Critical Statistics at the Mobility Edge of QCD
Critical Statistics at the Mobility Edge of QCD Dirac spectra S. M. Nishigaki Shimane Univ based on ongoing work with M. Giordano, T. G. Kovacs, F. Pittler MTA ATOMKI Debrecen Aug. 3, 2013 LATTICE 2013, Mainz
Introduction Wilson’s Lattice Gauge Theory : stochastic Dirac op. Boltzmann weight fixed const mq □ □ Anderson’s tight-binding Model : random Schrodinger op. Ux, ‘random’ SU(N) variable × analogy of localization? i. i. d. random variable Vx fixed const 1
Introduction Wilson’s Lattice Gauge Theory : stochastic Dirac op. Anderson’s tight-binding Hamiltonian : random Schrodinger op. “ ” Halasz-Verbaarschot ’ 95 critical statistics 2
PLAN slide nr. I II 01～ 02 Introduction 03～ 09 Basics: RMT & AH 10～ 13 Review: CS & deformed RM 14～ 15 LSD of CS & deformed RM SMN’ 98, ’ 99 16～ 17 Dirac sp. & chiral RM D-SMN’ 01, SMN’ 13 III 18～ 19 Review: Dirac sp. at high T 20～ 27 Dirac sp. at high T & deformed RM G-K-SMN-P’ 13
I. 1 RMT Random matrices {sparse, dimensionful} {dense, indep. random} sharing discrete symmetry Universality in local fluctuation of EVs ⇒ Gaussian Slater det : EVs = 1 D free fermions harmonic osc. WF (Hermite polyn. ) 3
I. 1 RMT Local EV correlation - bulk Level Spacing Distribution (LSD) Two-level Correlator =0 no corr =1 =2 RM =4 exp(-s) ~ s ~ exp(-c� s 2) 4
I. 2 AH vs RMT Anderson Hamiltonian Vx random Vx fixed t t W t xx 5
I. 2 AH vs RMT Anderson Hamiltonian Level Spacing Distribution (LSD) random Vi fixed t �d, w/o B vs GOE = �d, with B vs GUE = weak randomness : level statistics ⊂RM universality 6
I. 2 AH vs RMT NLs. M for Anderson H Wegner, Efetov ’ 80 s Gaussian av. over V(x) H-S transf diffusion cst e regime : 0 mode dominance : 0 d NLs. M ⇔ RM 7
I. 3 Localization NLs. M for Anderson H Wegner ’ 89 perturbative -function of NLs. Ms in d=2+e (g) ヨ fixed pt Insulator (localized) . g* conductance g d=2 (AII), d≥ 3 d=2 (AI, A) d=1 Metal (extended) 8
I. 3 Localization NLs. M for Anderson H : e regime, 0 mode dominance reduces to 0 D NLs. M ⇔ ergodic regime RMT ETh → ∞ : RMT √ diffusive regime ETh >> D : perturbation √ “mobility edge” ETh ~ D : perturbation × → phenomenological model desirable 9
II. 1 Critical Statistics EV density LSD of Anderson H Shklovskii et al ’ 93 example: 3 d, V=203, Nconf=104 randomness W/t =18. 1 mag. flux F=0. 4 p multifractal WF ~ L Scale Invariant Critical Statistics localized WF ≪ L no repulsion → Poisson 10
II. 1 Critical Statistics WFs and EVs at ME � d, with B Anomalous inverse part. ratio Chalker ’ 90 Zharekeshev-Kramer ’ 97 Sparse overlap distant levels becomes less repulsive level spacing Poisson-like level # variance “Level Repulsion without Rigidity” 11
II. 2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’ 94 Invariant RM spontaneously broken equivalent to free fermions at temp. T>0 U(N) inv → equivalent to Banded RM multifractal WF 12
II. 2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’ 94 Invariant RM spontaneously broken equivalent to free fermions at temp. T>0 U(N) inv “HCIZ integral” 12
II. 2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’ 94 : 1 D free fermions at T>0 T→ 0 : Fermi repulsion ⇒ RMT T→∞: classical, no repulsion ⇒ Poisson 0<T<∞ ⇒ intermediate statistics 13
II. 3 CS vs deformed RM = LSD : deformed RM SMN ’ 98 RM Poisson = properties of CS built-in ~ e-s/2 ~ s = deformation parameter 14
II. 3 CS vs deformed RM LSD : Anderson H at ME SMN ’ 99 3 d without B deformed RM = CS of AH → high-T QCD? a=3. 55 from tail fit s≫ 1 3 d with B 3 d with SOC 15
