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Critical Statistics at the Mobility Edge of QCD Dirac spectra

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 LATTICE2013, Mainz. Introduction. Wilson’s Lattice Gauge Theory : stochastic Dirac op.

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Critical Statistics at the Mobility Edge of QCD Dirac spectra

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  1. Critical Statistics at the Mobility Edge of QCD Dirac spectra S. M. NishigakiShimane Univ based on ongoing work with M. Giordano, T. G. Kovacs, F. Pittler MTA ATOMKI Debrecen Aug. 3, 2013 LATTICE2013, Mainz

  2. Introduction Wilson’s Lattice Gauge Theory :stochastic Dirac op. Boltzmann weight fixed const mq ‘random’ SU(N) variableUx, □ □ ×  analogy of localization? Anderson’s tight-binding Model : random Schrodinger op. i.i.d. random variable Vx fixed const

  3. Introduction Wilson’s Lattice Gauge Theory :stochastic Dirac op. Anderson’s tight-binding Hamiltonian : random Schrodinger op. “ ” Halasz-Verbaarschot ’95 critical statistics

  4. PLAN slide nr. 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 18～19 Review: Dirac sp. at high T 20～27 Dirac sp. at high T & deformed RM G-K-SMN-P’13 I II III

  5. I.1 RMT Random matrices {sparse, dimensionful} {dense, indep. random} sharing discrete symmetry Universality in local fluctuation of EVs ⇒ Gaussian Slater det : EVs = 1D free fermions harmonic osc. WF (Hermite polyn.)

  6. I.1 RMT Local EV correlation - bulk Two-level Correlator Level Spacing Distribution (LSD) =0no corr =1 =2 RM =4 exp(-s) ~ s ~ exp(-c s2)

  7. I.2 AH vs RMT Anderson Hamiltonian Vx random Vx W t t fixedt x x

  8. I.2 AH vs RMT Anderson Hamiltonian Level Spacing Distribution (LSD) random Vi fixedt d,w/o B d,with B vs GOE b=1vs GUE b=2 weak randomness:level statistics⊂ RM universality

  9. I.2 AH vs RMT NLsM forAnderson H Wegner, Efetov ’80s Gaussian av. over V(x) H-S transf diffusion cst e regime : 0 mode dominance :0d NLsM ⇔ RM

  10. I.3 Localization NLsM forAnderson H Wegner ’89 perturbativeb-function of NLsMs ind=2+e b(g) ヨ fixed pt . d=2 (AII), d≥3 g* conductance g d=2 (AI, A) d=1 Insulator (localized) Metal (extended)

  11. I.3 Localization NLsM forAnderson H : e regime, 0 mode dominance reduces to 0D NLsM ⇔ RMT ergodic regime ETh→ ∞: RMT √ diffusive regimeETh>>D: perturbation √ “mobility edge” ETh ~ D: perturbation × →phenomenological model desirable

  12. II.1 Critical Statistics Shklovskii et al ’93 LSD ofAnderson H example: 3d, V=203, Nconf=104 randomness W/t =18.1 mag. flux F=0.4p EV density localized WF ≪L no repulsion → Poisson multifractal WF  ~ L Scale Invariant Critical Statistics

  13. 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”

  14. II.2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’94 Invariant RM spontaneously broken equivalent to freefermions at temp. T>0 U(N) inv U(N) inv →equivalent to Banded RM multifractal WF

  15. II.2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’94 Invariant RM spontaneously broken equivalent to freefermions at temp. T>0 U(N) inv “HCIZ integral”

  16. II.2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’94 : 1D free fermions at T>0 T→0 : Fermi repulsion ⇒ RMT T→∞: classical, no repulsion ⇒ Poisson 0<T<∞ ⇒ intermediate statistics

  17. II.3 CS vs deformed RM SMN ’98 b=1 LSD : deformed RM RM Poisson b=2 properties of CS built-in ~ s ~ e-s/2 b=4 deformation parameter

  18. II.3 CS vs deformed RM SMN ’99 LSD : Anderson H at ME 3d without B deformed RM = CS of AH → high-T QCD? a=3.55 from tail fit s≫1 3d with SOC 3d with B

  19. III.0 Dirac spectrum Small Dirac EV fluctuation discretization garbage → wealth of physical info on cSB LEC global symm eregime : chRMT exact EV density, smallest EV distr, ... direct access to S, Fp , W8 ,…with probe l Splittorff, Lattice’12 plenary Verbaarschot, Lattice’13 7D

  20. III.0 Dirac spectrum kth Dirac EV distribution sample: U(1) Dirac spectrum vs chGUE at origin …not the subject of today’s talk →bulk of spectrum chiral condensate Damgaard-SMN’01 SMN’13 -th EV

  21. III.1 Dirac spectrum - previous Dirac spectra for high-T QCD   ？ → soft edge Airy hard edge Bessel soft edges Airy? Farchoni-deForcrand-Hip-Lang-Splittorff ’99 + too many other groups to list, sorry. other scenarios from RMT: Jackson-Verbaarschot ’96 Akemann-Damgaard-Magnea-SMN ’98

  22. 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

  23. 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

  24. 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: physical pt. determined by Budapest-Wuppertal #gauge: unimproved Wilson fermion: naive staggered * gauge: Symanzik improved fermion: 2-level stout-smeared staggered

  25. III.2 Dirac spectrum – current status Dirac spectra for high-T QCD at physical pt local EV window (2~10 evs) →LSD

  26. III.3 ME & deformed RM Dirac LSD for high-T QCD G-K-SMN-P ’13 deform. parameter vs EV window

  27. III.3 ME & deformed RM Dirac LSD for high-T QCD G-K-SMN-P ’13 deform. parameter vs EV window conclusion I dRM nicely fits low-lying Dirac spectra of high-T QCD in each EV window near ME, just as in Anderson H

  28. III.3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’13

  29. 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. scale inv M.E.

  30. 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.

  31. 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.

  32. 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.

  33. 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.

  34. 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.

  35. 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.

  36. 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.

  37. 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.

  38. III.3 ME & deformed RM deform. parameter vs EV window & size G-K-SMN-P ’13 larger spatial vol a=3.60 scale inv M.E. TDL : localized←ME→extended conclusion II finite fraction of small EVs exists & localizes even in presence of very light quarks

  39. III.3 ME & deformed RM profile of LSD G-K-SMN-P ’13 ● Poisson dRM RM ME ● path along which the system crosses over RM → Poisson is universal (indep of mq, T, a), almost follows 1-parameter deformed RM

  40. III.4 Physical implication Tpc from Mobility Edge Kovacs-Pittler ’12 universal, linear increase with T mobility edge 171MeV conclusion III Tpc consistent with disappearing localized mode

  41. 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

  42. Summary / QCD D onL3 × 1/T (<1/Tc) Anderson H onL3 ME : identical critical statistics EV density MNS deformed RM : exact? theory of Anderson loc. a=3.60 a=3.55

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