JHEP04(2014)149
Published o SISSA by Sp inge
Recei ed:Ma ch 14, 2014
Re ised:Ap il 3, 2014
Accep ed:Ap il 4, 2014
Published:Ap il 24, 2014
Holog aphic ze o sound a ini e empe a u e in he
Sakai-Sugimo o model
B andon S. DiNunno,aMa hias Ihl,bNiko Jokelacand Juan F. Ped azaa
aTheo y G oup, Depa men o Physics and Texas Cosmology Cen e ,
The Uni e si y o Texas a Aus in, Aus in, TX, 78712, U.S.A.
bCen o de F´ısica do Po o e Depa amen o de F´ısica e As onomia,
Faculdade de Ciˆencias da Uni e sidade do Po o,
Rua do Campo Aleg e 687, 4169-007 Po o, Po ugal
cDepa amen o de F´ısica de Pa ´ıculas e Ins i u o Galego de F´ısica de Al as Ene x´ıas (IGFAE),
Uni e sidade de San iago de Compos ela,
E-15782 San iago de Compos ela, Spain
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed],[email p o ec ed]
Abs ac : In his pape , we s udy he a e o he holog aphic ze o sound mode a ini e
empe a u e and non-ze o ba yon densi y in he decon ined phase o he Sakai-Sugimo o
model o holog aphic QCD. We es ablish he exis ence o such a mode o a wide ange
o empe a u es and in es iga e he dispe sion ela ion, quasi-no mal modes, and spec al
unc ions o he collec i e exci a ions in ou di e en egimes, namely, he collisionless
quan um, collisionless he mal, and wo dis inc hyd odynamic egimes. Fo su icien ly
high empe a u es, he ze o sound comple ely disappea s, and he low ene gy physics is
domina ed by an eme gen di usi e mode. We compa e ou indings o Landau-Fe mi
liquid heo y and o o he holog aphic models.
Keywo ds: Gauge-g a i y co espondence, AdS-CFT Co espondence, Holog aphy and
condensed ma e physics (AdS/CMT)
A Xi eP in : 1403.1827
Open Access,c
The Au ho s.
A icle unded by SCOAP3.doi:10.1007/JHEP04(2014)149
JHEP04(2014)149
Con en s
1 In oduc ion 1
2 The Sakai-Sugimo o model a ini e empe a u e 3
2.1 Gene al se up 3
2.2 Chemical po en ial 4
3 Ze o sound mode in he Sakai-Sugimo o model 5
3.1 DBI ac ion and longi udinal luc ua ions 5
3.2 Asymp o ic solu ion 7
4 Nume ical esul s 9
4.1 Quasi-no mal modes and dispe sion ela ions 10
4.1.1 Beha io o a enua ion a e ΓTa non-ze o empe a u e 12
4.2 Spec al unc ions 13
4.3 Absence o a Fe mi su ace 18
5 Conclusions and ou look 19
1 In oduc ion
Unde s anding he phase s uc u e o QCD a ini e empe a u e and non-ze o ba yon
densi y is a di icul ask and a majo ocus o cu en esea ch. A su icien ly la ge
ene gies, QCD is weakly in e ac ing and pe u ba ion heo y is adequa e. Howe e , a low
ene gies QCD is s ongly in e ac ing and pe u ba i e me hods become un eliable, c ea ing
a demand o new heo e ical ools. The disco e y o he gauge/g a i y co espondence [1–
3] has g an ed us access o he s udy o a la ge class o s ongly-coupled non-abelian gauge
heo ies.
One o he i s and mos success ul inca na ions o holog aphic QCD is he Sakai-
Sugimo o model [4,5], which is dual o a SU(Nc) gauge heo y wi h N chi al la o s.
This model is ob ained by conside ing N D8-D8-b anes embedded in he backg ound
o NcD4-b anes compac i ied on a ci cle (wi h an ipe iodic bounda y condi ions o he
e mions). I N Nc, he back eac ion o he la o b anes on he geome y can be
sensibly neglec ed and his, in he ield heo y, co esponds o wo king in a “quenched”
app oxima ion which dis ega ds qua k loops. Al hough many aspec s o he Sakai-Sugimo o
model closely esemble eal-li e QCD, his app oach is ul ima ely oubled by he la ge-
Nclimi inhe en in he holog aphic desc ip ion. In pa icula , o la ge-Nc, he low-
empe a u e low-densi y phase o holog aphic QCD becomes a c ys alline solid ins ead o
he Fe mi liquid o he Nc= 3 case. Mo eo e , conside ing he leading 1/Ncco ec ions
is a di icul ask because i equi es doing loop calcula ions in he bulk.
