Holographic Charge Oscillations
PPreprint typeset in JHEP style - HYPER VERSION
DAMTP-2014-89, DCPT-14/71
Holographic Charge Oscillations
Mike Blake , Aristomenis Donos and David Tong Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Cambridge, CB3 OWA, UK m.blake, [email protected] Centre for Particle Theory, Department of Mathematical Sciences,Durham University, Durham DH1 3LE, UK [email protected]
Abstract:
The Reissner-Nordstr¨om black hole provides the prototypical descriptionof a holographic system at finite density. We study the response of this system tothe presence of a local, charged impurity. Below a critical temperature, the inducedcharge density, which screens the impurity, exhibits oscillations. These oscillationscan be traced to the singularities in the density-density correlation function moving inthe complex momentum plane. At finite temperature, the oscillations are very similarto the Friedel oscillations seen in Fermi liquids. However, at zero temperature theoscillations in the black hole background remain exponentially damped, while Friedeloscillations relax to a power-law. a r X i v : . [ h e p - t h ] D ec ontents
1. Introduction and Summary 12. Charge Screening at Zero Density 43. Charge Screening at Finite Density 9A. Appendix 16
1. Introduction and Summary
The AdS Reissner-Nordstr¨om black hole offers a holographic description of a compress-ible phase of quantum matter at strong coupling [1]. However, despite much study,the nature of this phase remains somewhat mysterious. There is a chemical potentialin the boundary theory, yet this results in neither the condensation of bosons, nor theformation of a sharp Fermi surface.In recent years, there has been an (almost) exhaustive exploration of the momen-tum space structure of correlation functions in the Reissner-Nordstr¨om black hole andother holographic backgrounds. This has been prompted, in large part, by a desire tounderstand the fermionic physics captured by these geometries and, not least, how toreconcile the lack of a sharp Fermi surface with the Luttinger theorem [2, 3, 4]. At weakcoupling, the existence of a Fermi surface means that low-energy excitations of particlesand holes occur at finite momentum. These reveal themselves in correlation functionsas singularities at momentum k = 2 k F , twice the Fermi momentum. Holography pro-vides in a framework in which we can ask: what becomes of these 2 k F singularities atstrong coupling?In holographic theories, the spectrum of low-energy, charged excitations can be ex-tracted from the spectral density; that is the imaginary, dissipative part of the current-current correlation function. This was studied in some detail in [5, 6, 7]. No evidenceof singularities in momentum space was seen. Instead, the low-energy physics exhibits“local criticality”, an unusual form of scale invariance in which time scales but spacestands still [8]. The result is that there are low energy excitations over a range ofmomenta, k < µ , where µ is the chemical potential.1ensity-density correlation functions have been further explored in bulk geometriesassociated to other locally critical theories [7, 9], a number of probe brane models [10,11, 9] and gapless electron star geometries [12]. None of these exhibit 2 k F singularities.There are two exceptions where the bulk, bosonic geometry does exhibit 2 k F singular-ities. The first is somewhat exotic and involves a little string theory (a six-dimensionalnon-gravitational theory) with a finite density of strings [13]. It seems likely that thephysics underlying this behaviour is unrelated to the physics of Fermi surfaces. Thesecond exception occurs in AdS where the boundary theory is d = 1 + 1 dimensional.Here, the analog of the Reissner-Nordstr¨om background is the charged BTZ black hole.It was shown in [14] that tunnelling events in the bulk, associated to monopoles, giverise to the relevant singular behaviour. Arguments were also presented that suggestedsuch singularities may generically be associated to bulk magnetic degrees of freedom.However, while it is known that monopoles can give rise to a number of interestingeffects in higher dimensional holography [15, 16, 17], the emergence of 2 k F singulari-ties does not seem to be among them. At present, it appears that the beautiful effectdescribed in [14] is restricted to d = 1 + 1 dimensions. Charge Screening
A slightly different probe of the momentum structure of the system is offered by itsresponse to a charged impurity. The ground state becomes polarised and the charge isscreened. This effect is captured by the static charge susceptibility; that is, the real,reactive part of the density-density correlation function. Although this is also evaluatedat frequency ω →
