Entanglement entropy at large-N

JHEP01(2025)182
Published o SISSA by Sp inge
Recei ed: No embe 15, 2024
Accep ed: Decembe 25, 2024
Published: Janua y 29, 2025
En anglemen en opy a la ge-N
P. Tala e a
Depa men o Physics, Poly echnic Uni e si y o Ca alonia,
Diagonal 647, Ba celona, 08028, E, Spain
E-mail: [email p o ec ed]
Abs ac : I show ha a ea ly imes he e apo a ion p ocess o a s ack o NS5-b anes a
high ene gy is supp essed in he la ge-N limi . A much la e imes, he new saddles in he
g a i a ional ac ion a e no longe supp essed a la ge-N, and e apo a ion p oceeds as usual.
Keywo ds: 1/NExpansion, Black Holes in S ing Theo y, AdS-CFT Co espondence,
D-B anes
A Xi eP in : 2411.09427
Open Access,©The Au ho s.
A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP01(2025)182
JHEP01(2025)182
Con en s
1 Mo i a ions 1
2 NS5-b anes and LST backg ounds 2
2.1 K uskal coo dina es 3
3 En anglemen en opy 4
3.1 En anglemen en opy o a single in e al 5
3.2 En aglemen en opy o wo-disjoin in e als 7
4 In a iance 9
5 Some ema ks 10
6 Some gene aliza ions 10
1 Mo i a ions
In his sho no e, I would like o explo e a no el beha iou ha occu s in black holes
associa ed wi h a s ack o NS5-b anes a he Hagedo n empe a u e. This beha iou is a
di ec consequence o a la ge-N limi
1
and is oo ed in he peculia i y o he sys em. The
se up is qui e appealing, i s because Li le S ing Theo y (LST) is a non-local ield heo y
wi hou g a i y, and one hopes o use his knowledge as a p ecu so o unde s anding gene al
se ings o s ing heo y. And secondly, he he modynamic a iables indica e ha he sys em
is a he Hagedo n empe a u e, a poin whe e he pa i ion unc ion is poo ly de ined.
The heo e ical amewo k is de ined on he wo ld- olume o NS5-b anes in he limi o
anishing s ing coupling,
gα→
0
.
I was shown in [
1
] ha LST is holog aphically dual o
s ing heo y on CGHS black hole. We shall ely hea ily on his ac .
A Semi-classical analysis [
2
], shows ha he adia ion associa ed wi h he black hole
inside he LST is pu ely he mal and he e o e indis inguishable om whi e noise [
3
]. This
ac has deepe implica ions [
2
]: he p obabili y o emi ing a shell o ene gy
ω1
+
ω2
is equal
o he p obabili y o emi ing wo independen shells wi h he same o al amoun o ene gy,
ω1
+
ω2
. As a di ec consequence he adia ion always comes as a pu e s a e, he Hilbe
space can be ac o ized in o wo disjoin pa s,
H=Hin ⊕Hou ,(1.1)
co esponding o s a es loca ed a he inne and ou e sides o he e en ho izon espec i ely.
No e ha in
(1.1)
is missing an in e ac ion piece
Hin
which is manda o y i an ou side
obse e loca ed a he spa ial in ini y wan s o ex ac in o ma ion abou he sys em [
4
].
1F om now on, N will deno e he numbe o NS5 b anes.
– 1 –
JHEP01(2025)182
The aim o his wo k is o cla i y whe he his pa e n is due o he se up empe a u e
o o some deepe cause. To do his, I look a he en anglemen en opy in he spi i o [
5
].
Ou esul s a e unambiguous: he la ge-N limi supp esses semi-classical con igu a ions,
and as a consequence he sys em does no in e ac wi h he en i onmen . Co ec ions, i.e.
non- i ial saddles o he Euclidean pa h in eg al [
6
,
7
], a e no supp essed in his limi and
a e ully esponsible o he uni a y e apo a ion o he black hole. In he spi i o [
8
] his
new con igu a ions will be complex solu ions o he g a i a ional equa ions and co espond
o saddles ele an o he uni a y Page cu e.
