Numerical study on the entropic c-function
for SU(3) gauge theory
Etsuko Itou (KEK)
E.I., K. Nagata, Y. Nakagawa, A. Nakamura and V.I.Zakharov
PTEP 2016 (2016) no.6, 061B01
arXiv: 1512.01334 [hep-th]
supercomputers

YKIS conference:“Quantum Matter, Spacetime and Information”@ YITP, Kyoto University 2016/6/17

QCD
The lagrangian is given by SU(3) gauge theory

1 a 2 ¯
L = (F ) + (iD
4µ⌫µµm)rqq̄

Hadron“quenched” QCD
The lagrangian is given by SU(3) gauge theory

1 a 2 ¯
L = (F ) + (iD4µ⌫µµm)

rqq̄

HadronIt has the local SU(3) gauge invariance
Lattice reg. is only known nonperturbative and gauge
invariant reg. method from weak- to strong-coupling regimes

q q̄ potential V(r)interaction is very weakV (r) =c+ r
r

G. Bali Phys. Rept.343,(2001)11r > ⇤QCD

Entanglement entropy for quantum system(1+1)-dim. model

Holzhey,Larsen and Wilczek: NPB424 (1994) 443
Calabrese and Cardy: J.S.M.0406(2004)P06002

lB

Calabrese and Cardy, arXiv:0905.4013

A: correlation length of the systeml : subsystem(A) size
SA (l)lindependent of UV cutoﬀClllllcSA (l) = log + c1
3aSA (l)clog
3aQCD theory (T=0)
A color conﬁnement changes the eﬀective d.o.f of the system

microscopicallycolorful(gluons)

SA (l)l⇤QCD

colorless(singlet)O(Nc2 )

2D spin sys.

macroscopically

1 @SA (l)|@A| @l

4D QCDC⇠ 3lCll

O(1)?
1⇤QCDlTwo issues to obtain E.E. in 4d gauge theory
UV cutoﬀ dependence of 4d E.E.
2dSA (l) = c0 log(l/a)
4dAreaSA (l) = c0 2a0 Areac0 2 + c1 log(l/a) + (regular terms)

l1 @SA
Entropic C-function C(l) = l|@A| @l3cf) Holographic approach

Ryu and Takayanagi:PRL96(2006)181602
JHEP 0608(2006)045

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, T.Numasawa, N.Shiba, H.Tasaki 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, T.Numasawa, N.Shiba, H.Tasaki 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, T.Numasawa, N.Shiba, H.Tasaki arXiv:1502.04267
maximally gauge invariant reduced density matrix

red deﬁnitions are inadequate
for E.E. or rhoA

Strategy of our simulation(Gauge ﬁxing is unnecessary)gauge conﬁguration generation(on the n=2 replica lattice)measurement of observable(the action density)

SA (l)= lim
F [l, n] =n 1 l nl

BAtx

n=2 replica latticel

B
1d0Sl+a [U ]ASl [U ]Fodor (2009)

Period in temporal dir. on thereplica lattice depends on thex-coordinate.

Simulation results

Lattice results for SU(3) gauge theory
T=0, quenched QCD
103

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](1/|dA|) d S/dl [fm-3]

8

1 @SA (l) 6|@A| @l42000.2

0.4

0.6

0.8

1

l[fm]

4dSA (l) = c0Areaa2c00

Area+ c1 log(l/a) + (regular terms)
2lin short range, 1/l^3 scaling
conﬁrm the area law
leading term of l is dominatedEntropic C-function
1 @SAC(l) = l|@A| @lC(l)

3

0.4
0.3
0.2
0.1
0-0.1
-0.2
-0.3
-0.4
-0.5

In short l region, C is constant.
Best ﬁt value: C=0.2064(73)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]00.2

The discontinuity is not clear.
cf.)

1
⇤QCD

0.4

0.6

l [fm]

0.81

✓

1Tc◆

= 0.7

0.8[fm]Comparison with free ﬁeld approximation
Free ﬁeld calculation:(3+1)-dim. CFT

1
Nc2
SA (l) = c 2
| A|
aNc2
c 2
l

Estimation for non-abelian gauge theory
0Cgauge ⇠ 2c · 2 · 8 ⇠ 0.1568
0Ryu and Takayanagi:JHEP 0608(2006)045

c ⇠ 0.0049 for free real scalar theory
Cgauge ⇠ 0.1770(c ⇠ 0.00553...)

Our numerical result

Cgauge ⇠ 0.2064

a { a = 1, · · · , 8
Aµ µ = 1, · · · , 4

Casini and Huerta:
Phys. Rev. D 93, 105031 (2016), J. Phys. A42, 504007 (2009)

Almost consistentjust below QCD scale, the free ﬁeld app.
is well described.

Comparison with other results
SU(2) gauge theory
P.V.Buividovich and M.I.Polikarpov PLB670(2008)141

l<0.3[fm] C(l) = 0.07 ̶ 0.08
Our SU(3) result: C=0.2064(73)3Scaling down to SU(2): 0.2064 ⇥ ⇠ 0.0774
8enhancement around l ~0.5[fm]discontinuity exists# of conﬁgs. ~ 100 cf) SU(3):50,000

Hagedorn-type model
I. R. Klebanov, D. Kutasov and A. Murugan; Nucl.Phys. B796 (2008) 274

At large-Nc, conﬁning theory can beZdescribed by free glueballs
sum of all glueballs spectrum

Ctotal ⇠1

critical length lc ⇠ TH /2
discontinuity existsm e( H2l)mﬁnite Nc correction?
instanton eﬀect?
model is too naive?

Summary
This is the ﬁrst precise determination of E.E. for quenchedQCD
c-fn. in short regime correctly counts # of gluons
Discontinuity around lc is unclear as contrast with SU(2)/large-N results

Future directions
If “it looks very promising (or trivial)”
Real QCD including quarks
Conformal window in 4d SU(3) many Nf theory (a- /or c- fn. )Nf<9beta fn.

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

If not…

theoretical side

Lattice side

1/Nc correction
nonperturbative vacuumimproved Hagedorn model

UV cutoﬀ independence?
replica number (n ->1) dependencebackupﬁnite T for quenched SU(3)arXiv:1104.1011: Y.Nakagawa et al.

T< Tcthermal entropy

T >Tc

replica method

integration method (2% error)
Boyd et al.:NPB469,419(1996)

T

1.44Tcs

14.1(8)17T

2.02Tcs

61.8(15)56

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]

10505000.1 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)20

n=2
with n=3

8(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

replica number dependence
1010

n=2 Renyi entropy
n=3 Renyi entropy(1/|dA|) d S/dl [fm-3]

8-3(1/|dA|) d S/dl [fm ]8n=2with n=3642064
2000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]@lim
F [l, n] = (F [l, n = 2]
n!1 @n1

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

F [l, n = 1])/ n| n=1n(F [l, n = 3]22F [l, n = 2] + F [l, n = 1])| n=11