IImperial/TP/2011/JG/02
Holographic striped phases
Aristomenis Donos and Jerome P. Gauntlett
Blackett Laboratory, Imperial CollegeLondon, SW7 2AZ, U.K.
Abstract
We discuss new types of instabilities of D = 4 electrically charged AdS-Reissner-Nordstr¨om black branes that involve neutral pseudo-scalars. Theinstabilities spontaneously break translational invariance and are associ-ated with the dual three-dimensional CFTs, at finite temperature andfixed chemical potential with respect to a global abelian symmetry, ac-quiring striped phases. We show that such instabilities are present for theinfinite class of skew-whiffed AdS × SE solutions of D = 11 supergravity,albeit at a lower temperature than the known superfluid instabilities. a r X i v : . [ h e p - t h ] A ug Introduction
The AdS/CFT correspondence provides an important framework for studying thebehaviour of strongly coupled quantum field theories. Recently the possible appli-cations to condensed matter physics have received particular attention. One focushas been on the phase structure of conformal field theories when held at non-zerochemical potential with respect to a global abelian symmetry. At high temperaturesthe CFTs are described by electrically charged AdS-Reissner-Nordstr¨om (AdS-RN)black branes. As the temperature is lowered these black branes can become unstablegiving rise to new black brane solutions which then describe new phases of the dualfield theory.A well studied class of instabilities of the electrically charged AdS-RN black braneslead to the spontaneous breaking of the global abelian symmetry and hence to super-fluid phases. Such instabilities can occur when the metric and gauge field are coupledto charged fields and the superfluid phases are described by electrically charged AdSblack branes with additional charged hair. These instabilities and the back reactedblack branes were first studied in a bottom up context in [1][2][3] and then sub-sequently embedded into D=11 and D=10 supergravity in [4][5][6][7]. They werestudied using D-brane probes in [8].In this paper we will present a new class of instabilities of electrically charged AdS-RN black branes in D = 4 which lead to phases that spontaneously break the spatialtranslation invariance of the dual field theory. Such phases appear in condensedmatter physics in a variety of settings. More precisely, the phases spontaneouslybreak some or all of the symmetries of the underlying lattice. Examples includecharged density wave (CDW) phases and spin density wave (SDW) phases whichinvolve, as the names suggest, a spatial modulation of the charge density and thespin density, respectively (see [9][10] for reviews). Other orders include staggered fluxphases [11] and more general density waves with non-zero angular momentum [12].These orders have been discussed in the context of both heavy fermion and the cupratesuperconductors. In particular, it is well established that the cuprate superconductorsare characterised by a rich set of competing orders. The undoped materials areantiferromagnetic Mott insulators, but doping leads to superconductivity and stripedphases, in which there is unidirectional charge and spin density waves. There are alsoadditional ordered phases, including nematic phases and circulating current phaseswhich do not break the translational invariance of the lattice. A review of this areacan be found in [13] and a historical overview can be found in [14].1he new instabilities of the AdS-RN black branes that we discuss here will lead toblack brane solutions that are holographically dual to striped phases. More precisely,near the temperature T c at which these “striped black branes” appear, the d = 3 cur-rent of the global symmetry in the CFT, dual to the bulk gauge field, spontaneouslyacquires a spatially modulated vev of the form (cid:104) j t (cid:105) − ¯ j t ∝ cos(2 k c x ) , (cid:104) j x (cid:105) = 0 (cid:104) j y (cid:105) ∝ sin( k c x ) (1.1)where ¯ j t is a spontaneously generated constant component. Note that the translationinvariance in the x direction is spontaneously broken but it is preserved in the y direction. These stripes combine a CDW (corresponding to the spatially modulatedpart of j t ) with a “current density wave” (corresponding to the spatial modulationof j x , j y ). As one moves away from T c higher harmonics will appear, but the spatialmodulation of the CDW will have a period that is half of the