Fo hose amilia wi h QCD, his e ec has a close pa allel in he decay
ρ→ππ
a
la ge-N
c
. Inse ing he had onic spec um o la ge-N
c
QCD as a p oxy o he eal had onic
spec um p o ides a he good app oxima ion. In such a model he spec um o he heo y
in he la ge-N
c
limi consis s o an in ini e numbe o na ow s able meson s a es [
9
].
2 NS5-b anes and LST backg ounds
Holog aphy ela es LST o s ing heo y in he nea -ho izon geome y o NS5-b anes [
10
].
Ou s a ing poin is he supe g a i y solu ion o N coinciden nea -ex emal NS5-b anes
in he s ing ame [
11
]
ds2=− 1− 2
0
2!dx2
1+
6
X
j=2
dx2
j+1 + N
m2
s 2 d 2
1− 2
0/ 2+ 2dΩ2
3!,(2.1)
e2ϕ=g2
s1 + N
m2
s 2,
whe e
0
is he loca ion o he ho izon,
gs
is he asymp o ic s ing coupling cons an and
m2
s
is essen ially he s ing ension. The index
i
= 2
,··· ,
6co esponds o he la di ec ions
along he i e-b ane, and
d
Ω
3
is he line elemen o he uni 3 sphe e. The he modynamics
o he NS5-b ane is
ENS5
V5
=1
(2π)5(α′)3 N
g2
s
+ 2
0
g2
sα′!,(2.2)
βNS5 = 2πsN
m2
ss1 + m2
s 2
0
N.(2.3)
The i s e m, p opo ional o
N
g2
s
, in
(2.2)
is he ension be ween he ex emal NS5-b anes.
The e a e se e al limi s ha can be un on
(2.1)
. One o hem is a high ene gy limi , o
which we app oach he nea ho izon in he decoupling limi
0→0, gs→0, 2
0
g2
sα′≡ ixed .(2.4)
The esul ing heo y is conjec u ed o be dual o a Li le S ing Theo y [
10
]. To ake he
limi , i is mo e con enien o change he a iables o
u:=
gsl2
s
, u0:= 0
gsl2
s
,(2.5)
– 2 –
JHEP01(2025)182
a e which
(2.1)
becomes
ds2=− 1−u2
0
u2!dx2
1+
6
X
j=2
dx2
j+N
m2
su2 du2
1−u2
0/u2+u2dΩ2
3!, e2ϕ=Nm2
s
u2.(2.6)
The change
(2.4)
is mo i a ed as ollows: in Type IIB s ing heo y
u
co esponds o he
mass o a s ing s e ching be ween wo NS5-b anes, while in Type IIA
ul−1
s
is he s ing
ension o an open D2-b ane s e ched be ween wo D5-b anes. I is he he modynamics
associa ed wi h his se up
(2.6)
[
12
] ha will be mo e ele an o ou discussion below
ELST
V5
=1
(2π)5(α′)3N
g2
s
+u2
0l2
s, βLST = 2πsN
m2
s
,SLST
V5
=1
(2π)4(α′)2√Nu2
0ls.
(2.7)
No e ha he empe a u e is independen o he ene gy. This ac allows o une independen ly
bo h, ene gy and empe a u e, and makes he ee ene gy o anish. I is clea , om
(2.7)
,
ha i
u2
0l2
s≫N/g2
s
ollows ha
S
=
βLSTELST
which is he leading beha iou o he
en opy o a gas o weakly in e ac ing closed s ings a he Hagedo n empe a u e, he la e
iden i ied wi h ha in
(2.7)
[
13
].
Since we a e in e es ed in desc ibing he ime e olu ion ac oss he ho izon, which is
necessa y o s udy he wo-sided co ela ion unc ions, we shall adop K uskal coo dina es
o he desc ip ion o
(2.1)
and
(2.6)
. This is he ask o he es o his sec ion. Be o e
we s a , we shall w i e
(2.1)
and
(2.6)
gene ically as
ds2=− 1( )dx2
1+
6
X
j=2
dx2
j+Ai( ) d 2
1( )+ 2dΩ2
3!,(2.8)
wi h
1(u)=1−u2
0
u2,(2.9)
and wi h
Ai
(
)chosen p ope ly in each case.
