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Nuclea Physics B 909 (2016) 677–724
www.else ie .com/loca e/nuclphysb
Collec i e exci a ions o massi e la o b anes
Geo gios I sios a,d,e, Niko Jokela b,c,∗, Al onso V. Ramallo d,e
aDepa men o Physics, Uni e si y o O iedo, A da. Cal o So elo 18, 33007 O iedo, Spain
bDepa men o Physics, FIN-00014 Uni e si y o Helsinki, Finland
cHelsinki Ins i u e o Physics, P.O. Box 64, FIN-00014 Uni e si y o Helsinki, Finland
dDepa amen o de Física de Pa ículas, Uni e sidade de San iago de Compos ela,
E-15782 San iago de Compos ela, Spain
eIns i u o Galego de Física de Al as Ene xías (IGFAE), E-15782 San iago de Compos ela, Spain
Recei ed 25 Feb ua y 2016; ecei ed in e ised o m 29 May 2016; accep ed 6 June 2016
A ailable online 9 June 2016
Edi o : Leona do Ras elli
Abs ac
We s udy he in e sec ions o wo se s o D-b anes o di e en dimensionali ies. This con igu a ion is dual
o a supe symme ic gauge heo y wi h la o hype mul iple s in he undamen al ep esen a ion o he gauge
g oup which li e on he de ec o he un la o ed heo y de e mined by he di ec ions common o he wo
ypes o b anes. One se o b anes is dual o he colo deg ees o eedom, while he o he se adds la o o
he sys em. We wo k in he quenched app oxima ion, i.e., whe e he la o b anes a e conside ed as p obes,
and ocus speci ically on he case in which he qua ks a e massi e. We s udy he he modynamics and he
speeds o i s and ze o sound a ze o empe a u e and non- anishing chemical po en ial. We show ha he
sys em unde goes a quan um phase ansi ion when he chemical po en ial app oaches i s minimal alue
and we ob ain he co esponding non- ela i is ic c i ical exponen s ha cha ac e ize i s c i ical beha io . In
he case o (2 +1)-dimensional in e sec ions, we u he s udy al e na i e quan iza ion and he ze o sound
o he esul ing anyonic luid. We inally ex end hese esul s o non-ze o empe a u e and magne ic ield
and compu e he di usion cons an in he hyd odynamic egime. The nume ical esul s we ind ma ch he
p edic ions by he Eins ein ela ion.
©2016 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3.
*Co esponding au ho .
E-mail add esses: i siosgeo gios@unio i.es (G. I sios), [email p o ec ed] (N. Jokela), [email p o ec ed]
(A.V. Ramallo).
h p://dx.doi.o g/10.1016/j.nuclphysb.2016.06.008
0550-3213/©2016 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3.
678 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
1. In oduc ion
The e is hope ha he gauge/g a i y holog aphic duali y could se e o cha ac e ize new ypes
o comp essible s a es o ma e , i.e., s a es wi h non-ze o cha ge densi y which a y con inuously
wi h he chemical po en ial. Indeed, holog aphy p o ides g a i a ional desc ip ions o s ongly
in e ac ing sys ems wi hou long-li ed quasipa icles, si ua ions which canno be accommoda ed
wi hin he s anda d Landau’s Fe mi liquid heo y. Al hough he ield heo ies wi h known holo-
g aphic dual a e e y di e en om hose ound so a in Na u e, he e a e good easons o
belie e ha hese s udies could e eal gene ic uni e sal ea u es o s ongly in e ac ing quan um
sys ems [1].
In his pape we app oach his p oblem in a op-down model o in e sec ing b anes o di e -
en dimensionali ies. We will conside a s ack o Nccolo Dp-b anes which in e sec N la o
Dq-b anes (q≥p) along ncommon di ec ions. This con igu a ion, which we will deno e by
(n | p⊥q), is dual o a (p +1)-dimensional SU(Nc)gauge heo y wi h N undamen al hype -
mul iple s (qua ks) li ing on a (n +1)-dimensional de ec [2]. In he con ex o holog aphy, we
will wo k in he la ge Nc’ Hoo limi wi h N Nc. In his limi he qua ks a e quenched and
he Dq-b anes can be ea ed as p obes, whose ac ion is he Di ac–Bo n–In eld (DBI) ac ion, in
he g a i a ional backg ound c ea ed by he Dp-b anes. The embedding o he la o b anes is
pa ame e ized by a unc ion which measu es he dis ance be ween he wo ypes o b anes. The
ield heo y dual o his dis ance is he mass o he hype mul iple . Mo eo e , in o de o enginee
a sys em wi h non-ze o ba yonic cha ge densi y, we mus swi ch on a sui able gauge ield on he
wo ld olume o he la o b ane [3]. We will also s udy he in luence o a magne ic ield di ec ed
along wo o he spa ial di ec ions o he wo ld olume.
In [4] we s udied he collec i e exci a ions o gene ic b ane in e sec ions co esponding o
massless qua ks and we unco e ed a ce ain uni e sal s uc u e. The pu pose o his a icle is o
ex end he esul s o [4] o he case in which he qua ks ha e a non-ze o mass. We will s udy
i s he sys em a ze o empe a u e and non-ze o chemical po en ial. This is he so-called col-
lisionless quan um egime, in which he dynamics is domina ed by he ze o sound mode. This
mode is a collec i e exci a ion, i s ound in he holog aphic con ex in [5,6]. These esul s we e
gene alized o non-ze o empe a u e in [7,8] and o non- anishing magne ic ield in [9,10] (see
[11–28] o s udies on di e en aspec s o he holog aphic ze o sound). In [4] we de eloped a
gene al o malism which included all possible in e sec ions (n | p⊥q) and, in pa icula , we
ound an index λ(depending on n, p, and q) which de e mines he speed o ze o sound o mass-
less qua ks. This is in ima ely ela ed wi h he ac ha λde e mines he scaling dimension o
he cha ge densi y o o pu i sligh ly di e en ly, λac s as he poly opic index in he equa ion
o s a e o he holog aphic ma e .
In he case o massi e qua ks he embedding o he Dq-b ane is non- i ial and mus be
de e mined in o de o ex ac he di e en physical p ope ies. When he cha ge densi y is non-
anishing, he b ane eaches he ho izon o he geome y, i.e., we ha e a black hole embedding.
This embedding depends on a unc ion which pa ame e izes he shape o he la o b ane in he
backg ound geome y and, in gene al, mus be ound by nume ical in eg a ion o he equa ions
o mo ion o he p obe. Howe e , in he case o in e sec ions (n | p⊥q) which p ese e some
amoun o supe symme y a ze o empe a u e Tand chemical po en ial μsome ema kable sim-
pli ica ion occu s. Indeed, as shown in [29], in hese in e sec ions one can choose a sys em o
coo dina es such ha he embedding unc ion is a cyclic a iable o he DBI Lag angian when
T=0 and μ =0. As a consequence, he embedding unc ion and he physical p ope ies o he
con igu a ion, can be ound analy ically. In pa icula , one can s udy he ze o empe a u e he -
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 679
modynamics o hese sys ems and ind he speed o i s sound. This was done in e s. [12,13]
o he D3–Dqin e sec ions (3 | 3 ⊥7), (2 | 3 ⊥5), and (1 | 3 ⊥3). Mo eo e , by s udying he
quasino mal luc ua ion modes o he p obe, one can also compu e analy ically he speed o ze o
sound which, non- i ially, equals ha o he i s sound [8,11,30].
In his pape we gene alize hese esul s o any (n | p⊥q) in e sec ion wi h #ND =4, i.e.,
when n =(p +q−4)/2. These cases co espond o hose b ane in e sec ions which a e su-
pe symme ic in la space a low ene gies as he g a i a ional and Ramond–Ramond o ces
cancel ou . He e he index λcan only ake h ee di e en alues λ =2, 4, 6, co esponding o
codimension 2 (Dp–Dp), codimension 1 (Dp–D(p +2)), and codimension 0 (Dp–D(p +4)) in-
e sec ions, espec i ely. As in he con o mal D3-backg ound, he speeds o i s and ze o sounds
coincide. Mo eo e , we ind he same kind o uni e sali y as in he massless case: he speed is
he same o hose in e sec ions which ha e he same λindex, o codimension. Howe e , in he
massi e case he speed o sound depends con inuously on he chemical po en ial, i.e., on he
cha ge densi y, and anishes when he chemical po en ial eaches i s minimal alue, which co -
esponds o a anishing cha ge densi y d. Ac ually, as a gued in [30] o he D3–D7 and D3–D5
in e sec ions, he e is a quan um phase ansi ion as d→0 which exhibi s a non- ela i is ic scal-
ing beha io wi h hype scaling iola ion. A he ansi ion poin he black hole embeddings wi h
d=0degene a e in o a Minkowski embedding wi h ze o cha ge densi y. He e we will ind he
c i ical exponen s o he gene al #ND =4 in e sec ions, gene alizing he esul s o [30].
When he numbe no common dimensions o he colo and la o b anes is equal o wo,
he ma e hype mul iple s li e on a (2 +1)-dimensional heo y. In his case one can pe o m an
al e na i e quan iza ion o he quasino mal modes, which consis s in imposing mixed Di ichle –
Neumann bounda y condi ions a he UV. As shown in [31], his al e na i e quan iza ion amoun s
o ans o ming he cha ged exci a ions in o pa icles o ac ional s a is ics, i.e., anyons (see also
[28,32–34] o he analysis o di e en aspec s o he holog aphic anyonic sys ems). In [4] we
s udied he ze o sound mode as a unc ion o he cons an ha measu es he deg ee o mixing
he UV bounda y condi ions. We ound ha he anyonic ze o sound is gene ically gapped and
ha his gap can be ine- uned o ze o i a sui able magne ic ield is swi ched on. This choice
co esponds o he case, whe e he anyons expe ience no e ec i e magne ic ield. In his pape
we gene alize hese esul s o he case in which he qua ks a e massi e.
In his a icle we also s udy he hyd odynamic egime ha is eached when he empe a u e is
high enough. The dominan collec i e mode in his egime is a di usion mode, which has a pu ely
imagina y dispe sion ela ion cha ac e ized by a di usion cons an D. When he empe a u e is
non-ze o he embedding unc ion is no mo e a cyclic coo dina e o he DBI ac ion and canno
he e o e be ound analy ically. Thus, we s udy his T= 0 case by using nume ical me hods,
a e pe o ming a con enien change o a iables. Mo eo e , his nume ical analysis allow us o
check he analy ic esul s ound a ze o empe a u e, by aking he T→0limi . We also s udy
nume ically he sys em in he p esence o a magne ic ield B. We compa e he esul s o he
di usion cons an ob ained om he luc ua ion analysis a T=0 wi h he ones p edic ed by he
Eins ein ela ion, which gi es Din e ms o he DC conduc i i y σand he cha ge suscep ibili y
χ. Bo h σand χcan be ob ained om he embedding unc ion. We ind a e y good ag eemen
be ween he nume ical esul s o Dand he alue gi en by he Eins ein ela ion.
The es o his pape is o ganized as ollows. In sec ion 2we o mula e ou op-down holo-
g aphic model, sol e he equa ions o mo ion o he p obe a T=0 and μ =0, and s udy he
he modynamics a ze o empe a u e. In pa icula , in his sec ion we ind he speed o i s sound
and compu e he cha ge suscep ibili y a T=0. In sec ion 3we w i e he equa ions o mo ion o
he luc ua ions o he p obe a ze o empe a u e. In sec ion 4we analyze he ze o sound and ind
680 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
analy ically he dispe sion ela ion o his collec i e mode. Sec ion 5is de o ed o he s udy o
he scaling beha io nea he quan um c i ical poin . In sec ion 6we s udy he ze o sound mode
in an anyonic luid. Sec ion 7con ains ou esul s a non-ze o empe a u e and magne ic ield.
We summa ize ou esul s and discuss some possible u u e esea ch di ec ions in sec ion 8.
We complemen and gi e u he de ails o ou analysis in se e al appendices. Appendix A.1
con ains a de ailed de i a ion o he Lag angian o he luc ua ions a ze o empe a u e which
is used in sec ion 3. In Appendix A.2 we wo k ou he equa ions o mo ion o he luc ua ions
a T=0. In Appendix B we ind he co ela o o wo ans e se cu en s and ex ac he DC
conduc i i y in he absence o magne ic ield. Finally, in Appendix C we p o ide an al e na i e
de i a ion o he conduc i i y, alid also when B=0.
2. Massi e Dp–Dqsys ems wi h cha ge
Le us begin ou analysis by in oducing ou se up and s udying i s p ope ies a ze o empe -
a u e and magne ic ield. We will conside a gene ic Dp-b ane me ic a ze o empe a u e o he
ype:
ds2
10 =g ( ) d 2+gxx( ) (dx1)2+···+(dxp)2+g ( ) d y·dy, (2.1)
whe e y=(y1, ..., y9−p)a e he coo dina es ans e se o he Dp-b ane and he unc ions g ,
gxx, and g depend on he ans e se adial di ec ion =y·y. We now embed N Dq-b ane
p obes, wi h N Nc, ex ended along he di ec ions
( , x1,...,xn,y1,...,yq−n). (2.2)
We will e e o his con igu a ion as a (n | p⊥q) in e sec ion (nis he numbe o common
spa ial di ec ions o he Dpand Dq). This in e sec ion is ep esen ed by he a ay:
x1··· xnxn+1··· xpy1··· yq−nyq−n+1··· y9−p
Dp : × ··· × × ··· × − ··· − − ··· −
Dq : × ··· × − ··· − × ··· × − ··· −
We shall deno e by z he coo dina es y ans e se o he Dq-b ane:
z=(z1,...,z9+n−p−q), (2.3)
wi h zm=yq−n+m o m =1, ..., 9 +n −p−q. Mo eo e , we de ine ρas he adial coo dina e
o he subspace spanned by (y1, ..., yq−n):
ρ2=(y1)2+···+(yq−n)2.(2.4)
Le us make a sho commen on he global symme ies. The o iginal Dp-backg ound has
a o a ional symme y in he yidi ec ions, his co esponds o he SO(9 −p) R-symme y.
When we add N coinciden p obe Dq-b anes we in oduce U(N ) la o symme y. The
Dp–Dq-in e sec ion (n | p⊥q) b eaks he o iginal R-symme y, which can be easily ead o
om he isome ies. We end up wi h he global symme y SO(n, 1) ×U(N ) ×SO(p −n)p×
SO(q −n)q×SO(9 +n −p−q). The las g oup will be u he b oken when we conside
massi e Dq-b ane embeddings.
Since,
dy2=dρ2+ρ2d2
q−n−1+dz2,(2.5)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 681
he backg ound me ic in hese coo dina es can be w i en as:
ds2
10 =g ( ) d 2+gxx( ) (dx1)2+···+(dxn)2+(dxn+1)2+···+(dxp)2
+g ( ) dρ2+ρ2d2
q−n−1+dz2.(2.6)
Le us conside a s ack o Dq-b anes wi h a non- i ial p o ile in he ans e se space. We will
choose ou ans e se coo dina es in such a way ha his p o ile can be pa ame e ized as z=
(z1(ρ), 0, ..., 0). In wha ollows we jus w i e z(ρ) ins ead o z1(ρ) and we will deno e by
= (ρ) he unc ion:
(ρ) =ρ2+z(ρ)2.(2.7)
The induced me ic on he Dq-b ane wo ld olume a ze o empe a u e is:
ds2
q+1=g (ρ) d 2+gxx(ρ) (dx1)2+···+(dxn)2
+g (ρ)(1+z2)dρ2+ρ2d2
q−n−1,(2.8)
wi h z=dz/dρ. Le us compu e he DBI ac ion o he Dq-b ane in he case in which he e is a
wo ld olume gauge ield Fwi h componen s ρ . Thus, we will ake F o be gi en by:
F=A
dρ ∧d , (2.9)
whe e A
=∂ρA and we ha e chosen a gauge o Asuch ha Aρ=0. This means ha we aim o
s udy holog aphic ma e a non-ze o ba yon cha ge densi y by in oducing a chemical po en ial
o he diagonal U(1) ⊂U(N ). The DBI ac ion becomes:
SDq =−N TDq dq+1ξe
−φ−de (g +2παF)=d dnxdρLDBI ,(2.10)
wi h he Lag angian densi y LDBI gi en by:
LDBI =−Ne−φρq−n−1g
n
2
xx g
q−n−1
2
g |g |(1+z2)−(2πα)2A2
,(2.11)
whe e Nis he no maliza ion ac o
N=N TDq Vol(Sq−n−1), (2.12)
and whe e he ension o he Dq-b ane and he olume o he uni sphe e a e
TDq =1
(2π)q√αq+1gs
,Vol (Sq−n−1)=2πq−n
2
q−n
2.(2.13)
Fo a Dp-b ane backg ound a ze o empe a u e, he me ic and he dila on a e gi en by:
−g =gxx =
R7−p
2,g
=R
7−p
2,e
−2φ=R
(7−p)(p−3)
2.(2.14)
This backg ound sa is ies g |g | =1 and he Lag angian densi y LDBI can be w i en as:
LDBI =−Nρq−n−1
R(2n−p−q+4)(7−p)
41+z2−(2πα)2A2
.(2.15)
In he ollowing we will scale ou he cons an R, i.e., we will ake di ec ly R=1. To
a oid clu e , we also ede ine he gauge ield by abso bing he ac o s o he s ing leng h

