Collective excitations of massive flavor branes

A ailable online a www.sciencedi ec .com ScienceDi ec 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). 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