2.1 K uskal coo dina es
As is cus oma y we shall w i e he line elemen
(2.1)
in e ms o he
U
(
x1,
)and
V
(
x1,
)
K uskal coo dina es
2
U=−eci(Fi(x)−ax1), V =eci(Fi(x)+ax1),(2.10)
whe e
ci
,
3
is a cons an chosen on a case-by-case basis, see below, and
F
(
x
) e e s o he
To oise coo dina e in
(2.8)
Fi(x) = Zdx pAi(x)
1(x).(2.11)
2The pa ame e ais in oduced o make he a gumen s in he unc ions dimensionless.
3The subindex ideno es he model (2.1) o (2.6).
– 3 –
JHEP01(2025)182
A e using
(2.10)
one ob ains o
(2.8)
[
14
]
ds2=−e−2ciFi(a )| 1(a )|
a2c2
i
dU dV +
6
X
j=2
dx2
j+Ai( ) 2dΩ2
3=−Ω2( )dU dV +..., (2.12)
whe e he a gumen s inside he unc ions emphasise ha he ela ions
(2.10)
canno be
analy ically in e ed in mos si ua ions. Bo h solu ions,
(2.1)
and
(2.6)
, ha e a
S3
e m
which plays a i ial ole and we a e ee o igno e hese di ec ions o he ime being.
4
No e ha a his s age he p e ac o Ωin
(2.12)
anishes o becomes in ini e a he e en
ho izon, excep o a speci ic alue o he cons an
ci
, which is dic a ed p ecisely so ha Ω
does no anish in he en i e pa ch. The esul s o
cLST
and
cNS5
a e gi en in he equa ions
below. Fo bo h models we can pe o m he in eg al
(2.11)
analy ically, ob aining up o
an a bi a y addi i e cons an
5
LST
FLST(y) = b
2log(y2−1) , cLST =1
b,Ω2(y) = b2 2
0
y2.(2.13)
And
NS5
FNS5(y)=qy2+b2+1
2p1+b2log
√1+b2−py2+b2
√1+b2+py2+b2
, cNS5 =1
√b2+1 ,(2.14)
Ω2(y)= 2
0
y2e−2qb2+y2
b2+1 (1+b2)2
1+sb2+y2
b2+1 

2
,
whe e we ha e de ined
b2:= N
m2
s 2
0
.(2.15)
No e ha
F
(
y
)in
(2.13)
and
(2.14)
di e ge as we app oach he e en ho izon. This is
a consequence o he K e schmann scala o
(2.1)
o
(2.6)
ha ing a singula i y a
→
0.
Fu he mo e, Ω(
y
)is ini e on he ho izon o a la ge dis ances. As a consis ency check on
ou se ing we ha e e i ied ha in bo h cases
U V →
0,
(2.10)
, a he e en ho izon.
In igu e
(1)
we ep esen he con o mal diag am,
{U, V }
coo dina es
(2.10)
, o
(2.8)
.
3 En anglemen en opy
We a e going o ollow he now well-known s anda d analysis in o de o ind he en anglemen
en opy. We ake a wo-sided black hole, as shown in igu e
(1)
, which is ini ially in a pu e
4This is equi alen o conside ing only s-wa e emission.
5Hence o h ywill be a dimensionless a iable
y=: / 0.