0, the charge screening is not necessarily governed by the low-energyexcitations of the system. Instead, by the Kramers-Kronig relation, the susceptibilityis extracted from the spectral density by integrating over all frequencies. (This pointwas emphasised in the holographic context in [9]).Nonetheless, at least for weakly coupled systems, the screening of charge typicallyis dominated by the lowest energy modes. Indeed, the most visual manifestation ofthe 2 k F singularities of a Fermi surface occur in Friedel oscillations . This is the phe-nomenon in which the induced charge around a localised impurity varies as cos(2 k F r ),oscillating between positive and negative. Further, at zero temperature, this chargedecays as a power-law rather than the more typical exponential decay that arises inthe Debye or Thomas-Fermi approaches to screening. Heuristically, the origin of theFriedel oscillations lies in the fact that the lowest energy modes have finite size. Thesemodes enthusiastically cluster around the impurity but, unaware of their own cumber-some nature, over-screen the charge. The story is then repeated, over-exuberance piled2pon over-exuberance. The end result is a highly inefficient screening mechanism andthe wonderful rippling patterns of charge that are visible through scanning tunnellingmicroscopes.The purpose of this paper is to show that similar oscillations in the induced chargeoccur for the screening of an impurity in the Reissner-Nordstr¨om black hole. Theseoscillations exist despite the fact that there are no sharp features in the spectral density.Moreover, they arise in a rather surprising manner. At high temperatures, T (cid:29) µ ,there are no oscillations and a localised charge is exponentially screened in the familiarDebye fashion. Correspondingly, the leading poles in the charge susceptibility are onthe imaginary momentum axis. As the temperature is lowered, the response changes ina non-analytic manner, reminiscent of a second order phase transition. Below a criticaltemperature, T c ∼ µ , these poles move into the complex momentum plane. This hasthe effect that, at low temperatures, the induced charge oscillates. The form of theseoscillations is similar to the Friedel oscillations of a Fermi liquid. In particular, at finitetemperature, both oscillations are exponentially damped.However, at zero temperature, there is a difference between Friedel oscillations andthe charge oscillations we observe in the black hole background. As the temperatureapproaches zero, we find that many poles coalesce, resulting in a branch cut in thecomplex momentum plane. Importantly, this branch cut intersects neither the real norimaginary axis. Instead it terminates at a position in the complex momentum plane setby the chemical potential and appears to be associated to the AdS near-horizon regionof the geometry. This is in contrast to a Fermi liquid where the branch cut terminates atreal momentum k = 2 k F . It means that, at zero temperature, the oscillations aroundthe black hole remain exponentially damped, while the Friedel oscillations becomepower-law. This also explains why earlier studies of the spectral density, focussing onreal momentum and complex frequency, saw no hint of a sharp momentum structurein the Reissner-Nordstr¨om black hole: the momentum structure lies in the complexmomentum plane. Such complex momenta are relevant in the study of localised, staticperturbations, as opposed to physical, monochromatic excitations which are confinedto the real momentum axis.All the results in this paper can be understood within the regime of linear response.Recently, some very interesting work on impurities in the same theory explored thephysics beyond linear response, where an increase in the amplitude of the impurityresults in the nucleation of a black hole in the bulk [18].3 . Charge Screening at Zero Density Throughout this paper, we will work with the simplest holographic theory that candescribe charge screening; that is, a bulk d = 3 + 1 dimensional Einstein-Maxwelltheory with action S bulk = (cid:90) d x √ g (cid:20) R + 6 L − F µν F µν (cid:21) (2.1)Throughout the paper we set the AdS radius to be L = 1.The bulk action (2.1) provides a holographic description of a boundary conformalfield theory in d = 2 + 1 dimensions with a conserved U (1) current J µ dual to the gaugefield A µ . This current is associated to a global symmetry on the boundary theory. Inany pretence at modelling real materials, one would presumably identify this currentwith electric charge. The fact that the current is global means that we are neglectingthe effect of Coulomb forces between the charges, although the massless excitations ofthe CFT will mediate other forces.The neglect of Coulomb forces is a familiar deceit from computations of optical con-ductivity, both in