682 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
2παAμ→Aμ. Mo eo e , we will es ic ou sel es o he case in which he embedding unc-
ion z(ρ) is a cyclic a iable, i.e., when LDBI depends on zand no on z. The only dependence
on zin (2.15) is he one induced by he powe o mul iplying he DBI squa e oo . The e o e
z(ρ) is cyclic only when he ollowing condi ion be ween n, p, and qis sa is ied:
n=p+q−4
2.(2.16)
One can check ha his happens only in he supe symme ic in e sec ions wi h #ND =4: (p | p⊥
p+4), (p −1 | p⊥p+2), and (p −2 | p⊥p). In he ollowing we will es ic ou sel es o
hese cases. Le us de ine λas:
λ=2(q −n−1)=q−p+2.(2.17)
No ice ha λ =6, 4, 2 o he in e sec ions Dp–D(p +4), Dp–D(p +2), and Dp–Dp, espec-
i ely. We can hen w i e he Lag angian densi y as:
LDBI =−Nρλ
21+z2−A2
.(2.18)
The cyclic na u e o zand A implies he ollowing conse a ion laws:
1
N
∂LDBI
∂z=− ρλ
2z
1+z2−A2
≡−c
1
N
∂LDBI
∂A
=ρλ
2A
1+z2−A2
≡d, (2.19)
wi h cand dbeing cons an s o in eg a ion. These ela ions can be in e ed as:
z=c
ρλ+d2−c2,A

=d
ρλ+d2−c2.(2.20)
When c=d=0, bo h z(ρ) and A (ρ) a e cons an and we ha e a Minkowski embedding. Le us
suppose ha cdoes no anish. Then, i ollows om (2.20) ha A
and za e ela ed as:
A
=d
cz.(2.21)
When c2=d2=0 bo h z(ρ) and A (ρ) di e ge a ρ=0. The e o e, we disca d his con igu a-
ion and we will assume in he ollowing ha d2>c
2. In his case, om he exp ession o zand
A
w i en in (2.20) i is easy o conclude ha he poin ρ=0is eached. In wha ollows we
will assume ha his condi ion holds. We will in eg a e he equa ion o A (ρ) by imposing ha
A (0) =0. We ha e:
A (ρ) =d
ρ