– 4 –

JHEP01(2025)182
Singula i y
Singula i y
UV
−+→& −+→
− ∞ → & −+→
χ+
χ∞y∞
2y+
2
y+
1
y∞
1
I
Ho izon
Figu e 1. Con o mal diag am o ei he
(2.1)
o
(2.6)
. Each poin in he diag am ep esen s a
wo-dimensional Euclidean space, in ou case ex ending in he di ec ions
{x1, u}
o
(2.8)
. Al hough
he con o mal ac o s (2.13), (2.14) a e di e en , hei causal s uc u es a e he same. Red cu es
show he singula i y a
→
0. Dashed lines ep esen he ho izon,
→ 0
. Finally, solid black lines
ep esen he asymp o ic. One salien poin is ha he
→
0su aces bend he con o mal diag am,
as does he AdS Schwa zschild black hole [
21
]. The egion
χ
, wi h s a es iden i ied wi h he Haking
adia ion, is di ided in o wo pa s,
χ−
and
χ+
. The bounda y su aces o
χ±
a e
y±
2
espec i ely.
We show he con igu a ion con aining an island, I, ha ex ends ou side he ho izon. I s bounda ies
a e loca ed a y±
1. The goal is o compu e he en anglemen en opy o he union I∪χ+.
s a e. O e ime, i will become inc easingly en angled wi h he he mal ba h.
6
Thus i s
en opy is ini ially gi en by ha o he ma e en opy in he bulk while he black hole is
he e o p o ide he ixed cu ed backg ound. A he Page ime, he
O
(
G−1
N
)en anglemen
be ween he le and igh black holes is eplaced by he
O
(
G−1
N
)en anglemen be ween
he indi idual black holes and he ba h [
16
]. So he e a e a leas wo compe ing su aces
and he gene alised en opy is gi en by [
5
]
S(χ) = minIex IA(∂I)
4GN
+Sma e (χ∪I).(3.1)
The i s e m o
(3.1)
co esponds o he a ea o he edge o he island con ibu ion while
he second is he on Neumann en opy compu ed wi h he quan um ield heo y o malism
in he absence o g a i y.
7
This e m depends on he ela i e loca ion o he adia ion wi h
espec o he black hole ho izon.
3.1 En anglemen en opy o a single in e al
In a 2
d
CFT he e is a uni e sal exp ession o he en anglemen en opy o an in e al o
leng h
L
, whe e he edges a e e y a apa [
17
]. This exp ession, in u n, coincides wi h he
ma e en opy pe uni a ea o he me ic
(2.12)
i we igno e he ole o he
S3
[
18
],
8
Sb
ma e =c
3log L
ϵ≈c
6log "1
ϵ2
z2
y+y−
Ω(y+)Ω(y−)#(3.2)
6
I is e iden ha he Page cu e we shall ob ain is ha o a non-g a i a ing heo y which sa is ies he
spli p ope y o local QFT [15] and no ha o he black hole.
7See he inal ema k a he end o subsec ion 3.2.2.
8See igu e (1) o no a ion.
– 5 –
JHEP01(2025)182
whe e
ϵ
is a u egula o ,
c
is he cen al cha ge and
z2
ab = Ω2(a)Ω2(b) [U(b)−U(a)] [V(b)−V(a)] (3.3)
is he geodesic dis ance be ween wo bounda y poin s in K uskal coo dina es wi h
y+
= (
y, y
)
and
y−
= (
− y
+
iπ
2, y
).
9
Al hough
(3.2)
has been ound in 2
d
, he e a e ac ually se e al
nume ical checks on i s consis ency, unde some es ic ions, in highe dimensional space-
imes [
19
]. The exp ession
(3.2)
con ains bo h ul a iole and in a ed di e gences. The o me
a e aken in o accoun wi h he s anda d eno maliza ion echnique and he o e all ou come
is he eno maliza ion o he New on’s cons an
GN
. The la e di e gences a ise when one
o he
y±
app oaches he ho izon. Due o he gene al change o coo dina es
(2.10)
,
(3.2)
can be w i en as
Sb
ma e ≈c
3log eciFi(y)cosh ci
y
0+c
3log [2 Ω(y)] .(3.4)
The de ails o he se up, he o m o
F
(
x
)and Ω(
x
), a e no impo an a his poin . Wi h
(3.4)
a ace, i is no su p ising ha , wi h
ci
as a cons an , all he g a i a ional models will beha e
linea ly a la ge ime, once we ake in o accoun he p ope sub ac ions. In his egime, and
p o ided ha
(3.2)
is alid only o space ime poin s a om he ho izon, he emaining
e ms in
(3.4)
which do no di ec ly in ol e ime explici ly, a e subleading.
