SU(3)ゲージ理論におけるエントロピックC関数の測定
伊藤悦子 (KEK)collaboration with K. Nagata (KEK), Y.Nakagawa, A. Nakamura(Hiroshima U., RCNP) and V.I.Zakharov (Max Planck Inst.)
cf. arXiv:0911.2596 and 1104.1011:
Y.Nakagawa, A.Nakamura, S.Motoki and V.I.Zakharov

基研研究会「熱場の量子論とその応用」@ YITP, Kyoto University 2015/8/31

メインテーマ
エンタングルメント・エントロピー(エントロピックC関数)から閉じ込めに対する新しい知見を得る
４次元ゲージ理論における量子エンタングルメントの非摂動論的な振る舞いは？

cf) arXiv:1508.07132[hep-lat] today!
Lattice study on QCD-like theory with exact center symmetry
T.Iritani, E.I and T.Misumi

関連した研究: 9月2日（水）河野さん

Basic properties of E.E.
entanglement entropy for quantum system
how much a quantum state is entangled quantum mechanicallyd.o.f of the system
quantum properties of the ground state for the systemin ﬁnite T system, it gives a thermal entropy

QCD theory (T=0)
A color conﬁnement changes the d.o.f of the system

microscopicallycolorful(gluons)2O(Nc )

⇤QCD

macroscopicallycolorless(singlet)O(1)Deﬁnition of the entanglement entropy

Entanglement entropy (E.E.)
Stot =von Neumann entropy

tot = |Tr tot log tot||

decompose total Hilbert
space into two subsystems

Htot = HA

HB

reduced density matrix

A =density matrix

entanglement entropy

SA =:pure ground state

TrHB tot
TrA A log A

At ﬁnite T, it is equivalent to the thermal entropy.

ABSchematic picture of E.E.
Holzhey,Larsen and Wilczek: NPB424 (1994) 443(1+1)-dim. modelCalabrese and Cardy: J.S.M.0406(2004)P06002
Calabrese and Cardy, arXiv:0905.4013

Schematic picture of E.E.
(1+1)-dim. modellBA

Schematic picture of E.E.
(1+1)-dim. modell

BA

At the critical point,lc
SA (l) = log + c1
3ac is the central chargein 2d CFT.

: correlation length of the system

Schematic picture of E.E.
(1+1)-dim. model

BlA

In the non critical system,l
SA (l): correlation length of the system

lc
log
3a

Schematic picture of E.E.
(1+1)-dim. modell

BA

In the non critical case,l
SA (l)

lclog
3aSA (l)clog
3al: correlation length of the system

Schematic picture of E.E.
(1+1)-dim. modellBAt the critical point or l
noncritical system

Ain thelc
SA (l) = log + c13a
In the non-critical system,SA (l)lSA (l)lclog
3a

independent of UV cutoﬀCl: correlation length of the systemlDiﬃculties to obtain E.E. in 4d gauge theory
UV cutoﬀ dependence of 4d E.E.
dS
Entropic C-function C(l) = ldl

2dSA (l) = c0 log(l/a)
4dAreaSA (l) = c0 2a
Entropic C-function0 Areac0 2 + c1 log(l/a) + (regular terms)

l3 1 dSC(l) = l|@A| dlcf) Holographic approach

Ryu and Takayanagi:PRL96(2006)181602
JHEP 0608(2006)045
Nishioka and Takayanagi JHEP 0701(2007) 090

C is obtained by AdS and QFT(local) gauge invarianceE.E. for gauge theory•

P.V.Buividovich and M.I.Polikarpov PLB670(2008)141
extended Hilbert space•

H.Casini, M.Muerta and J.A.Rosabal arXiv:1312.1183
electric b.c.(electric center), magnetic center, trivial center•

D.Radicevic arXiv:1404.1391
magnetic center•

W.Donnelly PRD85 (2012) 085004
extended lattice construction•S.Ghosh, R.M.Soni,S.P.Trivedi arXiv:1501.02593•

S.Aoki, T.Iritani, M.Nozaki et.al. arXiv:1502.04267
maximally gauge invariant reduced density matrix

E.E. for gauge theory•

P.V.Buividovich and M.I.Polikarpov PLB670(2008)141
extended Hilbert space•

H.Casini, M.Muerta and J.A.Rosabal arXiv:1312.1183
electric b.c.(electric center), magnetic center, trivial center•

D.Radicevic arXiv:1404.1391
magnetic center•

W.Donnelly PRD85 (2012) 085004
extended lattice construction•

S.Ghosh, R.M.Soni,S.P.Trivedi arXiv:1501.02593•

S.Aoki, T.Iritani, M.Nozaki et.al. arXiv:1502.04267
maximally gauge invariant reduced density matrix

red deﬁnitions are inadequate
for E.E. or rhoA

E.E. for gauge theory•

P.V.Buividovich and M.I.Polikarpov PLB670(2008)141

Our work

extended Hilbert space•

H.Casini, M.Muerta and J.A.Rosabal arXiv:1312.1183
electric b.c.(electric center), magnetic center, trivial center•