holography and in more traditional settings, where it can be justifiedby the observation that electron-electron interactions are often not the most dominanteffect in charge transport. Here, however, we are interested in the screening of chargeand, in most materials, the dominant effect is due to the Coulomb force. This meansthat we are restricted to situations where the Coulomb force does not hold sway. Ofcourse, it is always possible to dress our results with Coulomb interactions, a procedurewhich is typically accomplished using the random phase approximation. We will notdo this in this paper, and the results we present are for the “bare” susceptibility .We start by describing the screening of charge in the conformal theory with vanishingchemical potential. We will find no surprises, but this gives us the opportunity torecapitulate some basic physics. We will model the impurity as a static chemicalpotential µ ( r ) such that µ ( r ) → r →
0. The exact nature of the impurity will notmatter too much for us. We only require that the fall-off is faster than 1 /r , so thatthis is an irrelevant deformation and does not change the infra-red of the theory. Forconcreteness, we choose a simple Gaussian profile µ ( r ) = Ce − r / R (2.2) Our interest in the screening of a global charge in the boundary is in contrast to the screeningof non-Abelian SU ( N ) gauge charge which has been studied holographically in a number of papers,starting with [19]. r (cid:29) R , ourresults will not be sensitive to this exact form of the impurity profile.Our goal is to compute the response of the charge density ρ ( r ) = J ( r ) due to thepresence of the impurity. Because the profile (2.2) is an irrelevant deformation, we canwork perturbatively in the strength of the impurity, given by the dimensionless combi-nation CR . We will restrict ourselves to the regime of linear response. In momentumspace, the charge density is controlled by the static susceptibility χ ( k ) = (cid:104) ρ ( k ) ρ ( − k ) (cid:105) and is given by ρ ( k ) ∼ χ ( k ) µ ( k )The response function is determined in the limit of frequency ω →
0, in which case χ ( k ) is real when evaluated on real momenta k . (We will later be interested in thebehaviour of the response function for complex k ). In the rest of this paper, we willdetermine how χ ( k ) depends on temperature T and chemical potential µ . Zero Temperature
We start at zero temperature where everything is dictated by conformal invariance. Ondimensional grounds, in any conformal field theory the charge susceptibility must begiven by χ ( k ) ∼ k .To determine the spatial profile of the induced charge density, we need only performthe Fourier transform, ρ ( r ) = (cid:90) d k (2 π ) e i k · r ρ ( k ) ∼ CR (cid:90) d k π χ ( k ) e i k · r e − R k / = CR (cid:90) ∞ dk kχ ( k ) e − R k / J ( kr ) (2.3)with J ( kr ) a Bessel function. The above expression for the charge density is a Hankeltransform. This can be performed analytically to obtain an exact expression for ρ ( r )in terms of Bessel functions. For χ ( k ) = k , we have ρ ( r ) = C R (cid:114) π e − r / R (cid:104) r R I ( r / R ) − (cid:16) r R − (cid:17) I ( r / R ) (cid:105) The resulting charge density is plotted in Figure 1.5 / R ( R / C ) · ρ Figure 1: T = 0, µ = 0: The charge density induced by an impurity in the AdS vacuum. The key features of the charge density are simple to understand. At large distances, r (cid:29) R , the charge falls off as a power-law, ρ ∼ − CR /r , as expected in a scaleinvariant theory. Perhaps the most striking feature is that the induced charge densitydips below zero. This too follows from scale invariance, albeit more indirectly. Thesusceptibility is χ ( k ) ∼ k which means that the zero-momentum Fourier mode vanishes: ρ ( k = 0) = 0. But this, in turn, means that the total, integrated charge, Q = (cid:82) d r ρ ( r ),vanishes. The induced charge must therefore dip below zero somewhere. Finite temperature
We now turn on finite temperature, T (cid:54) = 0. To study this situation, we turn to thegravitational description of the ground state given by the AdS Schwarzchild black hole,with metric ds = 1 z (cid:18) − f ( z ) dt + dz f ( z ) + dx + dy (cid:19) (2.4)where f ( z ) = 1 − (cid:18) zz + (cid:19) Here z is the radial, bulk coordinate such that the boundary lies at z = 0. The blackhole describes the boundary theory at temperature T = 3 / πz + .We perturb the background with the localised boundary chemical potential (2.2).This is, by now, the kind of calculation that is holographic bread and butter. (For areview of holographic linear response theory, see, for example, [20]). In this simple case,we need only consider the Maxwell equation for the temporal gauge field, δA t ( z, (cid:126)x ) =6 / T χ / T - - - -