current density wave.The simplest setting in which these striped instabilities can occur is when themetric and gauge field are coupled to a neutral pseudo-scalar ϕ . Indeed a key couplingdriving the instability is the coupling ϕF ∧ F , where F is the field strength of theabelian gauge field. In the context of Kaluza-Klein reductions of D = 10 and D =11 supergravity these types of couplings are commonplace and so we expect thatthe instabilities that we discuss, and straightforward generalisations thereof, will bewidespread. Here we will show that they are present in the context of the infiniteclass of skew-whiffed AdS × SE solutions of D = 11 supergravity, where SE is aseven-dimensional Sasaki-Einstein space. It has been shown in [15][5][6] that there isa consistent Kaluza-Klein truncation of D = 11 supergravity on an arbitrary SE to a D = 4 theory of gravity involving a metric, a gauge field, a neutral pseudo-scalar anda charged scalar. We show that this model exhibits spatially modulated instabilities,with vanishing charged scalar, and we determine the highest temperature at whichthey can occur. Although this demonstrates that such instabilities are indeed presentin a top down context, it should be noted that for this particular class of theories, theknown superfluid instability involving the charged scalar field, found in [5][6], alreadysets in at a higher temperature.We will also consider a more general class of models which couple the metric, agauge field and neutral pseudo-scalar to an additional massive vector field. Thesemodels naturally appear in N = 2 gauged supergravity models coupled to a vectormultiplet plus additional hypermultiplets, and we discuss some explicit examples. If one gauges the global symmetry in the boundary field theory, these currents would give riseto a spatially modulated magnetic field. D = 4 AdS-RN black brane solutions. A simple and powerful approach isto first consider linearised perturbations in the AdS × R background, which arisesas the IR limit of the AdS-RN geometry at zero temperature. Depending on theexplicit values of various couplings, we find that there are modes which violate theBF bound for a range of non-vanishing momentum k . This indicates that thereare associated instabilities of the AdS-RN black branes which set in at a non-zerotemperature whose value will depend on k . For the simple model with a single vectorfield, we explicitly construct the static normalisable zero modes and show that thereis a critical momentum k c which has the highest temperature, T c . At this temperaturea new branch of striped black branes will appear, spontaneously breaking translationinvariance with a spatial modulation set, at T c , by k c . These are dual to a stripedphase with spatial modulation of the vev of the dual d = 3 current ( j t , j x , j y ) given,near T c , by (1.1). As one moves away from T c higher harmonics will appear, but thespatial modulation of the CDW will have a period that is half of that of the currentdensity wave.Before presenting our new results we note that spontaneous breaking of translationinvariance has been found in the context of electrically charged AdS-RN black banes of D = 5 Einstein-Maxwell theory with a Chern-Simons term in [16][17]. Related earlierwork appears in [18] and subsequent work appears in [19][20]. Another holographicinvestigation of spontaneous breaking of translation invariance in the presence of amagnetic field was carried out for a D = 4 Einstein-Yang-Mills-Higgs model in [23]. We consider a class of D = 4 theories that couples a metric, a gauge field A and aneutral pseudo-scalar ϕ with Lagrangian given by L = 12 R ∗ − ∗ dϕ ∧ dϕ − V ( ϕ ) ∗ − τ ( ϕ ) F ∧ ∗ F − ϑ ( ϕ ) F ∧ F , (2.1) The existence of this striped phase obviously requires that the striped black branes appearingat T = T c are thermodynamically preferred. The thermodynamics of these black branes, as well asthe properties of other possible black brane solutions, will be investigated elsewhere. In [21][22] studies were made where translation invariance is explicitly broken by sources. F = dA . The corresponding equations of motion are given by R µν = ∂ µ ϕ∂ ν ϕ + g µν V − τ (cid:18) g µν F λρ F λρ − F µρ F ν ρ (cid:19) d ( τ ∗ F + ϑF ) = 0 d ∗ dϕ + V (cid:48) ∗ τ (cid:48) F ∧ ∗ F + 12 ϑ (cid:48) F ∧ F = 0 . (2.2)We will assume that the three functions V , τ and ϑ have the following expansions V = − m s ϕ + . . . , τ = 1 − n ϕ + . . . , ϑ = c √ ϕ + . . . . (2.3)The equations of motion (2.2) then admit the electrically charged AdS
