Quantum entanglement entropy
for SU(3) gauge theories
Etsuko Itou (KEK)collaboration with K. Nagata (KEK), Y.Nakagawa, A. Nakamura(Hiroshima U.) and V.I.Zakharov (Max Planck Inst.)
cf. arXiv:0911.2596 and 1104.1011:
Y.Nakagawa, A.Nakamura, S.Motoki and V.I.Zakharov

Hadrons and Hadron Interactions in QCD -- Eﬀective theories and Lattice -- (HHIQCD2015)@ YITP, Kyoto University 2015/3/6

entanglement entropy=?
a novel order parameter of the conﬁnement

Donʼt think....feel....

entanglement entropy=?
central charge in conformal ﬁeld theoryDonʼt think....feel....

Related works• H.Casini, M.Muerta and J.A.Rosabal arXiv:1312.1183
̀̀Remarks on entanglement entropy for gauge ﬁelds”• D.Radicevic arXiv:1404.1391
̀̀Note on Entanglement in Abelian Gauge Theories”• S.Ghosh, R.M.Soni,S.P.Trivedi arXiv:1501.02593
̀̀On the entanglement entropy for gauge theories”• S.Aoki, T.Iritani, M.Nozaki et.al. arXiv:1502.04267
̀̀On the deﬁnition of entanglement entropy in lattice gauge theories”• J.-W. Chen, S.-H. Dai and J.Y. Pang arXiv:1503.01766
̀̀Strong coupling expansion of the entanglement entropy of Yang-Millsgauge theories”

Outline- Introduction (For QCD and conformal theory in 4d )- Deﬁnition of entanglement entropy- Replica method- Results for the quenched QCD

Introduction

Basic properties of E.E.
entanglement entropy for quantum system- how 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

For QCD theory
A color conﬁnement changes the d.o.f of the systemdeconﬁnement phasecolorful(gluons)O(Nc2 )

conﬁnement phasecolorless(singlet)O(1)For conformal theories
Central charge is ...
- an important characteristic of CFT- roughly the d.o.f. of the conformal system

In 2d CFT,
A.A. Belavin, A.M.Polyakov, A.B.Zamolodchikov NPB241(1984)333
A. B. Zamolodchikov, JETP Lett. 43 (1986) 730

In 4d CFT,
xij = |xi

xj |J.I.Latorre and H.Osborn: hep-th/9703196
Y. Nakayama arXiv:1302.0884[hep-th]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.)
Stot =von Neumann entropy

density matrix

tot = |

decompose total Hilbert
space into two subsystems

Tr tot log tot||

Htot = HA

HB

reduced density matrix

A =entanglement entropy

SA =:pure ground state

TrHB tot
TrA A log AAAt ﬁnite T, it is equivalent to the thermal entropy.

B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. modell

BA

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. modelBl

A

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

lclog
3a

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

In the non critical case,lSA (l)lclog
3a

SA (l)

clog
3al: correlation length of the system

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

B

At the critical point or l
noncritical system

Ain thelc
SA (l) = log + c1
3a
In the non-critical system,

SA (l)

lclog
3aSA (l)lCl: correlation length of the systeml4-dimensional CFT and QFT
Ryu and Takayanagi:PRL96(2006)181602
JHEP 0608(2006)045

1SA (l)| A|
lHolographic approach(3+1)-dim. N = 4 SYM

1
Nc2
SA (l) = c 2| A|aNc2
l3

Nc2c 2lcʼ is obtained by AdS and QFT

lcl

Intuitively...
in short range: gluon dynamics ~ O(Nc^2) and dimensional analysis [mass^3]
in long range: only color singlet ~ O(1) and l-indepent because of conﬁnement

What we want to know?
- Nc dependence in the short l region- existence of discontinuity- value of lc (Lambda QCD?)- UV cutoﬀ (lattice cutoﬀ) dependence
1SA (l)| A|lCorrection terms exist in non-conformaltheory?