50 r / R l n [ ( R / C ) · ρ ] Figure 2: T (cid:54) = 0, µ = 0: The susceptibility χ ( k ) on the left. The finite temperatureresponse, log( | Rρ/C | ), is shown on the right for an impurity of width RT = 3 / π . δA t ( z ) e i(cid:126)k.(cid:126)x , which reads f ( z ) δA (cid:48)(cid:48) t − k δA t = 0 (2.5)We solve this equation numerically, subject to regularity at the horizon and extract thesusceptibility as the ratio χ ( k ) = − δA (cid:48) t /δA t as z → k/T . At highmomenta, k (cid:29) T it passes over to the scale invariant result χ ( k ) ∼ k as expected.However, it deviates from this result at low momenta and, in particular, χ ( k = 0) (cid:54) = 0.Correspondingly, the integrated charge is now non-vanishing. This is seen in the right ofFigure 2 where, because the induced charge density now decays exponentially, we haveplotted log( | Rρ/C | ) on the vertical axis. Of course, in a log-plot we must first take theabsolute value of ρ which means that sign of the induced charge is no longer obvious.Instead, the cusp reveals where ρ passes through zero. (In an analytic treatment, thecusp itself would reach down to negative infinity).At large distances, r (cid:29) R and T − (cid:29) R , the charge density decays exponentially,rather than the power-law that we saw at T = 0. In fact, one can check that theasymptotic induced charge takes the form ρ ( r ) ∼ CR e − r/λ √ rλ (2.6)where the screening length scales as λ ∼ /T . This is the normal Debye screeningbehaviour, expected of any relativistic quantum field theory at finite temperature.7 igure 3: T (cid:54) = 0, µ = 0: The absolute value of (cid:104) ρ ( k ) ρ ( − k ) (cid:105) correlator in the complex k x -plane. The bright spots are the poles. The asymptotic form (2.6) can be seen in the complex momentum plane. We align themomentum along the x -direction and evaluate χ ( k x ). The density-density correlationfunction exhibits a series of poles that lie strictly on the imaginary momentum axis,the first of which occurs at k = i/λ ∼ iT . This string of poles is shown in Figure 3. Atlarge distances, these poles dominate the Hankel transform of the susceptibility, givingrise to the exponential behaviour (2.6).Note that there is no analyticity in complex momentum: in contrast to complexfrequency, poles appear in both the upper and lower halves of the k x -plane. Thisreflects the fact that complex momentum is not relevant in a translationally invariantsystem since the resulting response will be of the form e ikx which, for complex k , growsin either the positive or negative x -direction. However, as is familiar from quantummechanics, the presence of a delta-function impurity allows us to pick up contributionsfrom the upper-half plane for x > x <
0, with theimpurity providing the necessary discontinuity in the derivative of the response at theorigin.The static screening properties of the AdS Schwarzchild black hole in AdS werepreviously determined in [21, 22] with broadly similar results. The charge susceptibilitywas also compared with weakly coupled N = 4 super Yang-Mills. Curiously, it wasclaimed that the screening is weaker at strong coupling and stronger at weak coupling.(For example, the Debye screening length λ was argued to increase as the ’t Hooftcoupling is increased). 8 . Charge Screening at Finite Density We now turn to our main interest: screening at finite density. We turn on a constantchemical potential, µ , on the boundary theory. The resulting bulk geometry is theReissner-Nordstr¨om black hole. The metric again takes the form (2.4), with f ( z ) = 1 − (cid:18) z µ (cid:19) (cid:18) zz + (cid:19) + z µ (cid:18) zz + (cid:19) This is accompanied by a profile for the Maxwell field, A = µ (cid:18) − zz + (cid:19) In this geometry, the boundary CFT field theory is warmed to temperature T = 14 πz + (cid:18) − z µ (cid:19) We once again perturb the boundary theory by a localised, charged impurity. Thechemical potential is taken to be µ ( r ) = µ + Ce − r / R As in the previous section, the linear response is computed via the susceptibility χ ( k ) = (cid:104) ρ ( k ) ρ ( − k ) (cid:105) .There is one aspect of the static susceptibility which is more easily computed withthe introduction of µ . This is the zero mode χ ( k = 0). As we saw above, this controlsthe integrated induced charge due to an impurity. It is given analytically by χ ( k = 0) = ∂Q ( T, µ ) ∂µ = 2 π T µ + 8 π T (cid:112) µ + 16 π T . (3.1)In the limit T (cid:29) µ , we have χ ( k = 0) = 4 πT /
3. This agrees with the k = 0 limit ofthe susceptibility shown on the left of Figure 2.To go beyond the k = 0 limit of the susceptibility we must work numerically. This issomewhat more involved than for the Schwarzchild black hole because the gauge fieldperturbation δA t now back-reacts on the metric. Working in radial gauge, δg zµ = 0and δA z = 0, we must consider the full set of perturbations δA t , δg tt , δg xx , δg yy - - - μ · r l n [ ρ / ( C · μ ) ] Figure 4:
T < T c , µ (cid:54) = 0. The induced charge density oscillates at large distances, shownhere for an impurity of width Rµ = 1 and temperature T /µ = 0 . The equations governing these perturbations were derived in [5, 6], although the staticsusceptibility was not calculated. We relegate details of this calculation to the Ap-pendix. In brief, it proceeds by first eliminating δg xx to give three coupled, ordinarydifferential equations for the static perturbations δA t , δg tt and δg yy . We solve thesenumerically and extract the static susceptibility χ ( k ).Armed with susceptibility, we can perform the Fourier transform and calculate theinduced charge density. For high temperatures, T (cid:29) µ , the response in charge densityis qualitatively similar to that of the Schwarzchild metric. In particular, the systemexhibits exponential Debye-like screening (2.6) at large distances. However, this be-haviour changes below a critical temperature which, numerically, we find to be T c ≈ . µ For
T < T c , the induced charge density oscillates. At long distances, the charge densityis given by ρ ( r ) = ρ + δρ ( r ) where ρ = µ /z + is the background charge, while δρ takes the form δρ ( r ) ∼ e − r/λ √ r cos( r/ξ ) (3.2)with the length scales λ and ξ set by µ and T . This charge density is shown in Figure 4.As before, nodes in the charge density appear as cusps in log( | ρ | /Cµ ). The appearanceof multiple cusps shows that the charge density is oscillating.The origin of these oscillations can again be understood by looking in the complexmomentum plane. At temperatures T ≥ T c , the susceptibility exhibits a string of polesalong the imaginary axis, just as we saw for the Schwarzchild black hole. This is the10 igure 5: T ≈ T c , µ (cid:54) = 0: The absolute value of the density-density correlator in thecomplex momentum plane above the phase transition at T = 0 . µ > T c (left) and belowthe phase transition at T = 0 . µ < T c (right). situation depicted in the left of Figure 5 where two of the poles are visible. This resultsin the now-familiar exponential damping of Debye screening of the form (2.6).As we lower the temperature, the two poles depicted in the figure get closer together.Eventually, at the critical temperature T = T c , these two poles merge. As the tempera-ture is lowered yet further, the poles move off the imaginary axis and into the complexmomentum plane, gaining a real part. This can be seen in right hand plot of Figure 5.The observed oscillations in the induced charge density for T < T c can be tracedto the position of the pole away from the imaginary axis. The Fourier transform isdominated by the pole, with the e i k · r factor now contributing an oscillatory piece. Ifwe write the location of the pole as k (cid:63) = k ( T ) + ik ( T ) then, for T < T c , the long-distance profile of charge density takes the form (3.2) with λ = 1 /k and ξ = 1 /k .Charge screening of the form (3.2) is very similar to that seen in Friedel oscillationsat finite temperature, where the damping factor takes the form e − T r . The differencebetween the two is only quantitative: for T (cid:28) µ , the Friedel oscillations are veryweakly damped while, for the black hole, the real part of k (cid:63) is always comparable tothe imaginary part as we lower the temperature. This will ultimately lead to a moredramatic difference between the oscillations identified above and Friedel oscillations atzero temperature. 11 - - - / μ R e [ k / μ ] Figure 6:
The trajectory tracedby the real part of the pole as thetemperature is lowered.
This qualitative change in the position of thepole at T = T c can be viewed as a kind of sec-ond order phase transition, with Re( k ) playing therole of the order parameter. (We stress, however,that this is not a real phase transition in the sensethat k = 0 thermodynamic quantities remain an-alytic). This is clearly seen if plot the trajectoryof Re( k ) for the lowest pole as we vary the tem-perature. (Note that, for T > T c , the trajectory isactually two poles on the imaginary axis). Numeri-cally we find that, close to the transition point, thebehaviour of Re( k ) is well modelled by the mean-field exponent, Re( k ) ∼ ( T c − T ) / T < T c A similar second order phase transition was observed previously in a phenomenologicalmodel of QCD, involving nucleons interacting with pions at finite density [23]. However,the phase transition appears not to occur in weakly coupled non-Abelian gauge theorieswhere poles in complex plane are observed at finite density [24], but their effects arewashed out at long distance by the more familiar Friedel oscillations.