Reissner-Nordstr¨om black brane solution ds = − f dt + dr f + r (cid:0) dx + dy (cid:1) A = (cid:16) − r + r (cid:17) dt , (2.4)with ϕ = 0, where f =2 r − (cid:18) r + 12 (cid:19) r + r + r r . (2.5)In particular, we note that the equation of motion for ϕ in (2.2) is satisfied with ϑ (cid:48) (cid:54) = 0 because for the purely electric AdS-RN solution F ∧ F = 0 (which would notbe true for a dyonic AdS-RN black brane). We also note that we have scaled thechemical potential to be unity, µ = 1, for convenience, and that the temperature ofthe black brane is T = (1 / πr + )(12 r − d = 3 CFTwith a global abelian symmetry, whose current, j , is dual to A , when held at non-vanishing chemical potential and at high temperatures. Phase transitions can arise ifthis solution becomes unstable at some critical temperature. We also note that ϕ isdual to an operator in the CFT with scaling dimension ∆ ± = (1 / ± (9 + 2 m s ) / ],with ∆ − only possible if − / ≤ m s < − / AdS × SE solutions of D = 11 supergravity. If we ignore the Einstein-Hilbert term in (2.1), the D = 4 models generalise the dimensionalreduction of the D = 5 Maxwell models with a Chern-Simons term considered in [16], by the additionof m s and n . In particular, the tachyonic instability in a uniform electric field in flat space at finitemomentum, considered in section 2 of [16], has an immediate extension to D = 4 massive axionscoupled to the electromagnetic field. SE = S in which case they preserve all of the supersymme-try. It was shown in [6], building on [15][5], that there is a consistent Kaluza-Kleinreduction on an arbitrary SE space to a theory involving a metric, a gauge field,a charged scalar and a pseudo-scalar. Any solution of this D = 4 theory can beuplifted to obtain an exact solution of D = 11 supergravity. In particular, there isan AdS vacuum solution which uplifts to the skew-whiffed AdS × SE solution. Itis consistent to further set the charged scalar to zero and the resulting D = 4 modelis then as in (2.1), (2.3) with n = 36, c = 6 √ m s = − AdS × R The near horizon limit of the extremal ( T = 0) AdS-RN black brane (2.4) is thefollowing AdS × R solution ds = − r dt + dr r + dx + dy F =2 √ dr ∧ dt , (2.6)with ϕ = 0 and we have scaled the spatial coordinates by a factor of 2 √ δg ty =2 √ rh ty ( t, r ) sin ( kx ) δg xy = h xy ( t, r ) cos ( kx ) δA y = a ( t, r ) sin ( kx ) δϕ = w ( t, r ) cos ( kx ) , (2.7)Note that the subscripts on h are simply to label specific functions of t and r .Substituting into the equations of motion (2.2) we find that the fluctuations (2.7)satisfy 2 √ k rh ty − k∂ t h xy − √ r (cid:0) ∂ r (cid:0) r ∂ r h ty (cid:1) + 2 r∂ r a (cid:1) =0 (2.8)2 √ ∂ t a + √ ∂ t h ty + r (cid:16) kr ∂ r h xy + √ ∂ r ∂ t h ty (cid:17) =0 (2.9) − √ kr ∂ t h ty + ∂ t h xy − r ∂ r (cid:0) r ∂ r h xy (cid:1) =0 (2.10) − r ∂ t a − k a + c kw + 12 (cid:0) h ty + ∂ r (cid:0) r ∂ r a (cid:1) + r∂ r h ty (cid:1) =0 (2.11) − r ∂ t w + c ka − (cid:0) m s + n + k (cid:1) w + 12 ∂ r (cid:0) r ∂ r w (cid:1) =0 . (2.12)5ne can check that equation (2.10) is implied by equations (2.8) and (2.9). Observethat m s and n only appear in the combination˜ m s ≡ m s + n . (2.13)It is illuminating to now introduce the field redefinition6 kφ xy = −√ ∂ r ( rh ty ) − √ a . (2.14)We then see that (2.9) implies r ∂ r h xy = ∂ t φ xy . (2.15)We also find that (2.11), (2.12) and the r derivative of (2.8) can now be packaged inthe following way. Defining the three vector v = ( φ xy , a, w ) we have (cid:3) AdS v − M v = 0 , (2.16)where (cid:3) AdS is the scalar Laplacian on the AdS space with radius squared equal to1 /
12, and the mass matrix M is given by M = k √ k √ k
