Entropic c-function of the SU(3) gauge theory in four …seminar/pdf_2015_kouki/...Entropic...

Entropic c-function of the SU(3) gauge theory

in four dimensions

Etsuko Itou (KEK)

Seminar @ Osaka University 2015/10/6

collaboration with K. Nagata (KEK), Y.Nakagawa, A. Nakamura (RCNP, RIKEN) and V.I.Zakharov (Max Planck Inst.)

cf. arXiv:0911.2596 and 1104.1011: Y.Nakagawa, A.Nakamura, S.Motoki and V.I.Zakharov

entanglement entropy =?

a novel approach to understand the confinement

Don’t think....feel....

entanglement entropy =?

central charge in conformal field theory

Don’t think....feel....

Outline

- Introduction (QCD and CFT )

- Definition of entanglement entropy

- Replica method

- Results for the quenched QCD

confinement

q-qbar potential

V (r) = � c

r+ �r

A nonperturbative property of QCD

center symmetry (only for pure YM)

r > ⇤�1QCD

r

q q̄

Hadron

interaction is very weak

G. Bali Phys. Rept.343,(2001)1

Basic properties of E.E.

A color confinement changes the d.o.f of the system

microscopically

colorful (gluons) � O(N2

c ) � O(1)colorless (singlet)

entanglement entropy for quantum systemhow much a quantum state is entangled quantum mechanically d.o.f of the system quantum properties of the ground state for the system in finite T system, it gives a thermal entropy

QCD theory (T=0)

macroscopically⇤QCD

For conformal theories

A.A. Belavin, A.M.Polyakov, A.B.Zamolodchikov NPB241(1984)333 A. B. Zamolodchikov, JETP Lett. 43 (1986) 730

In 2d CFT,

Central charge is ... - an important characteristic of CFT - roughly the d.o.f. of the conformal system

In 4d CFT, xij = |xi � xj |

J.I.Latorre and H.Osborn: hep-th/9703196 Y. Nakayama arXiv:1302.0884[hep-th]

cf) EMT on the lattice FlowQCD coll.

Phys.Rev. D90 (2014) 1, 011501

According to the conservation law of EMT, two coefficients are redundant.

The following ``a” is expected to be a monotonical decreasing fn. along the RG flow.

H.Osborn and A.C.Petkou, : hep-th/9307010

Definition of the entanglement entropy

Entanglement entropy (E.E.)

density matrix �tot = |��| |�� :pure ground state

von Neumann entropy

decompose total Hilbert space into two subsystems Htot = HA �HB

entanglement entropy SA = �TrA�A log �A

Stot = �Tr�tot log �tot

�A = �TrHB�tot

BA

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

reduced density matrix

Schematic picture of E.E.

(1+1)-dim. modelHolzhey,Larsen and Wilczek: NPB424 (1994) 443 Calabrese and Cardy: J.S.M.0406(2004)P06002 Calabrese and Cardy, arXiv:0905.4013


ABl

(1+1)-dim. model


ABl

�

� : correlation length of the system

(1+1)-dim. model At the critical point,

SA(l) =c

3log

l

a+ c1

c is the central charge in 2d CFT.


ABl

�


In the non critical system,(1+1)-dim. model

l� �

SA(l) � c

3log

l

aSA(l) � c

3log

l

a



ABl

�

(1+1)-dim. model

l� �

SA(l)� c

3log

�

a

l� �

SA(l) � c

3log

l

a

In the non critical case,


ABl

�


At the critical point or in the noncritical system

SA(l) =c

3log

l

a+ c1

SA(l)� c

3log

�

a

In the non-critical system,

l� �

(1+1)-dim. modell� �

�SA(l)�l

l

� C

l

�

independent of UV cutoff

Difficulties to obtain E.E. in 4d gauge theory

UV cutoff dependence of 4d E.E.

