Two-point Functions in a Holographic Kondo Model
Johanna Erdmenger, Carlos Hoyos, Andy O'Bannon, Ioannis Papadimitriou, Jonas Probst, Jackson M. S. Wu
OOUTP-16-27PFPAUO-16/16SISSA 61/2016/FISI
Two-point Functions in a Holographic Kondo Model
Johanna Erdmenger a,b, , Carlos Hoyos c, , Andy O’Bannon d, ,Ioannis Papadimitriou e, , Jonas Probst f, , Jackson M. S. Wu g a Institut f¨ur Theoretische Physik und Astrophysik, Julius-Maximilians-Universit¨at W¨urzburg,Am Hubland, D-97074 W¨urzburg, Germany. b Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, D-80805 Munich, Germany. c Department of Physics, Universidad de Oviedo, Avda. Calvo Sotelo 18, 33007, Oviedo, Spain. d STAG Research Centre, Physics and Astronomy, University of Southampton, Highfield,Southampton SO17 1BJ, United Kingdom. e SISSA and INFN - Sezione di Trieste, Via Bonomea 265, I 34136 Trieste, Italy. f Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road,Oxford OX1 3NP, United Kingdom. g Department of Physics and Astronomy, University of Alabama,Tuscaloosa, AL 35487, USA.
Abstract
We develop the formalism of holographic renormalization to compute two-point functions in aholographic Kondo model. The model describes a (0 + 1)-dimensional impurity spin of a gauged SU ( N ) interacting with a (1 + 1)-dimensional, large- N , strongly-coupled Conformal Field Theory(CFT). We describe the impurity using Abrikosov pseudo-fermions, and define an SU ( N )-invariantscalar operator O built from a pseudo-fermion and a CFT fermion. At large N the Kondo interac-tion is of the form O † O , which is marginally relevant, and generates a Renormalization Group (RG)flow at the impurity. A second-order mean-field phase transition occurs in which O condenses belowa critical temperature, leading to the Kondo effect, including screening of the impurity. Via holog-raphy, the phase transition is dual to holographic superconductivity in (1 + 1)-dimensional Anti-deSitter space. At all temperatures, spectral functions of O exhibit a Fano resonance, characteristicof a continuum of states interacting with an isolated resonance. In contrast to Fano resonances ob-served for example in quantum dots, our continuum and resonance arise from a (0 + 1)-dimensionalUV fixed point and RG flow, respectively. In the low-temperature phase, the resonance comes froma pole in the Green’s function of the form − i (cid:104)O(cid:105) , which is characteristic of a Kondo resonance. [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] M a r ontents The Kondo model of a magnetic impurity interacting with a Fermi liquid of electrons, proposed by JunKondo in 1964 [1], has been seminal for both experimental and theoretical physics. In experimentalphysics, the Kondo model explains the thermodynamic and transport properties of many systems,including certain types of quantum dots [2,3] and certain metals doped with magnetic impurities [1,4,5].Most famously, for doped metals the Kondo model successfully describes the logarithmic rise of theelectrical resistivity ρ with decreasing temperature T . In theoretical physics, the Kondo model providesperhaps the simplest example of a renormalization group (RG) flow exhibiting asymptotic freedom, thedynamical generation of a scale, namely the Kondo temperature, T K , and a non-trivial infra-red (IR)fixed point describing the screening of the impurity by the electrons. As a result, the Kondo model hasplayed a central role in the development of many techniques in theoretical physics: Wilson’s numericalRG [6–8], integrability [9–16], large- N limits [17–22], Conformal Field Theory (CFT) [23–28], andmore. For reviews of many of these, see for example refs. [29, 30].Indeed, given the successes of these techniques, the single-impurity Kondo model is often called a“solved problem.” However, in reality many fundamental questions about the Kondo model remainunanswered, such as how to measure (or even define ) the size of the Kondo screening cloud, howentanglement entropy (EE) depends on the size of a spatial subsystem, or how observables evolveafter a (quantum) quench, i.e. after the Kondo model is “kicked” far from equilibrium.Moreover, many generalizations of the original Kondo model remain impervious to the existing tech-1iques. For example, what if we replace the electron Fermi liquid with (strongly) interacting degrees offreedom, such as a Luttinger liquid? What if multiple impurities interact not only with the electrons,but also with each other? Answers to these questions are urgently needed to understand importantexperimental systems. For example, a heavy fermion compound can be described as a dense lat-tice of impurities in which the competition between the Kondo and inter-impurity interactions leadsto a quantum critical phase very similar to the “strange metal” phase of the cuprate superconduc-tors. Understanding the strange metal phase may be the key to understanding the mechanism ofhigh-temperature superconductivity. The Kondo lattice therefore remains a major unsolved problem.Motivated by these questions, in a series of papers we have developed an alternative Kondo model,based on holographic duality [31–34]. Holography equates certain strongly-interacting quantum fieldtheories (QFTs) with weakly-coupled theories of gravity in one higher dimension. Holography istherefore a natural tool for studying impurities coupled to strongly-interacting degrees of freedom,and is particularly well-suited for studying EE and far-from-equilibrium evolution.Our holographic model is based on the large- N [17–22,35,36] and CFT [20,23–28] approaches to Kondophysics. The large- N approach involves replacing the SU (2) spin symmetry with SU ( N ) and thensending N → ∞ , keeping T K fixed. Following many previous large- N Kondo models [13, 17, 20, 35, 36],we restrict to an impurity spin in a totally anti-symmetric representation of SU ( N ), whose Youngtableau is a single column with Q < N boxes, and describe the impurity spin using Abrikosov pseudo-fermions, χ , constrained to obey χ † χ = Q . The Kondo coupling between the impurity spin and theelectrons is then of the form λ O † O , where λ is the Kondo coupling constant and O = ψ † χ , with ψ an electron. At large N , the screening of the impurity appears as the formation of the condensate (cid:104)O(cid:105) (cid:54) = 0 below a critical temperature T c (cid:39) T K [13,17,35,36]. We thus refer to the phases with (cid:104)O(cid:105) = 0and (cid:104)O(cid:105) (cid:54) = 0 as “unscreened” and “screened,” respectively. Crucially, the logarithmic rise of ρ with T , which normally occurs when T (cid:29) T K , is absent at large N . However, the large- N limit is useful atlow temperatures, T ≤ T K , where λ is large and hence conventional perturbation theory in λ breaksdown. When T (cid:28) T K , ρ exhibits power-law scaling in T , with a power determined by the dimensionof the leading irrelevant operator about the IR fixed point [20, 27, 28].The CFT approach to Kondo physics begins with the observation that the impurity couples only tothe electron s -wave spherical harmonic, so non-trivial physics only occurs in the radial direction aboutthe impurity [23, 25, 28]. The low-energy physics is therefore effectively one-dimensional. Linearizingabout the Fermi momentum then produces a relativistic electron dispersion relation, with the Fermivelocity playing the role of the speed of light. The low-energy effective theory thus consists of free,relativistic fermions in one dimension, interacting with the impurity at the origin. That theory isa boundary CFT, which has an infinite number of symmetry generators, namely those of a singleVirasoro algebra, plus Kac-Moody algebras for charge, spin, and channel (or flavor) [23, 28]. Theseinfinite accidental symmetries make the CFT approach very powerful. For example, together with theboundary conditions these symmetries determine the IR spectrum completely [23–25, 28]. The CFTapproach also provides novel results for low- T scaling exponents [23–25, 27, 28].Our holographic model combines the large- N and CFT approaches, and adds two more ingredients.First, we gauge the SU ( N ) spin symmetry, so that the impurity spin becomes an SU ( N ) Wilsonline. Second, we make the SU ( N ) ’t Hooft coupling large, so that the gauge degrees of freedom(adjoint fields) are strongly-interacting. These two ingredients are necessary to produce a tractablegravitational dual, with a small number of light fields in a classical limit. Indeed, all holographicquantum impurity models to date use these two ingredients, as reviewed in refs. [31, 32].2o be specific, our holographic model includes four fields. First is an asymptotically AdS metric, withEinstein-Hilbert action with negative cosmological constant, which is dual to the stress-energy tensor.Second is a Chern-Simons gauge field, A , dual to Kac-Moody currents, J , representing our electrons ψ . Third is a Maxwell gauge field, a , restricted to a co-dimension one, asymptotically AdS brane,localized in the field theory direction, and dual to the Abrikosov pseudo-fermion charge j ≡ χ † χ .Fourth is a complex scalar field, Φ, also restricted to the brane, charged under both A and a , and dualto O = ψ † χ . In refs. [31, 32] we treated the matter fields as probes of a BTZ black brane.Our model is a novel impurity RG flow in both holography and condensed matter physics. In holog-raphy, our model is novel as a holographic superconductor [37, 38] in an AdS subspace of a higher-dimensional AdS space. Indeed, a general lesson of our model is that holographic superconductors in AdS describe impurity screening. In condensed matter physics, our model describes a novel impurityRG flow between two strongly-interacting fixed points, unlike the original Kondo model, where theUV fixed point is trivial and the IR fixed point may or may not be trivial, depending on the numberof channels [24, 25, 28, 39]. More specifically, our λ runs in the same way as the original Kondo model,but our model has a second coupling, the ’t Hooft coupling, which does not run, and is large. Ourholographic model not only reproduces expected large- N Kondo physics, such as condensation of O ,screening of the charge Q , power-law scaling of ρ with T at low T [31], etc., but also exhibits novelphenomena due to the large ’t Hooft coupling, as described below.Indeed, using our holographic model, we have begun to address some of the open questions aboutKondo physics. For example, in ref. [32], we introduced a second impurity in our holographic model,as a first step towards building a holographic Kondo lattice. We found evidence that the competitionbetween Kondo and inter-impurity (RKKY) interactions may lead to a quantum phase transition. Inref. [33] we calculated the impurity entropy in our holographic model, by calculating the change inEE due to the impurity, for an interval of length (cid:96) centered on the impurity. Calculating the EEholographically required calculating the back-reaction of the AdS matter fields on the metric [33, 40].The impurity screening reduced the impurity entropy, i.e. reduced the number of impurity degreesof freedom, consistent with the g -theorem [26, 41]. On the gravity side, the reduction in degrees offreedom appeared as a reduction in the volume of the bulk spacetime around the AdS brane, similarto the deficit angle around a cosmic string. Furthermore, at low T the EE decayed exponentially in (cid:96) as (cid:96) increased. The decay rate provides one definition of the Kondo screening length, which made aparticularly intuitive appearance in the gravity theory, as a distance the AdS brane “bends.”In this paper, we take a first step toward addressing another major open problem in Kondo physics:out-of-equilibrium evolution. In particular, we work in the probe limit, and compute response func-tions, namely the retarded Green’s functions involving O , j , and J , in the regime of linear response tosmall, time-dependent perturbations. We then compute the spectral functions, i.e. the anti-Hermitianparts of the retarded Green’s functions. We also separately calculate the poles in the Green’s functions,dual to the quasi-normal modes (QNMs) of the fields in the BTZ black brane background. Generically,these poles give rise to peaks in the spectral functions.We presented some of our results in a companion paper [42]. In this paper we will present full detailsof calculations and further results. In particular, we have three main results.Our first main result is technical: we perform the holographic renormalization (holo-ren) [43–53]of our model. The main challenge here is the well-known fact that a YM field diverges near theasymptotically AdS boundary, unlike YM fields in higher-dimensional AdS spaces. That divergencecan alter the asymptotics of fields coupled to the YM field, and indeed alters the asymptotics of our3eld Φ. The asymptotic region is dual to the UV of the field theory [54], so we learn that in ourholographic model j acts like an irrelevant operator, and in particular, changing the value of (cid:104) j (cid:105) ,which controls the impurity’s spin, changes the dimension of O at the UV fixed point. Such behaviordoes not occur in non-holographic Kondo models, and so, by process of elimination, must be dueto the strongly-interacting degrees of freedom we added. Strong-coupling effects can also appear inthe IR, for example the leading irrelevant operator about the IR fixed point likely has non-integerdimension [31].Our holo-ren draws from, and extends, several previous examples of holo-ren: for fields dual to irrel-evant operators [55, 56], for our holographic two-impurity Kondo model [32], and for asymptoticallyconical (rather than asymptotically AdS) black holes [57]. The holo-ren provides covariant boundarycounterterms, enabling us to compute renormalized correlators, including the thermodynamic freeenergy and two-point functions. The holo-ren also allows us to identify the Kondo coupling λ from aboundary condition on Φ [31–34].As in many large- N Kondo models, our holographic model exhibits a large- N , second-order, mean-field phase transition [31–34]. For all T , one class of static solutions obeying the boundary conditionsincludes Φ = 0, dual to the unscreened phase, with (cid:104)O(cid:105) = 0. When T ≤ T c , another class of solutionsappears, with Φ (cid:54) = 0, dual to the screened phase, with (cid:104)O(cid:105) (cid:54) = 0. For all T ≤ T c , the Φ (cid:54) = 0 solutionhas lower free energy than the Φ = 0 solution, so a phase transition occurs at T c , with mean-fieldexponent: for T just below T c , (cid:104)O(cid:105) ∝ ( T c − T ) / [31].In the unscreened phase, the holo-ren reveals that the only non-trivial retarded Green’s function in ourmodel is (cid:104)O † O(cid:105) , with all other one- and two-point functions completely determined by (cid:104)O(cid:105) , the Wardidentities for the currents j and J , and the particle-hole transformation Q → N − Q . For example, (cid:104)OO † (cid:105) can be obtained from (cid:104)O † O(cid:105) by taking
Q → N − Q . We denote (cid:104)O † O(cid:105) ’s Fourier transform as G O † O , which we compute as a function of complex frequency ω , and the associated spectral functionas ρ O † O ≡ − G O † O , which we compute for real ω . We are able to compute G O † O analytically, byobtaining an exact solution to Φ’s Klein-Gordon equation (with gauge covariant derivatives) in AdS ,with boundary condition involving the Kondo coupling λ .The defect’s asymptotic AdS isometry is dual to a (0 + 1)-dimensional conformal symmetry. When T > T c , the only breaking of that conformal symmetry is through T and the running of λ . For staticsolutions, we can approach the UV fixed point by sending T → ∞ , which also sends λ → λ →
0, Φ’s boundary condition reduces to Dirichlet [31, 32], guaranteeingthat G O † O indeed takes the form required by (0 + 1)-dimensional conformal symmetry [58, 59].More generally, our model falls into one of the three known classes of models whose large- N fixed pointsexhibit (0 + 1)-dimensional conformal symmetry. The first are holographic AdS models, such as ourmodel. The second are large- N quantum impurity models, including large- N Kondo models (withoutholography) [20]. The third are so-called Sachdev-Ye-Kitaev (SYK) models, namely fermions on alattice without kinetic terms and with long-range many-body interactions, in a large- N limit [59–73].For all three classes, (0 + 1)-dimensional conformal symmetry completely determines any Green’sfunction, such as G O † O , in terms of scaling dimension and global symmetry charges [59, 67, 74].However, in our model, as T decreases and λ grows, (0+1)-dimensional conformal symmetry is broken.As T → T c from above, in the complex ω -plane the lowest pole in G O † O , meaning the pole closestto the origin, which we denote ω ∗ , moves towards the origin. When T = T c , ω ∗ reaches the origin,and when T < T c , ω ∗ moves into the upper half of the complex ω plane, signaling the instability of4he unscreened phase when T < T c , as expected [31]. In contrast, in the standard (non-holographic)Kondo model, at large N and at leading order in perturbation theory in λ , the lowest pole sits exactlyat the origin of the complex ω plane for all T ≥ T c [75]. By process of elimination, our results for themovement of ω ∗ must arise from the additional degrees of freedom of our holographic model, and inparticular must be a strong coupling effect, since we do not rely on perturbation theory in either λ orthe ’t Hooft coupling.A pole in a retarded Green’s function (for complex ω ) leads to a peak in the associated spectralfunction (for real ω ). Our second main result is for ρ O † O in the unscreened phase: ω ∗ produces theonly significant feature in ρ O † O , namely a peak, and specifically a Fano resonance . Fano resonancesoccur when one or more resonance appears within a continuum of states (in energy). In such cases,scattering states have two options: they can either scatter off the isolated resonance(s) ( resonantscattering ), or they can bypass these resonances ( non-resonant scattering ). The classic example islight scattering off the excited states of an atom. In spectral functions, the interference betweenthe two options leads to a Fano resonance, which generically is asymmetric, with a minimum and amaximum (see fig. 2 (a)), and is determined by three parameters: the position, the width, and the
Fano or asymmetry parameter, q , which controls the distance between the minimum and maximum.In physical terms, q is proportional to the probability of resonant scattering over the probability ofnon-resonant scattering. For an introduction to Fano resonances, see for example ref. [76].In our case, the continuum comes from the (0 + 1)-dimensional fixed point dual to the AdS subspace,where the scale invariance implies any spectral function must be power law in ω , i.e. a continuum.Our resonance arises from our relevant deformation, i.e. our Kondo coupling, which necessarily breaksscale invariance. Moreover, the asymmetry of our Fano resonances is possible because particle-holesymmetry is generically broken when |Q − N/ | (cid:54) = 0.We expect asymmetric Fano resonances in any system with the same three ingredients, namely aneffectively (0 + 1)-dimensional UV fixed point, resonances that appear when scale invariance is broken,and particle-hole symmetry breaking. In fact, Fano resonances have appeared in such systems, thoughthey are often not identified as such. For example, Fano resonances appear in spectral functions ofcharged bosonic operators in the non-holographic large- N Kondo model [20] and in holographic dualsof extremal AdS-Reissner-Nordstrom black branes, whose near-horizon geometry is
AdS [58, 59, 77].Indeed, we expect Fano resonances in AdS models generically, such as Sachdev-Ye-Kitaev models [59–62, 66, 73], if some deformation breaks scale invariance and produces a resonance. Specifically (0 + 1)dimensions is special because any resonance must necessarily be immersed in a continuum, unlikehigher dimensions, where the two may be separated in momentum and/or real space.Fano resonances have been produced experimentally in side-coupled QDs [76, 78], that is, by couplingthe discrete states in a QD to a continuum of states in a quantum wire. Crucially, however, in thesecases (0 + 1)-dimensional scale invariance apparently plays no role: before the coupling between QDand quantum wire, spectral functions on the QD would be a sum of Lorentzians, not a scale-invariantcontinuum. Our Fano resonances therefore have a different physical origin from those in QDs, and aremore characterisitc of (0 + 1)-dimensional fixed points, as explained above.In the screened phase, the symmetry breaking condensate (cid:104)O(cid:105) (cid:54) = 0 induces operator mixing, so thatgenerically all two-point functions are non-trivial. However, the holo-ren shows that all four scalarcorrelators are equivalent: G O † O = G OO † = G OO = G O † O † , so we will discuss only G O † O , which wecompute numerically. Our third main result is: in the screened phase, the lowest pole in G O † O , ω ∗ ,is purely imaginary, and moves down the imaginary axis as T decreases. In fact, ω ∗ ∝ − i (cid:104)O(cid:105) for T T c . The spectral function ρ O † O then exhibits a Fano resonance symmetric under ω → − ω .This result is consistent with expectations from the standard (non-holographic) Kondo model. Atfinite N , an essential feature of the Kondo effect is the Kondo resonance , a peak in the spectralfunction of the conduction electrons, with five characteristic features. First, for all T the peak islocalized in energy exactly at the Fermi energy. Second, for all T the peak is localized in real spaceat the impurity. Third, as T approaches T K from above, the peak’s height rises logarithmically in T . Fourth, when T reaches T K , the peak’s height saturates and remains for all lower T at a valuefixed by the impurity’s representation (the Friedel sum rule). Fifth, as T drops below T K and thencontinues to decrease, the peak narrows, and at T = 0 has width ∝ T K . The Kondo resonance isa many-body effect ( i.e. is not obvious from the Kondo Hamiltonian) signaling the emergence of thehighly-entangled state in which the conduction electrons act collectively to screen the impurity. Formore details about the Kondo resonance, see for example the textbooks refs. [29, 75, 79].The features of the Kondo resonance change in the large- N limit, as explained in ref. [75] and referencestherein. In particular, the Kondo resonance is absent in the unscreened phase ( T > T c ), and appearsonly in the screened phase ( T < T c ). If we introduce Abrikosov pseudo-fermions χ , then due tooperator mixing induced by (cid:104)O(cid:105) (cid:54) = 0, the Kondo resonance can be transmitted from the electronspectral function to other spectral functions. In particular, in G O † O the Kondo resonance appears asa pole of the form ω ∝ − i (cid:104)O(cid:105) . As mentioned above, for T just below T c , we indeed find a pole in G O † O of precisely that form, providing compelling evidence for a Kondo resonance in our model.This paper is organized as follows. In section 2 we review our holographic Kondo model. In section 3we perform the holo-ren of our model. In section 4 we review Fano resonances. We present our resultsfor the unscreened phase in section 5, and for the screened phase in section 6. We conclude in section 7with discussion of our results and suggestions for future research. As mentioned above, our holographic model combines the CFT and large- N approaches to the Kondoeffect. In this section we will review these briefly and then introduce the action and equations ofmotion of our holographic model, and the transition between the unscreened to screened phases. Formore details on the CFT, large- N , and holographic approaches to the Kondo effect, see refs. [31–34].The CFT approach to the Kondo effect [23–28] begins with a (1 + 1)-dimensional effective description:relativistic fermions that are free except for a Kondo interaction with the impurity at the boundary ofspace. In that description, left-moving fermions “bounce off” the boundary and become right-moving,interacting with the impurity in the process. By extending the half line to the entire real line, reflectingthe right-movers to the “new” half of the real line, and re-labeling them as left-movers, we obtain asimpler description: left-movers alone, interacting with the impurity at the origin. The Hamiltonian(density) is then, in units where the Fermi velocity acting as speed of light is unity, H = 12 π ψ † α i∂ x ψ α + λ δ ( x ) S A ψ † α T Aαβ ψ β , (2.1)where ψ † α creates a left-moving electron with spin α , λ is the classically marginal Kondo coupling, T Aαβ are the generators of the SU (2) spin symmetry ( A = 1 , ,