24 + k − c k − c k k + ˜ m s . (2.17)Thus a diagnostic for an instability is if the mass matrix has an eigenvalue m that violates the AdS BF bound m ≥ − . (2.18)To be more precise, we have shown that (2.16), (2.17) are implied by (2.8)-(2.12). Inappendix A we show that, conversely, (2.16), (2.17) includes all of the perturbationsof interest. Thus, for fixed parameters ˜ m s , c we are looking for ranges of momenta k in which the smallest root of the characteristic polynomial of M , violates (2.18).In particular, we see that the off-diagonal term in M provides a mechanism, whenboth c (cid:54) = 0 and k (cid:54) = 0, to drive down the smallest eigenvalue.In figure 1 we show part of the domain in the ( ˜ m s , c ) plane for which there existsa range of momenta such that one of the corresponding AdS masses is below the BFbound (2.18). The characteristic polynomial of M is independent of the sign of k andthis leads to two cases. In the first case, marked in green in figure 1, the range of k with unstable modes is of the form ( − k max , k max ) and in particular contains unstable6 (cid:230)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) m (cid:142) s2 Figure 1: The shaded region in the ( ˜ m s , c ) plane has unstable perturbative modesin the AdS × R background. The green region has unstable modes including k = 0while the cyan region only has unstable modes with k (cid:54) = 0. The blue circle, lying justinside the cyan region, represents the model associated with skew-whiffed AdS × SE solutions. The other symbols are explained in the text.modes with k = 0. It is worth noting, though, that it is not necessarily the case thatthe k = 0 mode is the one with the smallest AdS mass squared. In the second case,marked in cyan in figure 1, all of the unstable modes have k (cid:54) = 0 and the range of k for which there are unstable modes consists of two disjoint regions: ( k min , k max ) andits reflection k → − k . The presence of perturbative instabilities in the
AdS × R background that we estab-lished in the last subsection suggests that the AdS-RN black brane (2.4), (2.5) shouldhave analogous perturbative instabilities appearing, for a given k , at some specifictemperature. In particular we shall look for the appearance of static normalisablezero modes which signal the onset of a dynamical instability. Inspired by the analysis7n the AdS × R background, we consider the fluctuation δg ty = λ [ r ( r − r + ) h ( r ) sin ( kx )] δA y = λ [ a ( r ) sin ( kx )] δϕ = λ [ w ( r ) cos ( kx )] . (2.19)Here λ is a small expansion parameter and note that we have not included a timeindependent perturbation for g xy , as suggested by (2.15).Expanding around the black brane background (2.4) we obtain a linear system ofordinary differential equations which we wish to numerically integrate from the outerhorizon, r = r + , to asymptotic infinity, r → ∞ . In order for the fluctuations (2.19) tobe regular, the functions h ty , a and w should remain finite on the horizon at r = r + h ( r ) = h + + O ( r − r + ) a ( r ) = a + + O ( r − r + ) w ( r ) = w + + O ( r − r + ) . (2.20)To see that the metric is regular at the horizon one can use the in-going Eddington-Finkelstein type coordinates v, r where v ≈ t + ln ( r − r + ).Unlike in the AdS × R background, the linear ODEs now depend on m s and n separately. For illustration we will focus on the case m s = − ϕ corresponds to an operator in the dual CFT with scaling dimension ∆ = 2.By varying n we can still vary ˜ m s . The most general asymptotic expansion of thefunctions in (2.19) as r → ∞ is h = h + · · · + h r + . . .a = a + . . . + a r + . . .w = w r + . . . + w r + . . . . (2.21)The parameters h , a and w correspond to deforming the dual field theory bythe operators dual to the fields, while h , a and w correspond to the operatorsacquiring vev’s. We are interested in looking for instabilities that spontaneouslybreak translational invariance so we set h = a = w = 0.We have three second order linear ODE’s to solve and so a solution is speci-fied by six integration constants. Now, for a given k , we have seven parameters r + , h + , a + , w + , h , a , w entering the ODEs (since λ drops out). However, since theequations are linear we can always scale one of these parameters to unity. Hence,8or a given k we expect a normalisable zero mode to appear, if at all, at a specifictemperature.We have numerically studied the existence of normalisable modes for various spe-cific values of n and c . We begin with a case with n = 0. We choose c = 4 . c . Note that this case is marked by the squarein the green area in figure 1. When k = 0 (and hence c k = 0), this case was al-ready analysed in [4] (∆ = 2 and q = 0 in their language). There it was shown thatnormalisable perturbative modes with k = 0 are present with critical temperature T < − . Here we find that perturbative static normalisable zero modes also existfor a range of k , with the critical temperatures depending on k as shown in figure 2a.The highest critical temperature occurs for wavenumber k ≈ .