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JHEP04(2014)149
On he o he hand, he decon ined phase o holog aphic QCD beha es as a di usi e
conduc o wi h es o ed chi al symme y and may be iden i ied wi h a s ongly coupled
liquid [6]. Indeed, he e a e some indica ions ha in his egime holog aphic QCD migh
be in a Fe mi liquid phase. Fi s , he densi y dependen pa o he hea capaci y a low
empe a u e is linea in T[7].1This beha io is expec ed o sys ems wi h a Fe mi su ace,
whe e only a ac ion o quasipa icles is exci ed a small empe a u es. Mo eo e , in he
phase whe e chi al symme y is es o ed, he ene gy densi y a ies as n5/3
B, which is he
expec ed powe o non- ela i is ic e mions [8].2
I is well known ha , a su icien ly low empe a u es, mos me als can be desc ibed by
Landau-Fe mi liquid heo y, a phenomenological model whe e he beha io o in e ac ing
e mions a e gi en in e ms o quasipa icle exci a ions o a Fe mi su ace. In pa icula , in
pe u ba i e QCD he exis ence o a Fe mi liquid phase a high enough chemical po en ial
is well es ablished [11–13], bu i is an in e es ing ques ion whe he one can s ill ind
signa u es o a sha p Fe mi su ace a s ong coupling. This seems unlikely a i s sigh ,
bu in he la ge ’ Hoo coupling limi , qua k-qua k in e ac ions a e supp essed and he e
is a chance o obse ing such beha io a leas in some egions o he pa ame e space.
One o he main ea u es eminiscen o Fe mi liquids is he exis ence o a gapless
exci a ion in he longi udinal cu en -cu en co ela o s [14]. This collec i e exci a ion,
known as he ze o sound, a ises om oscilla ions o he Fe mi su ace which change i s
shape bu no i s size. In he con ex o holog aphy he exis ence and cha ac e iza ion
o such mode has been s udied ex ensi ely in he li e a u e, s a ing wi h he seminal
pape [15], gene alized o non-ze o empe a u e [16] and o magne ic ield s eng h [17],
wi h an e e -g owing body o wo k ha includes [18–34]. These s udies ha e shed ligh on
he ques ion o whe he a ue Fe mi liquid desc ip ion can be eco e ed a low ene gies in
a holog aphic se up.3,4
In he decon ined phase o holog aphic QCD, in pa icula , i was ound ha a ze o
sound mode exis s a leas in he ze o empe a u e limi T/µ 1 [7]. Howe e , i is
well-known ha o such high densi ies he model is uns able owa ds o ma ion o ba yon
cha ge densi y wa es [39]. The eason behind his is ha he e a e Che n-Simons e ms
o he gauge ields which ypically lead o spa ially modula ed phases, simila o o he
#ND=6 cousins, D3/D7’ [40] and D2/D8’ [41] b ane in e sec ion models. An in e es ing
1In he Sakai-Sugimo o model, he decon ined phase occu s o T > Tc whe e Tc is a c i ical empe a u e
se by he Kaluza-Klein scale. He e, by low empe a u e we mean Tµ, whe e µis he ba yonic chemical
po en ial.
2In ac , his beha io holds o empe a u es below Tc and high enough ba yon densi ies [9,10]. In
his egime, he nuclea ma e is s ill in a solid phase and is cha ac e ized by a condensa e o ins an ons
on he p obe la o b anes.
3An in e es ing cons uc ion leading o a Fe mi liquid beha io was ecen ly p oposed in [35]. In his
pape , he au ho s conside ed a phenomenological model in which he bosonic sec o is go e ned by he
DBI ac ion, and whose cha ged sec o is pu ely e mionic. I would be in e es ing o s udy he longi udinal
cu en -cu en co ela o s o his se up and de e mine i i suppo s he exis ence o a ze o sound mode.
4The e has also been a lo o in e es in unde s anding non-Fe mi liquids and ma ginal Fe mi liquids in
he con ex o holog aphy [36–38]. These a e si ua ions wi h non- i ial IR ixed poin s whe e a Fe mi su ace
s ill exis s bu he quasipa icle desc ip ion b eaks down, and could play an impo an ole in unde s anding,
e.g., he s ange me al phase o high Tccup a es and hus high empe a u e supe conduc i i y.
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JHEP04(2014)149
ques ion we may ask he e is whe he his mode su i es a empe a u es and densi ies ha
a e s able agains his s iped phase o ma ion, and his pape is de o ed o answe ing his
ques ion.5,6Mo e speci ically, we will in es iga e he a e o he holog aphic ze o sound
mode in he decon ined phase o he Sakai-Sugimo o model in he ull space o pa ame-
e s Tand µ. We will cha ac e ize he di e en ansi ions, s a ing om he expec ed
quan um egime T/µ 1 [7] and ending wi h a high empe a u e egime T/µ 1 whe e
he ze o sound mode comple ely disappea s and he physics is domina ed by an eme gen
di usi e mode [6].
The pape is o ganized as ollows: in sec ion 2, we b ie ly e iew some impo an ea-
u es o he Sakai-Sugimo o D4/D8/D8 model ele an o he in es iga ion o he collec i e
exci a ions in conside a ion. In sec ion 3, we de i e he equa ions o mo ion o he longi u-
dinal luc ua ions o he gauge ield and we s udy in de ail he asymp o ic beha io o he
solu ions o bo h he nea -bounda y and nea -ho izon egions in he bulk. In sec ion 4, we
p esen ou nume ical esul s o he dispe sion ela ion, quasi-no mal modes, and spec al
unc ions o he a ious egions o he pa ame e space. Finally, in sec ion 5 we conclude
wi h a discussion o he impo an simila i ies and di e ences o he ze o sound mode in
he Sakai-Sugimo o model compa ed o o he holog aphic models and Landau-Fe mi liquid
heo y, and commen b ie ly on possible u u e di ec ions in he esea ch on holog aphic
quan um liquids.