3) in the fundamental representation, and S A is the spin of the impurity, which is localized at x = 0, hence the δ ( x ). The left-moving fermions6orm a chiral CFT, invariant under a single Virasoro algebra as well as SU (2) and U (1) Kac-Moodyalgebras, representing spin and charge, respectively (the U (1) acts by shifting ψ α ’s phase). With k > SU (2) k × SU ( k ) × U (1).The one-loop beta function for λ is negative. As a result, a non-trivial RG flow occurs only for ananti-ferromagnetic Kondo coupling, λ >
0. Due to asymptotic freedom, the UV fixed point is atrivial chiral CFT, namely free left-moving fermions and a decoupled impurity. The Virasoro andKac-Moody symmetries and (trivial) boundary conditions then determine the spectrum of eigenstatescompletely [23–25, 28]. The IR fixed point will again be a chiral CFT, whose spectrum of eigenstatescan be obtained from those in the UV by fusion with the impurity representation [24].Our holographic Kondo model will also employ a large- N limit [17–22], which is based on replacingthe SU (2) spin symmetry with SU ( N ) and then sending N → ∞ with N λ fixed. In particular,we will employ the large- N description of the Kondo effect as symmetry breaking at the impurity’slocation [13, 17, 35, 36], which begins by writing S A in terms of Abrikosov pseudo-fermions, S A = χ † α T Aαβ χ β , (2.2)where χ † α creates an Abrikosov pseudo-fermion. We construct a state in the impurity’s Hilbert spaceby acting on the vacuum with a number Q of the χ † α . Because the χ † α anti-commute, such a state willbe a totally anti-symmetric tensor product of the fundamental representation of SU ( N ) with rank Q .To obtain an irreducible representation, we must fix the rank Q by imposing a constraint, χ † α χ α = Q . (2.3)Due to the anti-commutation, Abrikosov pseudo-fermions can only describe totally anti-symmetricrepresentations of SU ( N ), so that Q ∈ { , , , . . . , N } . Following our earlier work [31–34], we willonly consider totally anti-symmetric impurity representations.Plugging eq. (2.2) into the Kondo interaction term in eq. (2.1), and using χ α ’s anti-commutationrelations as well as the completeness relation satisfied by the fundamental-representation SU ( N )generators, T Aαβ T Aγδ = 12 (cid:18) δ αδ δ βγ − N δ αβ δ γδ (cid:19) , (2.4)we can re-write the Kondo interaction as λ S A ψ † γ T Aγδ ψ δ = λ (cid:16) χ † α T Aαβ χ β (cid:17) (cid:16) ψ † γ T Aγδ ψ δ (cid:17) = 12 λ (cid:18) −O † O + Q − Q N (cid:16) ψ † α ψ α (cid:17)(cid:19) , (2.5)where the scalar operator O ≡ ψ † α χ α is (0 + 1)-dimensional, i.e. is a function of time t only, because χ α cannot propagate away from the impurity’s location, x = 0. Clearly, O is a singlet of the spin SU ( N ) k symmetry, is in the same SU ( k ) N × U (1) representation as ψ † α , and has the same auxiliary U (1) charge as χ α . Classically ψ α has dimension 1 / χ α has dimension zero, so O has dimension1 /
2. The Kondo interaction eq. (2.5) is thus classically marginal, i.e. λ is classically dimensionless.We can introduce Abrikosov pseudo-fermions for any N , but let us now take the large- N limit. Ineq. (2.5) the Q and ( Q /N ) ψ † α ψ α terms are then sub-leading in N relative to the O † O term, so theKondo interaction reduces to − λ O † O /
2. The solution of the large- N saddle point equations reveals a7econd-order mean-field phase transition: below a critical temperature T c , on the order of but distinctfrom T K , (cid:104)O(cid:105) (cid:54) = 0 [13, 17, 35, 36], spontaneously breaking the channel symmetry down to SU ( k − U (1) charge and U (1) auxiliary symmetry down to the diagonal U (1). Of course, spontaneoussymmetry breaking in (0 + 1) dimensions is impossible for finite N : the phase transition is an artifactof the large- N limit. Corrections in 1 /N change the phase transition to a smooth cross-over [13]. Thelarge- N limit describes many characteristic phenomena of the Kondo effect only when T ≤ T c , where (cid:104)O(cid:105) (cid:54) = 0, including the screening of the impurity by the electrons, and a phase shift of the electrons.As described in section 1, to obtain a classical Einstein-Hilbert holographic Kondo model, we want tocombine the CFT and large- N approaches and gauge the SU ( N ) k spin symmetry, which introducesthe ’t Hooft coupling, which we want to be large. Of course, the SU ( N ) k symmetry is anomalous,and so should not be gauged. To suppress the anomaly, we work in the probe limit: when N → ∞ we hold k fixed, so that k (cid:28) N , and then compute expectation values only to order N . In the probelimit the SU ( N ) k anomaly does not appear [31, 80], so that in effect SU ( N ) k → SU ( N ).Each SU ( N )-invariant, single-trace, low-dimension ( i.e. dimension of order N ) operator is dual to afield in the gravity dual. The stress-energy tensor is dual to the metric. The SU ( N ) currents are not SU ( N )-invariant, and hence have no dual fields. The SU ( k ) N × U (1) Kac-Moody currents are dual toan SU ( k ) N × U (1) Chern-Simons gauge field [81], which we call A . The U (1) charge j = χ † α χ α is dualto a U (1) gauge field, which we call a , localized to x = 0. The complex scalar O is bi-fundamentalunder SU ( k ) N × U (1) and the U (1) with charge j , and is dual to a complex scalar field, Φ, alsolocalized to x = 0, and bi-fundamental under A and a . For simplicity, following refs. [31–34] we willtake k = 1, so that the SU ( k ) N × U (1) Kac-Moody symmetry reduces to U (1). The Chern-Simonsgauge field A is then Abelian, with field strength F = dA . Similarly, a has field strength f = da .To describe a (1 + 1)-dimensional CFT with non-zero T , we use the BTZ black brane metric (withasymptotic AdS radius set to unity), ds = 1 z (cid:18) h ( z ) d z − h ( z )d t + d x (cid:19) , h ( z ) = 1 − z z H , (2.6)where z is the radial coordinate, with the boundary at z = 0 and horizon at z = z H , t and x arethe CFT time and space directions, and µ, ν = z, t, x . The CFT’s temperature is dual to the blackbrane’s Hawking temperature, T = 1 / (2 πz H ). The fields a and Φ are localized to x = 0, i.e. to thesubmanifold spanned by t and z , whose induced metric is asymptotically AdS , ds AdS = g mn dx m dx n = 1 z (cid:18) h ( z ) d z − h ( z )d t (cid:19) , (2.7)where m, n = t, z . The determinant of the metric in eq. (2.7) is g = − /z .The classical action of the holographic Kondo model of refs. [31–34] is the simplest action quadraticin the fields. We will split the bulk action into two terms, namely the Chern-Simons action for A , S CS , and the bulk terms for the fields in the asymptotically AdS submanifold, S AdS , S = S CS + S AdS , (2.8a) S CS = − N π (cid:90) AdS A ∧ d A, (2.8b)8 AdS = − N (cid:90) AdS d x √− g (cid:20) f mn f mn + ( D m Φ) † ( D m Φ) + M Φ † Φ (cid:21) , (2.8c)where D m is a gauge-covariant derivative, D m Φ = ( ∂ m + iA m − ia m ) Φ , (2.9)and M is Φ’s mass-squared. We will discuss the value of M , and the boundary terms that mustbe added to S for holo-ren, in section 3. We will also discuss the equations of motion following fromeq. (2.8), and their solutions, in detail in section 3. In the remainder of this section we will focus onfeatures of the equations of motion and their solutions relevant for our model’s phase structure.We split Φ into a modulus and phase, Φ = e iψ φ . Furthermore, throughout this paper we work in agauge with A z = 0 and a z = 0. As shown in refs. [31–34], a self-consistent gauge choice and ansatzthat can describe a static state with Q (cid:54) = 0 and possibly (cid:104)O(cid:105) (cid:54) = 0 includes A x ( z ), a t ( z ), and φ ( z ), withall other fields set to zero. The equations of motions for these fields are ∂ z A x = − πδ ( x ) √− g g tt a t φ , (2.10a) ∂ z (cid:0) √− g g zz g tt ∂ z a t (cid:1) = 2 √− g g tt a t φ , (2.10b) ∂ z (cid:0) √− g g zz ∂ z φ (cid:1) = √− g g tt a t φ + √− g M φ. (2.10c)Crucially, A x ( z ) does not appear in a t ( z ) or φ ( z )’s equation of motion, eqs. (2.10b) and (2.10c). As aresult, the only way that a t ( z ) and φ ( z ) “know” they live on a defect in a higher-dimensional spacetimeis through the blackening factor, h ( z ). In particular, if T = 0 then the defect’s metric is precisely thatof AdS . Moreover, A x ( z ) has trivial dynamics (as expected for a Chern-Simons gauge field): we onlyneed to solve for a t ( z ) and φ ( z ), and then insert those solutions into eq. (2.10a) to obtain A x ( z ).As mentioned in section 1, our holographic Kondo model exhibits a phase transition as T decreasesthrough a critical temperature T c , just like the standard (non-holographic) Kondo model at large N .For any T , eqs. (2.10b) and (2.10c) admit the solution a t ( z ) = µ − Q/z and φ ( z ) = 0. These solutionsare dual to states with (cid:104)O(cid:105) = 0. However, when T ≤ T c a second branch of solutions exists thathave φ ( z ) (cid:54) = 0. Given that φ ( z ) is dual to O † + O , these φ ( z ) (cid:54) = 0 solutions are dual to states with (cid:104)O † + O(cid:105) (cid:54) = 0, which implies (cid:104)O † (cid:105) = (cid:104)O(cid:105) = (cid:104)O † + O(cid:105) / (cid:54) = 0. We will therefore just refer to (cid:104)O(cid:105) (cid:54) = 0henceforth. To determine which state is thermodynamically preferred, we must determine which statehas lower free energy F , which we compute holographically from the on-shell Euclidean action: fordetails, see refs. [31–34]. Fig. 1 (a) shows F / ( N (2 πT )) as a function of T /T c for Q = 0 .
5, for the twobranches of solutions. Clearly the solution with φ ( z ) (cid:54) = 0 has lower F , and hence is thermodynamicallypreferred, for all T ≤ T c . Fig. 1 (b) shows our numerical results for κ/ (2 N ) (cid:104)O(cid:105) / √ T c as a functionof T /T c for Q = 0 .
5, where κ is our holographic Kondo coupling constant, defined in the boundaryterm eq. (3.61). Fig. 1 (b) also shows a numerical fit revealing second-order mean-field behavior: (cid:104)O(cid:105) ∝ ( T c − T ) / when T (cid:46) T c . Clearly our model exhibits a second-order mean-field transitionwhen T drops through T c . In section 5 we will show T c ∝ T K , where the proportionality constantdepends only on Q : see in particular fig. 6.As mentioned above, at large N the screening of the impurity, and other characteristic Kondo phe-nomena, such as a phase shift of the electrons, occurs only when T ≤ T c , where (cid:104)O(cid:105) (cid:54) = 0. We will thusrefer to states with (cid:104)O(cid:105) = 0 as the unscreened phase and states with (cid:104)O(cid:105) (cid:54) = 0 as the screened phase.9 .2 0.4 0.6 0.8 1.00.050.100.150.200.250.30 - - - - - - - - (a) (b) F N (2 ⇡T ) TT c TT c N hOip T c Figure 1: (a) The free energy F , normalized by 1 / ( N (2 πT )), as a function of T /T c for Q = 0 .
5. The solid line is for the unscreened phase, where (cid:104)O(cid:105) = 0, and which has F / ( N (2 πT )) = − Q / − . (cid:104)O(cid:105) (cid:54) = 0. Clearly the screened phase always has lower F , and henceis thermodynamically preferred, for all T ≤ T c . (b) The dots are our numerical results for κ N (cid:104)O(cid:105) / √ T c as a function of T /T c in the screened phase with Q = 0 .
5. The solid line isa numerical fit to 0 .
30 ( T c − T ) / . The agreement between the numerical results and thefit indicates second-order mean field behavior, (cid:104)O(cid:105) ∝ ( T c − T ) / .What does the screening look like on the gravity side of the correspondence? The flux of a t ( z ) controlsthe “size” of the impurity’s representation, by controlling the number of boxes in the associated Youngtableau. To see how, consider the a t ( z )’s general asymptotic form, a t ( z ) = µ − Q/z + . . . , where . . . represents terms that vanish as z →
0. The parameter µ acts as a chemical potential for j = χ † χ ,and in particular a non-zero µ breaks particle-hole symmetry. The particle-hole symmetric value ofthe charge is Q = N/
2, which thus corresponds to µ = 0. In general the parameter Q dependsmonotonically on µ . For example, for the solution a t ( z ) = µ − Q/z mentioned above, regularity of a t ( z ) at the horizon, a t ( z H ) = 0, requires Q = µz H . As a result, Q = 0 corresponds to Q = N/ Q > Q > N/
2, and
Q < Q < N/
2. A totally anti-symmetricrepresentation must have 0 ≤ Q ≤ N , which should translate to limits on Q . Our model is too crude todetermine the exact relation between Q and Q , and includes nothing to impose limits on Q , althoughthese features could potentially be incorporated, following similar models [82–85]. They only featurewe will need, however, is that Q is monotonically related to Q − N/ a t ( z ) at the boundary is Q . When φ ( z ) = 0, the flux of a t ( z ) is constantfrom the boundary to the horizon. However, when φ ( z ) (cid:54) = 0, the flux of a t ( z ) is transferred from a t ( z ) to A x ( z ), because Φ is bi-fundamental. Recalling that the holographic coordinate z correspondsto energy scale, where the boundary corresponds to the UV and increasing z corresponds to movingtowards the IR [86, 87], solutions with φ ( z ) (cid:54) = 0 thus describe an impurity whose size shrinks as wemove towards the IR [31]. In other words, the impurity is screened, as advertised.What does the phase shift look like on the gravity side of the correspondence? The phase shift isencoded in A x ( z ) [31]. In particular, eq. (2.10a) shows that ∂ z A x ( z ) (cid:54) = 0 if and only if both a t (cid:54) = 0 and10 ( z ) (cid:54) = 0. If we imagine compactifying x into a circle, then A x ( z ) (cid:54) = 0 implies a non-zero Wilson looparound the x direction, (cid:72) A (cid:54) = 0, which is dual to a phase shift for our strongly-coupled “electrons,”or more generally for any object charged under our U (1) channel symmetry. Non-zero ∂ z A x ( z ) meansthe phase shift grows as we move towards larger z , i.e. as we move towards the IR, as expected.In short, our holographic model captures some of the essential phenomena of the large- N Kondo effect,namely impurity screening and a phase shift at T ≤ T c , when (cid:104)O(cid:105) (cid:54) = 0. In the following we will showthat our holographic model also captures another essential phenomenon: the Kondo resonance. In this section we derive general expressions for the renormalized holographic two-point functions ofthe Kondo model described by the action in eq. (2.8), in both the unscreened and screened phases.Before we embark on the technical aspects of this calculation, it is instructive to outline the mainsteps involved, and to highlight several subtleties that this specific model presents.A particularly economical way of computing holographic two-point functions is to read them off directlyfrom the linearized fluctuation equations, bypassing the usual step of evaluating the on-shell actionto quadratic order in the fluctuations. This is possible due to the holographic identification of theradial canonical momenta, which on-shell become functions of the induced fields, with the one-pointfunctions of the dual operators in the theory with Dirichlet boundary conditions [51]. To obtainthe two-point functions it suffices to expand the canonical momenta to linear order in the inducedfields. As in standard linear response theory, the coefficients of the linear terms in this expansion areidentified with the corresponding response functions, i.e. the unrenormalized two-point functions [88].Inserting the covariant expansions of the canonical momenta to linear order in the fluctuations in thesecond order fluctuation equations results in a system of first order non-linear Riccati equations forthe response functions [88, 89]. Like the system of second order linear equations for the fluctuations,the system of Riccati equations for the response functions is generically coupled, and can only besolved numerically. However, in contrast to the second order linear equations, the general solution ofthe Riccati equations contains only one integration constant per response function, since the arbitrarysources have already been eliminated, which is determined by imposing regularity in the bulk of thespacetime. Generically, the fact that the arbitrary sources have been eliminated from the Riccatiequations renders them better suited for a numerical evaluation of the two-point functions.Both the on-shell action and the response functions obtained from the Riccati equations are gener-ically divergent and need to be evaluated with a radial cutoff near the AdS boundary. Moreover,local covariant boundary counterterms need to be determined in order to renormalize these quanti-ties. However, two important subtleties arise in obtaining the correct boundary counterterms in ourholographic Kondo model, both directly related to the special asymptotic behavior of the
AdS gaugefield. In contrast to gauge fields in AdS and above, in AdS and AdS the asymptotically leadingmode of an abelian gauge field is the conserved charge Q , instead of the chemical potential, µ [90]. Thesame phenomenon is observed with higher rank antisymmetric p -forms in higher dimensions [91]. Insuch cases, consistency of the boundary counterterms requires that they be a function of the canonicalmomentum conjugate to the gauge field, rather than the gauge field itself [57, 90].Moreover, the requirement that the charge Q be kept fixed leads to an asymptotic second class con-straint in phase space, which further complicates the computation of the boundary counterterms [90].11elaxing the constraint, i.e. changing the value of Q in this case, changes the form of the asymptoticsolutions for the scalar field. In order to have a well-defined space of asymptotic solutions, therefore,we must restrict the phase space asymptotically to the subspace defined by constant Q . However, if wewant to compute correlation functions of the operator dual to Q , which as we will discuss later is not alocal operator, then we must allow for infinitesimal deformations away from the asymptotic constraintsurface. The boundary counterterms then take the form of a Taylor expansion in the infinitesimaldeformation away from the constraint surface, with the coefficient of the n -th power renormalizing the n -point function of the operator dual to Q .In our holographic Kondo model, a further complication arises due to the double-trace boundaryconditions we need to impose on the scalar field in order to introduce the Kondo coupling. The responsefunctions obtained directly from the Riccati equations correspond to the two-point functions in thetheory defined by Dirichlet boundary conditions on the scalar field and Neumann boundary conditionson the AdS gauge field, i.e. keeping Q fixed. In the large- N limit, however, the renormalized two-point functions in the theory with double-trace boundary conditions on the scalar field are algebraicallyrelated to those in the theory with Dirichlet boundary conditions on the scalar field. The preciserelation is obtained by identifying additional finite boundary terms required to impose the double-trace boundary condition on the scalar field, and then carefully examining the variational problem.In this section we will address all the above subtleties as we go along. We start by reformulating theKondo model in eq. (2.8) in radial Hamiltonian formalism, which allows us to introduce the radialcanonical momenta, the linear response functions, and the Hamilton-Jacobi equation we must solvein order to determine the boundary counterterms. We then proceed to derive the Riccati equationsfor the linear response functions, determine their general asymptotic solutions in the UV ( i.e. nearthe asymptotically AdS boundary), and determine the most general regular asymptotic solution inthe IR ( i.e. deep in the bulk). The arbitrary integration constants appearing in the UV expansionsparameterize the renormalized two-point functions, and their value is determined by matching thesolution, numerically, to the regular asymptotic solution in the IR. Subsequently we determine theboundary counterterms necessary to renormalize the free energy, as well as the one- and two-pointfunctions in the theory with Dirichlet boundary conditions on the scalar field. Finally, the renormalizedtwo-point functions with a non-zero Kondo coupling are obtained by adding further boundary termsthat implement the double-trace boundary condition on the scalar field. To describe our holographic Kondo model in radial Hamiltonian language, we re-write the inducedmetric in eq. (2.7) in the form ds AdS = d r + γ d t , (3.1)where the radial coordinate z of eq. (2.7) is related to the canonical radial coordinate r of eq. (3.1) as r = log (cid:16) (cid:113) − z /z H (cid:17) − log(2 z ) , (3.2)with r ∈ [ r H , + ∞ ), and r H = − log(2 z H ), and asymptotically, γ = − e r + O (1) as r → + ∞ .12n these coordinates the action in eq. (2.8) may be written as S = − N π (cid:90) d x ¯ (cid:15) ij ( − A i ˙ A j + 2 A r ∂ [ i A j ] ) − N (cid:90) d t √− γ (cid:18) γ − f rt + | D r Φ | + γ − | D t Φ | + M Φ † Φ (cid:19) , (3.3)where ( i, j ) = ( t, x ), a dot denotes differentiation with respect to r , ˙ A j ≡ ∂ r A j , and ¯ (cid:15) ij ≡ (cid:15) zij . Fromeq. (3.3) we obtain the radial canonical momenta: π iA = δSδ ˙ A i = − N π ¯ (cid:15) ij A j , π ta = δSδ ˙ a t = − N √− γ γ − ( ˙ a t − ∂ t a r ) ,π Φ = δSδ ˙Φ = − N √− γ ( D r Φ) † , π Φ † = δSδ ˙Φ † = − N √− γ D r Φ . (3.4)In terms of the modulus and phase, Φ = e iψ φ , the scalar field’s canonical momenta become π φ = δSδ ˙ φ = − N √− γ ˙ φ, π ψ = δSδ ˙ ψ = − N √− γ φ ( A r − a r + ˙ ψ ) . (3.5)No radial derivatives of the components a r and A r appear in eq. (3.3), so they correspond to non-dynamical Lagrange multipliers. Moreover, the canonical momentum of the Chern-Simons field ineq. (3.4) amounts to a primary constraint, which implies that the canonical momentum and the gaugeconnection A i are not independent variables on phase space.The Legendre transform of the action in eq. (3.3) gives the radial Hamiltonian, H = (cid:90) d x ˙ A i π iA + (cid:90) d t ( ˙ a t π ta + ˙ φπ φ + ˙ ψπ ψ ) − S = (cid:90) d x A r (cid:18) − π ψ δ ( x ) + N π ¯ (cid:15) ij ∂ [ i A j ] (cid:19) + (cid:90) d t a r (cid:0) − ∂ t π ta + π ψ (cid:1) − N (cid:90) d t √− γ (cid:18) γ ( π ta ) + 14 π φ + 14 φ − π ψ (cid:19) + N (cid:90) d t √− γ (cid:16) γ − ( ∂ t φ ) + γ − φ ( A t − a t + ∂ t ψ ) + M φ (cid:17) . (3.6)Hamilton’s equations for the non dynamical fields a r and A r result in the first class constraints π ψ = i (Φ π Φ − Φ † π Φ † ) = ∂ t π ta , N π ¯ (cid:15) ij ∂ [ i A j ] = − ∂ i π iA = π ψ δ ( x ) , (3.7)which reflect the U (1) gauge invariances associated with the AdS and Chern-Simons gauge fields,respectively. We will see below that these constraints lead to Ward identities in the dual field theory.Hamilton-Jacobi theory connects the canonical momenta with the regularized on-shell action S throughthe relations π iA = δ S δA i , π ta = δ S δa t , π Φ = δ S δ Φ , π Φ † = δ S δ Φ † , (3.8) The expression in eq. (3.4) for the Chern-Simons momentum implies that S cannot be a local covariant functional of A i . This is consistent with the fact that A i parameterizes the full phase space, and only a particular component of A i ,depending on the boundary conditions, will be identified with the source of the dual current operator.
