Holographic Kondo and Fano resonances

Holog aphic Kondo and Fano esonances Johanna E dmenge ,1,* Ca los Hoyos,2,†Andy O’Bannon,3,‡Ioannis Papadimi iou,4,§ Jonas P obs ,5,∥and Jackson M. S. Wu6,¶ 1Ins i u ü Theo e ische Physik und As ophysik, Julius-Maximilians-Uni e si ä Wü zbu g, Am Hubland, D-97074 Wü zbu g, Ge many and Max-Planck-Ins i u ü Physik (We ne -Heisenbe g-Ins i u ), Föh inge Ring 6, D-80805 Munich, Ge many 2Depa men o Physics, Uni e sidad de O iedo, A da. Cal o So elo 18, 33007 O iedo, Spain 3STAG Resea ch Cen e, Physics and As onomy, Uni e si y o Sou hamp on, Sou hamp on SO17 1BJ, Uni ed Kingdom 4SISSA and INFN—Sezione di T ies e, Via Bonomea 265, I 34136 T ies e, I aly 5Rudol Peie ls Cen e o Theo e ical Physics, Uni e si y o Ox o d, 1 Keble Road, Ox o d OX1 3NP, Uni ed Kingdom 6Depa men o Physics and As onomy, Uni e si y o Alabama, Tuscaloosa, Alabama 35487, USA (Recei ed 8 Decembe 2016; published 12 July 2017) We use holog aphy o s udy a (1þ1)-dimensional con o mal ield heo y (CFT) coupled o an impu i y. The CFT is an SUðNÞgauge heo y a la ge N, wi h s ong gauge in e ac ions. The impu i y is an SUðNÞ spin. We igge an impu i y eno maliza ion g oup (RG) low ia a Kondo coupling. The Kondo e ec occu s only below he c i ical empe a u e o a la ge-Nmean- ield ansi ion. We show ha a all empe a u es T, impu i y spec al unc ions exhibi a Fano esonance, which in he low-Tphase is a la ge-N mani es a ion o he Kondo esonance. We hus p o ide an example in which he Kondo esonance su i es s ong co ela ions, and unco e a no el mechanism o gene a ing Fano esonances, ia RG lows be ween (0þ1)-dimensional ixed poin s. DOI: 10.1103/PhysRe D.96.021901 I. INTRODUCTION The Kondo e ec is he sc eening o an impu i y spin by a Landau Fe mi liquid (LFL) a low T[1,2]. A a ie y o echniques, such as Wilson’s RG, la ge-N, CFT, and mo e [3], ha e cap u ed many cha ac e is ic Kondo phenomena. Ne e heless, many ques ions esis solu ion, o example abou in e -impu i y in e ac ions, subsys em en anglemen en opy (EE), nonequilib ium p ocesses like quan um quenches, and mo e. In pa icula , wha happens when he LFL is eplaced wi h s ongly co ela ed elec ons? Fo example, how does he Kondo e ec change in a Lu inge liquid [4–8] o he Hubba d model [9,10]? In he la e case, expe imen s e eal d ama ic e ec s o s ong co ela ions, such as enhancemen o he Kondo empe a u e, TK[53]. On he heo y side, al hough special ools like bosoniza ion [4–8] and uncon olled mean- ield app oxima ions [9,11–17] ha e p o ided insigh , in gene al, eliable echniques do no ye exis o answe ques ions abou Kondo phenomena in s ongly-co ela ed sys ems. To add ess all o he abo e, we ha e de eloped an al e na i e Kondo model, based on holog aphic duali y [18–21]. Ou model eplaces he LFL by a (1þ1)- dimensional CFT in which spin SUð2Þis eplaced by gauged