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.

gluon quark

random SU(N) variable Un,µ 4-spinor & N-color d.o.f.

Andersonʼs tight-binding H = random Schrodinger op

i.i.d. random variable Vn fixed const

Wilsonʼs Lattice Gauge Theory = random Dirac op

× Dirac γµ

• tr U•U•U•U•

1 2 4 3 4

• □

□

quenched Boltzmann weight

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, Garcia2-SN-Verbaarschot PREʼ02

Part II : QCD transition & Dirac spectra

 chiral restoration by localization Diakonov-Petrov NPBʼ86  Dirac spectra at QCD transition Garcia2-Osborn NPAʼ06/PRDʼ07  level spacing : LGT vs dRMT Kato-SN *ʼ08

OUTLINE

Part I Critical Level Statistics & RMT

Anderson Hamiltonian

3D Orth. Braun-Montambaux 95

Two-level correlator

weak randomness : level statistics ⊂ RMT universality

i.i.d. random variable fixed const 3D Unitary

level |ψ (x) |2

delocalized ξ >> L localized ξ << L ξ ~ L

x

multifractal level repulsion → Wigner no repulsion → Poisson

Anderson Hamiltonian

scale invariant Critical Statistics

sparse overlap distant levels becomes less repulsive

i(x)

2 j(x) 2 x

• i j

(1D2 )/ d

s small s large P(s) s e s 2(S) logs S

Critical Level Statistics

level spacings level # variance Poisson-like

Chalker 90

“Level Repulsion w/o Rigidity”

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

• dep on dimensionality, b.c.

dep on dimensionality, b.c. conductance at fixed pt conductance at fixed pt g g＊

＊

AH (3D Orth.)

Critical Level Statistics

Zharekeshev-Kramer 97

Invariant RME in log2H potential

Muttalib et al 93 Moshe et al 94

Deformed RMT

phenomenological model for CLS

Invariant RME finite-T free fermions Banded RME

Mirlin et al 96 preferred basis common R2 for small deform.

R2(x) = sin x T 1sinhT x

• 2

TL liquid at T=1/g*

K(, ) = sin

• (

)

• before unfolding, inv RME always gives kernel

x() = av()d

• Deformed RMT

unfolding for and T small , av() = ( )d

• /2

+ /2

• ~

1 2a unusual unfolding K(x, x ) = sin(x x ) ( /a)sinha(x x )

• V(H) ~ 1

2a logH

( )

2

x = 1 2a log

~ sβ ~ e-s/2χ

Orth. Unit. Symp.

SN 98

Deformed RMT

level spacings : Tracy-Widom method for P

=2(s) = d 2 ds2 Det [0,s](1 K)

P

=1,4(s) similar

K(x, x ) = sin(x x ) ( /a)sinha(x x )

choose a=3.55 from tail fit s>>1

Level spacings : CLS vs dRMT

3D Unit. 3D Symp. 3D Orth.

SN 99

~ χ S ~ log S

2D Symp. 3D Orth.

number variance level spacings : 2D AH

CLS vs dRMT ~ s4 ~ e-s/2χ

Part II QCD transition & Dirac spectra

QCD transition

hadron phase ・ chiral symm breaking ・ color confinement

q q 0 Vqq (r) r

SQCD = d

1/T

• d 3r

x

• tr F

µ 2 +q

(iµ + Aµ) µ i / D 6 7 4 8 4 q+ m q q+µBq+q

localization of fermionic W.F. ⇒ chiral symm restoration

chiral quasi-zero mode Ψ on topological b.g.

=

Diakonov-Petkov scenario

: χSB needs energy band around origin

・ low T: Ψ on instanton b.g. extended ⇒ level repulsion ⇒ band around ・ high T : Ψ on periodic instanton b.g. localized in 3D ⇒ no level repulsion ⇒ collapse to no band I / D

• AI ~ 1

r3 = 0 I / D

• AI ~ eT r

= 0

q q = lim

0 lim V

() V

SU(N) variable Un,µ spinor & N-color

Lattice Gauge Theory

× Dirac γµ

• tr U•U•U•U•

1 2 4 3 4

• □

□

quenched Boltzmann weight

chiral SB confinement

・ what to measure q q = D(0) e

Fq /T tr U( r ,0),0LU( r ,Lt ),0

Lxa Lta

・ how to change T

fixed

Lx,y,z >> Lt

• a e /b0

T = 1 Lta

↗ ↘ ↗

Dirac spectra at QCD transition

SU(3) quenched LGT

• n 163~203×4, KS Dirac op.

Garcia2-Osborn 07

・ chiral symm restoration ・ deconfinement transition ・ localization simultaneous!

Dirac spectra at QCD transition

quenched ILM at T=ΛQCD, KS Dirac op.

Garcia2-Osborn 06 unitary dRMT a=3.2 SN 99

scale-inv critical statistics

Attn: ILM is a priori semiclassically biased not the real QCD

Dirac spectra at QCD transition

unitary dRMT a=3.2

SU(3) quenched LGT

• n 203×4, at β=7.93, KS Dirac op.

Garcia2-Osborn 07

Dirac spectra at QCD transition

quenched SU(2) LGT

• n (7×9×11)×4, KS Dirac op.*

cumulatitive EV distribution

si = i+1 i (i ) i+1 i xav

/ (i )1

x() = xav () + xosc ()

polynomial fit for each config. spectral unfolding Kato-SN 08

＊extensive study (T=0)

by Guhr et al 99 2nd order deconfinement

Poisson β=1.0 β=2.5 β=3.0 β=4.0 β=1.0 β=2.5 β=3.0 β=4.0 WD WD sympl dRMT a=.45 sympl dRMT a=.90

cumualtive distribution Kato-SN 08

SU(2) quenched LGT on (7×9×11)×4

Dirac spectra at QCD transition

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 works both in AH, QCD ~

thanks: Damgaard, Verbaarschot, Garcia2, Nagao, Kato - collaborator Zharekeshev, Schweizer, Kawarabayashi, Evangelou, Ohtsuki - AH data JSPS - grant phenomenologically modelled by