Back to Zero Temperature
Finally, we can ask: what becomes of these oscillations as we approach T → T c .(Indeed, for certain regimes of the parameters, we find that these higher poles can leadto one or two anomalous oscillations even at T > T c which occur at finite r before theybecome suppressed at large r ).Ultimately, it appears that these poles coalesce to form a branch cut. It is worthmentioning that the existence of a branch cut, lying roughly parallel to the imaginarymomentum axis, is reminiscent of the situation in Fermi liquids at zero temperature.There the branch cut extends down to the real axis which ensures the power-law fall-offof Friedel oscillations. A number of studies of 2 k F singularities in various non-Fermi12 igure 7: T < T c , µ (cid:54) = 0: The absolute value of the density-density correlator in thecomplex momentum plane, for T /µ = 0 .
21 (left) and
T /µ = 0 . liquid states find that the end of the branch-cut remains at a real value of the momentum[28, 29, 30, 31].In contrast, in our holographic model, the branch cut terminates in the complexplane and, consequently, the screening at zero temperature is not greatly changed from(3.2): the exponential suppression and oscillations both remain although, in principle,the accompanying power-law may change depending on the cut residue. This T = 0screening is plotted in Figure 8. The left-hand plot shows log( ρ/Cµ ), with the nodesappearing as cusps. To exhibit the oscillations more clearly, in the right-hand plot wehave rescaled the charge density by the exponential suppression factor e i Im( k (cid:63) ) r , with k (cid:63) the end of the branch cut.The existence of this branch cut at zero temperature should be viewed as the under-lying cause of our charge oscillations. There is compelling reason to believe that thisbranch cut is associated to the AdS × R near horizon region of the geometry. Thisis the regime of local criticality, which can be thought of as a scale invariant theorywith dynamical exponent z → ∞ , so that time and energies scale, while space andmomenta do not. This means that the current operators in the theory are labelled bytheir momentum k and, in the far infra-red, have dimension δ ± ( k ) = + ν ± ( k ), where ν ± ( k ) = 12 (cid:118)(cid:117)(cid:117)(cid:116) (cid:18) kµ (cid:19) ± (cid:115) (cid:18) kµ (cid:19) (3.3)13 - - - - - μ · r l n [ ρ / ( C · μ ) ] - - - μ · r e I m [ k ] r ρ / ( C · μ ) Figure 8: T = 0. The zero temperature response gives a series of oscillations (left). Thesecan be seen more clearly by factoring out the exponential damping (right). Both plots usean impurity of width Rµ = 1. In this paper we are interested in complex momenta. As we take the limit T → ν − ) = 0 (3.4)In particular, the location of the end point of the branch point lies at k/µ = 1 / √ i/ ν − . The branch cuts defined by Re( ν − ) = 0 lie in the complexplane. There are also additional branch cuts along the imaginary axes, associated tothe embedded square-root in (3.3), which do not correspond to poles in the Reissner-Nordstr¨om background.The agreement of the AdS and Reissner-Nordstr¨om branch cuts requires an expla-nation. It is certainly true that AdS correlation functions (at finite T or finite ω ) willexhibit such a branch cut in the complex momentum plane. Moreover, it is well knownthat the AdS region dominates the spectral density of the theory at low-frequencies[6, 25] and this can be understood using the kind of matching procedure pioneered in[26] between near- and far-horizon geometries. (See, for example, the appendix of [27]for a review of this procedure). However, it is unclear to us how to implement thisprocedure in the present context.The fact that the accumulation of poles appears to converge towards the locus (3.4) isstrong evidence that the oscillations are due to local criticality and there is an intuitiveway to understand this behaviour. The near-horizon region of the geometry supportslow-energy excitations with a range of momentum k < µ and therefore a minimum14 igure 9: T = 0, µ (cid:54) = 0: The argument of ν − in the complex momentum plane. The branchcuts along Re( ν − ) = 0 should be compared to the string of poles shown in Figure 7. wavelength which is roughly ∼ /µ . As we reviewed in the introduction, the essenceof Friedel oscillations is that the modes which do the screening have a finite size. Heretoo, we see that modes have a finite size, now with a range of wavelengths giving riseto the exponential behaviour rather than power-law. The lesson is similar to that in[25]: low-energy modes at finite momentum occur in both Fermi liquids and in locallycritical theories, ensuring that the two states share certain features. In [25], the focuswas on efficient umklapp scattering. Here we see that charge oscillations due to thescreening of an impurity can be added to this list.We finish with some speculation. It is natural to wonder if, in other models, withmore bells and whistles, one could move the branch cut so that it terminates on thereal momentum axis. This would then result in T = 0, Friedel-like oscillations, withpower-law decay. However, the fact that the branch cut lies at Re( ν − ) = 0 meansthat a new phenomenon accompanies the power-law oscillations: the onset of a finitewavelength instability. Such instabilities have been seen previously in a number ofdifferent holographic models [33, 34, 35]. In all of these cases, the infra-red dimension δ ± ( k ) of some operator violates the BF bound for some finite wavenumber which, atthe onset of the instability, translates into the requirement that ν ± ( k ) = 0 for somereal momentum k . This suggests that, within holography, the emergence of Friedeloscillations from the bosonic geometry would indicate that the system lives on the edgeof an instability. It would be interesting to explore this connection further.15 cknowledgements We’re grateful to David Berman, Jan de Boer, Sean Hartnoll, Gary Horowitz, NabilIqbal, Jorge Santos, Gonzalo Torroba and Benson Way for some combination of com-ment, discussion, correspondence, and sharing of [18]. We are supported by STFC andby the European Research Council under the European Union’s Seventh FrameworkProgramme (FP7/2007-2013), ERC Grant agreement STG 279943, Strongly CoupledSystems. MB is funded by Churchill College.