III. 0 Dirac spectrum Small Dirac EV fluctuation discretization garbage → wealth of physical info on SB LEC global symm ch. RMT e regime : exact EV density, smallest EV distr, . . . direct access to S Fp W 8 , …with probe l Splittorff, Lattice’ 12 plenary Verbaarschot, Lattice’ 13 7 D 16
III. 0 Dirac spectrum kth Dirac EV distribution sample: U(1) Dirac spectrum vs ch. GUE at origin chiral condensate Damgaard-SMN’ 01 SMN’ 13 …not the subject of today’s talk → bulk of spectrum -th EV 17
III. 1 Dirac spectrum - previous Dirac spectra for high-T QCD ？ → hard edge Bessel soft edge Airy soft edges Airy? Farchoni-de. Forcrand-Hip-Lang-Splittorff ’ 99 + too many other groups to list, sorry. other scenarios from RMT: Jackson-Verbaarschot ’ 96 Akemann-Damgaard-Magnea-SMN ’ 98
III. 1 Dirac spectrum - previous Dirac spectra for high-T QCD × soft edges Airy? Damgaard et al ’ 00 ・non-Airy behavior ・unfolding scale is different 18
III. 1 Dirac spectrum - previous Localization and QCD transition SU(3) quenched LGT on ~ × KS Dirac op. Garcia-Osborn 07 ・ chi symm restoration ・ localization ・ deconfinement simultaneous? . . . spectral averaing over a window too wide for Level Statistics 19
III. 2 Dirac spectrum – current status Dirac spectra for high-T QCD at physical pt Giordano-Kovacs-SMN-Pittler ’ 13 in prep. We have analyzed low-lying Staggered Dirac EVs for: gauge NF β mud ms a[fm] Ns SU(2) 0 2. 60 - - - SU(3) 2+1 3. 75. 001786. 05030. 125 Nt T Nconf NEV 16, 24, 32, 48 4 2. 6 Tc 3 k 256 24, 28, . . . , 48 394 Me. V 7 k~40 k 512~1 k 4 physical pt. determined by Budapest-Wuppertal # gauge: unimproved Wilson fermion: naive staggered * gauge: Symanzik improved fermion: 2 -level stout-smeared staggered 20
III. 2 Dirac spectrum – current status Dirac spectra for high-T QCD at physical pt gauge NF β mud ms a[fm] Ns SU(2) 0 2. 60 - - - SU(3) 2+1 3. 75. 001786. 05030. 125 Nt T Nconf NEV 16, 24, 32, 48 4 2. 6 Tc 3 k 256 24, 28, . . . , 48 394 Me. V 7 k~40 k 512~1 k 4 local EV window (2~10 evs) → LSD 22
III. 3 ME & deformed RM Dirac LSD for high-T QCD deform. parameter vs EV window G-K-SMN-P ’ 13 23
III. 3 ME & deformed RM Dirac LSD for high-T QCD deform. parameter vs EV window G-K-SMN-P ’ 13 conclusion I d. RM nicely fits low-lying Dirac spectra of high-T QCD in each EV window near ME, just as in Anderson H
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 larger spatial vol a=3. 60 larger spatial vol scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 LSD at ME larger spatial vol a=3. 60 scale inv M. E. 24
III. 3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’ 13 larger spatial vol a=3. 60 TDL : localized←ME→extended scale inv M. E. conclusion II finite fraction of small EVs exists & localizes even in presence of very light quarks 24
III. 3 ME & deformed RM profile of LSD G-K-SMN-P ’ 13 ● Poisson d. RM RM ME ● path along which the system crosses over RM → Poisson is universal (indep of mq, T, a), almost follows 1 -parameter deformed RM 25
III. 4 Physical implication Tpc from Mobility Edge Kovacs-Pittler ’ 12 mobility edge a re ea n i l, l a rs se th wi T nc i r e iv un 171 Me. V conclusion III Tpc consistent with disappearing localized mode 26
III. 4 Physical implications Origin of localized modes Bruckmann-Kovacs-Schierenberg ’ 11 conjecture: localized modes are associated w/ defects of Polyakov loop smeared SU(2) Polyakov loop ⇔ localized mode of DOV
Summary QCD D / on L 3 × 1/T (<1/Tc) Anderson H on L 3 EV density ME : identical critical statistics MNS deformed RM : exact? theory of Anderson loc. a=3. 60 a=3. 55
- Slides: 42