We pause o mo i a e he subsequen s eps in ou s udy be o e p oceeding wi h he
s anda d analysis, see o example [
20
]. No e ha he ele an quan i y wi hin he a gumen
o he ime-dependen pa o
(3.4)
is
ci
. In mos o he cases s udied in he li e a u e,
he cons an
ci
is a ha mless nume ical ac o
10
bu due o he p esence o he unc ion
Ai
(
)in
(2.12)
i now plays a c ucial ole, since i in oduces he ee pa ame e s ha de ine
he heo y. Thus, in he sequel, and in con as o p e ious s udies, i is no an ea ly/la e
epoch expansion wha cha ac e ises he s udy o he black hole emission bu he in e play o
he ac o s in he
ci
ela ion. Mo e speci ically, we a e in e es ed in he beha iou o he
en anglemen en opy wi h he dependence o
ci
on he numbe o b anes
ci
(
N
).
3.1.1 The NS5 and LST en anglemen en opies
To make he poin clea e , le us ha e a close look a he possible alues o
ci
(
N
) o he
(2.1)
and
(2.6)
models. Fi s o all, because hey a e a supe g a i y solu ion, he e ec i e s ing
coupling mus be bounded a i s maximum alue, he ho izon. This leads o he cons ain
N≪m2
s 2
0.(3.5)
Second, he scala cu a u e should be small i he classical geome y is o hold
R ∼ 1
N, N ≫1.(3.6)
Combining
(3.5)
and
(3.6)
we ge
1≪N≪m2
s 2
0.(3.7)
9
By his con en ion, he signs o
U
and
V
in he le wedge o he K uskal diag am picks and ex a
minus sign.
10We ha e checked his claim in se e al models ei he wi h single o mul i-ho izons.
– 6 –
JHEP01(2025)182
Wi h
(3.7)
a hand, we can conclude ha al hough he wo models a e o mally iden ical,
see
(2.8)
, he beha iou o he associa ed
ci
unc ions,
(2.13)
and
(2.14)
, is comple ely
di e en a la ge-N
cLST ≫1, cNS5 →1.(3.8)
Bea ing in mind ha app oxima ion he la e ime en anglemen en opies
(3.4)
a e
LST S(χ) = Sb
ma e ≈c
3
1
√N
y
ls
,(3.9)
NS5 S(χ) = Sb
ma e ≈c
3
y
0
.(3.10)
As expec ed, he en anglemen en opy g ows linea ly wi h ime in bo h models. A ew wo ds
a e in o de . Roughly speaking, en opy,
(2.7)
, gi es he amoun by which a sys em, le
us call i
A
, in e ac s wi h i sel . Whe eas he en anglemen en opy,
(3.9)
, cap u es he
in e ac ions be ween wo o i s subsys ems,
A1,A2∈ A
. Wha
(3.9)
shows is ha i you
ha e a pa ame ically la ge numbe o NS5-b anes,
O
(
N
)
≫ O
(1), bu wi h
(3.7)
s ill alid,
he sys em becomes non-in e ac ing, he Hilbe space ac o ises and he e o e we canno
eco e in o ma ion om ou side [
2
]. This is equi alen o saying ha he sys em becomes
non-in e ac ing wi h i s bounda y whe e he adia ion is collec ed. In addi ion, in his limi
he sys em can be conside ed as consis ing o non-in e ac ing s ings [
22
,
23
], whe e he s ing
ension is ela ed o he usual s ing ension as [
24
].
ˆτ=τ
N.(3.11)
Thus he Hagedo n s ing ension is quan ized o a ac ional uni o he o dina y s ing
ension and anishes a la ge-N.