SUðNÞ, wi h la ge Nand s ong gauge in e ac ions. Ou model has al eady e ealed no el s ong-coupling phenomena in RG [18,19], in e impu i y in e ac ions [19] and EE [20]. He e we ini ia e he s udy o nonequilib ium phenomena in ou model: we compu e linea esponse (G een’s) unc ions o a cha ged bosonic impu i y ope a o , O,in ou model. We ha e wo main esul s. Fi s , we ind a la ge-Nmani es a ion o he Kondo esonance [2,22,23], a signa u e o he Kondo e ec . As expec ed, ou Kondo esonance appea s only a Tbelow he c i ical empe a u e Tco a mean- ield ansi ion ha is common o la ge-NKondo models [23–27]:hOibecomes non-ze o when T≤Tc. We hus p o e unequi ocally ha ou holog aphic model ealizes a genuine Kondo e ec , as opposed o some o he impu i y physics, and u he mo e show ha a la ge-NKondo esonance can su i e s ong co ela ions essen ially in ac . Second, a all T,O’s spec al unc ion exhibi s a Fano esonance [28,29], which occu s when a Lo en zian es- onance is imme sed in a con inuum o s a es (in ene gy). A Fano esonance is cha ac e ized no only by i s posi ion, wid h, and heigh , like a Lo en zian, bu also by an asymme y pa ame e ,q, which measu es he ela i e s eng h o esonan e sus non- esonan sca e ing. Ou qinc eases as T→Tþ c. When T≤Tc, he Fano line- shape a ises om ou Kondo esonance, which mus be *[email p o ec ed]‑wue zbu g.de †[email p o ec ed] ‡[email p o ec ed] §ioannis.papadimi [email protected] ∥[email p o ec ed]c.uk ¶[email p o ec ed] PHYSICAL REVIEW D 96, 021901(R) (2017) 2470-0010=2017=96(2)=021901(6) 021901-1 © 2017 Ame ican Physical Socie y RAPID COMMUNICATIONS an isymme ic due o pa icle-hole symme y (PHS), and hence has he special alue q¼1. Al hough Fano esonances ha e been obse ed in many impu i y sys ems in one spa ial dimension [29–32], ou s a ise om a quali a i ely di e en mechanism. Fo in- s ance, in side-coupled quan um do s (QDs) [29,31,32] he Lo en zian esonances a e he disc e e s a es on he QD, and he con inuum comes om elec onic sca e ing s a es in he leads. Coupling he wo, o example by a Kondo coupling, can hen p oduce Fano esonances. Ou model also has an impu i y coupled o a con inuum in one spa ial dimension, i.e. he CFT. Howe e , ou model has wo couplings: he CFT’sSUðNÞgauge coupling and he Kondo coupling. The spec al unc ion o Oinhe i s (0þ1)-dimensional scale in a iance om he o me , and so exhibi s a con inuum o s a es, in con as o a QD’s disc e e s a es. The Kondo coupling hen igge s an RG low om ha (0þ1)-dimensional ixed poin , and c ea es a esonance ha canno escape he con inuum, hence p oducing a Fano line shape. To ou knowledge, such a mechanism o p oducing Fano esonances is no el, and mo eo e is easy o gene al- ize o any RG low be ween (0þ1)-dimensional ixed poin s, as ollows. Scale in a iance implies ha any spec al unc ion will be a ea u eless con inuum, which