2 The Sakai-Sugimo o model a ini e empe a u e
2.1 Gene al se up
Le us conside he nea ho izon geome y gene a ed by he NcD4-b anes
ds2=u
R3
2− (u)d 2+dx2
i+dx2
4+u
R−3
2du2
(u)+u2dΩ2
4,
eΦ=gsu
R3
4, F4=dC3=2πNc
Ω4
ω4,
whe e ,xi,i= 1,2,3,and x4 ep esen he dimensions o he wo ld olume o he D4-
b anes, uis he adial (holog aphic) coo dina e and dΩ2
4 he me ic o he uni ou -sphe e,
while Ω4and ω4deno e he olume and olume o m o he uni ou -sphe e, espec i ely.
The unc ion (u) := 1 −uH
u3is he usual emblackening ac o , whe e uH, he loca ion
o he ho izon, is ela ed o he empe a u e o he ield heo y ia
TH=3u1/2
H
4πR3/2.(2.1)
5The spa ially modula ed ins abili ies can be mi iga ed and e ec i ely washed ou by u ning on a
su icien ly la ge magne ic ield and/o a mass o he qua ks [16,17,30]. The o me is an ob ious
gene aliza ion o he p esen wo k, albei a e y in ol ed one due o a new phase opening up in he
heo y [42] ha leads o e y in e es ing physics [43]. We will lea e his issue o a u u e wo k.
6The con ining phase can also be uns able due o spa ial inhomogenei ies upon in oducing chemical
po en ials [44–46].
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JHEP04(2014)149
This desc ibes he high empe a u e, decon ined phase o he Sakai-Sugimo o model [4,5,
47] in Minkowski signa u e, app op ia e o eal ime dynamics.7,8
The adius o cu a u e Ris gi en by
R3=πgsNcl3
s=πλα0,(2.2)
whe e λis he ’ Hoo coupling cons an . The scale R4=2
3R3
uΛ1/2de e mines he
c i ical empe a u e Tc =1
2πR4a which a Hawking-Page (con inemen /decon inemen )
phase ansi ion occu s [47].9In o de o s ay wi hin he high empe a u e egime o he
model, we demand TH> Tc , o equi alen ly uH> uΛ.
In he ze o, o low empe a u e, con ined phase he N Ncp obe D8/D8-b anes
desc ibe a non- i ial p o ile in he x4(u) di ec ion and he wo b anches me ge a a adial
posi ion u0≥uΛ, which is he loca ion o he ip o he ciga -shaped subspace {x4, u}.
In he high empe a u e, decon ined phase ha we will be in e es ed in, he {x4, u}sub-
space will be cylinde -shaped and he s aigh embeddings, o which ∂ux4= 0, will be
ene ge ically a o ed.
2.2 Chemical po en ial
In oducing a non-ze o chemical po en ial amoun s o in oducing a lux F u 6= 0 along he
b ane wo ld olume [9,10,51] (see also [52–55]). The co esponding DBI ac ion eads
SD8=−NTZdu u5
2p1−(∂uA )2,(2.3)
whe e NT=N
Tand N=µ8
gsΩ4R3
2=√2
3(2π)−11
2NcN
√λ. He e we made a gauge choice o se
Au=0 and also escaled A →2πA . As usual, he conse ed cha ge associa ed wi h A eads
u5
2∂uA
p1−(∂uA )2=d. (2.4)
I ollows ha he elec ic ield sa is ies
∂uA =d
√d2+u5,(2.5)
om which one can ead o he chemical po en ial µas he asymp o ic alue o A ,
µ=A (u→ ∞) = Z∞
uH
du ∂uA =d
3πu3/2
H
2F11
2,3
10,13
10,−d2
u5
H.(2.6)
7In he D3/D7 model a ini e ba yon densi y, he p obe b ane embeddings a e “black hole” embeddings
ha all in o he ho izon [50]. This is he analogue o he high- empe a u e, decon ined phase we a e
in e es ed in he e. No e, howe e , ha in he decon ined phase o he Sakai-Sugimo o, he e exis s a egion
o pa ame e space whe e he (“sho cusp”) U-shaped embeddings a e a leas me a-s able, e en a non-ze o
ba yon densi y/chemical po en ial [9].
8Fo some ecen c i icism on he s anda d mechanism o he decon inemen /con inen phase ansi ion
in he Sakai-Sugimo o model, see [48,49].
9The pa ame e uΛse s he posi ion o he ip o he ciga in he ze o, o low empe a u e, con ined
phase o he Sakai-Sugimo o model and also de e mines he c i ical empe a u e a which he decon inemen
ansi ion happens.