0
d¯ρ
¯ρλ+d2−c2.(2.22)
This in eg al can be compu ed analy ically and exp essed in e ms o he hype geome ic unc ion
as:
A (ρ) =d
d2−c21
2−1
λ
ρ
ρλ+d2−c21
λ
F1
λ,1
2+1
λ;1+1
λ;ρλ
ρλ+d2−c2.(2.23)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 683
Simila ly, he embedding unc ion z(ρ) can be w i en as:
z(ρ) =c
d2−c21
2−1
λ
ρ
ρλ+d2−c21
λ
F1
λ,1
2+1
λ;1+1
λ;ρλ
ρλ+d2−c2.(2.24)
No ice ha when d2>c
2 he b ane eaches he Poinca é ho izon o he me ic a ρ=z=0 and
we ha e a black hole embedding. The wo cons an s dand ca e ela ed o he cha ge densi y and
condensa e o he dual heo y, espec i ely.
2.1. Ze o empe a u e he modynamics
Le us i s conside he in e sec ions wi h λ >2. We also es ic o T=0, as a non-ze o
empe a u e no much can be said analy ically. In his case he unc ions A (ρ) and z(ρ) in
(2.23) and (2.24) app oach a cons an alue in he UV egion ρ→∞. Acco ding o he s anda d
AdS/CFT dic iona y, he la o chemical po en ial μis he UV alue o A :
μ=A (ρ →∞)=d
d2−c21
2−1
λ
F1
λ,1
2+1
λ;1+1
λ;1=d
d2−c21
2−1
λ
γ, (2.25)
whe e γis he cons an
γ=1
√π1
2−1
λ1+1
λ(2.26)
and we used he iden i y F(A, B; C; 1) =(C) (C−A−B)
(C−A) (C−B).
The mass pa ame e mo he embedding is de ined as m =z(ρ →∞). I ollows om (2.24)
ha :
m=c
d2−c21
2−1
λ
γ. (2.27)
Le us in e (2.25) and (2.27) and compu e cand din e ms o μand m. Fi s , we no ice ha :
μ2−m2=d2−c22
λγ2.(2.28)
Since d2≥c2, eq. (2.28) implies ha μ ≥m o he embeddings we a e conside ing. Mo eo e ,
om (2.28) we ge d2−c2as a unc ion o μand mand, using his esul in (2.25) and (2.27),
we ob ain
c=mγ−λ
2μ2−m2λ−2
4,d=μγ−λ
2μ2−m2λ−2
4.(2.29)
When λ >2, μ =min (2.29) co esponds o c=d=0, i.e., o he Minkowski embeddings
wi h anishing densi y discussed abo e. Ac ually, as illus a ed in Fig. 1, he opology o he
embeddings changes when m →μ, whe e a quan um phase ansi ion akes place. The o de
pa ame e o his ansi ion is he cha ge densi y (see [29] o u he de ails).
Le us now e alua e he on-shell ac ion o he p obe. Using
1+z2−A2
on-shell =ρλ
2
ρλ+d2−c2,(2.30)
684 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
Fig. 1. In his igu e we plo he di e en embeddings o λ =4and m/μ =0.1, 0.8, 0.999 (bo om-up). The Minkowski
embeddings a ze o densi y and m/μ =1 would co espond o he cons an ho izon al line z/m =1.
we ind
Son-shell =−N
∞

0
ρλ
ρλ+d2−c2dρ , (2.31)
which is di e gen and mus be egula ed. We will do i by sub ac ing he same in eg al wi h he
in eg and e alua ed a he UV (ρ→∞). We a i e a
S eg
on-shell =−N
∞

0
ρλ
2ρλ
2
ρλ+d2−c2−1dρ =2N
λ+2d2−c21
λ+1
2γ. (2.32)
The ze o- empe a u e g and canonical po en ial is gi en by minus he egula ed on-shell ac-
ion:
=−S eg
on-shell =−2N
λ+2d2−c21
λ+1
2γ. (2.33)
In e ms o mand μ he g and canonical po en ial can be w i en as:
=−2N
λ+2γ−λ
2μ2−m2λ+2
4,(2.34)
whe e we ha e used (2.28). Mo eo e , he cha ge densi y is:
ρch =−∂
∂μ =μNγ−λ
2μ2−m2λ−2
4=Nd, (2.35)
which con i ms ou iden i ica ion o he cons an d. I is wo h no ing ha he o mulas ha we
will w i e down do no ha e he ac o o he (in ini e) olume o he gauge heo y di ec ions VRn,
a he all he modynamic quan i ies a e densi ies pe uni olume. Nex , we compu e he ene gy
densi y as:
=+μρ
ch =N
(λ +2)γ−λ
2μ2−m2λ−2
4(λμ2+2m2). (2.36)
To calcula e he speed o i s sound uswe make use o he equa ion
u2
s=∂P
∂ =∂P
∂μ∂
∂μ−1
,(2.37)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 685
whe e Pis he p essu e. Le us i s compu e he de i a i e appea ing in he nume a o . Since
P=−, we ge om (2.34):
∂P
∂μ =μNγ−λ
2μ2−m2λ−2
4.(2.38)
Mo eo e , om (2.36) we ha e:
∂
∂μ =μNγ−λ
2μ2−m2λ−6
4λ
2μ2−m2.(2.39)
These yield
u2
s=2μ2−m2
λμ
2−2m2,(2.40)
which is he esul we we e looking o . As a check no ice ha (2.40) gi es u2
s=2/λ o m =0,
which is he uni e sal esul ound in [4].1Mo eo e , he speed o sound (2.40) depends on
he in ege s (n, p, q) h ough he combina ion λ, i.e., usis he same o con o mal and non-
con o mal b ane backg ounds wi h he same index λ. In pa icula , o he D3–D7 and D3–D5
supe symme ic in e sec ions we ha e:
u2
s=μ2−m2
3μ2−m2, o D3–D7 ,
u2
s=μ2−m2
2μ2−m2, o D3–D5 .(2.41)
These esul s ag ee wi h he calcula ion in [12,13]. No ice ha he speed o sound anishes in he
ze o densi y limi wi h μ =m, which is a clea sign o a quan um phase ansi ion.
Le us now conside he case λ =2, which co esponds o he (p −2|p⊥p) in e sec ions. In
hese sys ems A (ρ) and z(ρ) g ow loga i hmically when ρ→∞and he AdS/CFT dic iona y
mus be adap ed acco dingly. Indeed, in his case he chemical po en ial and he mass a e ob ained
om he subleading e ms o A and zin he UV. Mo eo e , he on-shell ac ion has addi ional
loga i hmic di e gences, which mus be elimina ed wi h new coun e e ms [35,36]. As he esul
o his analysis one ge s ha he g and canonical po en ial o black hole embeddings akes he
o m  =−a(μ2−m2), whe e ais a posi i e cons an [37]. Repea ing he calcula ion o us
pe o med abo e, i is s aigh o wa d o e i y ha u2
s=1in his λ =2 case. No ice ha his
alue is exac ly he one ob ained by aking λ =2in (2.40).
3. Fluc ua ions
We now allow luc ua ions o bo h he gauge ield along he Minkowski di ec ions o he
in e sec ion and o he scala unc ion in he o m:
Aν=A(0)
ν+aν(ρ, xμ), z=z0(ρ) +ξ(ρ,xμ), (3.1)
1Fo he massless #ND =4 in e sec ions one can ew i e he global symme y in a sugges i e o m: SO(n, 1) ×
SU(N ) ×U(1) ×SO(3 −λ/2)p×SO(1 +λ/2)q×SO(5 −n). The SO(1 +λ/2)qpa o a es a sphe e o λ/2
dimensions, which cu iously coincides wi h he alue o he speed o sound (2.40) o m =0.
692 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
A i s -o de in δω, we ge :
δω =−2
p−3
2(5−p) cp
λd
γ
(d2−c2)
6−p
5−p−1
λ
(λ d2−2c2)
7−p
2(5−p) +1k
7−p
5−p.(4.32)
In e ms o m his exp ession becomes:
δω =−2
p−3
2(5−p) cp
λ
μ
(1−m2)
6−p
5−p−1
2
(λ −2m2)
7−p
2(5−p) +1k
7−p
5−p,(4.33)
whe e we used he ollowing ela ion o μ, d, and m:
μ=γd2
λ(1−m2)1
λ−1
2.(4.34)
Le us use he exp ession o cpin (4.9) and sepa a e he imagina y and eal pa s:
Im δω =−πλ
μ
(5−p)
p−3
5−p
1
5−p22
p−3
2(5−p) (1−m2)
6−p
5−p−1
2
(λ −2m2)
7−p
2(5−p) +1k
7−p
5−p
Re δω =πλ
μ
(5−p)
p−3
5−p
1
5−p2co π
5−p2
p−3
2(5−p) (1−m2)
6−p
5−p−1
2
(λ −2m2)
7−p
2(5−p) +1k
7−p
5−p.(4.35)
In pa icula , o p=3 he eal pa o Re δω anishes a he o de we a e wo king in (4.35) and
he comple e dispe sion ela ion is gi en by:
ωp=3=±√21−m2
λ−2m21
2
k−iλ
μ
1−m2
(λ −2m2)2k2.(4.36)
In o de o compa e wi h he esul s in [8,11], le us subs i u e μby i s exp ession in e ms o he
densi y d(eq. (4.34)). We ind
ωp=3=±√21−m2
λ−2m21
2
k−iλ2
d2
λ
1
2
1
2−1
λ1
λ
(1−m2)3
2−1
λ
(λ −2m2)2k2.(4.37)
In pa icula , o he D3–D5 sys em we ake λ =4 and a i e a he ollowing dispe sion ela ion:
ωD3–D5=±1−m2
2−m21
2
k−i4
d1
2
1
2
1
42
(1−m2)5
4
(2−m2)2k2.(4.38)
In Fig. 2 we check (4.38) by compa ing i wi h he esul s ob ained by nume ical in eg a ion
a non-ze o (bu small) empe a u e. As i can be app ecia ed in his igu e, he ag eemen is e y
good, bo h o he speed o ze o sound csand o he a enua ion (i.e., he imagina y pa o ω).