A. Appendix
In this Appendix we discuss in detail the calculation of the static susceptibility in theReissner-Nordstr¨om background. We perturb the solution by introducing δA t ( z, x ) = δA t ( z ) e ikx . As we discussed in the main text we will work in radial gauge δg zµ = 0 δA z = 0and so the other fields we must turn on are δg xx , δg yy and δg tt . The perturbationequations take a simpler form if we raise one index on these fields using the backgroundmetric. Then the linearised Maxwell equation reads f ( z ) δA (cid:48)(cid:48) t − k δA t + µ f ( z )2 z + (cid:16) ( δg t t ) (cid:48) − ( δg xx ) (cid:48) − ( δg yy ) (cid:48) (cid:17) = 0 (A.1)while we also have a host of Einstein equations f ( z )( δg yy ) (cid:48)(cid:48) + f (cid:48) ( z )( δg yy ) (cid:48) − f ( z ) z ( δg yy ) (cid:48) − f ( z ) z (cid:16) ( δg xx ) (cid:48) + ( δg t t ) (cid:48) (cid:17) (A.2) − µ z z + δA (cid:48) t − k δg yy − µ z z δg t t = 0 f ( z )( δg xx ) (cid:48)(cid:48) + f (cid:48) ( z )( δg xx ) (cid:48) − f ( z ) z ( δg xx ) (cid:48) − f ( z ) z (cid:16) ( δg yy ) (cid:48) + ( δg t t ) (cid:48) (cid:17) (A.3) − µ z z + δA (cid:48) t − k δg yy − (cid:16) k + µ z z (cid:17) δg t t = 0 f ( z )( δg t t ) (cid:48)(cid:48) − f ( z ) z ( δg t t ) (cid:48) + 3 f (cid:48) ( z )2 ( δg t t ) (cid:48) + (cid:16) f (cid:48) ( z )2 − f ( z ) z (cid:17)(cid:16) ( δg xx ) (cid:48) + ( δg yy ) (cid:48) (cid:17) (A.4)+ µ z z + δA (cid:48) t + (cid:16) µ z z − k (cid:17) δg t t = 0 f ( z ) (cid:16) ( δg t t ) (cid:48)(cid:48) + ( δg xx ) (cid:48)(cid:48) + ( δg yy ) (cid:48)(cid:48) (cid:17) − f ( z ) z (cid:16) ( δg t t ) (cid:48) + ( δg xx ) (cid:48) + ( δg yy ) (cid:48) (cid:17) (A.5)16 f (cid:48) ( z )2 (cid:16) δg t t ) (cid:48) + ( δg xx ) (cid:48) + ( δg yy ) (cid:48) (cid:17) + µ z z + δA (cid:48) t + µ z z δg t t = 02 f ( z )( δg t t ) (cid:48) + f (cid:48) ( z ) δg t t + 2 f ( z )( δg yy ) (cid:48) + 2 µ z z + δA t = 0 (A.6)Whilst this is a complicated system of equations, one can simplify the calculation ofthe susceptibility by finding a subsystem of equations that govern the fluctuationsof δA x , δg t t , δg yy . This is possible because the Einstein equations imply an algebraicexpression for ( δg xx ) (cid:48) in terms of the other fields and their derivatives. To see this onetakes the linear combination of equations (A.5) − (A.4) − (A.3) − (A.2). This yieldsthe relation( δg xx ) (cid:48) = z f ( z ) − zf (cid:48) ( z ) (cid:104) f (cid:48) ( z )( δg yy ) (cid:48) − f ( z ) z (cid:16) ( δg yy ) (cid:48) + ( δg t t ) (cid:48) (cid:17) (A.7) − µ z z + δA (cid:48) t − k δg yy − (cid:16) k + µ z z (cid:17) δg t t (cid:105) We can then construct a closed set of ordinary differential equations for δA t , δg yy and δg t t by using this identity to eliminate ( δg xx ) (cid:48) from equations (A.1), (A.2) and (A.6).The resulting equations are first order in δg t t and second order in the fields δA t and δg yy . This means that we need to fix five constants of integration to obtain a uniquesolution.At the horizon, we demand regular behaviour in general. At finite temperature thisis achieved by imposing the analytic expansion δA t ∼ c ( z − z + ) + O (( z − z + ) ) δg yy ∼ c + O ( z − z + ) (A.8) δg t t ∼ c ( z − z + ) + O (( z − z + ) )where c , c and c are constants. Plugging this expansion in the equations of motionfixes c as a linear combination of c and c . Imposing these boundary conditionsremoves the three modes that lead to singular behaviour near the horizon. Of thefive original modes, we are therefore left with two, corresponding to the unconstrainedconstants of integration c and c .At zero temperature, which happens when µ z (0)+ = √
