et.al. Bulk IPR X N = D2~2.3(origin) N D2 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 IPR, two massless flavors D2 ~ 1.5 (bulk) D2~2.3(origin) A A W = A A RMT P RMT A = s Ps ds 2 0 How to get information from a bunch of levels Spectrum Unfolding Spectral Correlators Quenched ILM, Bulk, T=200 Nuclear (quark) matter at finite temperature 1. Cosmology 10-6 sec after Bing Bang, neutron stars (astro) 3 Analytical, 4N=4 super YM ? 2. 1 Lattice QCD finite2size effects. 3. High energy Heavy Ion Collisions. RHIC, LHC Colliding Nuclei Hard Collisions QG Plasma ? Hadron Gas & Freeze-out sNN = 130, 200 GeV (center-of-mass energy per nucleon-nucleon collision) 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, Kravtsov, Chalker,Aoki, the wave function is multifractal. Schreiber,Castellani IPR I = ψ r d r L 4 2 Ld n d D2 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: 0 iD T T 0 IA AI 200MeV , N 1 fm V 4 ( ) T d x ( x z )iD ( x z ) ~ i(u Rˆ ) R 4 IA I I I A A 3 A 4 1. ILM explains the chiSB 2. Describe non perturbative effects in hadronic correlation functions (Shuryak,Schaefer,dyakonov,petrov,verbaarchot) QCD Chiral Symmetries L,R (1 5 ) Classical SU A (3) SUV (3) UV (1) U A (1) Quantum SUV (3) UV (1) U(1)A explicitly broken by the anomaly. SU(3)A spontaneously broken by the QCD vacuum qq (250 MeV ) 3 Dynamical mass Eight light Bosons (,K,), no parity doublets. Quenched lattice QCD simulations Symanzik 1-loop glue with asqtad valence 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 2 2 Metal n ( E ) = n n ~ log E β As 2 ( RMT ) P s ~ s e 2 Ec 1 / L 2 In units of the mean level spacing, the Thouless energy, g Ec d 2 L Eigenvalues in an insulator are not correlated. Insulator n ( E ) E ( Poisson ) P( s) e s 2 In the context of QCD the metallic region corresponds with the infrared limit (constant fields) of the Dirac operator" (Verbaarschot,Shuryak) 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 2 (s) ~ s J 0 (s) J1 (s) 2. RMT describes the eigenvalue correlations of the full QCD Dirac operator up to Ec. 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. Fπ2 Ec ~ 2 0 L L Ec g ~ Fπ2 L2 L 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