G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 693
Fig. 2. We depic he speed o ze o sound (le ) and he a enua ion di ided by momen um squa ed ( igh ) o he D3–D5
in e sec ion. The do s ha e been ob ained by in eg a ing nume ically he luc ua ion equa ions (A.48) and (A.49) a
ex emely small empe a u e (we use ˆ
d=106; ˆ
d, ˆω, and ˆ
ka e de ined in (7.17)). The con inuous cu e co esponds o
he analy ic exp ession (4.38).
4.4. The p=4case
As poin ed ou a ound (4.8), he p=4 case is special and we ha e o modi y ou analysis.
Indeed, he expansion o he Hankel unc ion H(1)
1(x) nea x=0 con ains loga i hmic e ms,
which implies ha E(ρ) and ξ(ρ) beha e nea he ho izon a low equency as:
E(ρ) =Aρ+Ac
42+A
2log ρ
2+···
ξ(ρ) =Bρ+Bc
42+B
2log ρ
2+··· ,(4.39)
whe e c4is he cons an :
c4=iπ+1−2γE.(4.40)
In (4.40) γE=0.577 ···is he Eule –Masche oni cons an . Le us now y o ob ain he expansion
(4.39) by pe o ming he limi s in he opposi e o de . As in [12], we ha e o compu e he nex
co ec ion o (4.17) and (4.19) nea he ho izon. Fi s we no ice ha he equa ions sa is ied by
E(ρ) and ξ(ρ) nea ρ=0a
e jus ob ained by aking p=4in (4.5):
E =−2
ρ3E, ξ
 =−2
ρ3ξ. (4.41)
Neglec ing he igh -hand side in (4.41) and in eg a ing wice, we a i e a a linea solu ion as
in (4.17) and (4.19). To go beyond his app oxima ion we plug he alues o Eand ξin o he
igh -hand side o (4.41) and pe o m he in eg a ion. In he low- equency limi ω2ρ, we
ha e:
E(ρ) =E(0)+b1C1+b2C2+(a1C1+a2C2)ρ +(a1C1+a2C2)2log ρ+···
ξ(ρ) =ξ(0)+˜
b1C1+˜
b2C2+(˜a1C1+˜a2C2)ρ +(˜a1C1+˜a2C2)2log ρ+··· .(4.42)
Le us now ma ch (4.39) and (4.42). By compa ing he linea and loga i hmic e ms o hese
equa ions we a i e a he same alues o Aand Bas hose w i en in (4.21). Mo eo e , using
hese alues o Aand Band iden i ying he cons an e ms, we ind he ollowing ma ix ela ion
be ween (E(0), ξ(0))and (C1, C2):
694 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
E(0)
ξ(0)=2(c4−log 2)a
1−b12(c4−log 2)a
2−b2
2(c4−log 2)˜a1−˜
b12(c4−log 2)˜a2−˜
b2C1
C2.(4.43)
As in he p<4 case, he sou ces anish non- i ially when he de e minan o he ma ix w i en
in (4.43) is ze o, namely:
(a
1˜a2+a2
2)
2(c4−log 2)2−(a1˜
b2+˜a2b1+2a2b2)
2(c4−log 2)
+b1˜
b2+b2
2=0.(4.44)
No ice ha (4.44) is ob ained om (4.23) by aking p=4 and changing cp→c4−log 2on
he la e . Using his obse a ion i is s aigh o wa d o ind he dispe sion ela ion encoded
in (4.44). A leading o de in ω∼k(4.44) educes o (4.25), which means ha he leading
dispe sion ela ion is jus gi en by (4.29) and (4.30). Mo eo e , he nex - o-leading con ibu ion
δω is:
δω =−√2λ
μ(c4−log 2)(1−m2)3
2
(λ −2m2)5
2
k3.(4.45)
The imagina y pa o δω is easily deduced om (4.45):
Im δω =−π√2λ
μ
(1−m2)3
2
(λ −2m2)5
2
k3.(4.46)
No ice ha (4.46) is he same as in he i s equa ion in (4.35) o p=4. Simila ly, he eal pa
o δω can be w i en as:
Re δω =√2λ
μ2γE−1+log 2(1−m2)3
2
λ−2m2k2 (1−m2)3
2
(λ −2m2)5
2
k3.(4.47)
4.5. The λ =2case
Fo λ =2 he in eg al J1(ρ), de ined in (4.13), is no con e gen and, he e o e, he exp es-
sions w i en in (4.15) o E(ρ) and ξ(ρ) a low equency a e no co ec . In o de o ob ain he
solu ion o (4.12) o λ =2, le us de ine he in eg al ¯
J1(ρ) as:
¯
J1(ρ) ≡
∞

ρ
d¯ρ¯ρ2
(¯ρ2+d2−c2)3
2−1
¯ρ=ρ
ρ2+d2−c2−1+log 2ρ
ρ2+d2−c2+ρ
.
(4.48)
Then, (4.12) o λ =2 can be in eg a ed as:
E(ρ) =E(0)−(ω2−k2)C
1[¯
J1(ρ) −log ρ]
−[(k2−ω2)c2+ω2d2]C1−cd k C2J2(ρ)
ξ(ρ)=ξ(0)+C2[¯
J1(ρ) −log ρ]+ddC
2−ckC
1J2(ρ) , (4.49)
whe e E(0)and ξ(0)a e cons an s. When ρis e y la ge he in eg als ¯
J1(ρ) and J2(ρ) anish by
cons uc ion and hus E(ρ) and ξ(ρ) beha e a he UV as:
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 695
E(ρ) =E(0)+(ω2−k2)C
1log ρ+···
ξ(ρ) =ξ(0)−C2log ρ+··· ,(ρ→∞). (4.50)
As a gued in [35], when he loga i hmic beha io displayed in (4.50) is p esen , he sou ces
a e iden i ied wi h he coe icien s o he loga i hms, which should anish. I is clea om he
beha io o ξ(ρ) in (4.50) ha we mus equi e ha C2=0. Mo eo e , he loga i hmic e m in
E(ρ) is absen ei he when C1=0o when:
ω=±k. (4.51)
I C1=C2=0i ollows om (4.49) ha he unc ions E(ρ) and ξ(ρ) a e cons an and also he
ma ching wi h he nea -ho izon esul s in (4.8) imply ha bo h Eand ξmus anish. The e o e,
he only non- i ial solu ion is gi en by he dispe sion ela ion (4.51), which co esponds o a
ze o sound mode wi hou dissipa ion and speed c2
s=1. No ice ha his esul coincides wi h
he alue o he speed o i s sound in (2.40) o λ =2. Mo eo e , when C2=0 and ω2=k2,
eq. (4.49) educes o:
E(ρ) =E(0)−ω2d2C1J2(ρ) , ξ(ρ) =ξ(0)−cdkC
1J2(ρ) . (4.52)
Taking ρ→0in (4.52) we can ma ch his esul wi h (4.8) and, as a consequence, we can show
ha E(0)and ξ(0)a e ela ed o he cons an C1as:
E(0)=C1
d2
d2−c2ω2+cpC1
d
d2−c2ω
2(6−p)
5−p
ξ(0)=C1
cd
d2−c2k+cpC1
c
d2−c2kω 2
5−p.(4.53)
No ice ha (4.53) coincides wi h (4.22) when λ =2, C2=0 and ω2=k2. In pa icula , hese
ela ions imply ha he a io o E(0)and ξ(0)is gi en by:
E(0)
ξ(0)=d
ck. (4.54)
The analysis pe o med so a in his sec ion is alid o p<4. When p=4we ha e o go
beyond he leading e m in ω, as in sec ion 4.4, in o de o ma ch he loga i hmic e ms in he
nea -ho izon expansion. I is easy o check ha he λ =2 solu ion w i en abo e can be co ec ed
o ma ch he ρ→0expansion in (4.39). The dispe sion ela ion is s ill gi en by (4.51) and (4.53)
con inues o hold in his case.
5. Hype scaling iola ion nea he c i ical poin
As al eady men ioned, he p obe D-b ane sys ems analyzed abo e unde go a quan um phase
ansi ion as μ →mand he densi y d anishes. I was shown in [30] ha he c i ical poin s
o he D3–D7 and D3–D5 in e sec ions a e desc ibed by a non- ela i is ic scale in a ian ield
heo y exhibi ing hype scaling iola ion. In his sec ion we ex end hese esul s o he case o
non-con o mal backg ounds (i.e., o p= 3) and we compu e he co esponding c i ical expo-
nen s.
Le us hus ollow he app oach o [30] and s udy he beha io o he sys em nea he quan um
c i ical poin a μ =m. Acco dingly, we conside a chemical po en ial o he o m:
μ=m+¯μ, (5.1)
696 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
whe e ¯μis conside ed o be small. A leading o de in ¯μwe can expand he di e en he mody-
namic unc ions o (2.34), (2.36), and (2.29) as:
=−P≈−2λ+6
4
λ+2γ−λ
2Nm¯μ)λ+2
4
= ≈2λ−2
4γ−λ
2Nmλ+6
4¯μλ−2
4
d≈2λ−2
4γ−λ
2mλ+2
4¯μλ−2
4,(5.2)
whe e is he ee ene gy densi y. The non- ela i is ic ene gy densi y eis de ined as in [30]:
e=−ρch m=−Ndm, (5.3)
whe e ρch =Ndis he physical cha ge densi y. By using (2.36) and (2.29) we ge :
e=Nγ−λ
2μ2−m2λ−2
4λμ2+2m2
λ+2−μm.(5.4)
Expanding a leading o de in ¯μ, we a i e a :
e≈2λ−2
4λ−2
λ+2Nγ−λ
2m¯μλ+2
4.(5.5)
Compa ing his esul wi h he one o he p essu e in (5.2), we ob ain he ollowing ela ion
be ween eand P:
e=λ−2
4P. (5.6)
Acco ding o he analysis in [30], he ela ion be ween eand Pa ze o empe a u e nea he
quan um c i ical poin is:
e=n−θ
zP, (5.7)
whe e θis he hype scaling iola ion exponen and zis he dynamical c i ical exponen . Eq. (5.7)
is a consequence o he scaling dimensions o e, P, ¯μ, and d, namely: [e] =[P] =z+n −θ,
[¯μ] =z, and [d] =n −θ. Thus, in ou case we ha e he ollowing ela ion be ween θand z:
θ=n−λ−2
4z. (5.8)
No ice ha he ela ion (5.8) be ween θand zcoincides wi h he ones ound in [30] o he D3–D7
sys em ( aking n =3 and λ =6) and o he D3–D5 in e sec ion ( aking n =2 and λ =4). In
o de o de e mine zwe look a he speed o sound (2.40) o μ ≈m. A i s -o de in ¯μi is
gi en by:
u2
s≈4
λ−2¯μ
m,(5.9)
and he co esponding dispe sion ela ion is:
ω≈4
λ−2¯μ
mk. (5.10)
Ma ching he scaling dimensions o bo h sides o (5.10) as in [30], using ha [ω] =zand [k] =1,
we conclude ha :
z=2.(5.11)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 697
The e o e θ akes he alue:
θ=n−λ
2+1.(5.12)
Taking in o accoun ha o he SUSY Dp–Dqin e sec ions we a e conside ing
n=p+q−4
2,λ=q−p+2,(5.13)
we can ew i e he exp ession o θsimply as:
θ=p−2.(5.14)
No ice ha o a D3–Dqin e sec ion he p e ious o mula gi es θ=1, in ag eemen wi h [30].
Eq. (5.14) is he gene aliza ion o his esul o any p.
Le us now conside he sys em a ini e empe a u e T. Acco ding o he analysis o [38],
when Tis small he ee ene gy densi y can be app oxima ed as:
(μ,m,T)= (μ,m,T =0)+πρ
ch T+O(T 2). (5.15)
Then, he non- ela i is ic ee ene gy densi y is gi en by:
non- el(μ,m,T)= (μ,m,T)−ρch m=e+πρ
ch T+O(T 2). (5.16)
A leading o de in ¯μwe ha e:
non- el(μ,m,T)=2λ−2
4λ−2
λ+2Nγ−λ
2m¯μλ+2
41+πλ+2
λ−2
T
¯μ+OT
¯μ2.(5.17)
In he quan um c i ical egion he non- ela i is ic ee ene gy densi y should scale as:
non- el ∼¯μ2−αgT
¯μνz ,(5.18)
whe e αis he exponen which cha ac e izes he scaling o he speci ic hea capaci y Cand νis
he exponen co esponding o he co ela ion leng h ξ(i.e., C∼(T −Tc)−αand ξ∼(T −Tc)−ν
nea a phase ansi ion a T=Tc). Compa ing (5.18) and (5.17) i ollows ha , in ou case, we
ha e:
2−α=λ+2
4,νz=1.(5.19)
Since z=2 o ou sys em, he exponen s αand νa e:
α=6−λ
4,ν=1
2.(5.20)
Using he exp ession o λin e ms o pand qw i en in (5.13), we can ecas αsimply as:
α=1−q−p
4.(5.21)
These esul s again coincide wi h he ones in [30] o he D3–D7 and D3–D5 in e sec ions.
Rema kably, he exponen s ob ained abo e sa is y he hype scaling- iola ion ela ion:
(n +z−θ)ν =2−α. (5.22)