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JHEP04(2014)149
3 Ze o sound mode in he Sakai-Sugimo o model
3.1 DBI ac ion and longi udinal luc ua ions
We wan o s udy he massless exci a ion coupled o he densi y ope a o in he
D4/D8/D8-sys em. This equi es analyzing he linea ized equa ions o mo ions ha ol-
low om he quad a ic ac ion desc ibing he luc ua ions o he gauge ields li ing on he
D8/D8-b anes.10 The DBI ac ion o he D8-b ane eads
SDBI,D8 =−T8ZdΩ4Zd4xZuB
uH
du e−Φp−de [Gmn +Fmn],(3.1)
whe e Gmn and Fmn a e he induced me ic and gauge ield s eng h, espec i ely.
Fo he in es iga ion o he ze o sound mode, we wan o s udy small (longi udinal)
luc ua ions o he gauge ield. We a e only in e es ed in luc ua ions which a e independen
o he S4coo dina es. The e o e we wo k in a gauge whe e Au≡0 and will se Ay(xµ, u) =
Az(xµ, u) = 0.11
Le us in oduce he ollowing use ul unc ions:
g(u) := pd2+u5, 1(u) := g(u) (u), 2(u) := g3(u)
u5, 3(u) := R3g(u)
u3 (u),(3.2)
in e ms o which he DBI ac ion o he longi udinal modes educes o
SDBI,D8 =−1
2NTZd4xZuB
uH
du  1(u) (∂uAx)2− 2(u) (∂uA )2− 3(u) (∂xA −∂ Ax)2.
(3.3)
The gauge ield is expanded as ollows
A (xµ, u) = A (u) + Zd4k
(2π)4eikµxµa (kµ, u),
Ax(xµ, u) = Zd4k
(2π)4eikµxµax(kµ, u),(3.4)
whe e we choose kµ= (−ω, k, 0,0) and some imes supp ess he dependence o aµon kµ.
Expanding he DBI ac ion o second o de in he luc ua ions yields he ollowing
equa ions o mo ion o he longi udinal modes,
∂u[ 2(u)∂ua (kµ, u)] −k2 3(u)a (kµ, u) + ω
kax(kµ, u)= 0,(3.5a)
∂u[ 1(u)∂uax(kµ, u)] + ω2 3(u)ax(kµ, u) + k
ωa (kµ, u)= 0.(3.5b)
Le us in oduce he gauge in a ian quan i y
E(kµ, u) := ka (kµ, u) + ωax(kµ, u),(3.6)
10The comple e D8 ac ion consis s o a Di ac-Bo n-In eld (DBI) and a Che n-Simons (CS) e m. Howe e ,
o he longi udinal pa o he gauge ield luc ua ions, i su ices o s udy he DBI pa o he ac ion. No e
also ha he gauge ield and me ic pe u ba ions decouple.
11This can be done consis en ly, since he longi udinal luc ua ions decouple om he ans e sal ones,
c . e.g., [6].
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JHEP04(2014)149
and mo eo e impose Gauss’ law, which is a cons ain equa ion ollowing om he equa ion
o mo ion o au(k, u),
ka0
x(u) + ωG(u)a0
(u) = 0,(3.7)
whe e he p ime deno es de i a ion wi h espec o uand G(u) := 2(u)
1(u)=g2(u)
u5 (u). Using
Gauss’ law i is possible o exp ess
a0
(u) = k
k2−ω2G(u)E0(u), a00
(u) = k
k2−ω2G(u)E00(u) + kω2G0(u)
(k2−ω2G(u))2E0(u),(3.8)
a0
x(u) = ωG(u)
ω2G(u)−k2E0(u), a00
x(u) = ωG(u)
ω2G(u)−k2E00(u)−ωk2G0(u)
(ω2G(u)−k2)2E0(u).(3.9)
Any one o he eqs. (3.5), oge he wi h he cons ain eq. (3.7), implies he emaining one.
De ining F(u) := k2−ω2G(u), we hus a i e a a single second o de equa ion o E(u)
(simila equa ions, albei in a di e en con ex , we e discussed in [56]),
E00(u) +  0
2(u)
2(u)−F0(u)
F(u)E0(u)− 3(u)F(u)
2(u)E(u)=0,(3.10)
o equi alen ly,
E00(u) + −5
u+15u4
2g2(u)+ω2G0(u)
k2−ω2G(u)E0(u)− R3u2k2−ω2G(u)
g2(u) (u)!E(u) = 0.
(3.11)
I u ns ou o be con enien o pe o m a change o a iables
y= 2 R3
u,(3.12)
and de ine bµ:= d1/5
2R3/2,12 which can be exp essed in e ms o µ aking in o accoun (2.6).