in (0þ1) dimensions means a powe law (o loga i hm) in equency. A ele an de o ma ion can hen explici ly b eak scale in a iance, igge an RG low o an IR ixed poin —in which case we expec he con inuum o su i e—and may also p oduce esonances. In highe dimensions, he eso- nances would no ha e o be wi hin he con inuum, o example he wo could be sepa a ed in momen um space. Howe e , in (0þ1) dimensions he esonances ha e no place o escape he con inuum, and hence mus p oduce Fano line shapes. In ac , such a mechanism was a wo k in some p e ious cases, such as he la ge-NKondo model a su icien ly low T[33], and holog aphic duals o T¼0cha ged black b anes [34–36]. Howe e , he esul ing Fano esonances wen uniden i ied, lea ing c ucial physics o e looked, namely he ela i e s eng h o esonan e sus non- esonan sca e ing, as measu ed by q. Ou esul s no only p o ide a no el pe spec i e on hese cases, bu also p edic Fano esonances in RG lows be ween o he (0þ1)- dimensional ixed poin s, such as Sachde -Ye-Ki ae ixed poin s [36–43]. Fu he esul s o ou model, including de ails o holo- g aphic eno maliza ion use ul o holog aphic impu i y models in gene al, will appea in [44]. II. HOLOGRAPHIC KONDO MODEL We i s b ie ly e iew some essen ial ea u es o he CFT and la ge-Napp oaches o he Kondo model, and how ou model builds upon and ex ends hem. The CFT app oach [45] is based on s-wa e educ ion o LFL e mions abou he impu i y, plus linea iza ion o he dispe sion ela ion. In/ou -going s-wa es become ela i is ic le / igh -mo ing e mions, ψLand ψR, in he adial di ec ion, . Re lec ing ψR o <0and elabeling ψR→ψLleads o ψLalone on he en i e axis, wi h impu i y a ¼0. The ψL o m a (1þ1)-dimensional chi al CFT wi h SUð2Þ1×Uð1Þspin and cha ge Kac- Moody cu en s, espec i ely. In he Hamil onian, he Kondo in e ac ion is δð ÞgKSAJA, wi h coupling cons an gK, impu i y spin SA, and spin cu en JA,A¼1,2,3. An an i e omagne ic coupling, gK>0, is ma ginally ele an , and igge s an RG low o an IR chi al CFT cha ac e ized by a phase shi o ψLand impu i y sc eening [45]. The la ge-Napp oach begins by eplacing spin SUð2Þ→SUðNÞ, ollowedbyN→∞wi h λK≡ NgK ixed [3,23,46,47]. We will only conside SAin o ally an isymme ic SUðNÞ ep esen a ions o ank Q, and in oduce Ab ikoso pseudo e mions χ ia SA¼ χ†TAχ, wi h SUðNÞgene a o s TA,A¼1;…;N2−1. Doing so in oduces an auxilia y Uð1Þac ing only on χ, bu wi h cha ge ixed by p ojec ing on o s a es wi h χ†χ¼Q.A la geN,SAJA¼−O†O=2wi h O≡ψ† Lχ [19], which is cha ged unde bo h he cha ge and auxilia y Uð1Þ’s. Ou holog aphic model [18] begins by gauging SUðNÞ, hus in oducing he ’ Hoo coupling, λ.We hen add deg ees o eedom o make he gauge heo y a (1þ1)-dimensional CFT wi h spa se ope a o spec um when Nand λbo h →∞, bu whose de ails o he wise a e i ele an . The heo y is hen holog aphically dual o Eins ein-Hilbe g a i y in (2þ1)-dimensional an i-de Si e space, AdS3[48]. The cha ge Uð1ÞKac-Moody cu en is dual o a Uð1ÞChe n-Simons gauge ield, A, he auxilia y Uð1Þis dual o a Maxwell ield aon an AdS2de ec a ¼0, and Ois dual o a complex scala ield Φalso in AdS2, cha ged unde bo h Aand a.As long as he s ess-ene gy enso is ini e, a la ge Nwe can neglec back eac ion o A,a,Φ(dual o undamen al ields) on he geome y (dual o adjoin ields). When T>0, he bulk me ic is hus he BTZ black b ane, ds2¼1 z2ðh−1ðzÞdz2−hðzÞd 2þd 2Þ; wi h hðzÞ¼1−z2=z2 Hwhe e zH¼1=2πT, and uni AdS adius. The ields aand Φa e localised o he asymp- o ically AdS2subspace a ¼0, wi h induced me ic gmn (m; n ¼z, ). We desc ibe he dynamics o A,a, and Φby he simple quad a ic ac ion [18], S¼−N 4πZBTZ A∧dAþSAdS2;ð1aÞ JOHANNA ERDMENGER e al. PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-2 RAPID COMMUNICATIONS SAdS2¼−NZx¼0 dzd ffiffiffiffiffiffi −g p1 4 mn mn þðDmΦÞ†ðDmΦÞþM2Φ†Φ;ð1bÞ wi h ield s eng h ¼da, co a ian de i a i e DmΦ¼ ð∂mþiAm−iamÞΦ, and mass-squa ed M2. A he ho i- zon z¼zHwe equi e egula i y o all ields. A he bounda y z¼0,a’s leading mode, a∼Q=z, is ela ed o Q:Q≠0b eaks χ’s PHS, so he PHS alue Q¼0is dual o he PHS alue Q¼N=2, and inc easing jQj co esponds o inc easing jQ−N=2j. The la ge-NKondo in e ac ion −λKO†O=2is classically ma ginal, hence Ohas UV dimension Δ¼1=2, which ixes M2and hence Φ’s nea -bounda y expansion, Φ∼ffiffiffi z pðαlog zþβÞ. In oducing he Kondo in e ac ion amoun s o adding a bounda y e m ∝−λKO†O=2 o S, which changes Φ’s bounda y condi ion om α¼0 o α¼ −λKβ[18,49,50]. Fo mo e de ails abou he bounda y e ms, see [19,44]. A holog aphic scaling analysis e eals ha λK uns loga i hmically, λK¼1=log ðT=TKÞ, di e g- ing a he dynamically-gene a ed Kondo empe a u e, TK≡Λe−1=λK=ð2πÞ, wi h λKe alua ed a he UV cu o , Λ. A holog aphic an i e omagne ic UV Kondo coupling, λK>0, is hus ma ginally ele an , b eaks con o mal in a iance, and igge s an RG low. As men ioned abo e, ou model has a mean- ield phase ansi ion [18]:hOi¼0(Φ¼0) when T>T cand hOi≠0 (Φ≠0) when T≤Tc. Condensa e o ma ion hOi≠0 b eaks he cha ge and auxilia y Uð1Þ’s o he diagonal, and signals he Kondo e ec , including a phase shi o ψL, dual o a Wilson line o A, and impu i y sc eening, dual o educ ion o lux be ween z¼0and z¼zH. We e e o he T>T cand T≤Tcphases as “unsc eened”and “sc eened,” espec i ely. In [18–21] we compu ed Tc nume ically. Below we ob ain an exac o mula o Tc. III. FANO RESONANCES I a e a ded G een’s unc ion o complex equency ω, GðωÞ, has a pole a ωp,GðωÞ∼Z ω−ωp, wi h complex esidue Z¼ZRþiZI, hen nea he pole he spec al unc ion ρðωÞ≡−2ImðGðωÞÞ will ha e a Fano esonance [28,29] (se ing ImðωÞ¼0), ρFanoðωÞ¼ ðω−ω0þqΓ=2Þ2 ðω−ω0Þ2þðΓ=2Þ2;ð2Þ wi h posi ion ω0¼ReðωpÞ, wid h Γ¼2jImðωpÞj, and asymme y pa ame e q¼−ZR=ZIþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þZ2 