698 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
6. Ze o sound in al e na i e quan iza ion
In his sec ion we will es ic ou sel es o he s udy o in e sec ions which a e (2 +1)-di-
mensional. In his case one can impose mixed Di ichle –Neumann bounda y condi ions o he
luc ua ion modes, i.e., one can adop an al e na i e quan iza ion scheme [39,40]. The equa ions
o mo ion a e he same o di e en quan iza ions, only he bounda y condi ions in he UV a e
di e en . On he dual ield heo y side his co esponds o ha ing an anyonic luid [31–34]. Le
us impose he ollowing bounda y condi ion a he UV:
lim
ρ→∞ nρλ
2 ρμ−1
2μαβ αβ =0,(6.1)
whe e nis a cons an ha cha ac e izes he bounda y condi ion ( he no mal quan iza ion condi-
ion conside ed so a co esponds o n =0). As in [4], i is s aigh o wa d o p o e ha (6.1) is
equi alen o equi e:
lim
ρ→∞ E=−inlim
ρ→∞ ρλ
2a
y,lim
ρ→∞ ay=in
ω2−k2lim
ρ→∞ ρλ
2E.(6.2)
No ice ha , e en i he equa ions o mo ion (3.13) and (3.14) o Eand aya e decoupled, he
mixed bounda y condi ions (6.2) in oduce a coupling be ween hem. The e o e, o implemen
(6.2) we ha e o s udy he equa ion o mo ion o ay, w i en in (3.14). Nea he ho izon ρ≈0
his equa ion educes o:
a
y+2
ρ7−pay=0,(6.3)
which is jus he same as (4.5). Fo p<5 he solu ion o (6.3) is gi en by he igh -hand-side
o (4.7). Mo eo e , o p<4 his solu ion beha es o low equencies as:
ay(ρ) =Cρ+Cc
p2
5−p+··· ,(p<4), (6.4)
wi h Cbeing a cons an . We now pe o m he wo limi s in he opposi e o de . Fo low equencies
(3.14) educes o:
∂ρρλ+d2−c2a
y=0,(6.5)
whose in eg a ion is s aigh o wa d:
ay(ρ) =a(0)
y−C3J3(ρ) , (6.6)
whe e a(0)
y=ay(ρ →∞), C3is a cons an o in eg a ion, and J3(ρ) is he ollowing in eg al
( o λ >2):
J3(ρ) =
∞

ρ
d¯ρ
(¯ρλ+d2−c2)1
2=2
λ−2ρ1−λ
2F1
2,1
2−1
λ;3
2−1
λ;−d2−c2
ρλ.(6.7)
Le us now expand ay(ρ) in powe s o ρ. Fi s , one can check ha , o small ρ, he in eg al J3(ρ)
can be app oxima ed as:
J3(ρ) ≈μ
d−ρ
√d2−c2,(6.8)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 699
whe e μis he chemical po en ial (2.25). The e o e, o small ρ, aycan be app oxima ed as:
ay(ρ) ≈a(0)
y−C3
μ
d+C3
ρ
√d2−c2.(6.9)
Le us now ma ch (6.4) and (6.9). F om he linea e ms, we ge he ollowing ela ion be ween
he cons an s Cand C3:
C=C3
√d2−c2.(6.10)
Using his ela ion, and iden i ying he cons an e ms in (6.4) and (6.9), we ge he ollowing
ela ion be ween a(0)
yand C3:
a(0)
y=μ
d+cp
√d2−c22
5−pC3.(6.11)
Le us now ew i e he bounda y condi ions (6.2) a low equency and momen um. F om he
exp essions o Eand ayin his egime (eqs. (4.12) and (6.6)), we conclude ha hey beha e in
he UV as:
Eρ→∞ ≈(ω2−k2)ρ−λ
2C1,a

yρ→∞ ≈ρ−λ
2C3.(6.12)
Taking his in o accoun , we can ecas he bounda y condi ions o he al e na i e quan iza ion
as a ela ion be ween he cons an s E(0), a(0)
y, C2, and C3. Indeed, le us de ine E(0)
nand a(0)
y,nas:
E(0)
n≡E(0)+inC3,a
(0)
y,n=a(0)
y−inC1.(6.13)
Then, (6.2) is equi alen o he condi ions:
E(0)
n=a(0)
y,n=0.(6.14)
The UV alues E(0)
n, ξ(0), and a(0)
y,ncan be ela ed o he cons an s C1, C2, and C3. In ma ix
o m his ela ion becomes:
⎛
⎜
⎝
E(0)
n
ξ(0)
a(0)
y,n
⎞
⎟
⎠=⎛
⎜
⎜
⎝
2
5−pcpa1−b12
5−pcpa2−b2in
2
5−pcp˜a1−˜
b12
5−pcp˜a2−˜
b20
−in0μ
d+cp
d2−c22
5−p
⎞
⎟
⎟
⎠⎛
⎝
C1
C2
C3⎞
⎠,(6.15)
whe e a1, a2, b1, b2and ˜a1, ˜a2, ˜
b1, ˜
b2a e gi en in (4.18) and (4.20), espec i ely. To ha e a
non- i ial solu ion o he condi ion E(0)
n=ξ(0)=a(0)
y,n=0we mus equi e ha he de e minan
o he ma ix in (6.15) be ze o. This leads o:
2
5−pcpa1−b12
5−pcp˜a2−˜
b2−2
5−pcpa2−b22
5−pcp˜a1−˜
b1
×μ
d+cp
√d2−c22
5−p+n2˜
b2−2
5−pcp˜a2=0.(6.16)
A leading o de in equency and momen um his equa ion simpli ies as:
b1˜
b2+b2
2+dn2
μ˜
b2=0.(6.17)
700 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
Fig. 3. We plo he dispe sions in he D3–D5 model (p =3, λ =4). In bo h plo s he ed poin s s and o nume ical esul s
( he nume ics we e pe o med a ex emely small empe a u e, ha is o alues o ˆ
d=106and ˆ
B=3 ·103in oduced
la e in (7.17)) whe eas he blue cu es a e he analy ic om (6.23); we emphasize ha he analy ic esul (6.23) is an
educa ed guess, bu ep oduces he nume ics p ecisely. (le ) We a y he quan iza ion pa ame e n =0, 1
2nc i , nc i
( op-down) a ixed m
μ=0.5. ( igh ) The quan iza ion pa ame e is chosen o be c i ical n =nc i . Di e en lines co -
espond o a ying m
μ=0.1, 0.5, 0.8 ( op-down). (Fo in e p e a ion o he e e ences o colo in his igu e legend, he
eade is e e ed o he web e sion o his a icle.)
Since:
b1˜
b2+b2
2=γ2
λd2−c22
λ−22(d2−c2)k2−(λd2−2c2)ω2
0,(6.18)
hen (6.17) implies he ollowing gapped dispe sion ela ion:
ω2=ω2
0=2(d2−c2)
λd2−2c2k2+dn
μ2
.(6.19)
In e ms o he educed mass pa ame e m, de ined in (4.27), we ha e
ω2
0=21−m2
λ−2m2k2+dn
μ2
.(6.20)
One can also calcula e he nex o de e m in he dispe sion ela ion. Indeed, one can check ha
ω=ω0+δω, whe e δω is gi en by:
δω =−2
p−3
2(5−p) cp
λ
μ
(1−m2)
6−p
5−p−1
2
(λ −2m2)
7−p
2(5−p) +1k
7−p
5−p−n2
λ−2
cp
kμ3γ
μλ
2ω
p−3
5−p
0
(1−m2)1
2+λ
4
.(6.21)
I was no iced in [4] o he massless embeddings ha he e ec o he al e na i e quan iza ion
is equi alen o swi ching on a magne ic ield a e sing he x1x2plane. Ac ually, i was ound
in [4] ha he e ec o a magne ic ield Be ec i ely changes he pa ame e nas
n→n−B
d.(6.22)
In he p esen massi e case we canno e i y analy ically he subs i u ion ule (6.22) since he
embedding unc ion z(ρ) is no a cyclic a iable in he p esence o a B ield. The e o e, we con-
jec u e ha he dispe sion ela ion o he ze o sound wi h gene al anyonic bounda y condi ions
and magne ic ield is gi en (a leading o de ) by:
ω2
0=21−m2
λ−2m2k2+1
μ2dn−B2.(6.23)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 701
Thus, he spec um is gene ically gapped o non- anishing Band n. Howe e , i can be made
gapless by adjus ing he al e na i e quan iza ion pa ame e n o he c i ical alue:
nc i ≡B
d.(6.24)
This pa icula case co esponds o one, whe e he anyonic luid expe iences ze o ne e ec i e
magne ic ield, hus he esul ing spec um is also gapless. In Fig. 3 we compa e he esul s ob-
ained om he nume ical in eg a ion o he luc ua ion equa ions o ou analy ic o mula (6.23).
We see ha he ag eemen is e y good and, in pa icula , he nume ics con i m ha he spec um
becomes gapless a n =nc i .
7. Fini e empe a u e
Le us now conside he Dp–Dqin e sec ions (n | p⊥q) a non-ze o empe a u e and mag-
ne ic ield. Fi s , we in oduce a mo e con enien sys em o coo dina es. Le us ep esen he
di e en componen s o he Ca esian coo dina es y ans e se o he Dp-b ane as:
ym= cos θη
m,m=1,···,q−n,
yl= sin θξl,l=q−n+1,···,9−p, (7.1)
whe e ηmand ξlsa is y:
q−n

