S. M. Nishigaki 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 Critical Statistics at the Mobility Edge of QCD Dirac spectra

Anderson’s tight-binding Model : random Schrodinger op. i.i.d. random variable V x fixed const Wilson’s Lattice Gauge Theory : stochastic Dirac op. □ □ Boltzmann weight analogy of localization? ‘random’ SU(N) variable U x,  fixed const m q ×   Introduction 1

Anderson’s tight-binding Hamiltonian : random Schrodinger op. Wilson’s Lattice Gauge Theory : stochastic Dirac op. Introduction “ Halasz-Verbaarschot ’95 ” critical statistics 2

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 PLAN I II III

{sparse, dimensionful} {dense, indep. random} sharing discrete symmetry Random matrices I.1 RMT Universality in local fluctuation of EVs ⇒ Gaussian harmonic osc. WF (Hermite polyn.) 3 Slater det : EVs = 1D free fermions

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

random V x fixed t I.2 AH vs RMT Anderson Hamiltonian x W VxVx x 5 t t

 d, w/o B  d, with B vs GOE  vs GUE  weak randomness : level statistics ⊂ RM universality I.2 AH vs RMT random V i fixed t 6 Level Spacing Distribution (LSD)

H-S transf Wegner, Efetov ’80s I.2 AH vs RMT NL  M for Anderson H Gaussian av. over V(x) diffusion cst  regime : 0 mode dominance : 0d NL  M ⇔ RM 7

perturbative  -function of NL  Ms in d=2+  (g)(g) Insulator (localized) Metal (extended) d=2 (AII), d≥3 d=2 (AI, A) g*g* ヨ fixed pt conductance g. d=1 I.3 Localization Wegner ’89 NL  M for Anderson H 8

:  regime, 0 mode dominance reduces to 0D NL  M ⇔ RMT ergodic regime : RMT √ ergodic regime E Th → ∞ : RMT √ diffusive regime E : perturbation √ diffusive regime E Th >>  : perturbation √ → phenomenological model desirable I.3 Localization NL  M for Anderson H 9 “mobility edge” : perturbation × “mobility edge” E Th ~  : perturbation ×

EV density localized WF  ≪ L no repulsion → Poisson multifractal WF  ~ L Scale Invariant Critical Statistics II.1 Critical Statistics example: 3d, V=20 3, N conf =10 4 randomness W/t =18.1 mag. flux  =0.4  Shklovskii et al ’93 LSD of Anderson H 10

Sparse overlap distant levels becomes less repulsive level spacing level # variance Poisson-like Chalker ’90 Zharekeshev-Kramer ’97 “Level Repulsion without Rigidity” II.1 Critical Statistics 11 Anomalous inverse part. ratio  d, with B WFs and EVs at ME

II.2 Deformed RM Invariant RM spontaneously broken equivalent to free fermions at temp. T>0 MNS model Moshe-Neuberger-Shapiro ’94 U(N) inv → equivalent to Banded RM multifractal WF 12

II.2 Deformed RM U(N) inv Invariant RM spontaneously broken equivalent to free fermions at temp. T>0 MNS model Moshe-Neuberger-Shapiro ’94 12 “HCIZ integral”

: 1D free fermions at T>0 T→0 : Fermi repulsion ⇒ RMT T→∞: classical, no repulsion ⇒ Poisson 0<T<∞ ⇒ intermediate statistics II.2 Deformed RM MNS model Moshe-Neuberger-Shapiro ’94 13

~ e -s /2     SMN ’98 LSD : deformed RM RM Poisson ~ s  properties of CS built-in deformation parameter II.3 CS vs deformed RM 14

3d with B 3d with SOC a=3.55 from tail fit s ≫ 1 deformed RM = CS of AH → high-T QCD? II.3 CS vs deformed RM SMN ’99 LSD : Anderson H at ME 15 3d without B

Small Dirac EV fluctuation discretization garbage → wealth of physical info on  SB discretization garbage → wealth of physical info on  SB  regime : exact chRMT LEC EV density, smallest EV distr,... direct access to  F   W 8  with probe III.0 Dirac spectrum global symm Splittorff, Lattice’12 plenary Verbaarschot, Lattice’13 7D 16

k th Dirac EV distribution k th Dirac EV distribution III.0 Dirac spectrum sample: U(1) Dirac spectrum vs chGUE at origin chiral condensate …not the subject of today’s talk → bulk of spectrum Damgaard-SMN’01 SMN’13 -th EV 17

Dirac spectra for high-T QCD soft edge Airy hard edge Bessel  ？→？→ III.1 Dirac spectrum - previous 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

Damgaard et al ’00  × ・ non-Airy behavior ・ unfolding scale is different soft edges Airy? Dirac spectra for high-T QCD III.1 Dirac spectrum - previous 18

SU(3) quenched LGT on   ~   ×  KS Dirac op. Garcia-Osborn 07 III.1 Dirac spectrum - previous... spectral averaing over a window too wide for Level Statistics ・ chi symm restoration ・ localization ・ deconfinement simultaneous? 19 Localization and QCD transition

# gauge: unimproved Wilson fermion: naive staggered We have analyzed low-lying Staggered Dirac EVs for: physical pt. determined by Budapest-Wuppertal Giordano-Kovacs-SMN-Pittler ’13 in prep. * gauge: Symanzik improved fermion: 2-level stout-smeared staggered III.2 Dirac spectrum – current status 20 Dirac spectra for high-T QCD at physical pt gaugeNFNF βm ud msms a[fm]NsNs NtNt TN conf N EV SU(2) ,24,32,4842.6T c 3k256 SU(3) ,28,...,484394Me V 7k~40 k 512~1 k

local EV window (2~10 evs) → LSD III.2 Dirac spectrum – current status 22 gaugeNFNF βm ud msms a[fm]NsNs NtNt TN conf N EV SU(2) ,24,32,4842.6T c 3k256 SU(3) ,28,...,484394Me V 7k~40 k 512~1 k Dirac spectra for high-T QCD at physical pt

III.3 ME & deformed RM Dirac LSD for high-T QCD 23 deform. parameter vs EV window G-K-SMN-P ’13

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

III.3 ME & deformed RM deform. parameter vs EV window & size 24 G-K-SMN-P ’13

scale inv M.E. larger spatial vol scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size G-K-SMN-P ’13 a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

scale inv M.E. III.3 ME & deformed RM 24 deform. parameter vs EV window & size LSD at ME G-K-SMN-P ’13 larger spatial vol a=3.60

TDL : localized←ME→extended III.3 ME & deformed RM finite fraction of small EVs exists & localizes even in presence of very light quarks conclusion II 24 deform. parameter vs EV window & size G-K-SMN-P ’13 larger spatial vol a=3.60 scale inv M.E.

path along which the system crosses over RM → Poisson is universal ( indep of m q, T, a), almost follows 1-parameter deformed RM III.3 ME & deformed RM profile of LSD 25 G-K-SMN-P ’13 dRM ME Poisson ● RM ●

T pc consistent with disappearing localized mode T pc from Mobility Edge Kovacs-Pittler ’12 universal, linear increase with T III.4 Physical implication conclusion III mobility edge 171MeV 26

conjecture: localized modes are associated w/ defects of Polyakov loop Origin of localized modes smeared SU(2) Polyakov loop ⇔ localized mode of D OV III.4 Physical implications Bruckmann-Kovacs-Schierenberg ’11

EV density QCD D on L 3 × 1/T (<1/T c ) Anderson H on L 3 Summary MNS deformed RM : exact? theory of Anderson loc. a=3.60 a=3.55 ME : identical critical statistics /