R=Z2 I p.Fano esonances a e an isymme ic when q¼1, meaning ρðωÞ is odd unde PHS, and symme ic when q¼0(an an i- esonance) o ∞(a Lo en zian), meaning ρðωÞis e en. Fano esonances a ise when a Lo en zian esonance is imme sed in a con inuum (in ene gy), due o in e e ence be ween he wo. The asymme y pa ame e qcon ains key dynamical in o ma ion, speci ically, q2∝ he a io o p obabili ies o esonan and non- esonan sca e ing. In ou model, he AdS2subspace inhe i s scale in a i- ance om AdS3, o in dual ield heo y language, he impu i y inhe i s scale in a iance om he CFT, so ρðωÞo impu i y ope a o s mus be a ea u eless con inuum. Ou ma ginally- ele an Kondo coupling hen b eaks scale in a iance and p oduces a esonance, while Q≠0b eaks PHS. We will show ha ρðωÞo O hen indeed gene ically exhibi s asymme ic Fano esonances. IV. SPECTRAL FUNCTIONS We compu e GðωÞholog aphically by sol ing o linea ized luc ua ions abou solu ions o he unsc eened and sc eened phases [44,51,52]. A all T, we ind ha he Kac-Moody cu en ’sGðωÞis una ec ed by he impu i y. In he unsc eened phase, we ind ha all cha ged GðωÞ anish, i.e. GOOðωÞ≡hOðωÞOð−ωÞi ¼ 0, while GO†OðωÞ≡hO†ðωÞOð−ωÞi ¼ N λK1−1 λKDðωÞ; DðωÞ≡H−1 2þiQ −iω 2πTþH−1 2−iQ þln2T TK; wi h Ha monic numbe H½x, and λKe alua ed a Λ. The o m o GOO†ðωÞis he same, bu wi h Q→−Q. Scale in a iance in (0þ1)-dimensions and Δ¼1=2imply a i ial UV con inuum: limω→∞ρO†OðωÞ¼0. Fo gi en Qand T,GO†OðωÞhas poles in ωwhen DðωÞ¼0. Figu e 1shows ou nume ical esul s o he FIG. 1. Posi ions o poles in he complex ω=ð2πTÞplane o Q¼1=2. Blue and pu ple deno e lowes and nex -lowes poles, espec i ely, o GO†OðωÞ(ReðωÞ>0) and GOO†ðωÞ (ReðωÞ<0), o T=Tc om 100 down o 1.001. Red and o ange deno e he same o GO†OðωÞ o T=Tc om 1 down o 0.2. A ows indica e mo emen o poles as Tdec eases. HOLOGRAPHIC KONDO AND FANO RESONANCES PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-3 RAPID COMMUNICATIONS posi ions o he lowes (closes o ω¼0) and nex -lowes poles o GO†OðωÞand GOO†ðωÞin he complex ω=ð2πTÞ plane, o Q¼1=2. O he Qgi e simila esul s. As T→Tþ c, he lowes pole mo es owa ds he o igin, a i es he e a Tc, and when T<T c, c osses in o he ImðωÞ>0 egion, signaling ins abili y (no shown). We hus iden i y Tcas he Twhe e Dðω¼0Þ¼0, Tc¼1 2TKexp −2ReH1 2þiQ: Figu e 2shows he no malized spec al unc ion ¯ ρO†OðωÞ≡−2λ2 K NImGO†OðωÞ e sus eal ω=ð2πTÞ o Q¼ 1=2and T=Tc¼16, 8, 4, 2. We ind a Fano esonance, as ad e ised, wi h asymme ic minimum and maximum. Nume ically, ω0≈ReðωpÞand Γ≈2jImðωpÞj,asin(2), whe e ωpis GO†OðωÞ’s lowes pole. As T→Tþ c,qg ows: q≈1.7a T¼16Tcwhile q≈4a T¼2Tc. Fo Tjus abo e Tc,T≳Tc, expanding in Tabou Tc and in ωabou ω¼0gi es, o GO†OðωÞ’s lowes pole, ωp 2πT¼−iT=Tc−1 ψ0½1 2þiQ;Z¼−iN λ2 K 2πTc ψ0½1 2þiQ;ð3Þ wi h digamma unc ion ψ½x. The esonance heigh hus g ows as ðT=Tc−1Þ−1and he wid h sh inks as T=Tc−1. I