m=1ηm2=
9−p

l=q−n+1ξl2=1.(7.2)
Clea ly, he ηm(ξl) a e he coo dina es o a (q −n −1)-sphe e ((8 +n −p−q)-sphe e). As:
9−p

l=q−n+1yl2= 2sin2θ,
q−n

m=1ym2= 2cos2θ, (7.3)
we iden i y he coo dina es zand ρused so a wi h:
z= sin θ, ρ= cos θ. (7.4)
I is s aigh o wa d o check ha
dy·dy=d 2+ 2dθ2+cos2θd
2
+sin2θd
2
⊥,(7.5)
whe e d2
=d2
q−n−1is he line elemen o he (q −n −1)-sphe e o he Dq-b ane wo ld-
olume and d2
⊥=d2
8+n−p−qis he me ic o he (8 +n −p−q)-sphe e ans e se o he
Dq-b ane. The en-dimensional me ic o a black Dp-b ane in hese coo dina es is:
ds2
10 =
R7−p
2− p( ) d 2+dx2
+R
7−p
2d 2
p( ) + 2dθ2+cos2θd
2
+sin2θd
2
⊥,(7.6)
whe e Ris a cons an adius and he blackening ac o pis:
p( ) =1− h
7−p
,(7.7)
708 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
luc ua ion equa ions couple he ans e se and longi udinal modes when B=0 and i is no clea
o us how o deal wi h his coupling. Fo his eason we ha e compu ed σby applying he me hod
o e . [41]. The de ails o his calcula ion a e explained in Appendix C. The inal esul o σis:
σ=N λ
h(1+ p−7
hB2)(cos θh)λ+d2
7−p
h+B2
7−p
2
h.(7.40)
I is now s aigh o wa d o w i e down he exp ession o Dwhich ollows om (7.30). Indeed,
le us de ine Bas:
B=H(cos θ)λ(1+ 2 p˙
θ2)
d2+H(cos θ)λ,(7.41)
whe e His he quan i y de ined in (7.12). Then, he Eins ein ela ion gi es he ollowing alue
o he di usion cons an :
D= λ
h(1+ p−7
hB2)(cos θh)λ+d2
7−p
2
h+
p−7
2
hB2
∞

h
d B
Hcos θλ
2
d2+Hcos θλ
×1+dλ
2 an θ∂θ
∂d + 2 p˙
θ
B
∂˙
θ
∂d.(7.42)
In Fig. 4 we compa e he p edic ions o (7.42) o he D2–D6 model and he nume ical esul s
ob ained by di ec in eg a ion o he coupled luc ua ion equa ions (A.48)–(A.50). As can be
app ecia ed in his igu e, he ag eemen be ween he wo me hods is e y good.
8. Summa y and conclusions
In his pape we s udied he collec i e exci a ions o la o Dq-b anes in he supe g a i y
backg ound gene a ed by colo Dp-b anes. The wo se o b anes a e sepa a ed in hei ans e se
di ec ions, which co esponds o adding massi e la o s in he dual ield heo y. We i s s udied
his Dp–Dqmodel a T=0and μ =0in he quenched app oxima ion. The non-ze o chemical
po en ial is gene a ed by a sui able wo ld olume gauge ield on he p obe. We hen gene alized
hese esul s o T=0 and non- anishing magne ic ield.
A ze o empe a u e and non- anishing chemical po en ial he supe symme ic Dp–Dqin-
e sec ions wi h #ND =4 can be s udied analy ically. We ob ained hei he modynamics and
i s and ze o sound, gene alizing p e ious esul s in he li e a u e o he con o mal cases wi h
p=3. These esul s allow o cha ac e ize he quan um phase ansi ion ha occu s when μ =m
and d=0. In his poin se e al he modynamic quan i ies anish and he sys em displays a
non- ela i is ic scaling beha io wi h hype scaling iola ion. We ha e been able o compu e he
co esponding c i ical exponen s.
We also analyzed he massi e la o b ane sys ems a non-ze o empe a u e and magne ic
ield. We e i ied nume ically ha , when he magne ic ield is non- anishing, he ze o sound
spec um becomes gapped, wi h he gap gi en by B/μ. Mo eo e , when Tis la ge enough he
sys em en e s in o a hyd odynamic egime, which is domina ed by a di usion mode. We de e -
mined nume ically he co esponding di usion cons an and e i ied he alidi y o he Eins ein
ela ion.

