Critical Level Statistics and QCD Phase Transition

Transcription

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

2 Wilsonʼs Lattice Gauge Theory = random Dirac op quenched Boltzmann weight # " tr U U U U spinor & N-color d.o.f. random SU(N) variable U n,µ Dirac γ µ Andersonʼs tight-binding H = random Schrodinger op i.i.d. random variable V n fixed const

3 OUTLINE Part I : Critical Level Statistics & RMT CLS at localization transition Shkhlovskii et al PRBʼ93 deformed RMT Muttalib et al PRLʼ93 level spacing : CLS vs drmt SN PREʼ98/99, Garcia 2 -SN-Verbaarschot PREʼ02 Part II : QCD transition & Dirac spectra chiral restoration by localization Diakonov-Petrov NPBʼ86 Dirac spectra at QCD transition Garcia 2 -Osborn NPAʼ06/PRDʼ07 level spacing : LGT vs drmt Kato-SN *ʼ08

4 Part I Critical Level Statistics & RMT

5 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

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

7 Critical Level Statistics sparse overlap "! i (x) 2! j (x) 2 %(1% D 2 )/ d # $ i % $ j x Chalker 90 distant levels becomes less repulsive level spacings level # variance s small s large P(s) " s # "e $% s & 2 (S) "logs " ' S Poisson-like Level Repulsion w/o Rigidity

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

9 Deformed RMT phenomenological model for CLS Invariant RME finite-t free fermions Moshe et al 94 common R 2 for small deform. Invariant RME in log 2 H potential Muttalib et al 93 preferred basis Banded RME 2 $ sin# x ' R 2 (x) = "& ) % T "1 sinh#t x ( Mirlin et al 96 TL liquid at T=1/g*

10 Deformed RMT before unfolding, inv RME always gives kernel unfolding " " # x(") = % $ av (")d" K(", "#) = ( " ' ' "# ) sin$ % & % " & "# #+& /2 for V(H ) ~ 1 ( and T small, " av (#) = ' "(# $ )d# $ ~ 1 #%& /2 2a log H ) 2 2a# unusual unfolding " # x = 1 log" 2a K(x, x ") =! sin#(x $ x ") (# /a)sinha(x $ x ")

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

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

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

14 Part II QCD transition & Dirac spectra

15 QCD transition 64 i 7 D / 48 1/T S QCD = # d" d 3 r # x $ tr F 2 µ% + q (i& µ + A µ )' µ q + m q q +µ B q + q 0 hadron phase chiral symm breaking color confinement q q " 0 V q"q (r)# r

16 Diakonov-Petkov scenario localization of fermionic W.F. chiral symm restoration = chiral quasi-zero mode Ψ on topological b.g. ( q q = "lim* ) lim #$0 V$% & '(#) V + -, : χsb needs energy band around origin low T: Ψ on instanton b.g. extended " I D / " AI # ~ 1 level repulsion band around " = 0 r 3 high T : Ψ on periodic instanton b.g. localized in 3D " I D / " AI # ~ e $%T r no level repulsion collapse to no band " = 0

17 Lattice Gauge Theory quenched Boltzmann weight # " tr U U U U how to change T L x,y,z >> L t chiral SB fixed " a " e #$ / b 0 T = 1 L t a what to measure q q = "# $ D (0) confinement e "F q /T # tr U ( r 0,0),0 LU ( r 0,L t ),0 L t a spinor & N-color SU(N) variable U n,µ Dirac γ µ L x a

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

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

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

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

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

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