13r for the modulus and phase of the scalar field, π φ = δ S δφ and π ψ = δ S δψ . The regularized on-shell action S , also known as Hamilton’s principal function in this context, is identified via the holographic dictio-nary with the regularized generating function of connected correlation functions in the theory definedby Dirichlet boundary conditions on the scalar and Chern-Simons fields, and Neumann boundaryconditions on the AdS gauge field. The canonical momenta, therefore, correspond to the regularizedone-point functions with arbitrary sources. The regularized two-point functions are thus obtained bydifferentiation of the canonical momenta with respect to the induced fields. As we will see in the nextsubsection, this property allows us to rewrite the fluctuation equations in terms of Riccati equations,which are first order, and whose solution gives directly the regularized two-point functions.Since S is identified with the regularized on-shell action as a function of the induced fields on aradial cutoff, its divergent asymptotic form determines the boundary counterterms that are requiredto renormalize the theory. The asymptotic form of S can be determined in covariant form by solvingthe radial Hamilton-Jacobi equation H + ∂ S ∂r = 0 ⇔ H + ˙ γ δ S δγ = 0 , (3.9)or more explicitly − N (cid:90) d t √− γ (cid:32) γ (cid:18) δ S δa t (cid:19) + 14 (cid:18) δ S δφ (cid:19) + 14 φ − (cid:18) δ S δψ (cid:19) (cid:33) + N (cid:90) d t √− γ (cid:16) γ − ( ∂ t φ ) + γ − φ ( A t − a t + ∂ t ψ ) + M φ (cid:17) + ˙ γ δ S δγ = 0 , (3.10)together with the two constraints δ S δψ = ∂ t (cid:18) δ S δa t (cid:19) , δ ( x ) δ S δψ = N π ¯ (cid:15) ij ∂ [ i A j ] = − ∂ i (cid:18) δ S δA i (cid:19) , (3.11)which reflect the U (1) gauge invariances associated with the AdS and Chern-Simons gauge fields,respectively. In this subsection we use the relation between the radial canonical momenta and the one-point func-tions in order to rewrite the second order fluctuation equations in the form of Riccati equations, whichare first order. For convenience, we will work with the complex scalar field Φ and its complex conju-gate Φ † , rather than its modulus and phase. In the coordinates of eq. (3.1), and in the radial gauge A r = a r = 0, the equations of motion associated with the action in eq. (3.3) are12 π ¯ (cid:15) ij ∂ i A j + δ ( x ) √− γ J r = 0 , (3.12a)12 π ¯ (cid:15) ij ˙ A j − δ ( x ) δ it √− γ γ − J t = 0 , (3.12b) ∂ r ( √− γ γ − ˙ a t ) + √− γ γ − J t = 0 , (3.12c) γ − ∂ t ˙ a t − J r = 0 , (3.12d) ∂ r ( √− γ ˙Φ) + √− γγ − ( ∂ t + i ( A t − a t )) Φ − √− γ M Φ = 0 , (3.12e)14here we have defined a current associated with Φ, J m ≡ − i (cid:16) Φ † D m Φ − ( D m Φ) † Φ (cid:17) . (3.13)We solve first for the Chern-Simons gauge field. Eliminating J r from eqs. (3.12a) and (3.12d) and J t from eqs. (3.12b) and (3.12c), results respectively in the two conditions ∂ r (cid:0) ¯ (cid:15) ij A j + 2 πδ ( x ) δ it √− γ γ − ˙ a t (cid:1) = 0 , (3.14a) ∂ i (cid:0) ¯ (cid:15) ij A j + 2 πδ ( x ) δ it √− γ γ − ˙ a t (cid:1) = 0 . (3.14b)The general solution for the Chern-Simons gauge field thus takes the form¯ (cid:15) ij A j = 2 πN π ta δ ( x ) δ it + ¯ (cid:15) ij A (0) j ( t, x ) , (3.15)where A (0) i ( t, x ) is a flat connection on the AdS boundary, i.e. ¯ (cid:15) ij ∂ i A (0) j = 0. This implies thatthe two components A (0) i ( t, x ) are not both arbitrary sources, in contrast to what happens for aMaxwell gauge field. As we shall see below, in order to obtain a well-defined variational problem forthe Chern-Simons gauge field, we must add the appropriate boundary term [81, 92–95].In our model, a key observation that will play a role in the choice of boundary conditions for theChern-Simons gauge field is that the AdS fields source only A x , while A t is independent of the radialcoordinate. This implies that we can use a residual U (1) gauge transformation, i.e. preserving theradial gauge A r = 0, to set A t to zero, so that the Chern-Simons gauge field decouples from theequations of motion for the AdS fields. In that choice of gauge, the Chern-Simons gauge field takesthe simple form A x = − πδ ( x ) √− γ γ − ˙ a t + A (0) x , A t = A (0) t = 0 , (3.16)where A (0) x is a function of x only, but is otherwise arbitrary. However, when we discuss the variationalproblem for the Chern-Simons gauge field, we will reinstate A (0) t .We now solve for the AdS fields. We want to find a real and static background solution, and thenconsider time-dependent fluctuations about that solution. The most generic real and static backgroundsolution includes a t ( r ) and φ ( r ), whose equations of motion are¨ a t − γ − ˙ γ ˙ a t − a t φ = 0 , (3.17a)¨ φ + 12 γ − ˙ γ ˙ φ − ( γ − ( a t ) + M ) φ = 0 . (3.17b)We have been able to solve these equations analytically ( i.e. without numerics) only for φ ( r ) = 0.Solutions with φ ( r ) (cid:54) = 0 were obtained numerically in refs. [31, 32].We now introduce fluctuations δa t , δ Φ, and δ Φ † about the static background solution, linearize theirequations of motion, and Fourier transform from time t to frequency ω via ∂ t → − iω , to obtain ωγ − δ ˙ a t = φ ( δ ˙Φ − δ ˙Φ † ) − ˙ φ ( δ Φ − δ Φ † ) (3.18a) δ ¨Φ + 12 γ − ˙ γδ ˙Φ − γ − ( ω + a t ) δ Φ − M δ Φ = γ − φ ( ω + 2 a t ) δa t , (3.18b) δ ¨Φ † + 12 γ − ˙ γδ ˙Φ † − γ − ( − ω + a t ) δ Φ † − M δ Φ † = γ − φ † ( − ω + 2 a t ) δa t . (3.18c)We will consider these equations in the unscreened and screened phases separately.15 .2.1 Response Functions in the Unscreened Phase In the unscreened phase, where φ = 0, eq. (3.18a) becomes trivial, and eqs. (3.18b) and (3.18c)decouple. These second-order equations for the fluctuations δ Φ and δ Φ † can be turned into first-order equations for the two-point functions as follows. The canonical momenta in eqs. (3.4) and (3.8)imply that on-shell the radial velocities become functions of the induced fields. To linear order in thefluctuations we thus have δ ˙Φ = R Φ † Φ δ Φ , δ ˙Φ † = R ΦΦ † δ Φ † , (3.19)where the response functions R Φ † Φ and R ΦΦ † depend only on the background a t and φ , as well as ω .Hermitian conjugation implies that R † ΦΦ † ( ω ) = R Φ † Φ ( − ω ). Inserting these expressions into the twodecoupled fluctuation equations, eqs. (3.18b) and (3.18c), leads to the two Riccati equations [88, 89]˙ R Φ † Φ + 12 γ − ˙ γ R Φ † Φ + R † Φ − γ − (cid:0) ω + a t (cid:1) − M = 0 , (3.20a)˙ R ΦΦ † + 12 γ − ˙ γ R ΦΦ † + R † − γ − (cid:0) ω − a t (cid:1) − M = 0 . (3.20b)Using eq. (3.2) to change the radial coordinate from r back to z , and using the solution for thebackground gauge field a t = Q (1 /z − /z H ) these Riccati equations become − zh / R (cid:48) Φ † Φ + 12 h − / (2 h − zh (cid:48) ) R Φ † Φ + R † Φ + z h − ( ω + Q (1 /z − /z H )) − M = 0 , (3.21a) − zh / R (cid:48) ΦΦ † + 12 h − / (2 h − zh (cid:48) ) R ΦΦ † + R † + z h − ( ω − Q (1 /z − /z H )) − M = 0 , (3.21b)where primes denote ∂ z , for example R (cid:48) Φ † Φ ≡ ∂ z R Φ † Φ .We want to solve eqs. (3.21) with in-going boundary conditions at the horizon. Eqs. (3.21) can besolved analytically, either directly, or by first transforming them into second-order linear homogeneousequations through the change of variables R Φ † Φ = − z h / y (cid:48) + /y + , R ΦΦ † = − z h / y (cid:48)− /y − , (3.22)where the functions y ± satisfy the second order equations y (cid:48)(cid:48)± + 2 zz − z H y (cid:48)± + (cid:32) ( ω ± Q (1 /z − /z H )) (1 − z /z H ) − ν + Q − / z (1 − z /z H ) (cid:33) y ± = 0 , (3.23)where ν ≡ (cid:112) M − Q + 1 /
4. The two linearly independent solutions of eq. (3.23) are y ± ( z, ω ; ν ) and y ± ( z, ω ; − ν ) where y ± ( z, ω ; ν ) = ( z/z H ) + ν (1 − z/z H ) iωzH (1 + z/z H ) + ν + iωzH F (cid:18)
12 + ν ∓ iQ + iωz H ,
12 + ν ± iQ, ν ; 2 zz + z H (cid:19) . (3.24) Eq. (3.23) is identical to the equation of motion in ref. [77] (after their eq. (5.20)), with the identifications ζ = z H , qe d = ± Q , mR = M . y ± ( z, ω ; ν ) and y ± ( z, ω ; − ν ) that satisfies in-going boundary condition atthe horizon is y in ± ( z, ω ; ν ) = 1 ν (cid:16) y ± ( z, ω ; ν ) + c ± ( ω ; ν ) y ± ( z, ω ; − ν ) (cid:17) , (3.25a) c ± ( ω ; ν ) ≡ − Γ(1 + 2 ν )Γ( − ν ± iQ − iωz H )Γ( − ν ∓ iQ )2 ν Γ(1 − ν )Γ( + ν ± iQ − iωz H )Γ( + ν ∓ iQ ) . (3.25b)The general in-going solutions of eqs. (3.21) are therefore R Φ † Φ = − z h / y (cid:48) in + ( z, ω ; ν ) y in + ( z, ω ; ν ) , R ΦΦ † = − z h / y (cid:48) in − ( z, ω ; ν ) y in − ( z, ω ; ν ) . (3.26)As explained in detail in refs. [31–34], to guarantee that O is dimension 1 /
2, and hence our Kondocoupling O † O is classically marginal, we must choose M = − / Q , so that ν = 0. In the limit ν →
0, the solution in eq. (3.25) has the asymptotic behavior y in ± ( z, ω ; 0) = 2 z / (log( z/z H ) + Θ ± ( ω )) + . . . , (3.27)where . . . represents terms that vanish faster than those shown as z →
0, andΘ ± ( ω ) ≡ H (cid:18) − ± iQ − iωz H (cid:19) + H (cid:18) − ∓ iQ (cid:19) + log 2 , (3.28)and H ( n ) denotes the n th harmonic number. The response functions’ asymptotic expansions are then R Φ † Φ = − − z/z H ) + Θ + ( ω ) + O ( z ) , R ΦΦ † = − − z/z H ) + Θ − ( ω ) + O ( z ) . (3.29)One of our main tasks in the remainder of this section is to determine how the coefficients in theasymptotic expansion in eq. (3.29) can be translated into the two-point functions of O and O † . In the screened phase, where φ (cid:54) = 0, eqs. (3.18) are three coupled equations for the three fluctuations.They can be turned into a system of coupled Riccati equations by introducing response functions as δ ˙Φ = R Φ † Φ δ Φ + R Φ † Φ † δ Φ † + γ − R Φ † a δa t , (3.30a) δ ˙Φ † = R ΦΦ δ Φ + R ΦΦ † δ Φ † + γ − R Φ a δa t . (3.30b)We could similarly introduce response functions for δ ˙ a t , however eq. (3.18a) implies that they arecompletely determined by the response functions in eq. (3.30). Inserting eq. (3.30) into the fluctuationequations eqs. (3.18b) and (3.18c) leads to a system of six coupled Riccati equations.Although the six response functions defined in eq. (3.30) will be useful for extracting the two-pointfunctions in the following, we will now show that in fact they can be mapped to only four independentresponse functions. The in-going boundary condition then forces one of these four to vanish identically,leaving only three non-trivial, independent response functions.17e first re-express eq. (3.18) in terms of the fluctuations of the modulus and phase, δφ and δψ ,respectively, which leads to two coupled second-order equations for the gauge invariant fluctuations,( δa t + iωδψ ) and δφ : ∂ r (cid:16) δ ˙ a t + iωδ ˙ ψ γ − φ − ω (cid:17) − γ − ˙ γ (cid:16) δ ˙ a t + iωδ ˙ ψ (cid:17) γ − φ − ω − φ ( δa t + iωδψ ) = 4 φ a t δφ, (3.31a) δ ¨ φ + 12 γ − ˙ γδ ˙ φ − γ − ω δφ − (cid:16) M + γ − (cid:0) a t (cid:1) (cid:17) δφ = 2 φ γ − a t ( δa t + iωδψ ) . (3.31b)Given a solution for ( δ ˙ a t + iωδ ˙ ψ ), we can extract δa t , and hence also δψ , by re-writing eq. (3.18a) as δ ˙ a t = 11 + γ − φ − ω (cid:16) δ ˙ a t + iωδ ˙ ψ (cid:17) . (3.32)We can turn eq. (3.31) into a system of Riccati equations by introducing four response functions, δ ˙ a t = R ( δa t + iωδψ ) + γ R δφ, δ ˙ φ = 12 ( R + γ R ) γ − ( δa t + iωδψ ) + 12 R δφ, (3.33)where, with the benefit of hindsight, we have parameterized δ ˙ φ so that R will satisfy a homogeneousequation. Using the identities δ Φ = δφ + iφ δψ and δ Φ † = δφ − iφ δψ , we can express the six responsefunctions introduced in eq. (3.30) in terms of only four independent response functions, namely thosein eq. (3.33), as advertised: R Φ a = 12 (cid:0) R + γ R − ωφ − R (cid:1) , R Φ † a = 12 (cid:0) R + γ R + ωφ − R (cid:1) (3.34a) R ΦΦ = 14 (cid:16) R − ω γ − φ − R − φ − ˙ φ + ωγ − φ − R (cid:17) , (3.34b) R Φ † Φ † = 14 (cid:16) R − ω γ − φ − R − φ − ˙ φ − ωγ − φ − R (cid:17) , (3.34c) R Φ † Φ = 14 (cid:16) R + 2 ωφ − R + ω γ − φ − R + 2 φ − ˙ φ + ωγ − φ − R (cid:17) , (3.34d) R ΦΦ † = 14 (cid:16) R − ωφ − R + ω γ − φ − R + 2 φ − ˙ φ − ωγ − φ − R (cid:17) . (3.34e)Inserting eq. (3.33) into eqs. (3.31) then leads to Riccati equations˙ R − γ − ˙ γ R + (cid:16) ω γφ (cid:17) R + 12 R ( R + γ R ) − φ = 0 , (3.35a)˙ R + 12 γ − ˙ γ R + (cid:16) ω γφ (cid:17) R R + 12 R R − φ γ − a t = 0 , (3.35b)˙ R + 12 γ − ˙ γ R + (cid:16) ω γφ (cid:17) R ( R + γ R ) + 12 R − (cid:0) M + γ − ( a t ) + γ − ω (cid:1) = 0 , (3.35c)˙ R − (cid:18) γ − ˙ γ − (cid:16) ω γφ (cid:17) R − R (cid:19) R = 0 . (3.35d)We can solve eq. (3.35d) by direct integration, R = C ( ω ) √− γ exp (cid:18) − (cid:90) d r (cid:48) (cid:20)(cid:16) ω γφ (cid:17) R + 12 R (cid:21)(cid:19) , (3.36)18here C ( ω ) is an integration constant. In appendix A we show that the in-going boundary conditionsfor the fluctuations on the horizon require C ( ω ) = 0, and hence R = 0. We have thus shown that onlythree non-trivial, independent response functions remain, as advertised. Setting R = 0, and usingeq. (3.2) to change the radial coordinate from r to z , eqs. (3.35) become − zh / R (cid:48) − (2 h − zh (cid:48) )2 h / R + (cid:18) − z ω hφ (cid:19) R − hz − R − φ = 0 , (3.37a) − zh / R (cid:48) + (2 h − zh (cid:48) )2 h / R + (cid:18) − z ω hφ (cid:19) R R + 12 R R + 4 φ z h − a t = 0 , (3.37b) − zh / R (cid:48) + (2 h − zh (cid:48) )2 h / R − (cid:18) − z ω hφ (cid:19) hz R + 12 R + 2 z h (cid:0) ( a t ) + ω (cid:1) − M = 0 . (3.37c)Using eqs. (3.37), we derive the near-horizon asymptotic expansions of R , R , and R in ap-pendix A, and the near-boundary asymptotic expansions in appendix B. Eqs. (3.37) are first-order,hence the solution for each response function has one integration constant, which we fix using thein-going boundary condition at the horizon (more specifically, by demanding that the near-horizonexpansion coincides with that in eq. (A.11)).In the screened phase we have been able to obtain the background solutions a t and φ only numerically.We have thus solved eqs. (3.37) only numerically, by integrating them from the horizon to the boundary,subject to the near-horizon behavior in eqs. (A.11). We then extract the two-point functions from thenear-boundary asymptotic expansions of the solutions, as we discuss in the next subsection. To extract the physical one- and two-point functions from the solutions for the background and the re-sponse functions, we must perform holographic renormalization (holo-ren) [43–52]. For a recent reviewof holo-ren, see ref. [53]. Holo-ren consists of deriving the appropriate boundary counterterms thatrender the variational problem well posed for the desired boundary conditions, as well as determiningthe resulting holographic dictionary, relating physical observables to the solutions in the bulk.As we mentioned in section 1 and at the beginning of this section, the holo-ren of our holographicKondo model involves a number of subtleties, stemming from the unusual form of the Fefferman-Graham (FG) expansion of gauge fields in
AdS and the related second class constraint eq. (B.2), aswell as the mixed boundary conditions we impose on the complex scalar Φ to introduce the Kondocoupling. In the remainder of this section we will address these issues systematically.We saw above that the functional S defined through eq. (3.8) coincides with the regularized on-shell action, which we will denote as S reg , and satisfies the Hamilton-Jacobi equation, eq (3.10). Inparticular, the divergent parts of S and S reg coincide, allowing us to determine the counterterms bysolving the Hamilton-Jacobi equation. Since we are only interested in the divergent part of S , wecan simplify the Hamilton-Jacobi equation eq. (3.10) by dropping terms that affect only the finiteparts of S . Using the leading form of the asymptotic expansions (B.4) in appendix B, and the generalsolution for the Chern-Simons field in eq. (3.15), a simple power counting argument shows that wecan ignore any terms that involve A t , ψ , or the time derivatives of any fields, and moreover, we cantake γ → − e r . To determine the counterterms, we can thus use the “reduced” Hamiltonian H reduced (cid:0) π ta , π φ , a t , φ ; γ (cid:1) = − N (cid:90) d t √− γ (cid:16) γ ( π ta ) + 14 π φ (cid:17) + N (cid:90) d t √− γ (cid:0) γ − a t + M (cid:1) φ , (3.38)19nd solve the simplified Hamilton-Jacobi equation H reduced (cid:16) π ta = δ S δa t , π φ = δ S δφ , a t , φ ; γ (cid:17) + 2 γ δ S δγ = 0 , (3.39)in order to determine the divergent part of S reg in the form S [ a t , φ ; γ ].At this point we encounter the first subtlety in the holo-ren of our model, namely, the leading termof the AdS gauge field’s FG expansion in eq. (B.4) is the charge term, Qe r , and not the chemicalpotential term, µ ( t ). This is a generic feature of gauge fields in AdS and AdS , as well as rank- p antisymmetric tensor fields in AdS d +1 with p ≥ d/ π ta , and not of the gauge potential a t .As a result, in practice we should determine not S , but its Legendre transform, (cid:101) S [ π ta , φ ; γ ] = S − (cid:90) d t π ta a t , (3.40)by solving the Legendre transform’s Hamilton-Jacobi equation, H reduced (cid:16) π ta , π φ = δ (cid:101) S δφ , a t = − δ (cid:101) S δπ ta , φ ; γ (cid:17) + 2 γ δ (cid:101) S δγ = 0 . (3.41)Our ansatz to solve eq. (3.41) is (cid:101) S G = N (cid:90) d t √− γ G ( u, v ) , (3.42)so that we now need to solve for G ( u, v ), where u ≡ (cid:0) π ta /N (cid:1) , v ≡ φ . (3.43)By construction, (cid:101) S G agrees with the Legendre transform of S reg , up to finite terms. Inserting ouransatz eq. (3.42) into eq. (3.41) gives us an equation for G ( u, v ), G + u − v (cid:0) G v + 2 u G u − M (cid:1) = 0 , (3.44)where G u ≡ ∂ u G and G v ≡ ∂ v G . Solving eq. (3.44) asymptotically near the boundary, subject tothe boundary conditions dictated by the FG expansions in appendix B, and specifically eq. (B.4),unambiguously determines the divergent part of the on-shell action, and hence the countertermsrequired to renormalize the theory.Moreover, knowing G ( u, v ) allows us to renormalize not only the on-shell action, but also the canonicalvariables, and hence the response functions through the identities a G t = − δ (cid:101) S G δπ ta = − N √− γ G u π ta , π G Φ = δ (cid:101) S G δ Φ = N √− γ Φ † G v , π G Φ † = δ (cid:101) S G δ Φ † = N √− γ Φ G v . (3.45)20inearizing these, and comparing to the definitions of the response functions in eq. (3.30), gives R G = − G u + 2 u G uu , R G ΦΦ † = R G Φ † Φ = 2 uv G uv G u + 2 u G uu − ( G v + v G vv ) , R G Φ a = G uv G u + 2 u G uu (cid:18) γπ ta Φ † N √− γ (cid:19) , R G ΦΦ = (cid:18) u G uv G u + 2 u G uu − G vv (cid:19) (Φ † ) , R G Φ † a = G uv G u + 2 u G uu (cid:18) γπ ta Φ N √− γ (cid:19) , R G Φ † Φ † = (cid:18) u G uv G u + 2 u G uu − G vv (cid:19) Φ , (3.46)where the superscript G on R G and the other response functions is merely a reminder that, byconstruction, they coincide with the exact response functions only asymptotically near the boundary.A second subtlety in the holo-ren concerns the form of the solution G ( u, v ) of eq. (3.44) and is relatedonce more to the asymptotic form of the AdS gauge field. The near-boundary asymptotic expansionsin appendix B imply that as r → ∞ , π ta ∼ N Q and hence u ∼ Q /