53 and has T c ≈ . AdS × SE solutions. The maximum critical tem-perature T c is very low for this case, so we illustrate what is going on by studying thesequence of values of n and c given by the rhombi and triangles in figure 1. Plotsof the critical temperatures for different k for which there is a static perturbativenormalisable mode are given in figures 2b and 2c. For all these cases, we find thatthe range of momenta k for which there is an unstable static zero mode does notinclude k = 0 as expected from the the analysis in the AdS × R background. Thefigures indicate that the critical temperature for the skew-whiffed case is going to bevery low, but certainly for a non-zero value of k . It is also worth noting that themaximum critical temperature is much lower than the critical temperature for theknown superfluid instability which is given by T ≈ . In the last subsection we constructed the normalisable zero modes in the AdS-RNblack brane background at leading order in perturbation theory. There is a criticalvalue of momentum k c which is associated with the highest critical temperature T c at which the zero modes appear (the maxima in figure (2)). When T = T c a newbranch of black branes will appear that spontaneously break translation invariance.In the leading order perturbations given in (2.20) and (2.21) we find that w , h and a are all non-zero, as the simple counting of integration constants above indicated. w (cid:54) = 0 implies that the scalar operator dual to ϕ has acquired a spatially modulated9 .2 0.4 0.6 0.8 1.0k0.0020.0040.0060.0080.0100.012T (a) n = 0, c = 4 . (b) n = 36, various c (c) c = 6 √
2, various n Figure 2: Plots of critical temperatures T versus k for the existence of normalisablestatic perturbations about the electrically charged AdS-RN black brane. All caseshave m s = −
4. With reference to figure 1, figure (a) corresponds to the square, figure(b) corresponds to the rhombi and figure (c) corresponds to the triangles.vev. Similarly, h (cid:54) = 0 implies that there is momentum transfer in the x direction.Finally, a (cid:54) = 0 implies that the dual current component (cid:104) j y (cid:105) is also acquiring aspatially modulated vev of the form (1.1). Thus, the new black branes are dual to acurrent density wave phase with spatial modulation given, at T c , by k c . As we willnow argue the phase is also a charge density wave (CDW) with (cid:104) j t (cid:105) of the form (1.1)and hence, near T c , a spatial modulation given by 2 k c .To see the CDW, we need to analyse the general structure of the equations arisingin the next to leading order in perturbation theory. One can show that a closed systemof equations is obtained if we take the second order perturbations to be given by δg tt = λ (cid:104) h (0) tt ( r ) + h (1) tt ( r ) cos (2 kx ) (cid:105) δg xx = λ (cid:2) h (0) xx ( r ) + h (1) xx ( r ) cos (2 kx ) (cid:3) δg yy = λ (cid:2) h (0) yy ( r ) + h (1) yy ( r ) cos (2 kx ) (cid:3) δA t = λ (cid:104) a (0) t ( r ) + a (1) t ( r ) cos (2 kx ) (cid:105) . (2.22)Plugging the total field expansion in the equations of motion (2.2) and expandingthem up to order O ( λ ) we obtain an inhomogeneous system of ordinary differentialequations, being sourced by the O ( λ ) zero mode solution. More specifically, thefunctions h ( α ) xx , h ( α ) yy and a ( α ) t satisfy second order equations, the function h (0) tt satisfiesa first order equation while h (1) tt satisfies an algebraic equation and hence can beeliminated from the system.We next need to impose regularity at the horizon and demand that the behaviouras r → ∞ corresponds to setting all source terms in the dual CFT to zero. Asimple count of parameters and integration constants then indicate that for a given10 , and in particular for k = k c , the zero mode found in the last subsection forms partof a one-parameter branch of spatially modulated black brane solutions, where theparameter can be taken to be the temperature. We will expand on the solutions tothese ODEs in more detail elsewhere, but the main point we wish to emphasise hereis that, generically, the asymptotic falloff of a (1) t ( r ) as r → ∞ , in particular, will beof the form a (1) t ( r ) = 0 + ¯ a (1) t r + . . . (2.23)implying that there is a spontaneous spatial modulation of the charge density with,from (2.22), characteristic wavenumber given by 2 k c , at T c , and in the same