12, the singular point at z = z (0)+ is irregular and we impose the power series expansion δA t ∼ c + ( z (0)+ − z ) / ν + ( k/µ ) (1 + O ( z (0)+ − z ))+ c − ( z (0)+ − z ) / ν − ( k/µ ) (1 + O ( z (0)+ − z ))17 g t t ∼ c + d +1 ( k/µ ) ( z (0)+ − z ) / ν + ( k/µ ) (1 + O ( z (0)+ − z ))+ c − d − ( k/µ ) ( z (0)+ − z ) / ν − ( k/µ ) (1 + O ( z (0)+ − z )) (A.9) δg yy ∼ c + d +2 ( k/µ ) ( z (0)+ − z ) / ν + ( k/µ ) (1 + O ( z (0)+ − z ))+ c − d − ( k/µ ) ( z (0)+ − z ) / ν − ( k/µ ) (1 + O ( z (0)+ − z ))with ν ± as in equation (3.3). The constants d ± i are determined by the indicial sys-tem of equations for the modes and are functions of k/µ , while the c ± are the twounconstrained modes near the horizon.In the UV, the asymptotic expansion of the modes is fixed in terms of the constantsof integration { s i , v j } δA t → s + v z + · · · δg yy → s + · · · + v z + · · · (A.10) δg t t → s + · · · . As expected, δg t t , which satisfies a first order equation, gives only one constant ofintegration close to the boundary. Proceeding naively, one might wish to set s = s = 0since these correspond to sources of the stress tensor of the boundary theory. At thesame time we would like to impose s = 1 which would then correspond to a source forthe charge density. However, it is clear we do not have enough freedom to apply all ofthe conditions. We have already fixed three constants of integration by our regularityconditions, (A.8) or (A.9), and so can only fix two more in the UV.This issue is a consequence of the fact we are working in radial gauge (an analogoussituation was found in holographic conductivity calculations in [32]). The gauge trans-formation which sets δg zµ = 0 is a large gauge transformation that acts on the boundarydata. One can therefore only apply boundary conditions that are gauge equivalent to s = s = 0. That is, we impose δA t → s δg t t − δg yy → s = s = s . Matching the solutions from the two boundariesnow will completely fix the remaining five constants { c i =1 , , v i =1 , , s } or { c ± , v i =1 , , s } .This solution can then be mapped to one satisfying s = s = 0 by performing a gaugetransformation of the form x → x + f ( x, z ) , z → z + h ( x, z ) such that f ( x, z ) → s ik e ikx + O ( z )18 ( x, z ) → s e ikx z + O ( z )close to z = 0 and are smooth close to the horizon. Of course, performing this gaugetransformation turns on non-zero perturbations in g zz and g zx and so breaks the radialgauge condition.Nevertheless, after performing this coordinate transformation, the metric dies offsuitably fast at the boundary. In these coordinates, the static susceptibility is thereforegiven by the radial derivative of δA t at the boundary. Using our expansion (A.10) wehave χ ( k ) = − v s + µ s z + s References [1] A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, “
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