2. Although the equations ofmotion allow Q ( t ) to be an arbitrary function of time, a well-defined space of asymptotic solutionsexits only when the constraint eq. (B.2) holds, which implies that Q / M / / ≡ u o on theconstraint surface. As a result, only Neumann boundary conditions are admissible for the AdS gaugefield a t , i.e. keeping the charge Q fixed. The solution of eq. (3.44) satisfying the boundary conditionsdictated by the near-boundary asymptotics in eq. (B.4) thus admits an expansion of the form G ( u, v ) = ∞ (cid:88) n =0 g n ( v )( u − u o ) n . (3.47)Crucially, the series in eq. (3.47) need not be convergent, and should be understood as an asymptoticexpansion only, truncated to a finite, but arbitrary, order.Eq. (B.4) also implies that asymptotically near the boundary, u − u o ∼ Q / − u o + O ( e − r r α ) ∼ QδQ + O ( e − r r α ), so u − u can receive two different potential contributions: δQ , which dominates ifnon-zero, and the mode α . When δQ (cid:54) = 0, the order m term in the expansion in eq. (3.47) encodes thenear-boundary divergences of the m -point function of the operator sourced by δQ . These divergencesenter two-point functions via the near-boundary asymptotic expansions of the response functions ineq. (B.10). If δQ = 0, however, then u − u o does not contribute to any such divergences, but also,no correlators of the operator sourced by δQ can be computed. In the latter case, therefore, thecounterterms come entirely from g ( v ). In that case, the near-boundary expansions of the responsefunctions appears in eq. (B.12), which encode the two-point functions of only O and O † .Inserting the expansion in eq. (3.47) into the equation for G , eq. (3.44), leads to a tower of differentialequations for the coefficients g n ( v ), the first three of which are g + u o − v ( g (cid:48) + 2 u o g − M ) = 0 , (3.48a) g + 1 − v (2 g (cid:48) g (cid:48) + 2 g + 8 u o g g ) = 0 , (3.48b) g − v ( g (cid:48) + 2 g (cid:48) g (cid:48) + 8 u o g + 12 u o g g + 8 g g ) = 0 , (3.48c) The two-impurity holographic Kondo model of ref. [32] involved a U (2) gauge field and a charged scalar in AdS . Mixedboundary conditions were imposed on the U (2) gauge field, which required the scalar mass M to change dynamically inorder to preserve the scalar field’s asymptotic form, and hence obtain a well defined variational problem. In the presentwork we treat M as a fixed parameter of the theory and so mixed boundary conditions on the AdS gauge field are notallowed. We stress that these types of problems do not arise in the absence of charged matter. For example in the modelof ref. [90], with a U (1) gauge field and dilatonic scalar in AdS , but no charged matter, Q was a strictly conservedquantity, and both Neumann and Dirichlet boundary conditions were permitted for the gauge field. ∂ v . We will only need to solve these equations asymptotically near the boundary,and only keeping terms up to a certain order, since higher orders will not contribute to the divergencesof an m -point function with fixed m . In particular, the near-boundary asymptotic expansions ineq. (B.4) allow us to parameterize g ( v ) and g ( v ) as g = − u o + h , g = − h , (3.49)where h and h behave as v times non-negative integer powers of log v as v →
0, as do g and g . Wepresent the explicit small- v expansions of h , h and g in appendix C.The near-boundary, or equivalently small- v , asymptotic solutions for g ( v ), g ( v ) and g ( v ) in ap-pendix C present yet another subtlety of the holo-ren of our model: our choice of the scalar field’smass, to guarantee that our Kondo coupling O † O is classically marginal, leads to powers of log v in the small- v expansions of g ( v ), g ( v ), and g ( v ). However, such non-analytic in v terms in thecounterterms amount to subtracting a non-analytic function of the source of the dual scalar operator,and hence violate the locality of the counterterms. To restore locality, we are forced to sacrifice theradial covariance of the counterterms [47, 88], i.e. the counterterms will exhibit explicit dependenceon the r cutoff, which is the holographic manifestation of a conformal anomaly. This is manifest, forexample, in the expressions for the counterterms in eq. (C.9) in appendix C.Given a near-boundary asymptotic solution G ct ( u, v ) of eq. (3.44), the counterterms are defined as (cid:101) S ct = − N (cid:90) d t √− γ G ct ( u, v ) , (3.50)and hence the renormalized action evaluated at the radial cutoff is (cid:101) S ren = (cid:101) S reg + (cid:101) S ct . (3.51)By construction, (cid:101) S ct has the same divergences as (cid:101) S reg , hence (cid:101) S ren remains finite as we remove thecutoff. Varying (cid:101) S ren gives then the renormalized canonical variables: δ (cid:101) S ren = (cid:90) d t (cid:16) − a ren t δπ ta + π renΦ δ Φ + π renΦ † δ Φ † (cid:17) + (cid:90) d x π i ren A δA i = (cid:90) d t (cid:0) − a ren t δπ ta + π ren φ δφ + π ren ψ δψ (cid:1) + (cid:90) d x π i ren A δA i , (3.52) a ren t = − δ (cid:101) S ren δπ ta = a t + 1 N √− γ G ct u π ta , π i ren A = δ (cid:101) S ren δA i = π iA = − N π ¯ (cid:15) ij A j ,π renΦ = δ (cid:101) S ren δ Φ = π Φ − N √− γ G ct v Φ † , π ren φ = δ (cid:101) S ren δφ = π φ − N √− γ G ct v φ,π renΦ † = δ (cid:101) S ren δ Φ † = π Φ † − N √− γ G ct v Φ , π ren ψ = δ (cid:101) S ren δψ = π ψ , (3.53)which are evaluated at the radial cutoff. As mentioned above, for the scalar field the canonicalmomentum is renormalized, while for the AdS gauge field, a t itself is renormalized instead, due tothe asymptotic behavior of gauge fields in AdS and the fact that the counterterms are local functionsof the canonical momentum π ta [90]. 22e now want to plug the FG expansions of the fields into the renormalized canonical variables ineq. (3.53). Crucially, however, we show in appendix B that background solutions and fluctuationshave different FG expansions, so we must treat them separately.The FG expansions for background solutions appear in eq. (B.4), reproduced here for convenience: a t = e r Q − Q (cid:16) α r + ( α − αβ ) r + (2 α − αβ + β ) r (cid:17) + µ + · · · , (3.54a) φ = e − r/ ( − αr + β ) + · · · , (3.54b) ψ = ψ − + ψ + r − + · · · , (3.54c)where µ , α , β and ψ − are arbitrary functions of time, while U (1) gauge invariance implies both that Q is independent of time and ψ + = 0. The . . . represent terms that vanish as r → ∞ faster than thoseshown, and which are completely determined by those shown, via the equations of motion. Insertingeq. (3.54) into eq. (3.53) and using the counterterms in eq. (C.9) allows us to remove the radial cutoff,and hence obtain the renormalized canonical variables in terms of the FG coefficients: (cid:98) a ren t ≡ lim r →∞ a ren t = µ + A (0) t − Q α (cid:0) β − αβ + 6 α β − α (cid:1) , (cid:98) π ta ≡ lim r →∞ π ta = N Q, (cid:98) π renΦ ≡ lim r →∞ ( re − r/ π renΦ ) = N βe − iψ − , (cid:98) Φ ≡ lim r →∞ ( r − e r/ Φ) = − αe iψ − , (cid:98) π renΦ † ≡ lim r →∞ ( re − r/ π renΦ † ) = N βe iψ − , (cid:98) Φ † ≡ lim r →∞ ( r − e r/ Φ † ) = − αe − iψ − , (cid:98) π ren φ ≡ lim r →∞ ( re − r/ π ren φ ) = 2 N β, (cid:98) φ ≡ lim r →∞ ( r − e r/ φ ) = − α. (3.55)We took µ → µ + A (0) t in the expression for (cid:98) a ren t , because the above asymptotic solutions for the AdS fields were obtained in the gauge of eq. (3.16), where A (0) t = 0. However, in order to identifythe correct one-point functions, the general dependence on all the sources must be reinstated. As weshall see, this contribution of A (0) t is crucial for obtaining the two-point functions.For the fluctuations, we determine the response functions by linearizing eq. (3.53) in the fields inducedat the radial cutoff. The complete analysis leading to the full set of renormalized response functionsis carried out in appendix C. As an illustration, we quote here the results for the renormalized scalarresponse functions only, which take the form R renΦΦ = R ΦΦ + G ct vv (Φ † ) , R renΦ † Φ † = R Φ † Φ † + G ct vv Φ , R renΦΦ † = R ΦΦ † + (cid:0) G ct v + v G ct vv (cid:1) , R renΦ † Φ = R Φ † Φ + (cid:0) G ct v + v G ct vv (cid:1) . (3.56)The FG expansions of the response functions appear in eq. (B.13), reproduced here for convenience: R Φ † Φ = −
12 + 1 r + (cid:98) R Φ † Φ r + · · · , R ΦΦ † = −
12 + 1 r + (cid:98) R ΦΦ † r + · · · , R ΦΦ = (cid:98) R ΦΦ r + · · · , (3.57)where (cid:98) R Φ † Φ , (cid:98) R ΦΦ † , and (cid:98) R ΦΦ are functions of frequency ω . The . . . represent terms that vanishas r → ∞ faster than those shown, and which are completely determined by those shown, via theequations of motion. Inserting eq. (3.57) into eq. (3.56) and using the counterterms in eq. (C.9) allows A (0) t can be reinstated by letting a t → a t + A (0) t , recalling that A t = A (0) t is constant and enters a (cid:48) t s equation ofmotion through the U (1) current J t , eq. (3.13), with gauge-covariant derivative in eq. (2.9).
23s to remove the radial cutoff, and hence obtain the renormalized response functions: (cid:98) R renΦΦ † = lim r →∞ (cid:0) r R renΦΦ † (cid:1) = (cid:98) R ΦΦ † , (cid:98) R renΦ † Φ = lim r →∞ (cid:0) r R renΦ † Φ (cid:1) = (cid:98) R Φ † Φ , (cid:98) R renΦΦ = lim r →∞ (cid:0) r R renΦΦ (cid:1) = (cid:98) R ΦΦ , (cid:98) R renΦ † Φ † = lim r →∞ (cid:0) r R renΦ † Φ † (cid:1) = (cid:98) R Φ † Φ † . (3.58)Eq. (3.58) is valid in both the unscreened and screened phases, although the values for (cid:98) R Φ † Φ , (cid:98) R Φ † Φ † , (cid:98) R ΦΦ and (cid:98) R ΦΦ † are different in the two phases. The renormalized action (cid:101) S ren cannot be identified with the generating function in the dual theory untilwe impose boundary conditions on the fields and add the corresponding finite boundary terms thatimpose these boundary conditions. The boundary conditions also dictate which combinations of therenormalized canonical variables in eq. (3.55) are identified with the sources in the dual field theory.In this subsection we will introduce the finite boundary terms of our model, and then identify thesources in the dual field theory. We will then determine the Ward identities of the dual field theory,and finally, determine the renormalized two-point functions of our model, in terms of coefficients inthe FG expansion of the response functions, eq. (B.13) or equivalently eq. (3.57).In our case, three finite boundary terms are required to have a well-posed variational problem thatcaptures the desired physics. First, for the Chern-Simons gauge field alone, with no AdS defect fields,a well-posed variational problem requires the boundary term [81, 92–95] S = N π (cid:90) d x √− ¯ γ ¯ γ ij A i A j = N π (cid:90) d x A + A − , (3.59)where A ± ≡ A x ± A t , and ¯ γ ij is the induced metric on a radial slice of AdS . Second, because thegeneral solution for the Chern-Simons gauge field in eq. (3.15) receives a contribution from the AdS fields, to guarantee a well-posed variational principle for the Chern-Simons gauge field in the presenceof the AdS defect we must add the finite boundary term S = − (cid:90) d t π ta A − , (3.60)which couples the Chern-Simons and AdS fields. Third, in order to introduce our Kondo coupling,we must add the finite boundary term [31, 32] S κ = κN (cid:90) d t (cid:98) π renΦ (cid:98) π renΦ † . (3.61)Putting everything together, the generating functional of the dual theory is W κ ≡ lim r →∞ ( (cid:101) S ren + S + S + S κ ) . (3.62) Changing the sign of the boundary term in eq. (3.59) simply interchanges the role of A + and A − in the following. Instead of eq. (3.61), refs. [31, 32] used the boundary term ( κ/N ) (cid:82) d t ( (cid:98) π ren φ ) = ( κ/ N ) (cid:82) d t ( (cid:98) π renΦ + (cid:98) π renΦ † ) , whichagrees with eq. (3.61) for background solutions, but not for fluctuations. Unlike the boundary term used in refs. [31, 32],eq. (3.61) preserves the U (1) gauge invariance associated with the AdS gauge field. The free energy obtained from W κ , that is with the Legendre transform in eq. (3.40) and the counterterms ineq. (3.50) with eq. (C.9), agrees with the free energy computed in refs. [31, 32].
24o identify the sources in the dual field theory, we consider the variational principle for W κ , δ W κ = (cid:90) d t (cid:16) − (cid:98) a t δ (cid:98) π ta + (cid:98) π renΦ δ (cid:98) Φ κ + (cid:98) π renΦ † δ (cid:98) Φ † κ (cid:17) + N π (cid:90) d x (cid:16) A (0)+ + πN (cid:98) π ta δ ( x ) (cid:17) δA (0) − , (3.63)where we have defined (cid:98) a t ≡ (cid:98) a ren t − A (0) t − A (0) − , (cid:98) Φ κ ≡ (cid:98) Φ + κN (cid:98) π renΦ † , (cid:98) Φ † κ ≡ (cid:98) Φ † + κN (cid:98) π renΦ . (3.64)A well-posed variational problem for W κ requires that we keep fixed (cid:98) π ta , (cid:98) Φ κ , (cid:98) Φ † κ , and A (0) − , hence weidentify these as the sources of the dual operators. Keeping these fixed corresponds to a Neumannboundary condition for the AdS gauge field, and a mixed (or Robin) boundary condition for thescalar field, in which α = κβ [31, 96]. Our holographic Kondo coupling is κ , related to the Kondocoupling λ of the Kondo Hamiltonian in eq. (2.1) as κ ∝ N λ . For more details about our holographicKondo coupling and its RG running, see ref. [31] and especially section 4 of ref. [32].The one-point functions of the dual operators are then defined via (cid:104)A t (cid:105) ≡ − δ W δ (cid:98) π ta = (cid:98) a t , (cid:104)J + (cid:105) ≡ δ W δA (0) − = N π (cid:16) A (0)+ + πN (cid:98) π ta δ ( x ) (cid:17) , (cid:104)O(cid:105) ≡ − δ W δ (cid:98) Φ κ = − (cid:98) π renΦ , (cid:104)O † (cid:105) ≡ − δ W δ (cid:98) Φ † κ = − (cid:98) π renΦ † , (cid:104) O (cid:105) ≡ − δ W δ (cid:98) φ κ = − (cid:98) π ren φ , (3.65)and are functions of the sources. The scalar operator O is defined as the conjugate to the real source (cid:98) φ κ = ( (cid:98) Φ κ + (cid:98) Φ † κ ) /
2. Using eq. (3.55), we can express these in terms of the FG expansion coefficients ineq. (B.4), or equivalently eq. (3.54), (cid:104)A t (cid:105) = µ − Q α (cid:0) β − αβ + 6 α β − α (cid:1) − A (0) − , (cid:104)J + (cid:105) = N π (cid:0) A (0)+ + πQδ ( x ) (cid:1) , (cid:104)O(cid:105) = − N βe − iψ − , (cid:104)O † (cid:105) = − N βe iψ − , (cid:104) O (cid:105) = − N β. (3.66)In general, the Ward identities for the U (1) currents dual to the Chern-Simons and AdS gaugefields depend on the choice of boundary conditions, since different boundary conditions may preservedifferent symmetries. Since the Kondo deformation in eq. (3.61) preserves the U (1) symmetry on theimpurity, the U (1) constraints in eq. (3.7) translate to the Ward identities (cid:98) Φ κ (cid:104)O(cid:105) − (cid:98) Φ † κ (cid:104)O † (cid:105) = ω (cid:98) π ta , ∂ − (cid:104)J + (cid:105) = N π ∂ + A (0) − + N Q ∂ − δ ( x ) , (3.67)where ∂ ± ≡ ∂ x ± ∂ t . The Ward identity for the Chern-Simons current J + is simply the condition ∂ − A (0)+ = ∂ + A (0) − , as in the absence of the AdS defect. Eqs. (3.67) are operator identities, i.e. they hold with arbitrary sources. Differentiating the Ward identities in eqs. (3.67) with respect to thesources leads to relations among higher-point functions.We are finally ready to compute the main result of this section, namely the two-point functions of ourmodel. To write the two-point functions involving J + , we introduce chiral coordinates x ± and theirFourier counterparts, the chiral momenta p ± . Varying our result for (cid:104)J + (cid:105) in eq. (3.65), and using theWard identity ∂ − A (0)+ = ∂ + A (0) − , we find (cid:104)J + ( p + , p − ) J + ( − p + , − p − ) (cid:105) = N π p + p − , (3.68)25hich is completely independent of the AdS fields, i.e. eq. (3.68) is identical to the previous resultsfor chiral currents in (1 + 1)-dimensional CFTs in refs. [81, 92, 93]. All two-point functions between J + and the impurity operators are zero, except for one: (cid:104)J + ( p + , p − ) A t ( − p + , − p − ) (cid:105) = − . (3.69)Since the two-point functions in eqs. (3.68) and (3.69) are completely insensitive to the transitionbetween the unscreened and screened phases, we will ignore them henceforth.In the unscreened phase, besides eqs. (3.68) and (3.69), the only non-trivial two-point function is theone between O and O † . To derive this two-point function we use the following identities, derivedin appendix C (in the unscreened phase the response functions (cid:98) R Φ π ta and (cid:98) R Φ † π ta vanish, and so theinfinitesimal source δ (cid:98) π ta does not contribute to these expressions): δ (cid:98) π Φ † = − N ( (cid:98) R Φ † Φ δ (cid:98) Φ + (cid:98) R Φ † Φ † δ (cid:98) Φ † ) , δ (cid:98) π Φ = − N ( (cid:98) R ΦΦ δ (cid:98) Φ + (cid:98) R ΦΦ † δ (cid:98) Φ † ) , (3.70)where (cid:98) R Φ † Φ , (cid:98) R Φ † Φ † , (cid:98) R ΦΦ and (cid:98) R ΦΦ † are the renormalized scalar response functions, which appearas coefficients in the FG expansions in eq. (3.57). The quantities in eq. (3.70) represent the renor-malized one-point functions, which in the regime of linear response are linearly proportional to thesources, where the proportionality factor is the renormalized two-point function. We thus need towrite eq. (3.70) in terms of the scalar sources. Using the scalar sources defined in eq. (3.64), we find δ (cid:98) Φ κ = (1 − κ (cid:98) R Φ † Φ ) δ (cid:98) Φ − κ (cid:98) R Φ † Φ † δ (cid:98) Φ † , δ (cid:98) Φ † κ = − κ (cid:98) R ΦΦ δ (cid:98) Φ + (1 − κ (cid:98) R ΦΦ † ) δ (cid:98) Φ † . (3.71)In the unscreened phase, in appendix B we find that (cid:98) R Φ † Φ † = 0 and (cid:98) R ΦΦ = 0, indicating that (cid:104)O ( ω ) † O † ( − ω ) (cid:105) = 0 and (cid:104)O ( ω ) O ( − ω ) (cid:105) = 0 respectively. Using that result, and by combining thevariations in eqs. (3.70) and (3.71) in the unscreened phase, we then find δ (cid:98) π Φ † = − N (cid:98) R Φ † Φ (1 − κ (cid:98) R Φ † Φ ) δ (cid:98) Φ κ , δ (cid:98) π Φ = − N (cid:98) R ΦΦ † (1 − κ (cid:98) R ΦΦ † ) δ (cid:98) Φ † κ , (3.72)from which we read off the two-point functions (cid:104)O † ( ω ) O ( − ω ) (cid:105) κ = N (cid:98) R Φ † Φ − κ (cid:98) R Φ † Φ , (cid:104)O ( ω ) O † ( − ω ) (cid:105) κ = N (cid:98) R ΦΦ † − κ (cid:98) R ΦΦ † . (3.73)We computed R Φ † Φ and R ΦΦ † in the unscreened phase in subsection 3.2.1, with the result in eq. (3.26)and asymptotic expansions in eq. (3.29). Indeed, comparing the asymptotic expansions in eq. (3.29)to the general FG expansions in eq. (3.57), we find (cid:98) R Φ † Φ = H (cid:18) −