D.Radicevic arXiv:1404.1391
magnetic center•

W.Donnelly PRD85 (2012) 085004
extended lattice construction•

S.Ghosh, R.M.Soni,S.P.Trivedi arXiv:1501.02593•S.Aoki, T.Iritani, M.Nozaki et.al. arXiv:1502.04267
maximally gauge invariant reduced density matrix

red deﬁnitions are inadequate
for E.E. or rhoA

Replica method
Calabrese and Cardy: J.S.M.0406(2004)P06002

Replica method
lZ=

B

A

entanglement entropy:N

1/T

SA =SA (l) =

limnn

1limn

1nln TrA Anln

Z(l, n)Zn

Replica method
lZ=

B

A

entanglement entropy:N

1/T

SA =SA (l) =B
Z(l, n) =NB ABnN

limnn

1

limn

1nln TrA Anln

Z(l, n)Zn

Replica methodlZ=

B

Aentanglement entropy:

N1/T

SA =limn

SA (l) =

B A

limn

1nln

Z(l, n)Znobservable:

BZ(l, n) =n1nln TrA A

SA (l)= lim
F [l, n] :free energy
n1ll nnNFodor (2009)

F [l + a, n = 2]a1=

dF [l, n = 2]

Sl+a [U ]Sl [U ]

0using the interpolation action

NB

Sint = (1)Sl [U ] + Sl+a [U ]we measure the diﬀ. of the action densitySimulation results

Simulation setupl

Wilson plaquette gauge action
Ns=Nt=16, 32
B A
l/a=2,3,4,5,(6)beta=5.70 - 5.87
# of conﬁguration 12,000~84,000
scale setting r̲0 =0.5 fm and ALPHA coll.

Lattice results for quenched SU(3)T=0, quenched QCD
108(1/|dA|) d S/dl [fm-3]

We measure ~84,000 conﬁgs.

3

C/x
a=0.1707[fm]a=0.1628[fm]a=0.1555[fm]a=0.1520[fm]a=0.1454[fm]a=0.1423[fm]a=0.1363[fm]a=0.1182[fm]64- in short range, 1/l^3 scaling- the coeﬃcient is C=0.2064(73)cf.) C~0.09 in SU(2)2The Nc dependence is

000.2

0.4

0.6l[fm]

0.8

10.2064
32= 2.29 ⇠ 20.09
2

Entropic C-function
independent of UV cutoﬀ

1 dS
C(l) = l|@A| dl
30.4

In short l region, C is constant.

0.3(l3/|dA|) d S/dl

0.2

The discontinuity is not clear.

0.1
0a=0.1707[fm]a=0.1628[fm]a=0.1555[fm]a=0.1520[fm]a=0.1454[fm]a=0.1423[fm]a=0.1363[fm]a=0.1182[fm]

-0.1
-0.2
-0.3
-0.4
-0.5
00.2cf.)

0.4

0.6
l[fm]

0.8

11
QCD

0.7[fm]Comparison with Ryu-Takayanagi results
Ryu and Takayanagi:PRL96(2006)181602
JHEP 0608(2006)045

Holographic (or ﬁeld theoretical) approach(3+1)-dim. CFT

1
Nc2
SA (l) = c 2
| A|
a

Nc2
c 2
lcʼ is obtained by AdS and QFT

0c ⇠ 0.0049

for free real scalar theorya = 1, · · · , 8
aEstimation for non-abelian gauge theory A{µ µ = 1, · · · , 40Cgauge ⇠ 2c · 2 · 8 ⇠ 0.1568

Our numerical result

Cgauge ⇠ 0.2064

Detail analyses
algorithm dependence ofthe numerical integrationﬁnite vol. eﬀect
10

Ns=Nt=16
Ns=Nt=32(1/|dA|) d S/dl [fm-3](1/|dA|) d S/dl [fm-3]

10

505000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]

1

UV cutoﬀ dependence
25

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]

1

replica number dependence
10

Ns=Nt=16
Ns=Nt=32 (half a)

20n=2
with n=38(1/|dA|) d S/dl [fm-3](1/|dA|) d S/dl [fm-3]

cubic polynomial
extended Simpron

1510564200-500.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]

100.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]

1

Summary
This is the ﬁrst precise determination of E.E. for quenched
QCD
Nc dependence in the short l region is Nc^2 as expected
by AdS/CFT and ﬁeld theoretical insights
No discontinuity exists as contrast with SU(2) results
Entropic C-function shows UV cutoﬀ independence
Value of C-function agrees with Ryu-Takayanagi work
replica number (n ->1) dependenceFuture directions for E.E. using the lattice• QCD at zero T• give a novel observation for conﬁnement• even in full QCD case

• QCD at ﬁnite T• gives the thermal entropy and the correlation length in QGP phase• conformal window in 4dim Nf ﬂavor QCD• would give the a-function and central chargeNf<9beta fn.

9<Nf<17
nontrivial IR ﬁxed point isfound by lattice simulationcf) E.I. PTEP(2013)083B01