directionas the current density wave. Note that a (0) t in (2.22) will have a similar asymptoticbehaviour and the non-vanishing 1 /r component will give rise to the ¯ j t piece in (1.1).Thus the new branches of black brane solutions, assuming that they are thermody-namically preferred, will be dual to striped phases incorporating both current densitywaves and CDWs.Note that the perturbative expansion parameter λ can be taken to be ( T − T c ) /T c .For small λ , i.e. near T c , we have argued that the wavenumber for the spatial modu-lation of the current density wave will be k c and for the CDW will be 2 k c . Followingan analogous discussion in [24], continuing to higher orders in the perturbative ex-pansion we expect that the wavenumber will receive corrections at order λ . However,the spatial modulation of the CDW will have a period that is always half that of thecurrent density wave and in the same direction. We now consider a more general class of D = 4 theories that couple a metric, apseudo-scalar and two vector fields with Lagrangian given by L = 12 R ∗ − ∗ dϕ ∧ dϕ − V ( ϕ ) ∗ − τ ( ϕ ) F ∧ ∗ F − ϑ ( ϕ ) F ∧ F − G ∧ ∗ G − m v ∗ B ∧ B + c ϕ √ F ∧ G , (3.1)where F = dA and G = dB . In the first line we take V = − m s ϕ + . . . , τ = 1 − n ϕ + . . . , ϑ = c √ ϕ + . . . , (3.2)as in the last section, while in the second line we have introduced two new parameters m v and c . In the AdS vacuum with A = B = ϕ = 0, the gauge field A is massless,11hile B has mass m v . These models admit the electrically charged AdS-RN blackbrane solution, (2.4), (2.5) with B = ϕ = 0, as a solution. This solution describesa dual d = 3 CFT at high temperature and finite chemical potential with respectto the global abelian symmetry whose current is dual to the gauge field A . Thescalar field, ϕ , is dual to a scalar operator with conformal dimension given beforeand the second vector field, B , is dual to a vector operator with conformal dimension∆ = (1 / m v ) / ].We are interested in perturbative instabilities of the electrically charged AdS-RNblack brane solution. We first point out that (3.1) can be generalised in a number ofobvious ways, including adding a f ( ϕ ) G ∧ G term, without affecting the linearisedanalysis that we cary out below. Furthermore, we can also couple additional matterfields. Thus the instabilities that we discuss for (3.1) will capture a large class ofexamples arising in string and M-theory. For example, it overlaps with the followingtwo cases that have been explicitly considered in the literature that involve N = 2gauged supergravity coupled to a vector multiplet plus additional hypermultiplets.The first case is the consistent truncation of D = 11 supergravity on a SE toa D = 4 theory whose AdS vacuum uplifts to the supersymmetric AdS × SE solution of D = 11 supergravity [15]. The matter content of the D = 4 theoryconsists of a metric, two vectors and six scalars, which package together into an N = 2 gravity multiplet, a vector multiplet and a hypermultiplet, and in particularthe vector multiplet contains a pseudo-scalar labelled as h in [15]. We find thatthe linearised perturbations for (3.1) that we consider below arise in a sector ofthe D = 4 theory of [15]. To see this one should identify the field strengths via H there = 1 / F − G/ √ F there = 1 / F + √ G ), the scalar h = − ( √ / √ ϕ andfinally rescale the metric g thereµν = 1 / g µν . One then finds that there are linearisedperturbations involving the pseudo-scalar which are exactly the same as those comingfrom (3.1) with m s = 20 , m v = 24 , n = 12 , c = 0 , c = 2 √ . (3.3)The second case is the consistent KK reduction of D = 11 supergravity on H × S where H is three-dimensional hyperbolic space [25] (in fact H can be replaced by anarbitrary quotient H / Γ). The resulting D = 4 theory has an AdS vacuum solutionwhich uplifts to a supersymmetric AdS × H × S solution of D = 11 supergravityand is dual to a d = 3 N = 2 SCFT that arises on M5-branes wrapping specialLagrangian 3-cycles H [26]. The matter content of this D = 4 theory consistsof a metric, two vectors and ten scalars, which package together into an N = 212ravity multiplet, a vector multiplet and two hypermultiplets, and in particular thevector multiplet contains a pseudo-scalar labelled as β in [26]. We find that thelinearised perturbations for (3.1) that we consider below also arise in a sector ofthe