is he e o e no ela ed o a Kondo esonance, which g ows loga i hmically as T→Tþ K[22]. Indeed, a la ge N we expec he Kondo esonance only in he sc eened phase [23]. Ou esonance is p esumably a bound s a e o ψL and χ, he alding he nascen sc eened phase. The Zin (3) gi es q ha depends only on Q, shown in Fig. 3. (An i)symme ic alues q¼1,0,∞occu when Q→0,∓∞, espec i ely. Indeed, Fig. 4shows ha e en o ela i ely modes Q¼1, he esonance is nea ly Lo en zian, he minimum ha ing p ac ically anished. In he sc eened phase, we ha e nume ical esul s o GO†OðωÞ[18–21,44]. Figu e 1shows ou nume ical esul s o he posi ions o he lowes and nex -lowes poles in GO†OðωÞ o Q¼1=2. O he Qgi e simila esul s. A T¼Tc he poles a e coinciden wi h hose o GO†OðωÞand GOO†ðωÞin he unsc eened phase. As Tdec eases below Tc,GO†OðωÞ’s lowes pole, ωp, mo es s aigh down he ImðωÞaxis. FIG. 2. The no malized spec al unc ion ¯ ρO†OðωÞ e sus eal ω=ð2πTÞ o Q¼1=2and, om sho es o alles , T=Tc¼16 ( ed), 8 (g een), 4 (o ange), and 2 (blue). FIG. 3. Asymme y pa ame e q e sus Q, o T≳Tc. FIG. 4. The no malized spec al unc ion, ¯ ρO†OðωÞ, e sus eal ω=ð2πTÞ o Q¼1and, om sho es o alles , T=Tc¼1.04 (g een), 1.02 (o ange), and 1.01 (blue). FIG. 5. The no malized spec al unc ion ¯ ρO†OðωÞ e sus eal ω=ð2πTÞ o Q¼1=2and, om alles o sho es , T=Tc¼0.998 (blue), 0.991 (o ange), and 0.964 (g een). JOHANNA ERDMENGER e al. PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-4 RAPID COMMUNICATIONS F om ou expe ience wi h he unsc eened phase, we expec ωp o p oduce a Fano esonance in he no malized spec al unc ion, ¯ ρO†OðωÞ. C ucially, ReðwpÞ¼0,soωp p ese es PHS, ReðωÞ→−ReðωÞ, so we expec an an i- symme ic Fano esonance a ReðωÞ¼0. Mo eo e , jImðωpÞjinc eases as Tdec eases, and so should he wid h Γ. Figu e 5con i ms ou expec a ions: ¯ ρO†OðωÞ’s only signi ican ea u e is a Fano esonance a ReðωÞ¼0 wi h q¼1, meaning pe ec ly an isymme ic minimum and maximum, and whose Γinc eases as Tdec eases. Addi ionally, he heigh dec eases, and indeed ou nume ics sugges limT→0¯ ρO†OðωÞ¼0. Figu e 6shows ou nume ical esul s o he posi ion o ωp e sus small λ2 K N2hOi2=ð2πTÞ, o equi alen ly, Tjus below Tc,T≲Tc, o Q¼1=2. Figu e 6also shows a linea i demons a ing ha .1 ωp∝−ihOi2:ð4Þ Ou model’s mean- ield beha io hOi∝ðTc−TÞ1=2 hen implies Γ∝Tc−T o T≲Tc. The beha io in (4) is in ac iden ical o ha in a LFL a la ge N. In a LFL, he Kondo esonance is o mally de ined in he LFL e mion spec al unc ion, and a la ge N appea s only in he sc eened phase, wi h Γ∝hOi2[23].Fo T≲Tc, he mean- ield beha io hOi∝ðTc−TÞ1=2 hen implies Γ∝Tc−T. C ucially, in he sc eened phase he Kondo esonance also appea s in o he spec al unc ions, due o ope a o mixing induced by he symme y b eaking [23]. In pa icula , a Kondo esonance should p oduce a pole in GO†OðωÞp ecisely o he o m in (4)2Ou esul (4) hus p o es he exis