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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
MNS modelMNS 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 modelMNS model Moshe-Neuberger-Shapiro ’94
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
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 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
III.3 ME & deformed RM
deform. parameter vs EV window
Dirac LSD for high-T QCDDirac LSD for high-T QCD
G-K-SMN-P ’13
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
24
deform. parameter vs EV window & deform. parameter vs EV window & sizesize
G-K-SMN-P ’13
a=3.60
scale inv M.E.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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.
III.3 ME & deformed RM
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
TDL : localized←ME→extended
III.3 ME & deformed RM
finite fraction of small EVs exists & localizeseven in presence of very light quarks
conclusion II
24
deform. parameter vs EV window & deform. parameter vs EV window & sizesize
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
III.3 ME & 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
/
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