In o de o acili a e he ansla ion be ween he yand ucoo dina es, we de ine
unc ions simila o he iabo e (by a sligh abuse o no a ion, we will con inue o use
(y) = 1 −y
yH6and g(y) = p1 + bµ10y10),
h1= (y)g(y)
y2, h2=g3(y)
y2, h3=g(y)
y2 (y),(3.14)
and no e he ollowing ans o ma ion ules:
u↔y,
1↔h1,
2↔h2,
3↔h3,
12No e ha he c i ical densi y, abo e which an ins abili y occu s in he ans e sal sec o coupled o he
Che n-Simons e m, was de e mined in [39], eq. (18). Fo he Che n-Simons coupling in he Sakai-Sugimo o
model, his yields
bµc i ≈3.7142π
3TH⇒eµc i .≈3.714.(3.13)
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JHEP04(2014)149
Mo eo e , in o de o ende eq. (3.11) dimensionless, we pe o m he ollowing escalings,
y→yHey, ω →eω
yH
, k →ek
yH
,bµ→eµ
yH
,(3.15)
so ha 13
˜ω=3
2π
ω
TH
,˜
k=3
2π
k
TH
,˜µ=3
2π
ˆµ
TH
.(3.17)
Eq. (3.10) hen becomes (in dimensionless a iables)
¨
E(ey) + ˙
h2(ey)
h2(ey)−˙
F(ey)
F(ey)!˙
E(ey)−h1(ey)F(ey)
h2(ey)E(ey) = 0.(3.18)
He e he do ˙ indica es de i a ion wi h espec o ey.
3.2 Asymp o ic solu ion
We p oceed o sol e eq. (3.18) by applying a s anda d F obenius se ies expansion. Fi s
no e ha eq. (3.18) has non-essen ial ( egula ) singula poin s a ey=−1,0,+1,∞,and a
he eal oo s o ey6ek2+eω2eµ10 ey4=ek2−eω2. The cha ac e is ic exponen s a he ho izon
(ey= 1) a e ±i˜ω
6, co esponding o solu ions nea he bounda y (ey= 0) wi h incoming-wa e
(−i˜ω
6) and ou going-wa e (+i˜ω
6) bounda y condi ions, espec i ely. On gene al g ounds,
he solu ion o eq. (3.18) ha sa is ies an incoming-wa e bounda y condi ion nea he
ho izon can be w i en as a linea combina ion o wo local solu ions nea he bounda y
(wi h exponen s 0 and 3) [57–59]:
E(ey) = AZI(ey) + BZII (ey),(3.19)
whe e
ZI(ey) = 1 + bI
1ey+bI
2ey2+. . . , (3.20)
ZII (ey) = ey31 + bII
1ey+bII
2ey2+. . ..(3.21)
The coe icien s bI,II
ia e de e mined by ecu sion ela ions ollowing om eq. (3.18). Fo
he case a hand, we ind
bI
1,3,5,... = 0, bII
1,3,5,... = 0,
bI
2=−1
2ek2−eω2, bII
2=1
10 ek2−eω2,
bI
4=−1
8ek2−eω22, bII
4=1
280 ek2−eω22, . . . . (3.22)
13Fo he p esen a ion o he nume ical compu a ions epo ed in sec ion 4, i is some imes ad an ageous
o wo k wi h he al e na i e dimensionless a iables
ω=ω
ˆµ, k =k
ˆµ.(3.16)
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JHEP04(2014)149
Close o he ho izon a ey= 1, we can ind an incoming-wa e solu ion o eq. (3.18) o
he o m
E(ey) = (˜y)−i˜ω
6R(ey),(3.23)
whe e R(ey) is egula a he ho izon.
In o de o ind he e a ded wo-poin co ela ion unc ions G
R(ω, k), Gxx
R(ω, k), and
G x
R(ω, k), we me ely need o compu e he pola iza ion unc ion Π(ω, k), which can be
easily achie ed using he me hod ou lined in [57,58] ( o a mo e gene al, and e y use-
ul, p esc ip ion, c . [60]): Essen ially, one needs o unc ionally di e en ia e he on-shell
bounda y ac ion wi h espec o he sou ces.
In e ms o A and Ax, he bounda y e m akes he ollowing o m
SB= lim
u→uBNZd4x (u)g(u)Ax∂uAx−g(u) (u)G(u)A ∂uA ,(3.24)
whe e uBis some adial cu -o . In e ms o E(kµ, u) (and E(kµ, y), esp.), we ob ain
SB= lim
u→uBNZd4k
(2π)4 (u)g(u)G(u)
k2−ω2G(u)E(−kµ, u)∂uE(kµ, u),(3.25)
= lim
y→02NR9/2Zd4k
(2π)4h2(y)
k2−ω2G(y)E(−kµ, y)∂yE(kµ, y).(3.26)
Now, close o he bounda y, we can expand
lim
y→0E(−kµ, y)∂yE(kµ, y) = A(−kµ)A(kµ)2bI
2y+ 3 B
A
y2
y3
H
+. . .,(3.27)
whe e we ha e w i en he sou ce o E(±kµ, y →0) as A(±kµ), in acco dance wi h
eq. (3.19). As usual, one needs o holog aphically eno malize he ac ion, i.e., emo e
he di e gen e ms om he ac ion by adding app op ia e coun e e ms:
SB, en.=SB+Sc ,
Sc = 2NR9/2Zd4k
(2π)4A(−kµ)A(kµ)
y.(3.28)
The ma ix o ( he mal) e a ded co ela o s is gi en by
G
R(ω, k) = k2Π(ω, k),
Gxx
R(ω, k) = ω2Π(ω, k),
G x
R(ω, k) = ω k Π(ω, k),(3.29)
wi h
Π(ω, k)≡δ2SB, en.