D = 4 theory of [26]. To see this one should identify the field strengths via˜ H there = 2 − / ( F + G/ √ F there = 2 / ( F − G/ √ β = (1 / √ ϕ andfinally choose the gauge coupling g there = 2 / . One then finds that there are linearisedperturbations involving the pseudo-scalar which are exactly the same as those comingfrom (3.1) with m s = 4 , m v = 8 , n = 12 , c = 0 , c = 2 √ c = 0. This was actually to be expectedfor these supersymmetric examples. First observe that c = 0 is required in orderto be able to consistently truncate the theory (3.1) to the Einstein-Maxwell sectorinvolving the metric and the gauge field A (otherwise there would be a F ∧ F sourceterm in the equation of motion for ϕ ). Second, we recall that for any AdS × M solution of D = 10 or D = 11 supergravity which is dual to an N = 2 SCFT in d = 3,it is conjectured that there is a consistent Kaluza-Klein truncation on M to N = 2 D = 4 minimal gauged supergravity, and this has been proven for several differentclasses [27]. The relevant gauge field is dual to a canonical “Reeb” Killing vectorof M . Since the bosonic sector of minimal gauged supergravity is simply Einstein-Maxwell theory, the more general N = 2 supergravity theories with A the canonicalgauge field must have c = 0.We now investigate instabilities of the electrically charged AdS-RN black branesolution of (3.1) by considering the following linearised perturbations in the AdS × R background: δg ty =2 √ r h ty ( t, r ) sin ( kx ) δg xy = h xy ( t, r ) cos ( kx ) δA y = a ( t, r ) sin ( kx ) δB y = b ( t, r ) sin ( kx ) δϕ = w ( t, r ) cos ( kx ) . (3.5) Recall that in section 2 we argued that associated with the skew-whiffed
AdS × SE solutions,which generically don’t preserve supersymmetry, there is a truncation with c = 6 √
2. For thespecial case when SE = S the models preserve N = 8 supersymmetry. This is consistent withthe discussion here because the gauge field being kept is not a canonical gauge field dual to a Reebvector compatible with the orientation of the S . v = ( φ xy , a, w, b ), the remaining independent equationsimply that (cid:3) AdS v − M v = 0 , (3.6)where (cid:3) AdS is the scalar Laplacian on the AdS space with radius squared equal to1 /
12, and the mass matrix M is given by M = k √ k √ k
24 + k − c k − c k ˜ m s + k c k c k m v + k (3.7)and ˜ m s ≡ m s + n .This mass matrix can have eigenvalues violating the AdS BF bound (2.18) fornon-zero values of k . The key features are the off-diagonal entries c k and c k . Thegeneral analysis is not that illuminating so we shall not present it here. Instead wemake the following observations. We first notice that when c = 0, the second vectorfield, B , decouples at the linearised level. Indeed when c = 0 the upper 3 × c = 0, which in particular arises inthe KK truncations of D = 11 supergravity associated with the supersymmetric AdS × H × S and AdS × SE solutions. Notice that the mass matrix (3.7) isnow block diagonal and that possible BF violating modes must appear in the lower2 × . The eigenvalues of this block are given by m ± = 12 (cid:18) k + ˜ m s + m v ± (cid:113) (2 c k ) + ( ˜ m s − m v ) (cid:19) . (3.8)One can easily show that for c > ( ˜ m s − m v ) the branch m − develops a minimumat a non-zero value of k given by k = ± (cid:113) c − ( ˜ m s − m v ) c , (3.9) In the context of
AdS × sphere solutions instabilities at finite momentum that are decoupledfrom gravitational perturbations have been studied in [28]. AdS mass given by m min = − c (cid:104) c + (cid:0) ˜ m s − m v (cid:1) − c (cid:0) ˜ m s + m v (cid:1)(cid:105) . (3.10)We now see that for sufficiently large c one can always have m min < − AdS × SE and AdS × H × S solutions thatwere given in (3.3) and (3.4), respectively, we see that these perturbations involvingthe pseudo-scalar do not violate the BF bound. It would be interesting to knowwhether or not there are supersymmetric top down models with c = 0 and largeenough c to give instabilities.Returning to the general analysis, as in the last section, we find that the insta-bilities in the AdS × R background are associated with static normalisable zeromodes appearing in the AdS-RN black brane at, for a given k , some critical tem-perature. We have investigated some specific examples and the results are similar tothose presented in figure (2). In this paper we have identified new instabilities of D = 4 electrically