Jus o comple eness, we oughly es ima e he so-called Page ime: as ime goes on,
(3.9)
has some con lic wi h he on Neumann en opy ini eness because he en opy adia ion
becomes la ge han ha o he black hole, which has only a ini e numbe o deg ees o
eedom. This happens a
y⪆Nm2
s 2
0,(3.12)
which p o es o be ex ao dina ily long o he
(2.6)
model.
3.2 En aglemen en opy o wo-disjoin in e als
Now i ’s ime o u n o he island con ibu ion and see i he p e ious pa e n has been
washed ou o s ill emains. We now look o non- i ial con igu a ions o he ba h a la e
imes. We conside he le
↔
igh symme ic island AB in igu e 1. Then he en anglemen
en opy is gi en by
S(χ) = A(∂I)
4GN
+Sa
ma e .(3.13)
These islands can be loca ed ei he inside o ou side he ho izon depending on he speci ic
g a i a ional model. This is equi alen o look o he en anglemen en opy o wo disjoin
– 7 –
JHEP01(2025)182
in e als inside
R2
[
25
] be ween he poin s (
y+
1, y−
1
)and (
y+
2, y−
2
)in acuum s a e. The
con ibu ion o he ma e can be ac o ised as ollows
Sa
ma e =c
3log "zy+
1y−
1zy+
2y−
2zy+
1y+
2zy−
1y−
2
ϵ4zy+
1y−
2zy−
1y+
2#.(3.14)
F om a physical pe spec i e
(3.14)
gi es he en anglemen en opy o he ma e ields
loca ed be ween he adia ion and he island egions.
Using
(2.10)
we again ob ain he same unc ional dependence on ime o bo h models,
assuming
ci
(
N
)as nume ical cons an ,
Sa
ma e ≈c
3log4eci(N)(F(y1)−F(y2))coshci(N) y1
0coshci(N) y2
0 (3.15)
+c
3log

cosh[ci(N)(F(y1)−F(y2))]−coshci(N)
0( y1− y2)
cosh[ci(N)(F(y1)−F(y2))]+coshci(N)
0( y1+ y2)
+c
3log[Ω(y1)Ω(y2)].
The ac ha
(3.15)
holds independen ly o any g a i a ional model, indica es ha he
adia ion-ba h coupling is somehow uni e sal in all o hem.
As we ha e al eady seen,
(3.10)
, he NS5-b ane model ollows he s anda d beha iou
wi h espec o he
N
dependence, and so om now on we shall concen a e mainly on
he LST model.
The nex s ep is o conside ha he island is o med nea he ho izon,
11 y1≈
1 +
ϵ
,
and ha
y2
is nea he bounda y,
y2≫
1. Then
(3.15)
becomes
12
Sa
ma e =c
6log32ϵy2
2cosh2cLST(N) y1
0cosh2cLST(N) y2
0
+c
3log

1−2√2ϵ
y2coshcLST(N)
0( y1− y2)
1+2√2ϵ
y2coshcLST(N)
0( y1+ y2)
+c
3log[Ω(y1)Ω(y2)].(3.16)
We shall now commen on he limi ing beha iou ,
(3.8)
, o
(3.16)
.
3.2.1 O(cLST(N)) ≲O( )
This i s limi is he s anda d one. We shall conside a la ge, bu ini e, ime e olu ion and
also a
cLST
(
N
)a mos o he same o de as ime wi h he cons ain
1≫2√2ϵ
y2
cosh cLST(N)
0
( y1− y2).(3.17)
Wi h his app oxima ion in mind and a li le o algeb a, we ge
Sa
ma e =2
3clogy2−2
3c√2ϵ
y2
coshcLST(N)
0
( y1− y2)+1
3clog[Ω(1+ϵ)Ω(y2)],(3.18)
which has an ex eme o he alues
y1= y2, ϵ =1
2y2
2
.(3.19)
11The ϵpa ame e , ϵ≪1, can be ei he posi i e o nega i e.
12The exp ession (3.16) coincides wi h he esul s o [26] once we se he unc ion Ω = 1 and cLST = 1.
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