12 + iQ − iωz H (cid:19) + H (cid:18) − − iQ (cid:19) − log( z H Λ / , (cid:98) R ΦΦ † = H (cid:18) − − iQ − iωz H (cid:19) + H (cid:18) −
12 + iQ (cid:19) − log( z H Λ / , (3.74)where 1 / Λ is the near-boundary cutoff in z . Plugging eq. (3.74) into eq. (3.73) then gives ourmain result for the unscreened phase, the renormalized two-point functions (cid:104)O † ( ω ) O ( − ω ) (cid:105) κ and (cid:104)O ( ω ) O † ( − ω ) (cid:105) κ as functions of the field theory parameters Q , T , and ω . We explore the physicsof these two-point functions in detail in section 5.26n the screened phase the variations of the renormalized one-point functions are δ (cid:98) π Φ † = − N ( (cid:98) R Φ † Φ δ (cid:98) Φ + (cid:98) R Φ † Φ † δ (cid:98) Φ † ) + (cid:98) R Φ † π ta δ (cid:98) π ta , (3.75a) δ (cid:98) π Φ = − N ( (cid:98) R ΦΦ δ (cid:98) Φ + (cid:98) R ΦΦ † δ (cid:98) Φ † ) + (cid:98) R Φ π ta δ (cid:98) π ta , (3.75b) δ (cid:98) a t = − (cid:16) (cid:98) R π ta Φ δ (cid:98) Φ + (cid:98) R (cid:98) π ta Φ † δ (cid:98) Φ † + (cid:98) R π ta π ta δ (cid:98) π ta (cid:17) − A (0) − . (3.75c)All response functions in these expressions are determined in appendix C, and their explicit forms interms of the coefficients of the FG expansions are shown in eq. (C.17).To evaluate the two-point functions involving the scalar operators we again need to determine theinfinitesimal sources δ Φ κ and δ Φ † κ in terms of the sources of the undeformed theory. From the defi-nitions in eq. (3.64), and the expressions for the response functions in terms of the FG coefficients inthe screened phase in eq. (C.17), we find δ (cid:98) Φ κ = (cid:16) − κ (cid:98) R (cid:17) δ (cid:98) φ − κ N (cid:16) (cid:98) R ∞ + ωα (cid:17) δ (cid:98) π ta , δ (cid:98) Φ † κ = (cid:16) − κ (cid:98) R (cid:17) δ (cid:98) φ − κ N (cid:16) (cid:98) R ∞ − ωα (cid:17) δ (cid:98) π ta , (3.76)where (cid:98) R ∞ is defined in eq. (C.10), and recall δ (cid:98) φ = ( δ (cid:98) Φ + δ (cid:98) Φ † ) /
2. However, linearizing the first Wardidentity in eq. (3.67) around a screened phase background solution gives δ (cid:98) Φ − δ (cid:98) Φ † = − κN ωα δ (cid:98) π ta , (3.77)so eq. (3.76) can be re-written as δ (cid:98) Φ κ = (cid:16) − κ (cid:98) R (cid:17) δ (cid:98) Φ − κ (cid:98) R δ (cid:98) Φ † − κ N (cid:98) R ∞ δ (cid:98) π ta , (3.78a) δ (cid:98) Φ † κ = − κ (cid:98) R δ (cid:98) Φ + (cid:16) − κ (cid:98) R (cid:17) δ (cid:98) Φ † − κ N (cid:98) R ∞ δ (cid:98) π ta . (3.78b)Eqs. (3.78) can be inverted to obtain δ (cid:98) Φ and δ (cid:98) Φ † in terms of δ (cid:98) Φ κ , δ (cid:98) Φ † κ and δ (cid:98) π ta : δ Φ = 11 − κ (cid:98) R (cid:16)(cid:16) − κ (cid:98) R (cid:17) δ (cid:98) Φ k + κ (cid:98) R δ (cid:98) Φ † κ + κ N (cid:98) R ∞ δ (cid:98) π ta (cid:17) , (3.79a) δ Φ † = 11 − κ (cid:98) R (cid:16) κ (cid:98) R δ (cid:98) Φ κ + (cid:16) − κ (cid:98) R (cid:17) δ (cid:98) Φ † k + κ N (cid:98) R ∞ δ (cid:98) π ta (cid:17) . (3.79b)Inserting eq. (3.79) into eq. (3.75) for the scalar one-point functions and making use of the linearizedWard identity eq. (3.77), we obtain δ (cid:98) π Φ = δ (cid:98) π Φ † = 11 − κ (cid:98) R (cid:18) − N (cid:98) R ( δ (cid:98) Φ κ + δ (cid:98) Φ † κ ) − (cid:98) R ∞ δ (cid:98) π ta (cid:19) , (3.80)from which we read off the two-point functions, (cid:104)O ( ω ) O ( − ω ) (cid:105) κ = (cid:104)O ( ω ) O † ( − ω ) (cid:105) κ = (cid:104)O † ( ω ) O ( − ω ) (cid:105) κ = (cid:104)O † ( ω ) O † ( − ω ) (cid:105) κ = N (cid:98) R / − κ (cid:98) R / , (3.81a) (cid:104)O ( ω ) A t ( − ω ) (cid:105) κ = (cid:104)O † ( ω ) A t ( − ω ) (cid:105) κ = (cid:98) R ∞ / − κ (cid:98) R / . (3.81b)27oreover, inserting eq. (3.79) in the gauge field one-point function in eq. (3.75) and using the expres-sions in eqs. (C.17) gives δ (cid:98) a t = 12 (cid:16) (cid:98) R ∞ + ω/α (cid:17) δ (cid:98) Φ + 12 (cid:16) (cid:98) R ∞ − ω/α (cid:17) δ (cid:98) Φ † − N (cid:98) R ∞ δ (cid:98) π ta − A (0) − (3.82)= (cid:98) R ∞ / − κ (cid:98) R / (cid:16) δ (cid:98) Φ κ + δ (cid:98) Φ † κ (cid:17) − N (cid:32) (cid:98) R ∞ + κ (cid:18) ωα (cid:19) − κ ( (cid:98) R ∞ ) − κ (cid:98) R / (cid:33) δ (cid:98) π ta − A (0) − , which reproduce the two-point functions in eqs, (3.69) and (3.81b), and from which we read off thetwo-point function (cid:104)A t ( ω ) A t ( − ω ) (cid:105) κ = 1 N (cid:32) (cid:98) R ∞ + κ (cid:18) ωα (cid:19) − κ ( (cid:98) R ∞ ) − κ (cid:98) R / (cid:33) . (3.83)As mentioned at the end of subsection 3.2.2, in the screened phase we have been able to obtain thebackground solutions a t and φ only numerically, and hence have only solved eq. (3.37) for R , R and R numerically. From those numerical solutions we then extract the FG expansion coefficients (cid:98) R , (cid:98) R and (cid:98) R using the near boundary expansions in eqs. (B.10), and thus obtain the two-pointfunction via eqs. (3.81) and (3.83). We present our numerical results for the scalar two-point functionsin the screened phase in section 6. A spectral function ρ is defined as the anti-Hermitian part of a retarded Green’s function, G : ρ ≡ i (cid:104) G − G † (cid:105) = − G. (4.1)In our system, we are interested in G O † O ≡ (cid:104)O † ( ω ) O ( − ω ) (cid:105) κ , (4.2)and the associated spectral function ρ O † O = − G O † O . Given the anti-Hermitian part of a Green’sfunction, a Kramers-Kroning relation completely determines the Hermitian part. The latter thereforecontains no additional information, so we will compute only the former, i.e. spectral functions. Ingeneral, for real ω , when ω > ω < ωρ O † O ≥ ω ∈ ( −∞ , ∞ ), so that ρ O † O ≥ ω > ρ O † O ≤ ω < i.e. they are oftenassociated with some “impurity”. Numerous examples of Fano resonances appear throughout physics,but a classic example is the scattering of light (the continuum) off the excited states of an atom (theresonant states). As mentioned in section 1, Fano resonances have also been observed in quantumimpurity models in one spatial dimension, including side-coupled QDs [76, 78]. For a brief review ofFano resonances, see for example ref. [76]. 28he Fano spectral function is ρ Fano = (cid:0) ω − ω + q Γ2 (cid:1) ( ω − ω ) + (cid:0) Γ2 (cid:1) , (4.3)where ω fixes the position of the Fano resonance, Γ fixes the width, q is called the “Fano” or “asym-metry” parameter, and we have fixed the normalization so that lim ω →±∞ ρ Fano = 1. The ρ Fano ineq. (4.3) can be re-written in an illuminating way: ρ Fano = 1 + (cid:0) q − (cid:1) (cid:0) Γ2 (cid:1) ( ω − ω ) + (cid:0) Γ2 (cid:1) + 2 q (cid:0) Γ2 (cid:1) ( ω − ω )( ω − ω ) + (cid:0) Γ2 (cid:1) , (4.4)where on the right-hand-side, the first term in the sum (the 1) represents the continuum, the secondterm is a Lorentzian representing the resonant state, and the third term is the “mixing” or “inter-ference” term arising from the interaction between the two. Indeed, the essential physics of Fanoresonances is that the incoming scattering states, from the continuum, have two paths through thesystem: they can either scatter off the resonant state (“resonant scattering”) or they can bypass theresonant state (“non-resonant scattering”). The interference between the two paths generically pro-duces an asymmetric resonance, the Fano resonance. The Fano parameter q characterizes the amountof mixing or interference. More precisely, q is proportional to a ratio of probabilities: q ∝ theprobability of resonant scattering over the probability of non-resonant scattering.Fig. 2 shows ρ Fano for some representative values of q . Fig. 2 (a) shows ρ Fano for generic q >
0, witha characteristic asymmetric Fano resonance. In these cases, ρ Fano has a minimum and maximum:minimum: ρ Fano = 0 at ω = ω − q Γ2 maximum: ρ Fano = 1 + q at ω = ω + q Γ2 .At ω = ω , which is between the minimum and maximum, ρ Fano = q . Taking q < ω = 0 axis, so we will restrict to q > q = 0, 1, and ∞ , the Fano resonance becomes symmetric. Fig. 2 (b) shows ρ Fano for q = 0, meaning purely non-resonant scattering. In this case, the maximum moves to ω = + ∞ while the minimum moves to ω = ω , leaving only a symmetric dip called an anti-resonance . Fig. 2(c) shows ρ Fano for q = 1, meaning equal probabilities of resonant and non-resonant scattering. In thiscase, the minimum and maximum are symmetric about ω = ω . Fig. 2 (d) shows ρ Fano /q for q → ∞ ,meaning purely resonant scattering. In this case, the minimum moves to ω = −∞ and the maximummoves to ω = ω , leaving the Lorentzian peak of the resonant state itself.Near a simple pole at ω ∗ = ω R + iω I in the complex ω plane, the retarded Green’s function is G = Zω − ω ∗ ,with residue Z . As is well-known, a real-valued Z leads to a Lorentzian resonance in ρ (where thelatter is restricted to real ω ). However, a complex-valued residue, Z = Z R + iZ I with Z I (cid:54) = 0, leadsto a Fano resonance: ρ = − G = − Z R ω I ( ω − ω R ) + ( ω I ) + − Z I ( ω − ω R )( ω − ω R ) + ( ω I ) = − ρ Fano , (4.5)where in the final equality we added and subtracted 1, and used the form of ρ Fano in eq. (4.4), withthe identifications ω = ω R and Γ / | ω I | and q − − Z R ω I , q = − Z I | ω I | . (4.6)29 (a) (b)(c) (d) Figure 2: The Fano spectral function, ρ Fano in eq. (4.3), as a function of ( ω − ω ) / (Γ / ω = ω , where ρ Fano = q ), for (a) a generic value of q , where the Fanoresonance is asymmetric, (b) q = 0, where the Fano resonance becomes a symmetric dipor anti-resonance , and (c) q = 1, where the minimum and maximum becomes symmetric.(d) Shows ρ Fano /q as a function of ( ω − ω ) / (Γ /
2) in the limit q → ∞ , where the Fanoresonance becomes a Lorentzian.The ratio of these two equations leads to q − sign ( ω I ) 2 Z R Z I q − ω I ) = − q are q = − Z R Z I ± (cid:115) Z R Z I , (4.7)or equivalently, using Z = | Z | e iθ , q = − cot θ ± csc θ. (4.8)We can obtain the solution with the minus (lower) sign from the solution with the plus (upper) signby shifting θ → θ + π , so we will restrict to the upper (plus) sign and to the interval θ ∈ [0 , π ], where q >
0. Fig. 3 shows q as a function of θ , and the table below shows how various limits of θ lead to thesymmetric Fano resonances in fig. 2. θ Z R Z I q Fig. 20 | Z | π/ | Z | π −| Z | ∞ (d)In sections 5 and 6 we will see that generically the spectral functions of O and O † exhibit Fanoresonances, in both the unscreened and screened phases, with various q . In our case, the continuum30 ✓ π π Figure 3: The Fano/asymmetry parameter q as a function of θ (solid black line), fromeq. (4.8), for a simple pole in a retarded Green’s function with complex residue Z = | Z | e iθ .The value q = 1 (dashed black line) produces a symmetric Fano resonance, as in fig. 2 (c).arises from the (0 + 1)-dimensional scale invariance associated with the AdS subspace, inherited fromthe (1 + 1)-dimensional scale invariance associated with AdS , and which forces any spectral functionto be a power law in ω , i.e. a continuum. Resonances can then only occur if scale invariance is broken,which we achieve via our marginally relevant Kondo coupling. In our model, the asymmetry is relatedto particle-hole symmetry breaking, that is, q will depend on Q . In this section we use the results of sections 2 and 3 to determine the excitation spectrum of our systemin the unscreened phase, by locating the poles of G O † O and G OO † in the plane of complex frequency ω (subsection 5.1), and the corresponding peaks in ρ O † O and ρ OO † for real ω (subsection 5.2).Some results for the poles appear already in refs. [31], in the unscreened phase and at small ω . Indeed,a key result of ref. [31] was that in the unscreened phase, and for any Q (including Q = 0), as T → T + c a pole moves towards the origin of the complex ω plane, reaching the origin at precisely T = T c . If wethen take T < T c but remain in the unscreened phase, then the pole moves into the upper half of thecomplex ω plane, Im ω >
0, signaling the instability towards the screened phase.Further results appeared in ref. [42], including in particular our central result, the analytic ( i.e. non-numerical) result for G O † O . In ref. [42], we discussed the movement of poles in G O † O as T → T + c ,presented an analytic formula for T c in terms of T K and Q , showed that ρ O † O generically has Fanoresonances, derived an analytic form for the pole producing the Fano resonance for T just above T c ,and showed that q → ∞ as Q → ∞ , producing symmetric Fano resonances (Lorentzians).In this section we will not only reproduce these results, but also extend them, in particular by exploringin far greater detail the T and Q dependence of the poles in G O † O and peaks in ρ O † O . Moreover,31e will present analytic results for poles in the T (cid:29) T c limit, which demonstrate conclusively theappearance of Fano resonances in ρ O † O when T (cid:29) T c .As derived in section 3, we have (cid:104)O ( ω ) † O † ( − ω ) (cid:105) = 0 and (cid:104)O ( ω ) O ( − ω ) (cid:105) = 0, and from eq. (3.73) G O † O = N ˆ R Φ † Φ − κ ˆ R Φ † Φ , (5.1)where from eq. (3.74) we haveˆ R Φ † Φ = H (cid:18) −
12 + iQ − iωz H (cid:19) + H (cid:18) − − iQ (cid:19) − ln( z H Λ / , (5.2)where H ( n ) denotes the n th harmonic number.We can write eqs. (5.1) and (5.2) in terms of field theory quantities using z H = 1 / (2 πT ) and byreplacing Λ with the Kondo temperature T K , following refs. [31, 32], as follows. In the metric ofeq. (2.6), we re-scale to produce dimensionless coordinates,( z/z H , t/z H , x/z H ) → ( z, t, x ) , (5.3)which leaves the metric in eq. (2.6) invariant, except for h ( z ) = 1 − z /z H → − z , so the boundaryremains at z = 0 but the horizon is now at z = 1. We also re-scale a t ( z ) z H → a t ( z ), which is thendimensionless. After the re-scaling, Φ( z )’s asymptotic expansion isΦ( z ) = α T z / ln z + β T z / + . . . , (5.4)where . . . represents terms that vanish faster than those shown when z →
0, and are completelydetermined by the terms shown, via the equations of motion. The boundary condition α = κβ discussed below eq. (3.64) is now α T = κ T β T , with κ T β T = z / H κβ , and where κ T ≡ κ κ ln ( z H Λ) , (5.5)is our running holographic Kondo coupling, with UV cutoff Λ. If κ <
0, then if T increases, meaning z H = 1 / (2 πT ) →
0, then κ T exhibits asymptotic freedom, κ T → − . We thus identify κ < κ < T decreases, so z H = 1 / (2 πT ) increases,then κ T diverges by definition at the Kondo temperature, T K ≡ Λ2 π e /κ . (5.6)Using eq. (5.6) in eq. (5.2) to replace Λ with T K , we thus find G O † O = − Nκ − Nκ H (cid:0) − + iQ − i ω πT (cid:1) + H (cid:0) − − iQ (cid:1) + ln (cid:16) TT K (cid:17) . (5.7)The form of G OO † is the same as G O † O , but with Q → − Q .32 .1 Unscreened Phase: Poles in the Green’s Function Clearly G O † O in eq. (5.7) has a pole whenever H (cid:18) −
12 + iQ − i ω πT (cid:19) + H (cid:18) − − iQ (cid:19) + ln (cid:18) TT K (cid:19) = 0 . (5.8)Given values for Q and T /T K , we can thus find the poles of G O † O by solving eq. (5.8) for ω/ (2 πT ).Because the form of G OO † is the same as G O † O , but with Q → − Q , if ω = Re ( ω ) + i Im ( ω ) is a poleof G O † O , then − ω = − Re ( ω ) + i Im ( ω ) will be a pole of G OO † . In other words, the poles of G O † O and G OO † come in pairs mirrored about the imaginary axis in the ω -plane.Fig. 4 shows our numerical results for the positions of poles of G O † O (black dots) and G OO † (graydiamonds) in the complex ω/ (2 πT ) plane, for the representative value Q = 0 .
5, for five temperatures:
T /T K = 4 . , . , . , . , . G O † O and G OO † has a sequence ofpoles descending down into the complex plane, i.e. with decreasing imaginary part, spaced apart fromone another by an amount ω/ (2 πT ) ≈
1, and with Re ( ω/ πT ) → Q as Im ( ω/ πT ) → −∞ .As T /T K decreases, the most significant change in fig. 4 occurs in the position of the “lowest” poles,meaning the poles nearest the origin at T /T K = 4 .
92 (fig. 4 (a). As
T /T K decreases, the lowest polesmove towards the origin (fig. 4 (b)), reach the origin at the critical temperature T /T K = 0 .
895 (fig. 4(c)), and then move into the upper half of the complex ω/ (2 πT ) plane (fig. 4 (d) and (e)), signalingan instability. For any other non-zero Q , the plots of the pole positions are qualitatively similar tothose in fig. 4. In particular, as T /T K decreases the lowest poles always pass through the origin andinto the upper half of the complex plane, signaling an instability.However, Q = 0 is slightly different. When Q = 0, so that H ( − / − iQ ) = H ( − /
2) = − . . . . isreal-valued, the only imaginary term in eq. (5.8) is in the argument of the harmonic number, whichis ∝ Re (cid:0) ω πT (cid:1) . As a result, solutions of eq. (5.8) must have Re (cid:0) ω πT (cid:1) = 0. Clearly, when Q = 0the particle-hole symmetry Re ( ω ) → − Re ( ω ) is restored. Fig. 5 shows our numerical results for thepositions of poles of G O † O (black dots) and G OO † (gray diamonds) in the complex ω/ (2 πT ) plane for Q = 0, for the temperatures T /T K = 44, 8, and 4. All the poles are now on the imaginary axis, butotherwise we observe similar behavior to the | Q | > T /T K decreases, the lowest poles infig. 5 (a) pass through the origin, now at a critical temperature T /T K = 8 in fig. 5 (b), and then crossinto the upper half of the complex plane in fig. 5 (c).Since the instability always appears as poles passing through the origin and into the upper half ofthe complex plane, we can determine the critical temperature T c easily, as the temperature where thepoles reach the origin: in eq. (5.8) we set ω = 0 and then solve for T /T K = T c /T K , with the resultln (cid:18) T c T K (cid:19) = − H (cid:18) −
12 + iQ (cid:19) − H (cid:18) − − iQ (cid:19) − ln 2 = − (cid:20) H (cid:18) −
12 + iQ (cid:19)(cid:21) − ln 2 . (5.9)Fig. 6 shows T c /T K as a function of Q , which has a maximum T c /T K = 8 at Q = 0, decreasesmonotonically as | Q | increases, and goes to zero as | Q | → ∞ .As mentioned in section 1, our results for the movement of ω ∗ differ dramatically from those of thestandard (non-holographic) Kondo model, at large N and at leading order in perturbation theory in λ [75]. In that model, the poles are determined by a condition identical to eq. (5.8), but without theln (2 T /T K ) term. As a result, the lowest pole sits exactly at ω = 0 for all T . The ln (2 T /T K ) term is33 ●●●●◆◆◆◆◆ - - - - - - ● ● ●●●◆◆◆◆◆ - - - - - - (a) (b) ● ● ●●●◆◆◆◆◆ - - - - - - ● ● ●●●◆◆◆◆◆ - - - - - - (c) (d) ● ● ●●●◆◆◆◆◆ - - - - - - (e) Figure 4: The positions of poles in G O † O (black dots) and G OO † (gray diamonds) in thecomplex ω/ (2 πT ) plane, determined by solving eq. (5.8) numerically, for Q = 0 . T /T K equal to (a) 4 .
92, (b) 1 .
34, (c) 0 . . . T /T K decreases,the “lowest” poles, meaning the poles closest to the origin at T /T K = 4 .
92 (a), movetowards the origin (b), reach the origin at
T /T K = 0 .