charged AdS-RN black branes in a broad class of models involving pseudo-scalars. Generically,the instabilities appear at non-vanishing spatial momentum. The static normalisablemodes that we identified correspond to the appearance of new branches of stripedblack brane solutions with the spatial modulation of the CDW being half that of thecurrent density wave and in the same direction. It will be interesting to investigatethe thermodynamics of the striped black branes and establish the conditions for whichthey are thermodynamically preferred over the AdS-RN black branes. We expect atleast in some cases that the phase transitions are second order and we will be ableto verify this by developing the perturbative expansion we have used in this paper.However, if they are first order we would need to construct the fully back reactedbranes by solving PDEs. This will also be necessary to be able to follow the stripedphases to their ultimate zero temperature ground states. Alternatively, perhaps it ispossible to construct candidate zero temperature ground states directly.We showed that spatially modulated instabilities are present in a D = 4 modelassociated with the skew-whiffed AdS × SE solutions of D = 11 supergravity. Weshowed that the maximal critical temperature at which the instabilities appear is15ower than the critical temperature associated with the superfluid instability. How-ever, it seems likely that the two instabilities will compete, possibly after a defor-mation by a relevant operator, and will lead to thermodynamically preferred stripedphases which may also be superconducting. We find this a particularly interestingdirection to pursue given the potential similarities with what is seen in the heavyfermion and the high temperature superconductors. Acknowledgements
We would like to thank Fay Dowker, Sung-Sik Lee, Subir Sachdev, Toby Wisemanand Jan Zaanen for helpful discussions. AD is supported by an EPSRC PostdoctoralFellowship. JPG is supported by an EPSRC Senior Fellowship and a Royal SocietyWolfson Award.
A A converse result
We showed that (2.8)-(2.12) imply (2.16), (2.17). Let us now prove a converse result.We start with a regular perturbation in the
AdS × R background satisfying (2.16),(2.17). We can then integrate (2.15) and (2.14) to define r h ty and h xy up to twoarbitrary integration functions of time, z ( t ) and z ( t ). Note that (2.9) is automati-cally satisfied. On the other hand, we find that demanding that (2.8) is satisfied, andfurthermore demanding that r h ty = 0 at r = 0 (i.e. the perturbation (2.7) is regularat r = 0), fixes z and z . Indeed we find that r h ty = − (cid:90) r dr (cid:16) a + 2 √ k ϕ xy (cid:17) ,h xy = (cid:90) dt (cid:20) r ∂ r ϕ xy − (cid:90) r dr (cid:16) √ k a + 12 k ϕ xy (cid:17)(cid:21) . (A.1)In order to make the asymptotic behaviour of (A.1) more transparent we will usethe three normal modes, s α , α = 1 , ,
3, coming from diagonalising the system ofequations (2.16). Specifically, if the eigenvalues of the mass matrix (2.17) are m α wehave (cid:3) AdS s α − m α s α = 0 (A.2)16e can then rewrite (A.1) as r h ty = − √ k (cid:88) α c α (cid:90) r dr m α s α h xy = 12 (cid:88) α c α (cid:90) dt (cid:20) r ∂ r s α − (cid:90) r dr m α s α (cid:21) (A.3)for some constants c α . We now focus on a specific mode, s , and assume a timedependence of the form e − iωt . The normalisable modes of (A.2) will behave as s ≈ r δ ( k ) (A.4)with δ ( k ) > / Re ( δ ( k )) = 1 / s in terms of Bessel functions we have the schematicexpansions rh ty ≈ e − iωt (cid:34) − √ k A + f r δ ( k ) − (cid:35) h xy ≈ e − iωt (cid:20) − iAω + f r δ ( k )+1 (cid:21) (A.5)where A is a finite constant. Finally, we point out that the dependence on A in (A.5)is actually a gauge artifact and hence is not modifying the AdS × R asymptotics.Indeed it can be eliminated by performing the (perturbative) coordinate change y → y + 12 ikω e − iωt A Q ( r ) sin ( kx ) (A.6)with Q a sufficiently smooth function such that the first few derivatives vanish at r = 0 and r = ∞ and also Q (0) = 0, Q ( ∞ ) = 1. Note that this change of coordinatesalso introduces a non-zero δg ry perturbation, but it has support only in the bulk andvanishes at both the origin and boundary and is thus benign. References [1] S. S. Gubser, “Breaking an Abelian Gauge Symmetry Near a Black HoleHorizon,”
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