ence o a Kondo esonance in ou model when T≲Tc, wi h de ining ea u es essen ially in ac despi e he s ong in e ac ions. V. CONCLUSION In a holog aphic model desc ibing he in e ac ion o a magne ic impu i y wi h a s ongly co ela ed CFT a la ge N, we disco e ed a no el mechanism o p oducing Fano esonances, namely ia RG lows be ween (0þ1)-dimen- sional ixed poin s. The o igin and consequences o such Fano esonances, in exis ing cases ha ha e gone uniden- i ied and in no el cases, dese e u he s udy, pa icula ly o he physics con ained in he asymme y pa ame e q. ACKNOWLEDGMENTS We would like o hank Ian A leck, Na han And ei, Pie s Coleman, Ma io Flo y, Hen ik Johannesson, And ew Mi chell, Max New zella, and Philip Phillips o help ul con e sa ions and co espondence. C. H. is suppo ed by he Ramon y Cajal ellowship RYC-2012-10370, he As u ian G an No. FC-15-GRUPIN14-108 and he Spanish na ional G an No. MINECO-16-FPA2015- 63667-P. A. O’B. is a Royal Socie y Uni e si y Resea ch Fellow. J. P. is suppo ed by he Cla endon Fund and S John’s College, Ox o d, and by he Eu opean Resea ch Council unde he Eu opean Union’s Se en h F amewo k P og amme (ERC G an ag eemen 307955). [1] J. Kondo, P og. Theo . Phys. 32, 37 (1964). [2] A. Hewson, The Kondo P oblem o Hea y Fe mions (Camb idge Uni e si y P ess, 1993). [3] D. L. Cox and A. Zawadowski, Ad . Phys. 47, 599 (1998). [4] D.-H. Lee and J. Tone , Phys. Re . Le . 69, 3378 (1992). [5] A. Fu usaki and N. Nagaosa, Phys. Re . Le . 72, 892 (1994). [6] P. F öjdh and H. Johannesson, Phys. Re . Le . 75, 300 (1995). [7] P. F öjdh and H. Johannesson, Phys. Re . B 53,3211(1996). [8] A. Fu usaki, J. Phys. Soc. Jpn. 74, 73 (2005). [9] P. Fulde, V. Ze in, and G. Zwicknagl, Z. Phys. B 92, 133 (1993). [10] T. Scho k and P. Fulde, Phys. Re . B 50, 1345 (1994). [11] G. Khaliullin and P. Fulde, Phys. Re . B 52, 9514 (1995). FIG. 6. Do s deno e he posi ion o GO†OðωÞ’s lowes pole e sus λ2 K N2hOi2=ð2πTÞ o Q¼1=2. The solid line is a linea i wi h slope ≈−17.6and in e cep a he o igin. 1In [44] we de i e (4) wi hou nume ics, ia a small-hOi expansion. 2Fo de ails, see o example chap e 18 o [23]. HOLOGRAPHIC KONDO AND FANO RESONANCES PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-5 RAPID COMMUNICATIONS [12] J. Iga ashi, K. Mu ayama, and P. Fulde, Phys. Re . B 52, 15966 (1995). [13] S. To now, V. Ze in, and G. Zwicknagl, a Xi :cond-ma / 9701137. [14] T. Scho k and S. Blawid, Phys. Re . B 56, 6559 (1997). [15] B. Da ido ich and V. Ze in, Phys. Re . B 57, 7773 (1998). [16] H. T. Duc and N. T. Thang, Mod. Phys. Le . B 13, 849 (1999). [17] W. Ho s e e , R. Bulla, and D. Vollha d , Phys. Re . Le . 84, 4417 (2000). [18] J. E dmenge , C. Hoyos, A. O’Bannon, and J. Wu, J. High Ene gy Phys. 12 (2013) 086. [19] A. O’Bannon, I. Papadimi iou, and J. P obs , J. High Ene gy Phys. 01 (2016) 103. [20] J. E dmenge , M. Flo y, C. Hoyos, M.