895 (c), and then pass into the upperhalf of the complex ω/ (2 πT ) plane (d and e), producing an instability.thus repsonsible for the non-trivial movement of ω ∗ , relative to the standard Kondo model. Indeed,the ln (2 T /T K ) term in eq. (5.8) can be viewed as arising from the renormalization of λ , i.e. as a strongcoupling effect arising from working non-perturbatively in both λ and the ’t Hooft coupling.We have been able to compute the position and residue of the poles analytically (without numerics)in two limits: T (cid:29) T c and T just above T c ( T (cid:38) T c ). In each case, we find a residue Z with non-zeroimaginary part, indicating that ρ O † O will exhibit Fano resonances, as we will confirm in subsection 5.2.In terms of T /T c (instead of T /T K ), G O † O takes a particularly simple form: using eq. (5.9) to re-write34 ●●●● - - - - - - ●●●●● - - - - - - ●●●●● - - - - - - (a) (b) (c) Figure 5: The positions of poles in G O † O (black dots) and G OO † (gray diamonds) in thecomplex ω/ (2 πT ) plane, determined by solving eq. (5.8) numerically for Q = 0 and T /T K equal to (a) 44, (b) 8, and (c) 4. Compared to the Q >
T /T K decreases, the lowest poles from (a) move up, reach theorigin at the critical temperature T /T K = 8 in (b) and then cross into the upper half ofthe complex ω/ (2 πT ) plane in (c), signaling an instability. - - Q T c /T K Figure 6: The critical temperature T c in units of T K , as a function of Q , from eq. (5.9).eq. (5.7), we find G O † O = − Nκ − Nκ H (cid:0) − + iQ − i ω πT (cid:1) − H (cid:0) − + iQ (cid:1) + ln (cid:16) TT c (cid:17) . (5.10)If T (cid:29) T c , or equivalently ln ( T /T c ) (cid:29)
1, then H (cid:0) − + iQ − i ω πT (cid:1) must also be large for G O † O tohave a pole. The Harmonic numbers H ( n ) grow large either when n → ∞ with | Arg ( n ) | < π , where H ( n ) → ln( n ), or when n approaches a negative integer, as apparent from the series representation H ( n ) = ∞ (cid:88) k =1 (cid:18) k − n + k (cid:19) . (5.11)35e are interested in poles near the origin of the complex ω -plane, rather than poles at large | ω | ,since the former have a larger effect on the spectral function, so we will only consider the poles where H (cid:0) − + iQ − i ω πT (cid:1) has argument equal to a negative integer. Explicitly, in the G O † O in eq. (5.10),near such a pole we use eq. (5.11) to take H (cid:18) −
12 + iQ − i ω πT (cid:19) − H (cid:18) −
12 + iQ (cid:19) ≈ − − + iQ − i ω πT + k , (5.12)with k = 1 , , , . . . . In that approximation, and with ln ( T /T c ) (cid:29)
1, the G O † O in eq. (5.10) becomes G O † O ≈ − Nκ − Nκ − − + iQ − i ω πT + k + ln ( T /T c ) . (5.13)The pole’s position ω ∗ = ω ∗ R + iω ∗ I and residue Z = Z R + iZ I are then given by ω ∗ πT = Q + i (cid:18) − k + 12 + 1ln ( T /T c ) (cid:19) , Z = − Nκ i (2 πT ) (cid:16) ln (cid:16) TT c (cid:17)(cid:17) , (5.14)where the lowest pole has k = 1, and the higher poles have k = 2 , , . . . . The residue Z in eq. (5.14)is purely imaginary, Z R = 0, so (recalling the table in section 4) we expect ρ O † O will have a q = 1symmetric Fano resonance.Eq. (5.10) makes obvious the pole at ω = 0 when T = T c . For T just above T c , T (cid:38) T c , we canobtain this pole’s position and residue by expanding eq. (5.10) in T around T c and simultaneously in ω around ω = 0. For the expansion in ω we use H (cid:18) −
12 + iQ − i ω πT (cid:19) − H (cid:18) −
12 + iQ (cid:19) = − ψ (cid:48) (cid:18)
12 + iQ (cid:19) iω πT + O (cid:18)(cid:16) ω πT (cid:17) (cid:19) , (5.15)where ψ (cid:48) ( n ) = ∂ n ψ ( n ) denotes the first derivative of the digamma function ψ ( n ). The pole’s position ω ∗ = ω ∗ R + iω ∗ I and residue Z = Z R + iZ I are then given by ω ∗ πT c = − iψ (cid:48) (cid:0) + iQ (cid:1) ( T /T c − , Z = − iψ (cid:48) (cid:0) + iQ (cid:1) (2 πT c ) Nκ , (5.16)as derived in ref. [42]. As T → T + c , both ω ∗ R and ω ∗ I vanish linearly, i.e. as T /T c −
1, with slopesdetermined by Q alone. Fig. 7 shows these slopes as functions of Q . In particular, fig. 7 (b) showsthat the magnitude of ω I ’s slope is largest when Q = 0 and decreases monotonically as | Q | increases.The residue Z in eq. (5.16) is in general complex-valued, so when T (cid:38) T c , the lowest pole in G O † O will produce a Fano resonance in ρ O † O . Plugging the Z in eq. (5.16) into eq. (4.7) gives us theFano/asymmetry parameter q as a function of Q , shown in fig. 8. Symmetric Fano resonances willoccur when Q → −∞ , 0, + ∞ , where q →
0, 1, and ∞ , respectively, corresponding to a Fano anti-resonance, symmetric Fano resonance, and Lorentzian resonance (figs. 2 (b), (c), and (d)), respectively. The spectral function ρ O † O in the unscreened phase is trivial to compute from G O † O in eq. (5.10): ρ O † O = − G O † O = 2 Nκ Im H (cid:0) − + iQ − i ω πT (cid:1) − H (cid:0) − + iQ (cid:1) + ln (cid:16) TT c (cid:17) , (5.17)36 ⇤ I ⇡T c T /T c - - - - - - - - - - - - - Q Q (a) (b) ! ⇤ R ⇡T c T /T c Figure 7: The slope of (
T /T c −
1) of the lowest pole in G O † O for T just above T c , asfunctions of Q , from eq. (5.16). (a) The slope of the real part of the pole, ω ∗ R , in units of2 πT c . (b) The slope of the imaginary part of the pole, ω ∗ I , in units of 2 πT c . q Q - Figure 8: The Fano/asymmetry parameter q as a function of Q for T (cid:38) T c , obtained byplugging the residue Z in eq. (5.16) into eq. (4.7) for q . The limits Q → −∞ , 0, + ∞ produce symmetric Fano resonances with q →
0, 1, + ∞ , respectively.where we now restrict to real-valued ω . In our case, ρ O † O vanishes when ω → | ω | → ∞ , inthe latter case vanishing as (ln | ω | ) − , ultimately because the Harmonic numbers are asymptoticallylogarithmic, as mentioned above. Such (ln | ω | ) − behavior means our ρ O † O cannot be exactly ρ Fano ineq. (4.3), since ρ Fano involves only powers of ω , with no logarithms. Nevertheless, we have shown insubsection 5.1 that the lowest pole in G O † O generically has residue with non-zero imaginary part, sowe expect Fano resonances in ρ O † O at ω near the real part of the lowest pole’s position, ω ∗ R .Fig. 9 shows ρ O † O / (cid:0) N/κ (cid:1) as a function of ω/ (2 πT ) for the representative value Q = 0 . (cid:29) T c regime, namely from T /T c = 10 (fig. 9 (a)) down to T /T c = 10 (fig. 9 (b)). From the T (cid:29) T c results in eqs. (5.13) and (5.14), we expect ρ O † O to have a q = 1 symmetric Fano resonancewhen ω equals the real part of the lowest pole’s position, ω ∗ R , which is ω ∗ R = Q when T (cid:29) T c . Sureenough, for sufficiently high T /T c , as in fig. 9 (a), ρ O † O has an approximately q = 1 symmetric Fanoresonance at ω ≈ ω ∗ R ≈ Q . As T /T c decreases through twelve orders of magnitude, the asymmetryof the resonance appears to increase, although the position changes by only ≈ ω ∗ R ≈ . T /T c = 10 , while ω ∗ R ≈ .
475 when
T /T c = 10 . We have confirmed numerically that as T /T c decreases through the values in fig. 9, the peak value of the resonance grows as 1 / (ln( T /T c )) ,consistent with the T (cid:29) T c results for ω ∗ I and Z I in eq. (5.14). Crucially, the resonance in fig. 9 is notat the particle-hole symmetric value ω = 0, and so is not the Kondo resonance—as expected, sincethe Kondo resonance is generically absent at large- N in the unscreened phase. - - - - - - - - - - - - - - (a) (b) Figure 9: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for the repre-sentative value Q = 0 . T (cid:29) T c regime, namely for (a) T /T c = 10 and (b) T /T c = 10 (dotted), 10 (dot-dashed), 10 (dashed), and 10 (solid).Fig. 10 shows ρ O † O / (cid:0) N/κ (cid:1) as a function of ω/ (2 πT ) for Q = 0 . T /T c = 10 (fig. 10 (a))down to T /T c = 1 . T /T c = 5 .
5, corresponding to
T /T K = 4 .
92 (fig. 4 (a)),and
T /T c = 1 .
5, corresponding to
T /T K = 1 .
34 (fig. 4 (b)). In fig. 10, as
T /T c decreases, we see fourchanges in the resonance. First, the peak of the resonance moves towards ω = 0, following the positionof the lowest pole in G O † O . For example, compare the position of the peak in ρ O † O at T /T c = 5 . . G O † O infig. 4 (a) or (b), respectively. Second, the resonance grows taller, by about an order of magnitude forthe values of T /T c in fig. 10. Third, the peak grows narrower, also by about an order of magnitudefor the values of T /T c in fig. 10. Fourth, the Fano/asymmetry parameter q increases. For example, q ≈ T /T c = 10 (fig. 10 (a)) and q ≈ T /T c = 1 . ρ O † O / (cid:0) N/κ (cid:1) as a function of ω/ (2 πT ) for Q = 0 . T (cid:38) T c regime, namelyfor T /T c = 1 . T /T c = 1 .
001 (fig. 11 (b)). The four trends observed in fig. 10appear again in fig. 11. First, the resonance moves towards ω = 0, following the real part of theposition of the lowest pole in G O † O in the T (cid:38) T c regime, given by ω ∗ in eq. (5.16), which in particularhas ω ∗ R ∝ ( T /T c − T (cid:38) T c results for ω ∗ and Z of eq. (5.16) into eq. (4.5) reveals that the peak of the resonance increases as ( T /T c − − .Such power-law growth, rather than logarithmic growth, again indicates that the resonance is not a38 - - - - - - - - - - - - - - - (a) (b) Figure 10: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for Q = 0 . T /T c = 10, (b) 7 . . . . G O † O in the T (cid:38) T c regime, which from eq. (5.16) has ω ∗ I ∝ ( T /T c − q increases. For example, q ≈ . T /T c = 1 . q ≈ . T /T c = 1 .
001 (fig. 10 (b)). - - (a) (b) - - Figure 11: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for Q = 0 . T /T c = 1 .
1, (b) 1 .
01 (dotted), 1 . .
005 (dashed), and 1 .
001 (solid).In the T (cid:38) T c regime, we expect symmetric Fano resonances when Q → −∞ , 0, ∞ , as discussed beloweq. (5.16) and in fig. 8. We indeed find such behavior, already at relatively small values of | Q | . Fig. 12shows ρ O † O / (cid:0) N/κ (cid:1) as a function of ω/ (2 πT ) for T /T c = 1 .
01 and (a) Q = −
1, (b) Q = 0, and (c) Q = +1. We clearly see symmetric Fano (anti-)resonances with (a) q ≈ . q = 1, and (c) q ≈ .
9, respectively, all consistent with eq. (5.16) and fig. 8.For the special value Q = 0 nothing breaks the particle-hole symmetry Re ω → − Re ω , and all polesof G O † O have vanishing real part, as shown for example in fig. 5. When Q = 0 we thus expect a q = 1 symmetric Fano resonance at ω = 0 for all T /T c . Fig. 13 shows ρ O † O / (cid:0) N/κ (cid:1) as a function of ω/ (2 πT ) for Q = 0 and T /T c from T /T c = 100 down to 2 . T /T c = 1 . .
01 (fig. 13 (b)). We indeed find q = 1 symmetric Fano resonances at ω = 0 for all T /T c .39 - - - - - - - - - - - - - - - - (a) (b) (c) Figure 12: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for T /T c = 1 . Q = −
1, (b) Q = 0, and (c) Q = +1. - - - - - - - - (a) (b) Figure 13: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for Q = 0 and(a) T /T c = 100 (dotted), 10 (dot-dashed), 5 (dashed), and 2 . T /T c = 1 . .
05 (dot-dashed), 1 .
03 (dashed), and 1 .
01 (solid).We can also consider ρ O † O in the unscreened phase when T < T c , bearing in mind that the unscreenedphase is unstable when T < T c because G O † O has a pole with Im ω ∗ >
0, as discussed above. Fig. 14(a) shows ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for Q = 0 . T < T c , namely for T /T c = 0 . .
5, corresponding to
T /T K = 0 .
671 and 0 . T > T c in figs. 10 and 11. First, the resonancemoves away from ω = 0, with peak position at ω ≈ ω ∗ R . Second, the resonance grows shorter. Third,the resonance grows wider. Fourth, the value of q decreases. In particular, q ≈ .
11 for
T /T c = 0 . q ≈ .
06 for
T /T c = 0 .
5. Fig. 14 (b) shows ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for Q = 0 and T /T c = 0 .
75 and 0 .
5. In that case, as expected we find a q = 1 symmetric Fano resonance at ω = 0whose height decreases as T decreases. All of these behaviors are consistent with the motion of thelowest pole in G O † O in the complex ω plane described in subsection 5.1.In summary, we have learned two key lessons from the poles in G O † O and corresponding resonancesin ρ O † O in the unscreened phase. First, we do not see a Kondo resonance, consistent with theexpectations of large- N Kondo models, where the Kondo effect (screening, phase shift, etc.) occursonly in the screened phase. Second, the resonances we find are all Fano resonances, consistent with ourinterpretation that (0 + 1)-dimensional scale invariance implies a continuum, and our Kondo couplingthen breaks scale invariance and produces a resonance that is necessarily immersed the continuum.40 - - - - - - - - - - (a) (b) Figure 14: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for T /T c = 0 . . Q = 0 . Q = 0. In this section we use the results of sections 2 and 3 to determine the excitation spectrum of oursystem in the screened phase (
T < T c ) by locating the poles in G O † O in the plane of complex ω (subsection 6.1), and the corresponding peaks in ρ O † O for real ω (subsection 6.2).The main results of this section appeared in ref. [42], namely that for T just below T c ( T (cid:46) T c ), apole of the form ω ∗ ∝ − i (cid:104)O(cid:105) appears in G O † O , giving rise to a q = 1 symmetric Fano resonancein ρ O † O , which is a signature of a Kondo resonance at large N . In this section we will present someadditional details about these results. Moreover, in appendix D we show, without using numerics,that ω ∗ ∝ − i (cid:104)O(cid:105) , but only for Q = − /
2, although our methods should easily generalize to any Q .As derived in eq. (3.81), in the screened phase G O † O = N ˆ R / − κ ˆ R / , (6.1)and G O † O = G OO † = G OO = G O † O † , so we will henceforth discuss only G O † O . In the unscreenedphase we had the analytic ( i.e. non-numerical) result for ˆ R Φ † Φ in eq. (5.2), however, in this sectionour solutions for ˆ R will be numerical. Clearly G O † O in eq. (6.1) has a pole whenever 1 − κ ˆ R / G OO † , G OO , G O † O † , (cid:104)O ( ω ) A t ( − ω ) (cid:105) κ and (cid:104)O † ( ω ) A t ( − ω ) (cid:105) κ .) Given values of Q and T /T K , we can thus find the poles in G O † O by solving theequation 1 − κ ˆ R / ω/ (2 πT ), which we have done numerically. Our numerical results for thepositions of the poles appear in fig. 15, for Q = 0 . T /T c = 1 in fig. 15 (a), 0 .
588 (b), 0 . .
200 (d). When
T /T c = 1, the poles’ positions agree with those we found in the unscreenedphase in subsection 5.1, including in particular the lowest pole, ω ∗ , sitting at the origin of the complex ω/ (2 πT ) plane. As T /T c decreases the most significant change occurs in ω ∗ , which moves straight41own the imaginary axis. For any other non-zero Q , the plots of the pole positions are qualitativelysimilar to those in fig. 15, except for Q = 0, where all the higher poles are on the imaginary axis. Inparticular, for all Q , including Q = 0, as T decreases the most significant change occurs in ω ∗ , whichmoves straight down the imaginary axis.For T just below T c , T (cid:46) T c , we find that ω ∗ is determined by (cid:104)O(cid:105) . More specifically, fig. 16 showsthat ω ∗ ∝ − i (cid:104)O(cid:105) when T (cid:46) T c . In appendix D, for the case Q = − / i.e. without numerics) that ω ∗ ∝ − i (cid:104)O(cid:105) for T (cid:46) T c . Given the mean-field scaling discussed in sec. 2, (cid:104)O(cid:105) ∝ ( T c − T ) / when T (cid:46) T c , we thus have ω ∗ ∝ − i | T − T c | when T (cid:46) T c . (a) (b)(c) (d) ● ● ●●◆◆◆◆ - - - - - ● ● ●●◆◆◆◆ - - - - - ● ● ●●◆◆◆◆ - - - - - ● ● ●●◆◆◆◆ - - - - - Figure 15: Our numerical results for the positions of poles in G O † O in the complex ω/ (2 πT )plane, for Q = 0 . T /T c equal to (a) 1, (b) 0 . . . T /T c decreases, the most significant change occurs in the position of the lowest pole, whichmoves straight down the imaginary axis.As mentioned in section 1, a pole in G O † O of the form ω ∗ ∝ − i (cid:104)O(cid:105) is precisely the manifestation ofthe Kondo resonance that we expect at large N [75]. In other words, in addition to the dynamicallygenerated scale T K , impurity screening, a phase shift, and so forth, our holographic Kondo model alsocorrectly captures an essential spectral feature of the Kondo effect, namely the Kondo resonance. Knowing the result of subsection 5.2, that our spectral function ρ O † O generically exhibits a Fanoresonance associated with the lowest pole ω ∗ in G O † O , and knowing the result of subsection 6.1, thatin the screened phase ω ∗ is purely imaginary and simply moves down the imaginary axis as T decreases,42 hOi N (2 ⇡T ) Im ✓ ! ⇤ ⇡T ◆ ●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● - - - - Figure 16: In the screened phase, the lowest pole in G O † O , ω ∗ , is purely imaginary (seefig. 15). The black dots denote Im ω ∗ / (2 πT ) as a function of κ (cid:104)O(cid:105) / ( N (2 πT )) for Q = 0 .
5. The solid black line is a numerical linear fit with slope ≈ − . ≈ × − . The agreement between the data and the fit shows that ω ∗ ∝ − i (cid:104)O(cid:105) .we can anticipate how ρ O † O will behave in the screened phase. Given that ω ∗ is purely imaginary,and hence does not break particle-hole symmetry Re ω → − Re ω , we expect ρ O † O to exhibit a q = 1symmetric Fano resonance at ω = 0. Moreover, given that ω ∗ moves straight down the imaginary ω axis as T decreases, we expect the Fano resonance’s width to increase as T decreases.Our numerical results for ρ O † O in the screened phase confirm these expectations. Fig. 17 shows ournumerical results for ρ O † O / ( N/κ ) in the screened phase as a function of real-valued ω/ (2 πT ) for Q = 0 . T /T c ≈ . . . q = 1 symmetric Fano resonanceswhose width increases as T decreases. We also find that the resonance’s height decreases rapidly as T decreases: in fig. 17, T /T c decreases by only about 4%, from T /T c ≈ .
998 down to
T /T c ≈ . T decreases further (not shownin fig. 17), ρ O † O continues to flatten, and indeed, as T approaches zero, ρ O † O appears to approachzero for all ω . All of these features of ρ O † O appear for other values of Q as well, including Q = 0.In the standard (non-holographic) large- N Kondo model with Abrikosov pseudo-fermions, the Kondoresonance has width ∝ (cid:104)O(cid:105) [75]. For T (cid:46) T c , the mean-field behavior (cid:104)O(cid:105) ∝ ( T c − T ) / then impliesthe width is ∝ T c − T . When T → (cid:104)O(cid:105) reaches a finite value ∝ T / K at the minimum of itswine-bottle effective potential. The Kondo resonance then has width ∝ T K , similarly to finite N .Our model also exhibits mean-field behavior, and hence a width ∝ T c − T when T (cid:46) T c . However, inour screened phase, as T decreases our manifestation of the Kondo resonance, i.e. the q = 1 symmetricFano resonance in ρ O † O , flattens out, and ultimately disappears, so that at T = 0 apparently ρ O † O is featureless. What accounts for the difference? In our model, (cid:104)O(cid:105) ’s effective potential is apparentlyunbounded: we found numerically that (cid:104)O(cid:105) grows without bound as T decreases, because Φ growswithout bound. Indeed, as T decreases, eventually the solutions for a t ( z ) and Φ( z ) violate the probelimit: the stress-energy tensor grows without bound, and eventually cannot be neglected in Einstein’sequation. That is unsurprising, given that in our bulk action eq. (2.8), Φ’s potential is unbounded,being only a mass term, M Φ † Φ. Presumably, stabilizing Φ’s potential, for example with a (Φ † Φ) term, would stabilize (cid:104)O(cid:105) , and hence stabilize the width of our resonance.43 ●●●●●●●●● ● ● ● ● ● ●●●●●●●●●●●●●●●●● ◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ■■■■■■■■■■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■ - - - - Figure 17: The spectral function, ρ O † O / (cid:0) N/κ (cid:1) , as a function of ω/ (2 πT ) for the rep-resentative value Q = 0 . T /T c = 0 .
998 (dots), 0 . .