G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 709
When he in e sec ion is (2 +1)-dimensional we pe o med an al e na i e quan iza ion o he
luc ua ions, which co esponds o adding deg ees o eedom wi h ac ional s a is ics (anyons).
In hose sys ems he ze o sound is gene ically gapped, al hough i becomes gapless i he mag-
ne ic ield is chosen app op ia ely. In ac , his choice co esponds o a luid o anyons expe-
iencing ze o e ec i e magne ic ield, hus he occu ence o gapless mode was expec ed. Ou
unde s anding o he anyonic luid is s ill lacking, hough. In o de o desc ibe i s p ope ies be -
e one would need o make a de ini e choice o he SL(2, Z) ans o ma ion as his is needed
o make an iden i ica ion o he esul ing cha ge densi y o he anyons. Mo eo e , as he e is a
esidual gauge eedom in adding bounda y e ms o he ac ion, he calcula ion o he ee en-
e gy depends c ucially on he chosen SL(2, Z) ans o ma ion. The a ia ional p inciple is s ill
well-de ined, which allowed us in he cu en analysis o in es iga e he anspo p ope ies and
collec i e phenomena o he anyon luid in e ms o he s a is ics, p opo ional o he quan iza ion
pa ame e n.
The e a e se e al o he open opics which dese e u he in es iga ion. The Dp-b ane me -
ics wi h p= 3 iola e hype scaling [42] wi h θ=−(p −3)2/(5 −p). I would be wo h o
explo e he ela ion be ween his scaling o he backg ound and he one ound abo e o he
p obe. Ano he in e es ing p oblem o he u u e would be he analysis o mo e gene al Dp–Dq
in e sec ions. Con a y o he supe symme ic cases s udied he e, he massi e embeddings o a
gene al Dp–Dqmodel a e gene ically uns able and one mus u n on luxes on he wo ld olume
o he p obe o s abilize hem (see, o example [43–45]). These addi ional wo ld olume gauge
ields gi e an impo an con ibu ion o he Wess–Zumino e m o he p obe ac ion.3I would be
e y in e es ing o de elop a gene al o malism o he collec i e exci a ions o he p obe b ane
which could inco po a e all he pa icula cases s udied in he li e a u e.
I would also be in e es ing o analyze he sys ems in which he backg ounds a e no gene a ed
by b anes in la space. Le us men ion he cases o b anes on he coni old (as in he Klebano –
Wi en model [48]) and he ABJM model [49]. Since he massi e embeddings depend on he
pa icula model, i is expec ed ha he esul s will no be comple ely uni e sal. I is in e es ing,
howe e , o de e mine he ea u es common o all he cases.
The collec i e exci a ions o b ane in e sec ions analyzed so a in he li e a u e ha e been
ca ied ou in he p obe app oxima ion. The e o e, i is qui e na u al o explo e he e ec s on he
esul s o ha ing dynamical qua ks. In o de o p o ide an answe o his p oblem we need o ha e
supe g a i y backg ounds which include he back eac ion o he la o b anes. By employing
di e en app oxima ions, hese backg ounds can be ound o some sys ems. Le us men ion he
case o ABJM wi h smea ed la o b anes [50–53], which a e geome ies ee o pa hologies,
al hough hey do no inco po a e he e ec o non-ze o densi y. This e ec is included in he
geome y ecen ly ound in [54], which is dual o h ee-dimensional supe Yang–Mills heo y
wi h comp essible ma e . In he nea u u e we in end o s udy he collec i e exci a ions o he
la o b anes o some o hese sys ems.
Acknowledgemen s
We hank Yago Bea and Ca los Hoyos o discussions and c i ical eadings o he manusc ip .
N.J. is suppo ed by he Academy o Finland G an No. 1268023. A.V.R. and G.I. a e unded
3An in e es ing al e na i e iewpoin wi hou luxes is discussed in [46,47]. In his con ex oo, howe e , one would
need o ake o he Wess–Zumino e ms in o accoun ( oge he wi h modi ying he UV asymp o ics) and ou esul s a e
no di ec ly applicable.
710 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
by he MINECO and FEDER g an FPA2014-52218-P, by he Consolide -Ingenio 2010 P o-
g amme CPAN (CSD2007-00042), by Xun a de Galicia (GRC2013-024). G.I. is also unded by
FPA2012-35043-C02-02.
Appendix A. Fluc ua ion equa ions o mo ion
In his appendix we ob ain he Lag angian densi y, and he co esponding equa ions o mo ion,
o he luc ua ions o he embedding scala and he gauge ields a non- anishing cha ge densi y
d=0 and magne ic ield B=0. As was he case o he backg ound equa ions, i is use ul o
ea he analysis o T=0and T=0using di e en pa ame iza ion.
A.1. Fluc ua ions a ze o empe a u e
In his subsec ion we ocus on T=0 case. Le us conside a luc ua ion o he gauge ield and
embedding as in (3.1) and (3.2). The induced me ic g akes he o m:
g=¯g+ˆg, (A.1)
whe e ¯gis he ze o h-o de me ic and ˆgis he pe u ba ion. Le us spli ˆgin he o m:
ˆg=ˆg(1)+ˆg(2).(A.2)
The non-ze o elemen s o ˆg(1)a e:
ˆg(1)
ρxμ=z
0
7−p
2
∂μξ, ˆg(1)
ρρ =2z
0
7−p
2
∂ρξ, (A.3)
whe eas ˆg(2)has he o m:
ˆg(2)
ab =1
7−p
2
∂aξ∂
bξ(A.4)
(we a e aking he adius R=1in (2.14)). In o de o expand he DBI Dq-b ane ac ion we no ice
ha he Bo n–In eld de e minan can be w i en as:
−de (g +F)=−de ¯g+F(0)de (1+X) , (A.5)
whe e he ma ix Xis gi en by:
X≡¯g+F(0)−1ˆg+ .(A.6)
To e alua e he igh -hand side o eq. (A.5), we shall use he expansion:
de (1+X) =1+1
2T X−1
4T X2+1
8T X2+O(X3). (A.7)
Mo eo e , in he in e se ma ix ¯g+F(0)−1we will sepa a e he symme ic and an isymme ic
pa s:
¯g+F(0)−1=G−1+J,(A.8)
whe e Jis he an isymme ic componen and he symme ic ma ix Gis he open s ing me ic.
The ele an componen s o Ga e:
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 711
G =− ¯g (1+z2
0)
¯g |¯g |(1+z2
0)−A(0)2
,Gxixj=δij
¯gxx
Gρρ =− ¯g
¯g |¯g |(1+z2
0)−A(0)2
.(A.9)
Using he ac ha ¯g |¯g | =1, and elimina ing z
0and A(0)
, we ge :
G =−ρλ+d2
|¯g |ρλ=− ρλ+d2
(ρ2+z2
0)7−p
4ρλ
Gρρ =ρλ+d2−c2
¯g ρλ=(ρ2+z2
0)7−p
4ρλ+d2−c2
ρλ
Gxixj=δij
(ρ2+z2
0)7−p
4
,(A.10)
which a e jus he componen s w i en in (3.3). The elemen s o he an isymme ic ma ix Ja e:
J ρ =−Jρ =− A(0)
¯g |¯g |(1+z2
0)−A(0)2
=−dρλ+d2−c2
ρλ.(A.11)
By explici calcula ion one can e i y ha T Xis gi en by:
T X=2z
0
7−p
2
Gρρ ∂ρξ+2J ρ ρ +Gab
7−p
2
∂aξ∂
bξ, (A.12)
while T X2is:
T X2=−Gac Gbd cd ab +Gac Gbd ˆg(1)
ab ˆg(1)
cd
+2(J ρ)2(ˆg(1)
ρ )2+( ρ)2−4J ρ Gab ˆg(1)
ρa b .(A.13)
This las exp ession can be w i en mo e explici ly as:
T X2=−Gac Gbd cd ab +2(z
0)2
7−pGρρ Gab ∂aξ∂
bξ+2(z
0)2
7−p(Gρρ )2(∂ρξ)2(A.14)
+2(J ρ)2(z
0)2
7−p(∂ ξ)2+( ρ)2−4z
0
7−p
2
J ρGab∂aξ
b −4z
0
7−p
2
J ρGρρ ∂ρξ
ρ .
F om hese exp essions we ge ha :
1
2T X−1
4T X2+1
8T X2=z
0
7−p
2
Gρρ ∂ρξ+J ρ ρ +1
4Gac Gbd cd ab
+Gab
2 7−p
21−(z
0)2Gρρ
7−p
2∂aξ∂
bξ−(z
0)2
2 7−p(J ρ)2(∂ ξ)2+z
0
7−p
2
J ρGab ∂aξ
b .
(A.15)
Le us now ob ain he Lag angian densi y om hese esul s. Fi s o all, we can check ha he
i s -o de e ms do no con ibu e o he equa ions o mo ion and, he e o e, we jus d op hem.
Mo eo e , in he second-o de e ms we can subs i u e by 0(ρ), gi en by:
0(ρ) =ρ2+z0(ρ)2.(A.16)
712 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
Taking in o accoun he ze o h-o de Lag angian and ha :
1−(z
0)2Gρρ
7−p
2
0
=1−A(0)2
1+(z
0)2−A(0)2
,(A.17)
we ge :
L=−Nρλ
21+(z
0)2−A(0)2
×1
4Gac Gbd cd ab +1
2
7−p
2
0
1−A(0)2
1+(z
0)2−A(0)2
Gab ∂aξ∂
bξ
−(z
0)2
2 7−p
0
(J ρ)2(∂ ξ)2+z
0
7−p
2
0
J ρGab ∂aξ
b .(A.18)
Subs i u ing he alues o z
0and A(0)
(w i en in (2.20)), he Lag angian densi y o he luc ua-
ions a ze o empe a u e can be w i en as in (3.4).
A.2. Fluc ua ions a non-ze o empe a u e
In his subsec ion we ocus on T=0 and B=0, by luc ua ing he scala and he gauge ields
(7.33). Fi s we compu e he a ia ion o he induced me ic. By using he expansions
dθ2=˙
θ2
0d 2+2˙
θ0∂aζd dx
a+∂aζ∂
bζdx
adxb+···
cos2θ=cos2θ0−sin(2θ0)ζ −cos(2θ0)ζ2+··· ,(A.19)
whe e xa=(xμ, ) =( , xi, ), we can ep esen he induced me ic gin he o m:
g=¯g+ˆg, (A.20)
whe e ¯gis he ze o h-o de me ic and ˆgis he pe u ba ion. We will expand ˆgup o second o de
in he luc ua ions. Acco dingly, le us spli ˆgin he o m:
ˆg=ˆg(1)+ˆg(2),(A.21)
whe e ˆg(1)(ˆg(2)) a e he i s (second) o de e ms o ˆg. The non-ze o elemen s o ˆg(1)a e:
ˆg(1)
=2 p−3
2˙
θ0˙
ζ, ˆg(1)
xμ= p−3
2˙
θ0∂μζ, ˆg(1)
mn =− p−3
2sin(2θ0)ζγ
mn ,(A.22)
whe eas hose o ˆg(2)a e:
ˆg(2)
ab = p−3
2∂aζ∂
bζ, ˆg(2)
mn =− p−3
2cos(2θ0)ζ2γmn ,(A.23)
whe e m, na e indices along he in e nal (q −n −1)-sphe e and γmn is he me ic o a uni
Sq−n−1. Le us now de ine he open s ing me ic Gand he an isymme ic enso Jas in (A.8),
wi h F(0)being he gauge ield s eng h (7.10). The componen s o he in e se o he open s ing
me ic in his case a e:
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 713
G =− ¯g ( −1
p+ 2˙
θ2
0)
|¯g |¯g (1+ 2 p˙
θ2
0)−˙
A(0)2
,G =|¯g | p
|¯g |¯g (1+ 2 p˙
θ2
0)−˙
A(0)2
,
Gx1x1=Gx2x2=¯gxx
¯g2
xx +B2,Gxixj=δij
¯gxx
,(i,j=3,4,...) ,
Gmn =γmn
2¯g cos2θ0
,(A.24)
whe e A(0)is he gauge po en ial o he ield s eng h F(0). Using hese explici equa ions o
he me ic and elimina ing ˙
A(0)
, we ge :
G =− 1
7−p
2 p1+d2
H(cos θ0)λ,G = 7−p
2 p
1+ 2 p˙
θ2
01+d2
H(cos θ0)λ,
Gx1x1=Gx2x2=¯gxx
¯g2
xx +B2≡Gxx ,Gxixj=δij
¯gxx
,(i,j=3,4,...) ,
Gmn =γmn
2¯g cos2θ0
.(A.25)
The only non-ze o elemen s o he an isymme ic ma ix Ja e:
J =−J =− ˙
A(0)
|¯g |¯g (1+ 2 p˙
θ2
0)−˙
A(0)2
Jx1x2=−Jx2x1=− B
¯g2
xx +B2.(A.26)
Mo e explici ly:
J =−J =− d
H(cos θ0)λH(cos θ0)λ+d2
1+ 2 p˙
θ2
0
Jx1x2=−Jx2x1=− B
¯g2
xx +B2≡Jxy .(A.27)
We nex de ine he ma ix Xas in (A.6) and we pe o m he expansion (A.7) o he DBI
de e minan . The aces o Xneeded a e:
T X=GMN ˆgMN −JMN MN ,(A.28)
and
T X2=GMN GPQ−JMN JPQ(ˆgMP ˆgNQ − MP NQ)−4GMN JPQ ˆgMP NQ .
(A.29)
In hese o mulas he indices M, N, P, and Q un o e all wo ld olume di ec ions (including
he angula ones). The Lag angian densi y o he luc ua ions is gi en by:
L=L01+1
2T X−1
4T X2+1
8T X2+O(X3),(A.30)