-N. New zella, and J. M. S. Wu, Fo sch . Phys. 64, 109 (2016). [21] J. E dmenge , M. Flo y, C. Hoyos, M.-N. New zella, A. O’Bannon, and J. Wu, Fo sch . Phys. 64, 322 (2016). [22] P. Phillips, Ad anced Solid S a e Physics (Camb idge Uni e si y P ess, 2012). [23] P. Coleman, In oduc ion o Many-Body Physics (Camb idge Uni e si y P ess, 2015). [24] P. Coleman and N. And ei, J. Phys. C 19, 3211 (1986). [25] P. Coleman, Phys. Re . B 35, 5072 (1987). [26] T. Sen hil, S. Sachde , and M. Voj a, Phys. Re . Le . 90, 216403 (2003). [27] T. Sen hil, M. Voj a, and S. Sachde , Phys. Re . B 69, 035111 (2004). [28] U. Fano, Phys. Re . 124, 1866 (1961). [29] A. E. Mi oshnichenko, S. Flach, and Y. S. Ki sha , Re . Mod. Phys. 82, 2257 (2010). [30] V. Madha an, W. Chen, T. Jamneala, M. F. C ommie, and N. S. Wing een, Science 280, 567 (1998). [31] V. Madha an, W. Chen, T. Jamneala, M. F. C ommie, and N. S. Wing een, Phys. Re . B 64, 165412 (2001). [32] J. Gö es, D. Goldhabe -Go don, S. Heemeye , M. A. Kas ne , H. Sh ikman, D. Mahalu, and U. Mei a , Phys. Re . B 62, 2188 (2000). [33] O. Pa colle , A. Geo ges, G. Ko lia , and A. Sengup a, Phys. Re . B 58, 3794 (1998). [34] T. Faulkne , H. Liu, J. McG ee y, and D. Vegh, Phys. Re . D83, 125002 (2011). [35] T. Faulkne , N. Iqbal, H. Liu, J. McG ee y, and D. Vegh, Phil. T ans. R. Soc. A 369, 1640 (2011). [36] S. Sachde , Phys. Re . X 5, 041025 (2015). [37] S. Sachde and J.-W. Ye, Phys. Re . Le . 70, 3339 (1993). [38] A. Ki ae , in KITP S ings semina and En anglemen 2015 p og am, (2015). [39] J. Polchinski and V. Rosenhaus, J. High Ene gy Phys. 04 (2016) 001. [40] A. Je icki, K. Suzuki, and J. Yoon, J. High Ene gy Phys. 07 (2016) 007. [41] J. Maldacena and D. S an o d, Phys. Re . D 94, 106002 (2016). [42] A. Je icki and K. Suzuki, J. High Ene gy Phys. 11 (2016) 046. [43] E. Wi en, a Xi :1610.09758. [44] J. E dmenge , C. Hoyos, A. O’Bannon, I. Papadimi iou, J. P obs , and J. M. S. Wu, J. High Ene gy Phys. 03 (2017) 039. [45] I. A leck, Ac a Phys. Pol. B 26, 1869 (1995). [46] N. Bicke s, Re . Mod. Phys. 59, 845 (1987). [47] P. Coleman, in Handbook o Magne ism and Ad anced Magne ic Ma e ials: Fundamen als and Theo y, edi ed by H. K onmulle and S. Pa kin (John Wiley and Sons, New Yo k, 2007), Vol. 1, p. 95. [48] O. Aha ony, S. S. Gubse , J. M. Maldacena, H. Oogu i, and Y. Oz, Phys. Rep. 323, 183 (2000). [49] E. Wi en, a Xi :hep- h/0112258. [50] I. Papadimi iou, J. High Ene gy Phys. 05 (2007) 075. [51] D. T. Son and A. O. S a ine s, J. High Ene gy Phys. 09 (2002) 042. [52] P. K. Ko un and A. O. S a ine s, Phys. Re . D 72, 086009 (2005). [53] T. B ugge , T. Sch eine , G. Ro h, P. Adelmann, and G. Czjzek, Phys. Re . Le . 71, 2481 (1993). JOHANNA ERDMENGER e al. PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-6 RAPID COMMUNICATIONS