δE(−k, 0)δE(k, 0) .(3.30)
The e o e, we a i e a
Π(ω, k) = 6NR9/2
y3
H
1
k2−ω2B
A=16π3µ8Ω4
9gs
R6T31
k2−ω2B
A.(3.31)
– 8 –
JHEP04(2014)149
0.300
0.305
0.310
0.315
0.320
0.325
0.330
0.335
0
500 000
1.0 ´106
1.5 ´106
Ω
ΧΩ
2
k
=0.5
2
4
6
8
10
2
4
6
8
10
Ω
ΧΩ
2
k
=0.5
Figu e 7. Spec al unc ion χxx(˜ω) (no malized by 6NR9/2
y3
H˜ω2) in egion II, collisionless he mal
egime, o ek= 0.5 and ˜µ= 11 ( ed), ˜µ= 13 (blue), ˜µ= 15 (o ange). No e ha he no maliza ion
ac o enhances he heigh o he peak o ˜ω < 1. Le panel: Zoomed in close o he p oximi y o
he peaks. Righ panel: A sligh ly highe ˜ω, he e is a second peak de eloping as he empe a u e
inc eases and he mal exci a ions become impo an .
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0
2´107
4´107
6´107
8´107
Ω
ΧΩ
2
k
=0.01
5
10
15
20
25
0
5
10
15
20
25
Ω
ΧΩ
2
k
=0.01
Figu e 8. Spec al unc ion χxx(˜ω) (no malized by 6NR9/2
y3
H˜ω2) in egion III, hyd odynamic
egime, o ek= 0.01 and ˜µ= 5 ( ed), ˜µ= 10 (blue), ˜µ= 15 (o ange). Le panel: The ze o sound
mode decays u he and mo es close o he o igin. Righ panel: The hyd odynamic di usi e
con ibu ion becomes dominan compa ed o he emnan s o he ze o sound mode.
– 15 –

JHEP04(2014)149
0.0
0.5
1.0
1.5
2.0
0
500 000
1.0 ´106
1.5 ´106
2.0 ´106
Ω
ΧΩ2
Μ
=15
2
3
4
5
6
7
4
6
8
10
12
Ω
ΧΩ2
Μ
=15
Figu e 9. Spec al unc ion χxx(ω) (no malized by 6NR9/2
y3
Hω2) in egion I, collisionless quan um
egime, o eµ= 15 and k= 0.4 ( ed), k= 0.6 (blue), k= 1.0 (o ange), k= 1.5 (pink), k= 2.0
(g ey), and k= 3.0 (pu ple). Le panel: The sha p, dis inc peaks co espond o he ze o sound
mode. Righ panel: Zoomed in close o he p oximi y o he peaks.
0.0
0.1
0.2
0.3
0.4
0.5
0
5000
10 000
15 000
20 000
25 000
Ω
ΧΩ2
Μ
=3
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0
2000
4000
6000
8000
Ω
ΧΩ2
Μ
=3
Figu e 10. Spec al unc ion χxx(ω) (no malized by 6NR9/2
y3
Hω2) in egion II, collisionless he mal
egime, o eµ= 3 and k= 1/30 ( ed), k= 1/15 (blue), k= 1/10 (o ange), k= 2/15 (pink), k= 1/6
(g ey), and k= 1/5 (pu ple). Le panel: The ze o sound mode decays and b oadens. Righ panel:
Zoomed in close o he p oximi y o he peaks.
– 16 –
JHEP04(2014)149
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0
2´107
4´107
6´107
8´107
Ω
ΧΩ2
k=0.01
0.000
0.005
0.010
0.015
0.020
0
2.0´106
4.0´106
6.0´106
8.0´106
1.0´107
1.2´107
1.4´107
Ω
ΧΩ2
k=0.01
0.000
0.005
0.010
0.015
0
200 000
400 000
600 000
800 000
1.0 ´106
1.2 ´106
1.4 ´106
Ω
ΧΩ2
k=0.01
Figu e 11. Spec al unc ion χxx(ω) (no malized by 6NR9/2
y3
Hω2) in egion III, hyd odynamic
egime, o k= 0.01 and eµ= 15 ( ed), eµ= 10 (blue), eµ= 5 (o ange), eµ= 2.5 (pink), eµ= 1.25
(g ey), eµ= 1 (pu ple). Going om he la ges o he smalles peak co esponds o inc easing he
empe a u e. No e he di e en scales on he e ical axis. Each subsequen igu e zooms in o an
a ea o he p e ious igu e (c . [25], igu e 10). The ze o sound mode des abilizes as we mo e deepe
in o he hyd odynamic egime.