2Nc
l3

1
Nc2
SA (l) = c 2| A|allcNc2c 2l00⇥(1+c log(a/l))

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

1n
ln TrA An

ln

Z(l, n)Zn

Replica method
lZ=

B

A

entanglement entropy:

N

1/T

SA =
SA (l) =BZ(l, n) =N

B A
BnN

limnn1limn

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:

B
Z(l, n) =n

1nln TrA ASA (l)= lim
F [l, n] :free energy
n1ll nF [l + a, n = 2]anN

1=dF [l, n = 2]

Sl+a [U ]Sl [U ]

0using the interpolation action

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

What we want to know?
- Nc dependence in the short l region- existence of discontinuity- value of lc (Lambda QCD?)- UV cutoﬀ (lattice cutoﬀ) dependence- replica number dependence(add second line)@lim
F [l, n] = (F [l, n = 2] F [l, n = 1])/ n| n=1
n!1 @nn(F [l, n = 3] 2F [l, n = 2] + F [l, n = 1])| n=12

Simulation results

Lattice results for quenched SU(2)Buividovich and Polikarpov:
NPB802(2008)458

Lattice results for quenched SU(2)Buividovich and Polikarpov:
NPB802(2008)458- in short range, 1/l^3 scaling- discontinuity is clear!?

# of conﬁgs. is 18 -141

Entropic C-function

1 dS
C(l) = l|@A| dl
3

should be constant in short l region

But the data is enhanced
Lattice artifact?

Our result

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 12,000~30,000- scale setting r̲0 =0.5 fm and ALPHA coll.

Lattice results for quenched SU(3)T=0, quenched QCD
103C/xβ=5.70(cubic.)
β=5.75(cubic.)
β=5.80(cubic.)
β=5.72(cubic.)
β=5.74(cubic.)
β=5.77(cubic.)
β=5.78(cubic.)
β=5.87(cubic.)(1/|dA|) d S/dl [fm-3]

864

We measure ~30,000 conﬁgs.
- in short range, 1/l^3 scaling- the coeﬃcient is roughly C=0.2cf.) C~0.09 in SU(2)2The Nc dependence is000.2

0.4

0.6
l[fm]

0.8

10.232⇠ 20.092

What we want to know?
- Nc dependence in the short l region
Nc2 as expected by AdS/CFT and ﬁeld theoretical insights- existence of discontinuity- value of lc (Lambda QCD?)- UV cutoﬀ (lattice cutoﬀ) dependence- replica number (n ->1) dependenceEntropic C-function
1 dS
C(l) = l|@A| dl3

0.5
0.4

In short l region, C is constant.

(l3/|dA|) d S/dl

0.3
0.2

The discontinuity is not clear yet.

0.1

0.20
β=5.70(cubic.)
β=5.75(cubic.)
β=5.80(cubic.)
β=5.72(cubic.)
β=5.74(cubic.)
β=5.77(cubic.)
β=5.78(cubic.)
β=5.87(cubic.)

0-0.1
-0.2
-0.3
-0.4
-0.5
00.2cf.)

0.4

0.6
l[fm]

0.8

11
QCD

0.7[fm]What we want to know?
- Nc dependence in the short l region
Nc2 as expected by AdS/CFT and ﬁeld theoretical insights- existence of discontinuity

still unclear, but there is a signal- value of lc (Lambda QCD?)
seems to be the same with Lambda QCD- UV cutoﬀ (lattice cutoﬀ) dependence- replica number (n ->1) dependenceUV cutoﬀ dependence
Ns = 16, l/a = 2 ! Ns = 32, l/a0 = 4
a = 2a025

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

20
15
1050-500.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]

1

What we want to know?
- Nc dependence in the short l region
2Nc as expected by AdS/CFT and ﬁeld theoretical insights- existence of discontinuity

still unclear, but there is a signal- value of lc (Lambda QCD?)
seems to be the same with Lambda QCD- UV cutoﬀ (lattice cutoﬀ) dependence
still unclear- replica number (n ->1) dependenceReplica number dependence@limF [l, n] = (F [l, n = 2]
n!1 @nF [l, n = 1])/ n| n=1n
(F [l, n = 3]
2102F [l, n = 2] + F [l, n = 1])| n=1

n=2
with n=3-3(1/|dA|) d S/dl [fm ]864
2000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
l [fm]

1

What we found…
- Nc dependence in the short l region
2Nc as expected by AdS/CFT and ﬁeld theoretical insights- existence of discontinuity

still unclear, but there is a signal- value of lc (Lambda QCD?)
seems to be the same with Lambda QCD- UV cutoﬀ (lattice cutoﬀ) dependence
still unclear- replica number (n ->1) dependence
seems to be negligible

Future directions for E.E. using the lattice• QCD at zero T• would give the Lambda QCD• even in full QCD case

• QCD at ﬁnite T• gives the thermal entropy and the correlation length in QGP phase• conformal window in 4dim many ﬂavor QCD• would give the a-function and central charge

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)17

T

2.02Tcs

61.8(15)

56

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)17

T

2.02Tcs

61.8(15)56