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,

S. M. NishigakiS. M. Nishigaki Shimane Univ

based on ongoing work with

M. Giordano, T. G. Kovacs, F. PittlerM. Giordano, T. G. Kovacs, F. Pittler

MTA ATOMKI Debrecen

Aug. 3, 2013 LATTICE2013, Mainz

Critical Statistics at the Critical Statistics at the

Mobility Edge of QCD Dirac spectraMobility Edge of QCD Dirac spectra

Anderson’s tight-binding Model

: random Schrodinger op.

i.i.d. random variable Vx

fixed const

Wilson’s Lattice Gauge Theory: stochastic Dirac op.

□□

Boltzmann weight

analogy of localization?

‘random’ SU(N) variable Ux,

fixed const mq

×

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

PLANPLAN

I

II

III

{s{sparsparse, dimensionful} {dense, indep. random}e, dimensionful} {dense, indep. random}

sharing discrete symmetry

Random matricesRandom 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 CorrelatorTwo-level Correlator

exp(-s)

Level Spacing Distribution (LSD) Level Spacing Distribution (LSD)

=0 no corr

=1=2 RM

=4

~ s ~ exp(-c s2)

Local EV correlation - bulkLocal EV correlation - bulk

I.1 RMT

4

random Vx

fixed t

I.2 AH vs RMT

Anderson HamiltonianAnderson Hamiltonian

x

W

Vx

x

5

t t

Anderson HamiltonianAnderson Hamiltonian

d, w/o B d, with B vs GOE vs GUE

weak randomness : level statistics ⊂ RM universality

I.2 AH vs RMT

random Vi

fixed t

6

Level Spacing Distribution (LSD) Level Spacing Distribution (LSD)

H-S transf

Wegner, Efetov ’80s

I.2 AH vs RMT

NLNLM forM for Anderson HAnderson H

Gaussian av. over V(x)

diffusion cst

regime : 0 mode dominance

: 0d NLM ⇔ RM7

perturbative perturbative -function of NL-function of NLMs inMs in d=2+d=2+

(g)

Insulator(localized)

Metal (extended)

d=2 (AII), d≥3

d=2 (AI, A)g*

ヨ fixed ptconductance g.

d=1

I.3 Localization

Wegner ’89NLNLM forM for Anderson HAnderson H

8

: regime, 0 mode dominance

reduces to 0D NLM ⇔ RMT

ergodic regime ergodic regime ETh → ∞ : : RMT √RMT √

diffusive regimediffusive regime EETh >> : : perturbation √perturbation √

→ phenomenological model desirable

I.3 Localization

NLNLM forM for AndersonAnderson H H

9

““mobility edge” mobility edge” ETh ~ : perturbation ×: perturbation ×

　　　

EV

de

nsi

ty

localized WF ≪ L

no repulsion → Poisson

multifractal WF ~ L

Scale InvariantCritical Statistics

II.1 Critical Statistics

example:

3d, V=203, Nconf=104

randomness W/t =18.1

mag. flux =0.4

Shklovskii et al ’93LSD ofLSD of Anderson Anderson HH

10

Sparse overlap

distant levels becomes less repulsive

level spacing

level # variance

Poisson-like

Chalker ’90Zharekeshev-Kramer ’97

“Level Repulsion without Rigidity”

II.1 Critical Statistics

11

Anomalous inverse part. ratio

d, with B WFs and EVs at MEWFs and EVs at ME

II.2 Deformed RM

Invariant RM spontaneously broken

equivalent to free fermions at temp. T>0

MNS modelMNS model Moshe-Neuberger-Shapiro ’94

U(N) invU(N) inv

→ equivalent to Banded RM

multifractal WF12

II.2 Deformed RM

U(N) inv

Invariant RM spontaneously broken

equivalent to free fermions at temp. T>0


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


13

~ e-s/2

SMN ’98

LSD : dLSD : deformed Reformed RMM RM

Poisson

~ sproperties ofCS 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 ’99LSD : Anderson H at MELSD : Anderson H at ME