(local) gauge invariance

Entropic C-function

SA(l) = c0Area

a2� c00

Area

l2+ c1 log(l/a) + (regular terms)

SA(l) = c0 log(l/a)2d

4d

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

Nishioka and Takayanagi JHEP 0701(2007) 090

Entropic C-function

C(l) = l31

|@A|@SA

@l

C(l) = l@SA

@l

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

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





magnetic center

• W.Donnelly PRD85 (2012) 085004





E.E. for gauge theory

red definitions are inadequate for E.E. or rhoA

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





magnetic center

• W.Donnelly PRD85 (2012) 085004





Our work

E.E. for gauge theory

Replica method

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

Replica method

ABZ =SA = � lim

n�1

�

�nlnTrA�n

A

l

SA(l) = � limn�1

�

�nln

�Z(l, n)

Zn

�

entanglement entropy:

N� � 1/T

Z(l, n) =A

B

B

B

Replica method

ABZ =SA = � lim

n�1

�

�nlnTrA�n

A

l


�

�nln

�Z(l, n)

Zn

�


N� � 1/T

nN�

N�

Replica method

ABZ =SA = � lim

n�1

�

�nlnTrA�n

A

l


�

�nln

�Z(l, n)

Zn

�

� F [l + a, n = 2]� F [l, n = 2]a

=� 1

0d��Sl+a[U ] � Sl[U ]��

observable:

:free energy�SA(l)�l

= limn�1

�

�l

�

�nF [l, n]


N� � 1/T

nN�

we measure the diff. of the action density

N� Sint = (1� �)Sl[U ] + �Sl+a[U ]using the interpolation action

Z(l, n) =A

B

B

B

Fodor (2009)

Simulation results

Lattice results for pure SU(2) gaugeBuividovich and Polikarpov:

NPB802(2008)458

- in short range, 1/l^3 scaling - discontinuity is clear!?

Entropic C-functionshould be constant in short l region

# of configs. is 18 -141

But the data is enhanced Lattice artifact?

# of configs. is 18 -141 Error estimation would be

insufficient?

C(l) = l31

|@A|@SA

@l

Our result

Simulation setup

Wilson plaquette gauge action Ns=Nt=16, 32 l/a=2,3,4,5,(6) beta=5.70 - 5.87 # of configuration 12,000~84,000 scale setting r_0 =0.5 fm and ALPHA coll.

AB

l

Lattice results for pure SU(3) gauge

T=0, quenched QCD

- in short range, 1/l^3 scaling - the coefficient is C=0.2064(73)

cf.) C~0.09 in SU(2)

We measure ~84,000 configs.

The Nc dependence is

0.2064

0.09= 2.29 ⇠ 32

22

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1

(1/|d

A|) d

S/d

l [fm

-3]

l[fm]

C/x3a=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]

Entropic C-function

1�QCD

� 0.7[fm]cf.)

In short l region, C is constant. (first evidence!) The discontinuity is not clear.

independent of UV cutoff

-0.5-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4

0 0.2 0.4 0.6 0.8 1

(l3 /|dA|

) d S

/dl

l[fm]

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]

C(l) = l31

|@A|@SA

@l

Comparison with Ryu-Takayanagi results

Field theoretical approach

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

(3+1)-dim. CFT1

|�A|SA(l) = cN2

c

a2� c� N

2c

l2

c’ is obtained by AdS and QFT

c0 ⇠ 0.0049 for free real scalar theory

Estimation for non-abelian gauge theory

Our numerical result

Aaµ

Cgauge ⇠ 0.2064

a = 1, · · · , 8µ = 1, · · · , 4

Cgauge ⇠ 2c0 · 2 · 8 ⇠ 0.1568

{

Detail analysesfinite vol. effect algorithm dependence of

the numerical integration

UV cutoff dependence replica number dependence

0

5

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(1/|d

A|) d

S/d

l [fm

-3]

l [fm]

Ns=Nt=16Ns=Nt=32

0

5

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(1/|d

A|) d

S/d

l [fm

-3]

l [fm]

cubic polynomialextended Simpron

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(1/|d

A|) d

S/d

l [fm

-3]

l [fm]

n=2with n=3

-5

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(1/|d

A|) d

S/d

l [fm

-3]

l [fm]

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

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

Summary

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 charge

beta fn. nontrivial IR fixed point is found by lattice simulation cf) E.I. PTEP(2013)083B01

Nf<9 9<Nf<17

finite T for quenched SU(3)

T< Tc T >Tc

thermal entropy

s � 14.1(8)

s � 61.8(15) 56

17

replica method integration method (2% error)

arXiv:1104.1011: Y.Nakagawa et al.

T � 2.02Tc

T � 1.44Tc

Boyd et al.:NPB469,419(1996)

Entropic c-function of the SU(3) gauge theory in four …seminar/pdf_2015_kouki/...Entropic...

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