714 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
whe e L0is he ze o h-o de Lag angian densi y, gi en by:
L0=−NH(cos θ0)λ1+ 2 p˙
θ2
0
d2+H(cos θ0)λ.(A.31)
No ice ha he equa ion o he embedding θ0( ) can be w i en as:
∂ L0 2¯g G ˙
θ0=−λ
2 an θ0L0.(A.32)
Le us now conside he i s -o de con ibu ions o L. They o igina e om he T X e m in
(A.30). The e o e:
L(1)=L01
2GMN ˆg(1)
MN −1
2JMN MN .(A.33)
By using he alues o he i s -o de me ic w i en in (A.22), we ge ha he i s e m in (A.33)
can be w i en as:
L0
2GMN ˆg(1)
MN =L0 2¯g G ˙
θ0˙
ζ−λ
2 an θ0ζ.(A.34)
In eg a ing by pa s he i s e m in (A.34) and using (A.32) one can easily check ha (A.34)
educes o a o al de i a i e and, he e o e, can be d opped om he Lag angian. Mo eo e , he
second e m in (A.33) can be w i en as:
−1
2L0JMN MN =Nd
+L0
B
¯g2
xx +B2 x1x2,(A.35)
and clea ly does no con ibu e o he equa ions o mo ion o he luc ua ions. Le us now con-
cen a e on he second-o de e ms in L. A e some wo k, we ge :
L=L01
4GabGcd −JabJcd +1
2JacJbd  ac bd
+ 2¯g
21− 2¯g G ˙
θ2
0Gab∂aζ∂
bζ−λ
41+1−λ
2 an2θ0ζ2
−λ
2 2¯g G an θ0˙
θ0ζ˙
ζ− 4¯g2
2J 2˙
θ2
0(∂ ζ)2+λ
4 an θ0Jab ζ
ab
+ 2¯g ˙
θ0J Gab∂aζ b +JabG ∂aζ b −1
2JabG ∂ ζ ab.(A.36)
Le us in eg a e by pa s he ζ˙
ζ e m on he second line o (A.36). In his p ocess we gene a e
he ollowing con ibu ion o L:
λ
4∂ L0 2¯g G an θ0˙
θ0ζ2=−λ2
8L0 an θ02ζ2+λ
4L0 2¯g G ˙
θ2
0
cos2θ0
ζ2,(A.37)
whe e we ha e used he embedding equa ion (A.32). Plugging his esul in o (A.36) we ge he
inal o m o he Lag angian o he luc ua ions, which is gi en by:
L=L01
4GabGcd −JabJcd +1
2JacJbd  ac bd (A.38)
+1− 2¯g G ˙
θ2
0 2¯g
2Gab∂aζ∂
bζ−λ
4 cos2θ0
ζ2− 4¯g2
2J 2˙
θ2
0(∂ ζ)2
+λ
4 an θ0Jab ζ
ab + 2¯g ˙
θ0J Gab∂aζ b +JabG ∂aζ b −1
2JabG ∂ ζ ab.
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 715
Le us now wo k ou he equa ions o mo ion de i ed om his Lag angian densi y. We will
assume ha all ields only depend on , and one o he Ca esian coo dina es (say x). Fi s o
all, we w i e he equa ion o a in he a =0 gauge. We ge he ollowing Gauss’ law:
G ∂ ˙a +Gxx ∂i˙ai= 2¯g J ˙
θ0∂ ˙
ζ+¯
λ
2
J
G an θ0∂ ζ. (A.39)
The equa ion o a becomes:
∂ L0G G ˙a − 2¯g J ˙
θ0˙
ζ−L0J λ
2 an θ0ζ+Jxy ∂xay
+L0GxxG ∂x x − 2¯g J ˙
θ0∂2
xζ+L0J Jxy ∂x˙ay=0.(A.40)
The equa ion o axis:
∂ L0G Gxx ˙ax+J Jxy ∂ ay+L0G Gxx ∂ x
+L0Gxx 2¯g J ˙
θ0∂ ∂xζ−L0J Jxy ∂ ˙ay=0.(A.41)
Taking in o accoun ha L0J =cons an , his las equa ion can be ew i en as:
∂ L0G Gxx ˙ax+L0G Gxx∂ x +L0Gxx 2¯g J ˙
θ0∂ ∂xζ=−L0J ∂ Jxy∂ ay.
(A.42)
Mo eo e , a e some simpli ica ions, he equa ion o mo ion o aycan be w i en as:
∂ L0G Gxx y+L0Gxx G ∂ y +Gxx ∂x xy 
=L0∂ JxyJ x − 2¯g G ˙
θ0∂xζ.(A.43)
Finally, le us w i e he equa ion o mo ion o he scala luc ua ions. We ge :
∂ L0 2¯g G 1− 2¯g G ˙
θ2
0˙
ζ−J ˙
θ0˙a +λ
2 cos2θ0
L01− 2¯g G ˙
θ2
0ζ
+λ
2 an θ0L0J ˙a +L0 2¯g 1− 2¯g G ˙
θ2
0G ∂2
ζ+Gxx∂2
xζ
−L0 4¯g2
(J )2˙
θ2
0∂2
ζ+L0 2¯g J Gxx ˙
θ0∂x x =L0∂ Jxy 2¯g ˙
θ0G xy .(A.44)
Le us nex Fou ie ans o m he gauge ield and he scala o momen um space as in (7.34) and
le us de ine he elec ic ield Eas he gauge-in a ian combina ion:
E=ka
+ωa
x.(A.45)
In momen um space he Gauss law (A.39) becomes:
ωG ˙a −kGxx ˙ax=ω 2¯g J ˙
θ0˙
ζ+¯
λ
2ωJ
G an θ0ζ. (A.46)
We can combine (A.46) and (A.45) o ge ˙a and ˙axin e ms o he gauge-in a ian combina ion
Eand he scala ield ζ:
˙a =Gxx k˙
E+ω2 2¯g J ˙
θ0˙
ζ+ω2¯
λ
2
J
G an θ0ζ
G ω2+Gxxk2
˙ax=G ω˙
E−kω 2¯g J ˙
θ0˙
ζ−kω ¯
λ
2
J
G an θ0ζ
G ω2+Gxxk2.(A.47)
716 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
Mo eo e , using (A.47) one can demons a e ha (A.40) and (A.42) a e equi alen o he ollow-
ing equa ion o he elec ic ield E:
∂ L0G Gxx
G ω2+Gxxk2G ˙
E−k 2¯g J ˙
θ0˙
ζ−kλ
2
J
G an θ0ζ
−L0G Gxx E+kL0Gxx 2¯g J ˙
θ0ζ=iL0J ∂ (Jxy)a
y,(A.48)
whe e Gxx has been de ined in (A.25). Simila ly, we can wo k ou he equa ion o he scala ζ
in e ms o E. In momen um space his equa ion becomes:
∂ L0 2¯g G 1− 2¯g G ˙
θ2
0˙
ζ−J ˙
θ0˙a +λ
2 cos2θ0
L01− 2¯g G ˙
θ2
0ζ
+λ
2 an θ0L0J ˙a −L0 2¯g 1− 2¯g G ˙
θ2
0G ω2+Gxxk2ζ
+L0 4¯g2
(J )2˙
θ2
0ω2ζ+L0 2¯g J Gxx ˙
θ0kE =ikL0˙
θ0 2¯g G ∂ (Jxy)ay,(A.49)
whe e i should be unde s ood ha ˙a is gi en by he i s equa ion in (A.47). Finally, he equa ion
o mo ion o he ans e se luc ua ion ayis:
∂ L0G Gxx ˙ay−L0GxxG ω2+Gxxk2ay=
−iL0J ∂ (Jxy)E −ikL0˙
θ0 2¯g G ∂ (Jxy)ζ . (A.50)
Appendix B. T ans e se co ela o s and he conduc i i y
Le us conside he case in which he magne ic ield anishes, B=0. In his case, he equa ion
o mo ion (A.50) o he ans e se luc ua ion ayis:
∂ L0G Gxx ˙ay−L0Gxx G ω2+Gxx k2ay=0.(B.1)
This equa ion can be ew i en as:
¨ay+∂ log L0G Gxx˙ay−G ω2+Gxx k2
G ay=0.(B.2)
Mo e explici ly, he equa ion o mo ion o ayis:
¨ay+∂ log d2+ λcos θ0λ
1+ 2 p˙
θ2
0
p˙ay
+1+ 2 p˙
θ2
0
7−p 2
p
(ω2− pk2) λ(cos θ0)λ+ω2d2
d2+ λcos θ0λay=0.(B.3)
We now s udy he equa ion o mo ion (B.3) o ayin he low equency egime in which k∼O()
and ω∼O(2). Le us i s s udy (B.3) nea he ho izon = h. Wi h his pu pose we expand
θ0( ) nea = h:
θ0( ) ≈θh−λ
2(7−p)
λ−1
hcos θhλ an θh
d2+ λ
hcos θhλ( − h)+··· .(B.4)
G. I sios e al. / Nuclea Physics B 909 (2016) 677–724 717
We also expand he coe icien s o he equa ion o he ans e se luc ua ions:
∂ log d2+ λcos θ0λ
1+ 2 p˙
θ2
0
p=1
− h+d1+···
1+ 2 p˙
θ2
0
7−p 2
p
(ω2− pk2) λ(cos θ0)λ+ω2d2
d2+ λcos θ0λ=A
( − h)2+c2
− h+··· ,(B.5)
whe e A, d1, and c2a e gi en by:
A=ω2
(7−p)2 5−p
h
d1=1
2 h
(p −8)d2+(p +λ−8) λ
hcos θhλ
d2+ λ
hcos θhλ+λ2
8(7−p)
2λ−1
hcos θh2λ
d2+ λ
hcos θhλ2 an2θh
c2=− 1
7−p
p+λ−6
hcos θhλ
d2+ λ
hcos θhλk2+1
(7−p)2 6−p
h
ω2
+λ2
4(7−p)3
p+2λ−6
hcos θh2λ an2θh
d2+ λ
hcos θhλ2ω2.(B.6)
Le us now sol e o ayin F obenius se ies a ound = h:
ay( ) =( − h)α(1+β( − h)+...) , (B.7)
whe e he exponen s αand β, a o de 2, a e gi en by:
α=− iω
(7−p)
5−p
2
h
,β≈−(α d1+c2). (B.8)
F om he exp essions o d1and c2w i en in (B.6) we ind ha βis gi en by:
β=i1
2(7−p)
7−p
2
h
(p −8)d2+(p +λ−8) λ
hcos θhλ
d2+ λ
hcos θhλ
+λ2
8(7−p)2
5−p
2
h
2λ−1
hcos θh2λ
d2+ λ
hcos θhλ2 an2θhω
+1
7−p
p+λ−6
hcos θhλ
d2+ λ
hcos θhλk2.(B.9)
Le us now ake he nea -ho izon and low equency limi s in opposi e o de . Fi s , we w i e (B.3)
as:
¨ay+˙
G
G˙ay+Qa
y=0,(B.10)
724 G. I sios e al. / Nuclea Physics B 909 (2016) 677–724
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