964 (squares). For all
T /T c we find a q = 1 symmetric Fano resonanceat ω = 0 whose height decreases and width increases as T /T c decreases. We studied the poles in retarded Green’s functions and the associated peaks in spectral functions inthe holographic Kondo model of refs. [31–34]. We had three main results. First was the holo-renof our model, which provided the covariant counterterms required to compute the renormalized freeenergy and one- and two-point functions in our model. Second, at all T , we found that genericallythe poles in our Green’s functions have residue with non-zero imaginary part, giving rise to Fanoresonances in spectral functions. Fano resonances occur when a resonance appears in a continuum(in energy) of states. Our continuum comes from (0 + 1)-dimensional scale invariance, inherited from(1 + 1)-dimensional scale invariance of our holographic CFT. Our resonances are possible becausewe break scale invariance via our marginally-relevant Kondo coupling. Third, in the screened phase,where (cid:104)O(cid:105) (cid:54) = 0, and with T just below T c , we found a pole in G O † O of the form ω ∗ ∝ − i (cid:104)O(cid:105) ,precisely as expected for the Kondo resonance at large N [75]. In contrast, in the unscreened phase ω ∗ passed through the origin as T decreased through T c , which was clearly a strong coupling effect: inthe standard (non-holographic) Kondo model at large N and at leading order in perturbation theoryin λ , in the unscreened phase ω ∗ sits at the origin of the complex ω plane for all T [75].For the future, some obvious, immediate tasks involve improvements to our model. For example, givingour bulk scalar Φ a quartic self-interaction could not only prevent Φ from diverging at low T , andhence maintain the validity of the probe limit at low T , but could also prevent our Kondo resonancefrom disappearing as T decreases, as we discussed in subsection 6.2. Indeed, adding a quartic termwould introduce an additional dimensionful parameter in our model, which could presumably be fixedby demanding that our Kondo resonance has width ∝ T K when T = 0.However, as discussed in refs. [31, 32], all holographic quantum impurity models to date, including44urs, have a fundamentally worrying issue: the spin symmetry group is the gauge group, SU ( N ).Holography provides direct access only to gauge-invariant quantities. As a result, many importantquantities that are not spin singlets, such as the magnetization and spin susceptibility, are prohibitivelydifficult if not impossible to calculate using holography. The obvious route to address this issue is todevelop holographic quantum impurity models in which spin is a global symmetry.We have seen that even a minimal holographic quantum impurity model can produce Fano resonances.Indeed, Fano resonances require simple, common ingredients, and thus are very generic. We there-fore expect Fano resonances in practically any holographic quantum impurity model, under the keycondition that conformal symmetry is broken at the impurity. (Otherwise, all two-point functions atthe impurity are determined by the conformal symmetry, as we mentioned in the section 1.) In fact,more generally we expect asymmetric Fano resonances in practically any holographic system with aUV fixed point, breaking of scale invariance, and breaking of particle-hole symmetry.Most importantly, we expect our holographic Kondo model, other similar holographic quantum im-purity models, and variations of SYK models, to be useful in addressing many of the open questionsmentioned in section 1, about EE, quantum quenches, etc. We expect Fano resonances in particular toplay a crucial role in developing a precise “dictionary” between theoretical models and experiments. Acknowledgments
We thank Ian Affleck, Natan Andrei, Piers Coleman, Mario Flory, Antal Jevicki, Henrik Johannesson,Andrew Mitchell, Max Newrzella, and Philip Phillips for helpful conversations and correspondence.A. O’B. is a Royal Society University Research Fellow. J. P. is supported by the Clarendon Fundand St John’s College, Oxford, and by the European Research Council under the European Union’sSeventh Framework Programme (ERC Grant agreement 307955). C.H. is supported by the Ramon yCajal fellowship RYC-2012-10370, the Asturian grant FC-15-GRUPIN14-108 and the Spanish nationalgrant MINECO-16-FPA2015-63667-P.
AppendicesA Near-Horizon Expansions in the Screened Phase
In this appendix we determine the near-horizon expansions of the response functions R , R , R ,and R defined in eq. (3.34) in the screened phase. We use these expansions to fix in-going boundaryconditions at the horizon when we solve eq. (3.37) numerically for the response functions.In this appendix we will switch from the holographic radial coordinate z in eq. (2.6) to the coordinate ζ ≡ z H − z , so that near the horizon, and using eq. (3.2) to translate from z to r of eq. (3.1), γ = − z H ζ + O ( ζ ) , ∂ r = (cid:112) z H ζ (1 + O ( ζ )) ∂ ζ , γ − ˙ γ = (cid:114) z H ζ (1 + O ( ζ )) . (A.1)45ear the horizon, eqs. (3.17) for the background fields a t and φ thus become ∂ ζ a t − z H ∂ ζ a t − φ z H ζ a t = 0 , ∂ ζ φ + 1 ζ ∂ ζ φ + (cid:0) a t (cid:1) z H ζ φ − M z H ζ φ = 0 , (A.2)with regular solutions a t = a (1) (cid:16) ζ + (2 + φ )2 z H ζ + O ( ζ ) (cid:17) , φ = φ (0) (cid:16) M z H ζ + O ( ζ ) (cid:17) , (A.3)with integration constants a (1) and φ (0) , which we determine in our numerical solutions by matchingwith the integration constants in the near-boundary expansions.Using the above, we can determine the near-horizon expansions of the fluctuations δa t , δφ , and δψ ineq. (3.18). Near the horizon, the constraint eq. (3.18a) becomes − z H ζ iω∂ ζ δa t + 2 φ ∂ ζ δψ = 0 . (A.4)with solutions δa t = c ζ c (1 + O ( ζ )) , δψ = c iωz H (2 + c )4 φ (1 + c ) ζ c (1 + O ( ζ )) , (A.5)with integration constants c and c . Inserting eq. (A.5) into eqs. (3.18b) and (3.18c) gives us (cid:18) ζ ∂ ζ + (2 + c ) z H c ) ω (cid:19) δa t = 2 z H φ (0) a (1) ζ δφ, (A.6a) (cid:18) ζ ∂ ζ + ζ∂ ζ + z H ω (cid:19) δφ = a (1) ω z H (2 + c )8 φ (0) (1 + c ) δa t , (A.6b)to leading order, with the linearly-independent solutions δa t = c ζ c (1 + O ( ζ )) , (A.7a) δφ = c z H φ (0) a (1) (cid:18) c + 2 c + 1 (cid:19) (cid:18) ( c + 1) + (cid:16) z H ω (cid:17) (cid:19) ζ c (1 + O ( ζ )) , c = ± i z H ω , (A.7b) δa t = c ζ c (1 + O ( ζ )) , (A.7c) δφ = c z H a (1) ω φ (0) (cid:18) c + 2 c + 1 (cid:19) (cid:18) ( c + 2) + (cid:16) z H ω (cid:17) (cid:19) − ζ c (1 + O ( ζ )) , c = − ± i z H ω . (A.7d)The most general in-going solution is a linear combination of the solutions with c = − i z H ω ≡ c (1)in , c = − − i z H ω ≡ c (2)in . (A.8)Near the horizon, the definitions of the response functions in eq. (3.33) become δ ˙ a t = R ( δa t + iωδψ ) − ζz H R δφ, (A.9a) δ ˙ φ = 12 R δφ + 12 (cid:16) R − z H ζ R (cid:17) ( δa t + iωδψ ) , (A.9b) δ ˙ ψ = iωz H ζφ δ ˙ a t . (A.9c)46nserting the two linearly-independent in-going solutions of eq. (A.7) into eq. (A.9) leads to fouralgebraic equations for the leading near-horizon behavior of the response functions, ω z H φ R + ( c (1)in + 1) + (cid:0) z H ω (cid:1) z H φ (0) a (1) R = −√ z H ( c (1)in + 1) ζ , (A.10a) ω z H φ R + z H ω a (1) φ (0) (cid:18) ( c (2)in + 2) + (cid:16) z H ω (cid:17) (cid:19) − ζ R = −√ z H ( c (2)in + 1) ζ , (A.10b)12 R − a (1) z H ω φ (0) (cid:18) ( c (1)in + 1) + (cid:16) z H ω (cid:17) (cid:19) − ζ (cid:16) R − z H ζ R (cid:17) = √ z H ζ − c (1)in , (A.10c)12 R − ( c (2)in + 2) + (cid:0) z H ω (cid:1) z H φ (0) a (1) ζ − (cid:16) R − z H ζ R (cid:17) = √ z H ( c (2)in + 2) ζ − , (A.10d)with solutions R = i √ φ ωz / H (cid:112) ζ (cid:40) ζ (cid:32) z H − iφ ωz H (1 − iωz H ) − iM ω − iωz H (cid:33) + O ( ζ ) (cid:41) , (A.11a) R = − i √ ωz / H √ ζ (cid:26) − ζ (cid:18) z H − iM ωz H (1 − iωz H ) (cid:19) + O ( ζ ) (cid:27) , (A.11b) R = − √ z / H a (1) φ (0) − iωz H (cid:112) ζ (cid:40) − ζ (cid:32) iω − iωz H ) + M ω (3 i + ωz H )2(1 − iωz H )(2 − iωz H )+ (cid:0) (2 i + ωz H )(1 + ω z H ) + 2 i (cid:1) φ ωz H (1 − iωz H )(2 − iωz H ) (cid:33) + O ( ζ ) (cid:41) , (A.11c)which are the main results of this appendix. Inserting eq. (A.11) into the general solution for R ineq. (3.36) then gives us R = C ( ω ) √ z / H (cid:16) ζ + iz H ω + O ( ζ / ) (cid:17) , (A.12)and hence in-going boundary conditions require that C ( ω ) = 0 and thus R = 0, as advertised insubsection 3.2.2. As a result, the Riccati equations in eq. (3.35) simplify to those in eq. (3.37). B Near-Boundary Expansions
In this appendix we determine the general Fefferman-Graham (FG) asymptotic expansions of the
AdS fields in our model. As mentioned at the beginning of section 3 these FG expansions involve a numberof subtleties, related to the special form of the FG expansion of gauge fields in AdS . In particular,the leading asymptotic mode of the gauge field is the charge Q instead of the chemical potential µ ,unlike gauge fields in higher-dimensional AdS spacetimes, and moreover the value of Q affects theFG expansion of the scalar field Φ. As a result, a well-defined space of asymptotic solutions requireskeeping Q fixed, which corresponds to an asymptotic second class constraint on the space of solutions.47uch a constraint is unusual, compared to many holographic systems, although the constraint requiredfor Lifshitz asymptotics in Einstein-Proca theory [97, 98] is analogous.A direct result of the constraint is that, if we allow fluctuations about a background solution to havenon-zero variation of Q , then the background and fluctuations need not have the same FG expansions.Indeed, in that case, higher order fluctuations are increasingly dominant asymptotically, relative toboth the background solutions and to the lower order fluctuations. As a result, the small fluctuationapproximation breaks down asymptotically, and we are forced to work with a cut-off near the boundary,until fluctuations proportional to δQ are set to zero. In addition, generically no well-defined asymptoticsolutions to the full non-linear equations of motion exist, so we must consider the FG expansions ofthe background and of the fluctuations separately. Below we determine the FG expansions both forthe background and the fluctuations, discussing separately fluctuations with δQ (cid:54) = 0 and δQ = 0. Note about Notation:
In this appendix and in appendix C, O log ( x ) denotes a quantity thatasymptotes to zero like x log k ( x ) as x → + , with k a non-negative integer. B.1 Expansions of the Background and the Second Class Constraint
Upon choosing a gauge with A t = 0, the equations of motion for a t , φ , and ψ , eqs. (3.12), become¨ a t − γ − ˙ γ ˙ a t − φ ( a t − ∂ t ψ ) = 0 , (B.1a)¨ φ + 12 γ − ˙ γ ˙ φ − ˙ ψ φ + γ − ∂ t φ − γ − ( a t − ∂ t ψ ) φ − M φ = 0 , (B.1b) ∂ r ( φ ˙ ψ ) + 12 γ − ˙ γφ ˙ ψ − γ − ∂ t (cid:0) φ ( a t − ∂ t ψ ) (cid:1) = 0 , (B.1c) γ − ∂ t ˙ a t = 2 φ ˙ ψ. (B.1d)Given the asymptotic form of the metric, γ ∼ − e r as r → + ∞ , as long as φ → i.e. the dual operator is relevant), then the gauge field’s leading asymptotic behavior is a t ∼ e r Q ( t ),with Q ( t ) an arbitrary function of time t . Moreover, Q enters φ ’s equation as a mass term, so that φ has an “effective mass” M − Q , hence Q affects the FG expansion of φ . A well-defined space ofasymptotic solutions thus requires the (second class) constraint that Q is fixed. The charge Q is notautomatically conserved by the equations of motion, due to the coupling to the charged scalar field.Charge conservation, therefore, can only be imposed as a boundary condition.As in ref. [31], we fix Q such that O has dimension 1 /
2, so that our Kondo coupling O † O is classicallymarginal. The scalar’s effective mass must thus saturate the AdS Breitenlohner-Freedman bound: M − Q = − . (B.2)We want to determine the FG expansions with Q satisfying the constraint eq. (B.2). Crucially, in thefirst three equations in (B.1), terms containing time derivatives affect only sub-leading orders in theFG expansion: for the leading non-normalizable and normalizable orders, we can thus ignore all timederivatives in eqs. (B.1). For similar reasons, we can take γ = − e r for the purpose of determining48he FG expansions. With these simplifications, eqs. (B.1) become¨ a t − ˙ a t − φ a t = 0 , (B.3a)¨ φ + ˙ φ − ˙ ψ φ + e − r a t φ − M φ = 0 , (B.3b) ∂ r ( φ ˙ ψ ) + φ ˙ ψ = 0 , (B.3c)and hence the FG expansions of the AdS fields are a t = e r Q − Q (cid:16) α r + ( α − αβ ) r + (2 α − αβ + β ) r (cid:17) + µ ( t ) + · · · , (B.4a) φ = e − r/ ( − α ( t ) r + β ( t )) + · · · , (B.4b) ψ = ψ − ( t ) + ψ + ( t ) r − + · · · , (B.4c)where µ ( t ), α ( t ), β ( t ) and ψ ± ( t ) are arbitrary functions of time, and . . . represent terms that vanishas r → ∞ faster than those shown, and are completely determined by those shown, via the equationsof motion. Inserting eqs. (B.4) into eq. (B.1d), which is the constraint imposed by the AdS U (1)gauge invariance, and using eq. (B.2), we find ψ + = 0 and α − ∂ t Q = 0. The FG expansions are thusparameterized by the arbitrary functions µ ( t ), α ( t ), β ( t ) and ψ − ( t ). Moreover, µ ( t ) is defined only upto a U (1) gauge transformation, µ ( t ) → µ ( t ) + ∂ t λ ( t ). We will refer to eqs. (B.4) as “background FGexpansions,” because Q is required to satisfy eq. (B.2). Fluctuations are allowed to violate eq. (B.2),which leads to different FG expansions, as we will see. B.2 Expansions of the Response Functions
In the unscreened phase, we want to find the FG expansions of the response functions R Φ † Φ and R ΦΦ † , using the Riccati equations in eq. (3.20). As above, to do so we may ignore terms involvingtime derivatives, i.e. frequency ω , and we may set γ = − e r , in eq. (3.20), leading to˙ R Φ † Φ + R Φ † Φ + R † Φ + 14 = 0 , ˙ R ΦΦ † + R ΦΦ † + R † + 14 = 0 , (B.5)and hence the FG expansions of R Φ † Φ and R ΦΦ † are R Φ † Φ = −
12 + 1 r − (cid:98) R Φ † Φ = −
12 + 1 r + (cid:98) R Φ † Φ r + · · · , (B.6a) R ΦΦ † = −
12 + 1 r − (cid:98) R ΦΦ † = −
12 + 1 r + (cid:98) R ΦΦ † r + · · · , (B.6b)where (cid:98) R Φ † Φ and (cid:98) R ΦΦ † are functions of ω , and . . . represent terms that vanish as r → ∞ faster thanthose shown, and are completely determined by those shown, via eq. (3.20).In the screened phase, we instead need to solve instead the Riccati equations eqs. (3.35), with R = 0,as required by in-going boundary conditions at the horizon, as shown in appendix A. Again ignoring49erms involving time derivatives, and setting γ = − e r , eqs. (3.35) become˙ R − R + R − e r R − φ = 0 , (B.7a)˙ R + R + R R + 12 R R + 4 e − r φ a t = 0 , (B.7b)˙ R + R − e r R + 12 R + 12 = 0 . (B.7c)These equations admit two distinct classes of asymptotic solutions, depending on whether δQ ( t ) (cid:54) = 0or δQ ( t ) = 0. We present both of these solutions in turn.For fluctuations with δQ (cid:54) = 0, the defining relations in eqs. (3.33) and the asymptotic solution for a t in eq. (B.4) imply that asymptotically R ∼
1. Eqs. (B.7) then determine the leading asymptoticbehavior of the response functions: R = 1 + O log ( e − r ), R = O log ( e − r/ ), R = − O (1 /r ). Ineq. (B.7c), the term ∝ R is exponentially subleading relative to the other terms, and hence can beignored. The resulting equation for R then admits an exact solution, with asymptotic expansion R = − r − (cid:98) R / O log ( e − r ) , (B.8)where (cid:98) R is an undetermined function of ω . Eqs. (B.7a) and (B.7b) then determine R = − e − r/ r − (cid:98) R / (cid:90) d r (cid:16) r − (cid:98) R / (cid:17) e − r/ φ ( r ) a t ( r ) + O log ( e − r/ ) , (B.9a) R = 1 + e − r (cid:90) d re r (cid:16) e r R + 2 φ (cid:17) + O log ( e − r ) . (B.9b)Expanding these then leads to the FG expansions R = 1 + e − r (cid:18) Q α r − Q α (cid:98) R α + 6 β ) r + 118 (cid:16) (12 − Q (cid:98) R ) α + 12 Q ( (cid:98) R α + β ) β (cid:17) r + (cid:16) Qα
36 (24 (cid:98) R + Qα (cid:98) R ) − α β − Q β (cid:98) R (cid:17) r + 172 (cid:16) Qα (cid:98) R (24 (cid:98) R + Qα (cid:98) R ) − Q (12 (cid:98) R + Qα (cid:98) R ) β + 36(4 + Q (cid:98) R ) β (cid:17) r + (cid:98) R + O (1 /r ) (cid:19) + O log ( e − r ) , (B.10a) R = e − r (cid:16) Qα r − Q α (cid:98) R + 6 β ) r − Q (cid:98) R α (cid:98) R − β ) + (cid:98) R r + (cid:98) R (cid:98) R r + (cid:98) R (cid:98) R r + O (1 /r ) (cid:17) + O log ( e − r/ ) , (B.10b) R = − r + 1 r (cid:98) R + 12 r (cid:98) R + 14 r (cid:98) R + 18 r (cid:98) R + O (1 /r ) , (B.10c)where (cid:98) R , (cid:98) R and (cid:98) R are undetermined functions of the frequency ω . If we plug eqs. (B.10) intothe defining relations eqs. (3.33), then these asymptotic expansions lead to linear fluctuations that areasymptotically more divergent that the background solutions in eqs. (B.4)—an effect of the asymptotic50econd class constraint eq. (B.2), which is violated infinitesimally by the linear fluctuations with δQ (cid:54) = 0. The second class constraint also causes the integration constant (cid:98) R to enter the asymptoticexpansions of R and R before their corresponding integration constants (cid:98) R and (cid:98) R . We musttherefore determine the asymptotic expansions of R and R beyond the order where (cid:98) R and (cid:98) R appear linearly, since these terms enter in the expansions of R and R .While fluctuations with δQ (cid:54) = 0 have three integration constants, (cid:98) R , (cid:98) R and (cid:98) R , fluctuations with δQ = 0 have only one, as we will now show. For fluctuations with δQ = 0, the three response functionshave the leading order behavior R = O log ( e − r ), R = O log ( e − r/ ), and R = − O (1 /r ).Eq. (B.7) then implies that R is again given by eq. (B.8), while R = − e r ∞ (cid:90) r d r (cid:48) e − r (cid:48) (cid:18) e r (cid:48) R + 2 φ (cid:19) + O log ( e − r ) , (B.11a) R = 4 e − r/ r − (cid:98) R / ∞ (cid:90) r d r (cid:48) e − r (cid:48) / (cid:16) r (cid:48) − (cid:98) R / (cid:17) φ ( r (cid:48) ) a t ( r (cid:48) ) + O log ( e − r/ ) . (B.11b)Expanding eq. (B.11) using eq. (B.4) then gives the FG expansions R = e − r (cid:16) − (1 + 4 Q ) α r + α (cid:0) Q ) β − (1 + 20 Q ) α (cid:1) r (B.12a)+ (cid:16) (1 + 28 Q ) α β − (1 + 4 Q ) β −
12 (1 + 4(2 (cid:98) R + 21) Q ) α (cid:17) + O (1 /r ) (cid:17) + O log ( e − r ) , R = e − r (cid:16) − Qα r + 4 Q ( β − α ) + 4 Qβ − (cid:98) R + 4) Qα r + O (cid:16) r (cid:17)(cid:17) + O log ( e − r ) , (B.12b) R = − r + 1 r (cid:98) R + 12 r (cid:98) R + 14 r (cid:98) R + 18 r (cid:98) R + O (1 /r ) . (B.12c)Inserting the expansions for R , R and R for either δQ (cid:54) = 0 or δQ = 0 into eqs. (3.34), then gives R Φ † Φ = −
12 + 1 r + (cid:98) R Φ † Φ r + · · · , R ΦΦ † = −
12 + 1 r + (cid:98) R ΦΦ † r + · · · , R ΦΦ = (cid:98) R ΦΦ r + · · · , (B.13)which is of the same form as the unscreened case, eq. (B.6), but now with the constraints (cid:98) R Φ † Φ = (cid:98) R ΦΦ † = (cid:98) R ΦΦ + 1 /κ = 14 ( (cid:98) R + 2 /κ ) , (B.14)where κ = β /α comes from the background solution for the scalar, as discussed below eq. (3.64). C Further Details of Holographic Renormalization