15

3d without B

Small Dirac EV fluctuationSmall Dirac EV fluctuation

discretization garbage → wealth of physical info on discretization garbage → wealth of physical info on SBSB

regime : exact

chRMT

LEC

EV density, smallest EV distr, ...

direct access to FW8 …with probe

III.0 Dirac spectrum

global

symm

Splittorff, Lattice’12 plenary

Verbaarschot, Lattice’13 7D

16

kkthth Dirac EV distribution Dirac EV distribution

III.0 Dirac spectrum

sample: U(1) Dirac spectrum vs chGUE at origin

chira

lco

nden

sate …not the subject

of today’s talk

→ 　　 bulk of spectrum

Damgaard-SMN’01

SMN’13-th EV

17

Dirac spectra for high-T QCDDirac spectra for high-T QCD

soft edgeAiry

hard edgeBessel

？→

III.1 Dirac spectrum - previous

soft edgesAiry?

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 edgesAiry?

Dirac spectra for high-T QCDDirac spectra for high-T QCD


18

SU(3) quenched LGT

on ~× KS Dirac op. Garcia-Osborn 07


... spectral averaing over a window too wide for Level Statistics

・ chi symm

restoration

・ localization

・ deconfinement

simultaneous?

19

Localization and QCD transition 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 ptDirac spectra for high-T QCD at physical pt

gauge NF β mud ms a[fm] Ns Nt T Nconf NEV

SU(2) 0 2.60 - - - 16,24,32,48 4 2.6Tc 3k 256

SU(3) 2+1 3.75 .001786 .05030 .125 24,28,...,48 4 394MeV

7k~40k 512~1k

local EV window (2~10 evs) →　　　 LSD

III.2 Dirac spectrum – current status

22

gauge NF β mud ms a[fm] Ns Nt T Nconf NEV

SU(2) 0 2.60 - - - 16,24,32,48 4 2.6Tc 3k 256

SU(3) 2+1 3.75 .001786 .05030 .125 24,28,...,48 4 394MeV

7k~40k 512~1k

Dirac spectra for high-T QCD at physical ptDirac spectra for high-T QCD at physical pt

III.3 ME & deformed RM

Dirac LSD for high-T QCDDirac 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 ineach EV window near ME, just as in Anderson H

conclusion I


deform. parameter vs EV window

Dirac LSD for high-T QCDDirac LSD for high-T QCD

G-K-SMN-P ’13


deform. parameter vs EV window & deform. parameter vs EV window & sizesize

24

G-K-SMN-P ’13

scale inv M.E.

larger spatial vol

larger spatial vol

scale inv M.E.


24


G-K-SMN-P ’13

a=3.60

scale inv M.E.


24

deform. parameter vs EV window & deform. parameter vs EV window & sizesizeLSD at ME

G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.


24


G-K-SMN-P ’13

larger spatial vol

a=3.60

TDL : localized←ME→extended


finite fraction of small EVs exists & localizeseven in presence of very light quarks

conclusion II

24


G-K-SMN-P ’13

larger spatial vol

a=3.60

scale inv M.E.

path along which the system crosses over RM → Poisson isuniversal (indep of mq, T, a), almost follows 1-parameter deformed RM


profile of LSDprofile of LSD

25

G-K-SMN-P ’13

dRM

ME

Poisson

●

RM

●

Tpc consistent with disappearing localized mode

TTpcpc from Mobility Edge from Mobility Edge Kovacs-Pittler ’12

univers

al, linear i

ncrease

with

T

III.4 Physical implication

conclusion III

mo

bili

ty e

dg

e

171MeV

26

conjecture:

localized modes are associated w/ defects of Polyakov loop

Origin of localized modesOrigin of localized modes

smeared SU(2) Polyakov loop ⇔ localized mode of DOV

III.4 Physical implications

Bruckmann-Kovacs-Schierenberg ’11

EV

den

sity

QCD D on L3 × 1/T (<1/Tc) Anderson H on L3

Summary

MNS deformed RM : exact? theory of Anderson loc.

a=3.60 a=3.55

ME : identicalcritical statistics

/

Documents

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,