– 17 –
JHEP04(2014)149
0
1
2
3
4
5
0
10
20
30
40
k

Re@GRD
Ω
=0
Figu e 12. The eal pa o GR(eω= 0,ek) o a ious choices o eµ, om bo om o op, eµ= 0
(blue), eµ= 1 (pu ple), eµ= 5 (o ange), eµ= 10 (g een), eµ= 20 ( ed), eµ= 100 (black). No e ha
he GR(eω= 0,ek= 0) in e cep s ha e been shi ed o imp o e p esen abili y.
4.3 Absence o a Fe mi su ace
This ques ion was o iginally posed in e . [7] in he con ex o he ze o empe a u e limi
o he Sakai-Sugimo o model. He e we will e iew hei a gumen s and ex end he in es-
iga ion o non-ze o empe a u e. F om he heo y o Fe mi liquids, we know ha he
exis ence o a sha p Fe mi su ace is associa ed wi h a discon inui y in he dis ibu ion
unc ion which in u n is di ec ly obse able as a singula i y a k= 2kFin he e a ded
cu en -cu en G een’s unc ions in he small equency limi ,
GR(ω→0, k)∼k
2kF−1ln k
2kF−1.(4.3)
We nume ically e alua e GR(ω, k)∼B
Ain his limi . Figu e 12 shows a plo o GR(0,ek)
o a wide ange o ekand a ious empe a u es. O cou se, since [7] did no ind any
e idence o a sha p Fe mi su ace a ze o empe a u e, we do no expec o ind aces
o a Fe mi su ace in he small equency limi o he e a ded G een’s unc ions a non-
ze o empe a u e ei he , and me ely in end o co obo a e hei esul . Indeed, we do no
obse e any cha ac e is ic s uc u e consis en wi h a singula i y in he G een’s unc ions.
In [42], he au ho s explo ed i he model exhibi s quan um oscilla ions associa ed wi h
a se ies o c ossings o he Landau le els wi h he Fe mi su ace. They concluded ha
he model does no possess de Haas- an Alphen oscilla ions, e en a ze o empe a u e.
Simila ly, ou analysis does no seem o lead o u he e idence o an obse able Fe mi
su ace, despi e he exis ence o a collec i e exci a ion co esponding o ze o sound. Some
possible implica ions a e lis ed in he conclusions.
– 18 –
JHEP04(2014)149
5 Conclusions and ou look
The p incipal line o esea ch ollowed in he p esen wo k was o cla i y he na u e o
holog aphic quan um liquids, in pa icula , hei low ene gy e ec i e desc ip ion a ini e
densi y, and o compa e hem wi h p ope ies o eal ma e ials such as liquid Helium-3,
which a su icien ly low empe a u es a e well unde s ood in e ms o Landau’s heo y o
Fe mi liquids.
We ocused on he decon ined phase o he Sakai-Sugimo o model o holog aphic QCD,
expanding on p e ious esul s ega ding he exis ence o a ze o sound mode in he longi-
udinal channel o he cu en -cu en co ela o s. Mo e speci ically, ollowing [16,25], we
gene alized he ze o- empe a u e (o in ini e ba yonic densi y) esul s o [7] o he ze o
sound mode o a bi a y alues o Tand µand we s udied in de ail i s beha io as T/µ
was inc eased.
Le us summa ize some o he impo an indings o he Sakai-Sugimo o model a ini e
empe a u e and ba yon densi y/chemical po en ial:
•Some he modynamic p ope ies o he decon ined phase o he Sakai-Sugimo o model
appea o be consis en wi h a s ongly-coupled Fe mi liquid in e p e a ion. We mus
emphasize ha he exis ence o such a phase a high enough ba yon densi y is well
es ablished in pe u ba i e QCD [11–13], which means ha he exis ence o a Fe mi
su ace seems o nai ely ex apola e o he s ong coupling egime. In pa icula ,
he speci ic hea a low empe a u es scales like C ∼T[7], in ag eemen wi h he
p edic ions o Landau-Fe mi heo y, while in he s anda d D3/D7 case C ∼T6[25].
On he o he hand, he ene gy densi y a ies as n5/3
B, which is he expec ed equa ion
o s a e o non- ela i is ic e mions [8].
•The model ea u es a s able ze o sound mode wi h a non-s anda d k3dependence o
he a enua ion a e a low and mode a e empe a u es (collisionless quan um and
collisionless he mal egimes).
•Fo la ge enough empe a u es (hyd odynamic egime), he ze o sound mode is e -
ec i ely damped ou by he mal luc ua ions. The low ene gy physics in his case is
domina ed by an eme gen di usi e mode, in ag eemen wi h he esul s o [6].
•The beha io o he a enua ion a e a ini e empe a u e in he h ee egimes dis-
cussed abo e is eminiscen o Landau-Fe mi liquid heo y, albei wi h di e en em-
pe a u e dependences. In pa icula , we ound ha ΓT∼cons an in he collisionless
quan um egime and ΓT∼T3in he collisionless he mal egime. In he hyd ody-
namic egime, we can dis inguish be ween a cold and a ho phase whe e ΓT∼T−3
and ΓT∼T−1, espec i ely. In he la e egime, he a enua ion a e e e s o he
di usion mode, while in he i s wo cases, o he ze o sound mode.