In this appendix we summarize some technical results related to the holo-ren in subsection 3.3. Inparticular, we determine asymptotically the functions g ( v ), g ( v ), and g ( v ), defined in eq. (3.47), upto the relevant order for renormalizing the two-point functions, and we obtain explicit expressions forthe renormalized response functions that enter in the two-point functions.51 .1 Determining the Boundary Counterterms We write g and g as in eq. (3.49): g = − u o + h and g = − h . Plugging these into eq. (3.48)and expanding in v , and using the fact that h , h , g and g are all O log ( v ) as v →
0, we find h − v ( h (cid:48) + 1 /
4) = O log ( v ) , h − v (2 h (cid:48) h (cid:48) + 2) = O log ( v ) , g − v ( h (cid:48) + 2 h (cid:48) g (cid:48) ) = O log ( v ) , (C.1)where primes denote ∂ v (see appendix B for the definition of O log ). A simple power-counting argumentusing the near-boundary asymptotic expansion of the scalar field in eq. (B.4) suffices to show that ingeneral only terms up to order O log ( v ) can potentially contribute to near-boundary divergences, so wecan neglect all the right-hand-sides in eqs. (C.1). The resulting equations can then be solved exactly.The most general solution for h ( v ) can be expressed implicitly in the form11 − λ ( v ) + log(1 − λ ( v )) = q + log 2 −
12 log v, λ ( v ) ≡ (cid:114) h ( v ) v − , (C.2)where q is an integration constant. Expanding this solution for small v , we obtain h ( v ) = v (cid:16)
12 + 1log v + 2 q + 1(log v ) + 4 q (log v ) + 8 q ( q − v ) + 16 q (cid:0) q − q + 1 (cid:1) (log v ) + 32 q (cid:0) q − q + 4 q − (cid:1) (log v ) + · · · (cid:17) , (C.3)where q ≡ log( − log v ) + c . The equations for h ( v ) and g ( v ) are linear, with general solutions h = ϑ ( v ) (cid:16) q − v (cid:90) d¯ vϑ (¯ v ) h (cid:48) (¯ v ) (cid:17) , g = ϑ ( v ) (cid:16) q − v (cid:90) d¯ v h (cid:48) (¯ v ) ϑ (¯ v ) h (cid:48) (¯ v ) (cid:17) , ϑ ≡ exp (cid:16) v (cid:90) d¯ v vh (cid:48) (¯ v ) (cid:17) , (C.4)where ¯ v is a dummy integration variable, and q and q are integration constants. Expanding thesesolutions at small v gives us ϑ ( v ) = v (log v ) (cid:16) q log v + 4 q (3 q − v ) + 8 q (cid:0) q − q + 2 (cid:1) (log v ) + 16 q (cid:0) q − q + 12 q − (cid:1) (log v ) + 80 q (cid:0) q − q + 238 q − q + 12 (cid:1) v ) + · · · (cid:17) , (C.5a) h ( v ) = − v log v (cid:16) − q log v + 4 q (log v ) + 4 q ( q − v ) + 8 q (cid:0) q − q + 2 (cid:1) (log v ) + 8 q (cid:0) q − q + 36 q − (cid:1) v ) + · · · (cid:17) + q ϑ ( v ) , (C.5b) g ( v ) = − v (log v ) (cid:16) − q − log v + 12 q + 2 q − v ) − q + 26 q − q −
10 + q (log v ) (C.5c)+ 16 q + 52 q − q − − q )(log v ) + 2 q (cid:0) q + 4 q − q + 45 q − (cid:1) v ) + · · · (cid:17) + q ϑ ( v ) . q , q , q correspond respectively to the constants (cid:98) R , (cid:98) R and (cid:98) R in thenear-boundary expansions of the response functions in eq. (B.10). This can be deduced as follows.Combining (3.34) and (3.46), and using the expansion in eq. (3.47) and eqs. (C.1), we obtain R G = 1 + h + 2 Q g + O log ( e − r ) , (C.6a) R G = − h (cid:48) e − r φa t + O log ( e − r/ ) , (C.6b) R G = − φ h (cid:48)(cid:48) = − h (cid:48) + O log ( e − r ) , (C.6c)where the last equality in eq. (C.6c) follows from the first in eq. (C.1). As in eq. (3.46), the superscript G indicates that these response functions are obtained from eq. (3.42), not the full on-shell action.Moreover, taking π G φ = π G Φ + π G Φ † (see eq. (3.45)) with the π φ in eq. (3.5) gives˙ v = − vh (cid:48) + O log ( e − r ) . (C.7)Eqs. (C.6) and (C.7), together with eqs. (C.1), suffice to show that R G , R G and R G satisfy thecorresponding eqs. (B.7), with the important caveat that φ in eqs. (B.7) is replaced by φ , i.e. thesolution that satisfies the first order eq. (C.7). Since φ and φ have the same asymptotic behavior,apart from the values of the coefficients α and β , R G , R G and R G have near-boundary expansionsof the same form as those of R , R and R , and hence they should have the same integrationconstants. This implies that q , q and q are related to (cid:98) R , (cid:98) R and (cid:98) R , respectively, although theexplicit map between these integration constants is rather complicated.However, the fact that R G , R G and R G satisfy eqs. (B.7) with φ replaced by φ , does have implica-tions for the boundary counterterms. We have just argued that the near boundary expansion of R G isof the same form as that of R in eq. (B.10), but with some integration constant (cid:98) R G that is relatedto q . Since v = φ , eq. (C.7) implies that φ has a near-boundary expansion of the form in (B.4) with β = (cid:16) (cid:98) R G / − (cid:17) α. (C.8)If we want to use R G , R G and R G as counterterms to renormalize R , R and R , respectively,we must set β = β and α = α , since these are the values appearing in the near boundary expansionsin eq. (B.10). However, eq. (C.8) then forces us to set (cid:98) R G = 4 + 2 /κ . This poses no problem forrenormalizing R , but as we pointed out earlier, an unusual feature of the asymptotic expansions ineq. (B.10) is that R and R contain divergences that involve (cid:98) R , which is a dynamical quantitydetermined by the near-horizon conditions. Setting (cid:98) R G = 4 + 2 /κ will thus not renormalize R and R . This is similar to cases where a source for an irrelevant operator is turned on perturbatively, muchlike our δQ , and additional multi-trace counterterms are required [55]. In our case this means R G ( v )and R G ( v ) should be considered functions of R ren22 = R + 2 − h ct0 ) (cid:48) as well, i.e. R G ( v ; R ren22 ) and R G ( v ; R ren22 ), where v = φ should evaluated on the background. These functions can be determinedby demanding they satisfy exactly the same equations as R and R , eq. (B.7).As discussed in section 3, an additional complication arises due to the logarithmic dependence of thefunctions g ( v ), g ( v ), and g ( v ) on v , which forces us to introduce explicit cutoff dependence in thecounterterms, to ensure they are local functions of the scalar source. For example, keeping only termsthat contribute to the near-boundary divergences we set g ct ( v ) = v (1 / − /r ) − u o , (C.9)53hich suffices to renormalize the on-shell action (evaluated with δQ = 0), as well as R .We will not give the explicit expressions for the counterterms R ct12 and R ct11 here, but they can beconstructed as outlined above, and they allow us to obtain the renormalized quantities R ∞ ≡ lim r →∞ (cid:0) e r ( R + R ct11 ) (cid:1) = (cid:98) R + C ( (cid:98) R , α , β ) , (C.10a) R ∞ ≡ lim r →∞ (cid:16) re r/ ( R + R ct12 ) (cid:17) = (cid:98) R + C ( (cid:98) R , α , β ) , (C.10b) R ∞ ≡ lim r →∞ (cid:0) r ( R + R ct22 ) (cid:1) = (cid:98) R , (C.10c)where C ( (cid:98) R , α , β ) and C ( (cid:98) R , α , β ) are determined by the specific choice for the countertermfunctions. C.2 Renormalized Response Functions
To determine the renormalized response functions, and hence the corresponding two-point functions,we need to consider the variation of the one-point functions. Moreover, if we want to allow δQ (cid:54) = 0,then the variations of the one-point functions must be considered at a radial cutoff, and the cutoffshould be removed only in the end.A general variation of the AdS gauge field momentum at a radial cutoff yields δπ ta = − N √− γγ − δ ˙ a t = − N √− γγ − (cid:16) R aa δa t + γ R a Φ δ Φ + γδ Φ † R a Φ † (cid:17) , = − N √− γγ − (cid:16) R ren aa δa ren t + γ R ren a Φ δ Φ + γδ Φ † R ren a Φ † (cid:17) , (C.11)where we have used the definitions in eqs. (3.53) and introduced the renormalized response functions R ren aa = R aa R aa ( G ct u + 2 u G ct uu ) , R ren a Φ = R a Φ − R aa N √− γ γ − G ct uv π ta Φ † R aa ( G ct u + 2 u G ct uu ) , R ren a Φ † = R a Φ † − R aa N √− γ γ − G ct uv π ta Φ1 + R aa ( G ct u + 2 u G ct uu ) . (C.12)Using the fact that R = 0 for solutions that satisfy ingoing boundary conditions at the horizon, weeasily find that the response functions R aa , R a Φ and R a Φ † are related to those introduced in eqs. (3.30)and (3.33) as R aa = R , R a Φ = γ − R Φ † a , R a Φ † = γ − R Φ a .However, since the one-point function associated with the AdS gauge field is given by a ren t , we needto express δa ren t in terms of the variations of the other variables. Namely, δa ren t = − γ R ren aa (cid:18) R ren a Φ δ Φ + R ren a Φ † δ Φ † + 1 N √− γ δπ ta (cid:19) = − (cid:16) R ren π ta Φ δ Φ + R ren π ta Φ † δ Φ † + R ren π ta π ta δπ ta (cid:17) , (C.13)where R ren π ta Φ = γ R ren a Φ R ren aa = (cid:16) γ R a Φ − N √− γ G ct uv π ta Φ † (cid:17) (1 + O log ( e − r )) , (C.14a) R ren π ta Φ † = γ R ren a Φ † R ren aa = (cid:16) γ R a Φ † − N √− γ G ct uv π ta Φ (cid:17) (1 + O log ( e − r )) , (C.14b) R ren π ta π ta = − √− γN R ren aa = − √− γN (cid:16) R aa (cid:0) G ct u + 2 u G ct uu (cid:1) (cid:17) (1 + O log ( e − r )) , (C.14c)54nd we have used that R aa = 1 + O log ( e − r ). These renormalized response functions at the radialcutoff are directly related with the physical two-point functions in section 3.Similarly, the generic variation of the renormalized scalar canonical momenta at the radial cutoff gives δπ renΦ † = − N √− γ (cid:16) δ ˙Φ + δ ( G ct v Φ) (cid:17) = − N √− γ (cid:16) R Φ † Φ δ Φ + R Φ † Φ † δ Φ † + γ − R Φ † a δa t + δ ( G ct v Φ) (cid:17) = − N √− γ (cid:16) R renΦ † Φ δ Φ + R renΦ † Φ † δ Φ † (cid:17) + R renΦ † π ta δπ ta , (C.15a) δπ renΦ = − N √− γ (cid:16) δ ˙Φ † + δ ( G ct v Φ † ) (cid:17) = − N √− γ (cid:16) R ΦΦ δ Φ + R ΦΦ † δ Φ † + γ − R Φ a δa t + δ ( G ct v Φ † ) (cid:17) = − N √− γ (cid:16) R renΦΦ δ Φ + R renΦΦ † δ Φ † (cid:17) + R renΦ π ta δπ ta , (C.15b)where R renΦΦ = R ΦΦ + G ct vv (Φ † ) + O log ( e − r ) , R renΦ † Φ † = R Φ † Φ † + G ct vv Φ + O log ( e − r ) , (C.16) R renΦΦ † = R ΦΦ † + (cid:0) G ct v + v G ct vv (cid:1) + O log ( e − r ) , R renΦ † Φ = R Φ † Φ + (cid:0) G ct v + v G ct vv (cid:1) + O log ( e − r ) , R renΦ π ta = (cid:16) R Φ a − √− γN G ct uv π ta Φ † (cid:17) (cid:0) O log ( e − r ) (cid:1) , R renΦ † π ta = (cid:16) R Φ † a − √− γN G ct uv π ta Φ (cid:17) (cid:0) O log ( e − r ) (cid:1) . These renormalized response functions at the radial cutoff are also directly related with the physicaltwo-point functions in section 3.Finally using eq. (3.34) and the limits in eq. (C.10), we can remove the radial cutoff to obtain therenormalized response functions (cid:98) R renΦΦ † = lim r →∞ (cid:0) r R renΦΦ † (cid:1) = (cid:98) R ΦΦ † = 14 ( (cid:98) R + 2 β /α ) , (C.17a) (cid:98) R renΦ † Φ = lim r →∞ (cid:0) r R renΦ † Φ (cid:1) = (cid:98) R Φ † Φ = 14 ( (cid:98) R + 2 β /α ) , (C.17b) (cid:98) R renΦΦ = lim r →∞ (cid:0) r R renΦΦ (cid:1) = (cid:98) R ΦΦ = 14 ( (cid:98) R − β /α ) , (C.17c) (cid:98) R renΦ † Φ † = lim r →∞ (cid:0) r R renΦ † Φ † (cid:1) = (cid:98) R Φ † Φ † = 14 ( (cid:98) R − β /α ) , (C.17d) (cid:98) R renΦ π ta = lim r →∞ (cid:16) re − r/ R renΦ π ta (cid:17) = (cid:98) R Φ π ta = − (cid:16) (cid:98) R ∞ − ω/α (cid:17) , (C.17e) (cid:98) R renΦ † π ta = lim r →∞ (cid:16) re − r/ R renΦ † π ta (cid:17) = (cid:98) R Φ † π ta = − (cid:16) (cid:98) R ∞ + ω/α (cid:17) , (C.17f) (cid:98) R ren π ta Φ † = lim r →∞ (cid:16) re − r/ R renΦ π ta (cid:17) = (cid:98) R π ta Φ † = − (cid:16) (cid:98) R ∞ − ω/α (cid:17) , (C.17g) (cid:98) R ren π ta Φ = lim r →∞ (cid:16) re − r/ R renΦ † π ta (cid:17) = (cid:98) R π ta Φ = − (cid:16) (cid:98) R ∞ + ω/α (cid:17) , (C.17h) (cid:98) R ren π ta π ta = lim r →∞ (cid:16) R ren π ta π ta (cid:17) = (cid:98) R π ta π ta = 1 N (cid:98) R ∞ , (C.17i)55here (cid:98) R ∞ and (cid:98) R ∞ are defined in eq. (C.10). Eqs. (C.17) are valid for the screened phase only. In theunscreened phase, the scalar’s response functions (cid:98) R Φ † Φ and (cid:98) R ΦΦ † are integration constants determinedby imposing boundary conditions on the horizon, while all other response functions vanish. D Analytic Derivation of the Lowest Pole in the Screened Phase
In this appendix we present an analytic ( i.e. non-numerical) derivation of the behavior ω ∗ ∝ − i (cid:104)O(cid:105) of the lowest pole in the screened phase, for T (cid:46) T c .In this appendix we use the metric in eq. (2.6), but with the re-scaling in eq. (D.1) to producedimensionless coordinates, ( z/z H , t/z H , x/z H ) → ( z, t, x ) , (D.1)which leaves the metric in eq. (2.6) invariant, except for h ( z ) = 1 − z /z H → − z , so the boundaryremains at z = 0 but the horizon is now at z = 1. We also re-scale a t ( z ) z H → a t ( z ), which is thendimensionless. After the re-scaling, Φ( z )’s asymptotic expansion is that of eq. (5.4),Φ( z ) = α T z / ln z + β T z / + . . . , (D.2)where here and below . . . represents terms that vanish faster than those shown when z →
0, and theboundary condition α = κβ becomes α T = κ T β T . We additionally re-scale to produce a dimensionlessfrequency: ωz H = ω/ (2 πT ) → ω . Moreover, in this appendix we exclusively use Q = − / a t ( z, t ) = a t ( z ) + δa t ( z, t ), where a t ( z )is the background solution and δa t ( z, t ) is the fluctuation, and similarly Φ( z, t ) = Φ ( z ) + δ Φ( z, t ),and Φ † ( z, t ) = Φ † ( z ) + δ Φ † ( z, t ). In the screened phase, Φ ( z ) (cid:54) = 0 and Φ † ( z ) (cid:54) = 0. In this appendixwe will assume the background solution is real, Φ ( z ) = Φ † ( z ). Next we Fourier transform using ∂ t → − iω , and use the same notation for the Fourier transforms of the fluctuations, for example δa t ( z, ω ). Linearizing the equations of motion about in the fluctuations then gives the fluctuationequations (the equivalent of eq. (3.18), but in the coordinates of eq. (D.1)), δ Φ (cid:48)(cid:48) + h (cid:48) h δ Φ (cid:48) + ( ω + a t ) h δ Φ + ω + 2 a t h Φ δa t = 0 , (D.3a) δ Φ † (cid:48)(cid:48) + h (cid:48) h δ Φ † (cid:48) + ( ω − a t ) h δ Φ † − ω − a t h Φ † δa t = 0 , (D.3b) δa (cid:48)(cid:48) t + 2 z δa (cid:48) t − † Φ z h δa t + Φ δ Φ † z h (cid:0) ω − a t (cid:1) − Φ † δ Φ z h ( ω + 2 a t ) = 0 , (D.3c) ωz δa (cid:48) t + h (cid:104) Φ ( δ Φ (cid:48) − δ Φ † (cid:48) ) − Φ (cid:48) ( δ Φ − δ Φ † ) (cid:105) = 0 , (D.3d)where prime denotes ∂ z , for example Φ (cid:48) ≡ ∂ z Φ.We want the QNMs, that is, solutions for the fluctuations that are normalizable at the boundary z = 0 and in-going at the horizon z = 1, which exist only for particular ω [99, 100]. The asymptoticexpansions of the fluctuations are δa t = δQz + δµ + . . . , δ Φ = δα T z / log z + δβ T z / + . . . . (D.4)56o guarantee normalizability, and specifically to guarantee that the asymptotic expansions of thefluctuations do not have terms more divergent than the asymptotic expansions of the backgroundsolutions, we must impose δQ = 0, which requires δα T = κ δβ T , with the same value of κ as thebackground solution Φ ( z ).We parameterize the solutions of eq. (D.3) as δ Φ( z, ω ) = h − iω/ p ( z ) y ( z, ω ) , δ Φ † ( z, ω ) = h − iω/ p ( z ) y † ( z, ω ) , δa t ( z, ω ) = h − iω/ a ( z, ω ) , (D.5)where the powers of h are determined by the in-going boundary condition at the horizon, p ( z ) is thebackground solution Φ ( z ) with α = 1, so that asymptotically p ( z ) = z / log z + 1 κ T z / + . . . , (D.6)and now we must solve for y ( z, ω ), y † ( z, ω ), and a ( z, ω ), which must be regular at both the boundary z = 0 and the horizon z = 1.We want the QNM solutions for T near T c , where the condensate (cid:104)O(cid:105) ∝ α/κ is small, or equivalentlyΦ ( z ) is negligible. We thus treat p ( z ) as a small correction to the solution in the unscreened phase,that is, we use the background solution with Φ ( z ) = 0 and Q = − /
2, where a t ( z ) = − (cid:18) − z (cid:19) , (D.7)and then determine p ( z ) by solving the equation of motion for the scalar, linearized about the solutionwith Φ ( z ) = 0 and eq. (D.7), which gives p ( z ) = − (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:114) zz + 1 P i − (cid:18) z − z + 1 (cid:19) , (D.8)where P ν is a Legendre function of the first kind.When T (cid:46) T c , we know from subsection 6.1 that the lowest QNM frequency ω ∗ is near the origin ofthe complex ω plane, and hence is also small. We thus expand y ( z, ω ), y † ( z, ω ), and a ( z, ω ) in both ω and also α ∝ κ (cid:104) O(cid:105) , y ( z, ω ) = ∞ (cid:88) n,m =0 ω n α m y nm ( z ) , y † ( z, ω ) = ∞ (cid:88) n,m =0 ω n α m y † nm ( z ) , a ( z, ω ) = ∞ (cid:88) n,m =0 ω n α m a nm ( z ) , (D.9)so that now we must solve for the coefficients y nm ( z ), y † nm ( z ), and a nm ( z ). For n = 0 and m = 0, y (cid:48)(cid:48) + (cid:20) p (cid:48) p + h (cid:48) h (cid:21) y (cid:48) = 0 , ωz (cid:20) a (cid:48) + h (cid:48) h a (cid:21) = 0 , (D.10)and y † ( z ) obeys the same equation as y ( z ). The only solutions regular at both the boundary z = 0and the horizon z = 1 are y (cid:48) ( z ) = 0, y † (cid:48) ( z ) = 0, and a ( z ) = 0. For higher values of n and m , theequations for the coefficients are inhomogeneous, y (cid:48)(cid:48) nm + (cid:20) p (cid:48) p + h (cid:48) h (cid:21) y (cid:48) nm = I nm , z (cid:20) a (cid:48) nm + h (cid:48) h a nm (cid:21) = A nm , (D.11)57here y † nm obeys the same equation as y nm , but with source I † nm . The sources I nm and A nm dependonly on solutions at lower order in n and m . For example, I n = I † n = − a t h a ( n − , which implies y (cid:48) n = y † (cid:48) n , which in turn implies A n = 0. Furthermore, A m = 0 so that a m = 0. Determiningthe sources I nm , I † nm , and A nm is straightforward but unilluminating, so we will not present explicitresults for them. However, the most singular behavior possible at the horizon z = 1 is I nm ∝ ( z − − ,and similarly for I † nm and A nm . As a result, solutions regular at the horizon z = 1 have the form y (cid:48) nm ( z ) = − h ( z ) p ( z ) (cid:90) z d ¯ z h (¯ z ) p (¯ z ) I nm (¯ z ) , a nm ( z ) = − h ( z ) (cid:90) z d ¯ z h (¯ z )¯ z A nm (¯ z ) , (D.12)where ¯ z is a dummy variable, and y † nm (cid:48) ( z ) obeys the same equation as y (cid:48) nm ( z ), but with I nm → I † nm .Regularity of a ( z, ω ) at the boundary requires (cid:90) d ¯ z h (¯ z ) p (¯ z ) (cid:16) I (¯ z ) − I † (¯ z ) (cid:17) = 0 , (D.13)and a second condition, identical to eq. (D.13), but with I → I and I † → I † . Regularity of y ( z, ω ) at the boundary requires (cid:90) d ¯ z h (¯ z ) p (¯ z ) (cid:2) ω I (¯ z ) + ω I (¯ z ) + α I (¯ z ) (cid:3) = 0 , (D.14)while regularity of y † ( z, ω ) at the boundary requires a condition identical to eq. (D.14), but with I → I † , I → I † , and I → I † . However, using I = I † , as mentioned above, and the secondregularity condition for a ( z, ω ), we can show that the regularity condition for y † ( z, ω ) is equivalentto that for y ( z, ω ) in eq. (D.14). We are thus left with only eq. (D.14), which will be satisfied onlyfor certain values of ω . In particular, in our regime of interest, with small ω and α , the solution ofeq. (D.14) gives the lowest QNM frequency, ω ∗ ≈ − α (cid:82) d ¯ z h (¯ z ) p (¯ z ) I (¯ z ) (cid:82) d ¯ z h (¯ z ) p (¯ z ) I (¯ z ) . (D.15)Performing the integrals in eq. (D.15) numerically, we find ω ∗ ≈ − i α . Given α ∝ κ (cid:104)O(cid:105) , we havethus shown that for Q = − /
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