Critical Level Statistics and QCD Phase Transition S.M. Nishigaki - PowerPoint PPT Presentation

Critical Level Statistics and QCD Phase Transition S.M. Nishigaki Dept. Mat. Sci, Shimane Univ. quark gluon

Wilson ʼ s Lattice Gauge Theory = random Dirac op quenched Boltzmann weight 4- spinor & N- color d.o.f. � � tr U • U • U • U • 1 2 4 3 4 random SU( N ) variable U n, µ □ □ × Dirac γ µ Anderson ʼ s tight-binding H = random Schrodinger op i.i.d. random variable V n fixed const

OUTLINE Part I : Critical Level Statistics & RMT  CLS at localization transition Shkhlovskii et al PRB ʼ 93  deformed RMT Muttalib et al PRL ʼ 93  level spacing : CLS vs dRMT SN PRE ʼ 98/99, Garcia 2 -SN-Verbaarschot PRE ʼ 02 Part II : QCD transition & Dirac spectra  chiral restoration by localization Diakonov-Petrov NPB ʼ 86  Dirac spectra at QCD transition Garcia 2 -Osborn NPA ʼ 06/PRD ʼ 07  level spacing : LGT vs dRMT Kato-SN * ʼ 08

Part I Critical Level Statistics & RMT

Anderson Hamiltonian i.i.d. random variable fixed const Two-level correlator Braun-Montambaux 95 3D Orth. 3D Unitary weak randomness : level statistics ⊂ RMT universality

Anderson Hamiltonian level delocalized ξ >> L ξ ~ L localized ξ << L multifractal | ψ (x) | 2 x level repulsion no repulsion scale invariant → Wigner → Poisson Critical Statistics

Critical Level Statistics 2 � (1 � D 2 )/ d 2 � j ( x ) � � i ( x ) sparse overlap � � i � � j Chalker 90 x distant levels becomes less repulsive s small s large P ( s ) � s � � e � � s level spacings Poisson-like � 2 ( S ) � log s � � S level # variance “Level Repulsion w/o Rigidity”

Critical Level Statistics AH (3D Orth.) Zharekeshev-Kramer 97 level spacings number variance • indep of scale indep of scale IR fixed pt • IR fixed pt indep of randomness type • • indep of randomness type quasi-universality quasi-universality g ＊ ＊ • • dep on dimensionality, b.c. dep on dimensionality, b.c. conductance at fixed pt conductance at fixed pt g

Deformed RMT phenomenological model for CLS Invariant RME Invariant RME finite- T free fermions in log 2 H potential common R 2 Moshe et al 94 for small deform. Muttalib et al 93 preferred basis 2 � sin � x � R 2 ( x ) = � Banded RME � � T � 1 sinh � T x � � Mirlin et al 96 TL liquid at T =1/ g *

Deformed RMT ( � � ) � sin � � � � � � K ( � , � � ) = before unfolding, inv RME always gives kernel � � � � � unfolding � � � x ( � ) = � av ( � ) d � 1 V ( H ) ~ 1 � + � /2 2 � av ( � ) = � � ( � � ) d � ~ � for and T small , ( ) 2 a log H � �� /2 2 a � � � x = 1 2 a log � unusual unfolding � sin � ( x � � x ) K ( x , � x ) = ( � / a )sinh a ( x � � x )

Deformed RMT level spacings : SN 98 Orth. Tracy-Widom method for � = 2 ( s ) = d 2 P ds 2 Det [0, s ] (1 � K ) P � = 1,4 ( s ) similar Unit. ~ e - s /2 χ ~ s β Symp. sin � ( x � � x ) K ( x , � x ) = ( � / a )sinh a ( x � � x )

Level spacings : CLS vs dRMT 3D Orth. SN 99 choose a =3.55 from tail fit s >>1 3D Unit. 3D Symp.

CLS vs dRMT number variance level spacings : 2D AH 3D Orth. 2D Symp. ~ e - s /2 χ ~ χ S ~ s 4 ~ log S

Part II QCD transition & Dirac spectra

QCD transition i / D 6 7 4 4 8 d 3 r 2 + q 1/ T ( i � µ + A µ ) � µ q + µ B q + q � � S QCD = d � x � tr F q + m q µ � 0 hadron phase ・ chiral symm breaking q q � 0 ・ color confinement V q � q ( r ) � r

Diakonov-Petkov scenario localization of fermionic W.F. ⇒ chiral symm restoration = chiral quasi-zero mode Ψ on topological b.g. � � � ( � ) � : χ SB needs energy band around origin q q = � lim � � 0 lim � � � V � V �� ・ low T : Ψ on instanton b.g. AI ~ 1 ⇒ level repulsion ⇒ band around extended � I / D � � � = 0 r 3 ・ high T : Ψ on periodic instanton b.g. AI ~ e � � T r ⇒ no level repulsion ⇒ collapse to localized in 3D � I / D � � � = 0 no band

↗ ↘ ↗ Lattice Gauge Theory ・ how to change T L x , y , z >> L t fixed T = 1 a � e � � / b 0 � quenched L t a Boltzmann weight � � tr U • U • U • U • ・ what to measure 1 2 4 3 4 □ □ q q = � � � D (0) chiral SB � F q / T � tr U ( ,0),0 L U ( e confinement r r 0 0 , L t ),0 spinor & N- color L t a SU( N ) variable U n, µ × Dirac γ µ L x a

Dirac spectra at QCD transition SU(3) quenched LGT Garcia 2 -Osborn 07 on 16 3 ~ 20 3 × 4, KS Dirac op. ・ chiral symm restoration ・ deconfinement transition ・ localization simultaneous!

Dirac spectra at QCD transition Attn: ILM is a priori semiclassically biased not the real QCD Garcia 2 -Osborn 06 quenched ILM at T = Λ QCD , KS Dirac op. unitary dRMT a =3.2 SN 99 scale-inv critical statistics

Dirac spectra at QCD transition SU(3) quenched LGT Garcia 2 -Osborn 07 on 20 3 × 4, at β =7.93 , KS Dirac op. unitary dRMT a =3.2

Dirac spectra at QCD transition quenched SU(2) LGT Kato-SN 08 ＊ extensive study ( T =0) on ( 7 × 9 × 11) × 4 , KS Dirac op.* by Guhr et al 99 2nd order deconfinement EV distribution cumulatitive s i = � i + 1 � � i � � i + 1 � � i x ( � ) = x av ( � ) + x osc ( � ) / ( � i ) � 1 � ( � i ) x av polynomial fit for each config. spectral unfolding

Dirac spectra at QCD transition SU(2) quenched LGT on ( 7 × 9 × 11) × 4 Kato-SN 08 Poisson β =1.0 cumualtive distribution β =2.5 WD β =3.0 sympl sympl β =4.0 dRMT dRMT a =.45 a =.90 WD β =1.0 β =2.5 β =3.0 β =4.0

Summary ・ Diakonov-Petkov scenario: Localization of Fermionic WF ⇒ QCD Phase Transition confirmed via Dirac spectra ・ Muttalib conjecture: phenomenologically Critical Level Statistics Deformed RMT ~ modelled by at Mobility Edge works both in AH, QCD Damgaard, Verbaarschot, Garcia 2 , Nagao, Kato - collaborator thanks: Zharekeshev, Schweizer, Kawarabayashi, Evangelou, Ohtsuki - AH data JSPS - grant