•We did no disco e any e idence o a di ec signa u e o a sha p Fe mi su ace a
ze o o non-ze o empe a u e. One possible explana ion is ha he Fe mi su ace
s uc u e expec ed in he spec al unc ions a low equencies migh only appea a
– 19 –
JHEP04(2014)149
o de N /Nc, in which case we would need o include back eac ion o ha e a isible
e ec .14 I his is he case, he i s s ep would be o cons uc he back eac ed
backg ound a ini e chemical po en ial wi h he luc ua ions u ned o . On op o
his backg ound we would need o conside small luc ua ions. Gauge luc ua ions will
gene ically mix wi h me ic luc ua ions and we would ha e o conside an app op ia e
gauge in a ian combina ion o he sound channel as in [20]. The second possibili y
is ha he ze o sound mode ha we obse e is no ac ually due o he p esence o a
Fe mi su ace, bu a he , should be in e p e ed as a Golds one boson a ising om he
spon aneous b eaking o a speci ic symme y, see e.g. [22,33]. Bo h op ions would be
in e es ing in hei own igh . Ei he we con i m he exis ence o a Fe mi liquid phase
a s ong coupling o we disco e a new low ene gy desc ip ion o s ongly-co ela ed
elec ons wi h no Fe mi su aces.
•A la ge chemical po en ial o empe a u e a io µ/T he model is expec ed o decay
o a spa ially inhomogeneous phase [39]. The c i ical alue ˜µ∼3.7 o he on-se
o he ins abili y is o he same o de as he h eshold o he ze o sound o be
comple ely damped by he mal e ec s and ceasing o p opaga e, hus lea ing only a
ai ly small ange o pa ame e space o he ze o sound. Howe e , o p ecisely add ess
his ques ion one should i s cons uc he s iped solu ion ou o he equa ions o
mo ion ollowing om he DBI+CS ac ion, and compa e he ee ene gies o deciphe
which phase domina es.
Thus, in conclusion, al hough some o he ea u es o he model a e compa ible wi h he
Landau heo y o Fe mi liquids, u he esea ch is necessa y in o de o de e mine whe he
his model o holog aphic QCD has a ue low ene gy desc ip ion in e ms o quasipa icle
exci a ions o a Fe mi su ace o i i cons i u es a no el ype o s ongly-coupled quan um
liquid phase wi hou a Fe mi su ace a all.
As men ioned in he in oduc ion, he inclusion o a la ge magne ic ield in he wo ld-
olume o he la o b anes would be an in e es ing ex ension o he p esen wo k because
such a case may no be uns able o he o ma ion o a spa ially modula ed phase. The
ze o sound mode is expec ed o de elop a gap in he dispe sion ela ion, and would lead o
in e es ing physics ega ding, e.g, he a e o Kohn’s heo em o non ela i is ic e mions
wi h pai wise in e ac ions. Ano he in e es ing possibili y o u u e esea ch would be o
in es iga e he case o non-supe symme ic la o D7/D7- and D5/D5-b ane se ups in he
Klebano -Wi en and Klebano -S assle backg ounds [62–70]. Bo h cases a e cu en ly
unde conside a ion.
Acknowledgmen s
The au ho s would like o acknowledge e y use ul con e sa ions and co espondence
wi h Al onso Ballon Bayona, Richa d Da ison, Mohammad Edala i, Ma hias Kaminski,
Ma hew Lippe , Al onso Ramallo, Go don Semeno , and Dimi ios Zoakos. M.I. is g a e-
ul o he Theo y G oup o he Depa men o Physics a he Uni e si y o Texas a Aus in,
14Some back eac ed con igu a ions in he Sakai-Sugimo o model we e s udied in [61].
– 20 –

JHEP04(2014)149
he Depa men o Physics a he Uni e si y o Washing on, he Depa men o Physics a
he Uni e si y o B i ish Columbia, and he Theo y G oup, Ins i u o Galego de F´ısica de
Al as Ene x´ıas, Uni e sidade de San iago de Compos ela o hospi ali y du ing in e me-
dia e s ages o his wo k; mo eo e , M.I. would like o exp ess his g a i ude o Es ibali z
Uka and Ch is oph Sachse o hospi ali y du ing his ex ended s ay in Aus in, TX. The
wo k o B.D and J.P is pa ially suppo ed by he Na ional Science Founda ion unde g an
G an No. PHY-1316033 and by he Texas Cosmology Cen e . M.I. is unded by he FCT
ellowship SFRH/BI/52188/2013. The Cen o de F´ısica do Po o is pa ially unded by
FCT h ough he p ojec s PTDC/FIS/099293/2008 and CERN/FP/116358/2010. N.J. is
unded by he Spanish g an FPA2011-22594, by he Consolide -Ingenio 2010 P og amme
CPAN (CSD2007-00042), by Xun a de Galicia (GRC2013-024), and by FEDER. N.J. is
also suppo ed by he Juan de la Cie a p og am.
Open Access. This a icle is dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (CC-BY 4.0), which pe mi s any use, dis ibu ion and ep oduc ion in
any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
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