aa r X i v : . [ h e p - t h ] M a r PUPT-2264
Colorful horizons with charge in anti-de Sitter space
Steven S. Gubser
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Dated: March 2008)An abelian gauge symmetry can be spontaneously broken near a black hole horizon in anti-deSitter space using a condensate of non-abelian gauge fields. There is a second order phase transitionbetween Reissner-Nordstrom-anti-de Sitter solutions, which are preferred at high temperatures, andsymmetry breaking solutions, which are preferred at low temperatures.
PACS numbers: 11.15.Ex, 04.25.Dm, 11.25.Tq
I. INTRODUCTION
I have previously argued that an abelian gauge sym-metry can be spontaneously broken by a condensate ofa charged scalar that forms near the horizon of a non-extremal charged black hole, presumably indicating someform of superfluidity or superconductivity [1, 2]. Unpub-lished numerical calculations based on the model pro-posed in [2] indicate that “superconducting” horizons arethermodynamically preferred below some non-zero crit-ical temperature. But there are enough parameters inthe lagrangians considered in [1, 2] that it is challengingto characterize the degree of universality or robustnessof such numerical results. I will therefore consider ananalogous phenomenon in a theory whose lagrangian ismostly determined by symmetry principles: I = 12 κ Z d x √− g L (1)where L = R + 6 L −
14 ( F aµν ) , (2)and F aµν = ∂ µ A aν − ∂ ν A aµ + gǫ abc A bµ A cν (3)is the field strength of an SU (2) gauge field. I use mostlyplus signature, and ǫ abc is the totally antisymmetric ten-sor with ǫ = 1. I aim to persuade the A µ and A µ gaugebosons to leap out of the horizon and condense near it,thereby breaking a U (1) symmetry associated with A µ .This should be possible if the gauge coupling g is largeenough.It is convenient to represent the gauge field as a matrix-valued one form: A = A aµ τ a dx µ , (4)where τ a = σ a / i , so that[ τ a , τ b ] = ǫ abc τ c . (5) (By σ a I mean the usual Pauli matrices.) I will restrictattention to the following ansatz: ds = e a (cid:0) − hdt + dx + dx (cid:1) + dr e a h (6a) A = Φ τ dt + w ( τ dx + τ dx ) , (6b)where a , h , Φ, and w are functions only of r . The gaugefield (6b) is a slight simplification of the ansatz consideredin [3, 4]. A substantial literature has grown up aroundsimilar solutions of Einstein-Yang-Mills theory: see [5, 6]for reviews.The electrostatic potential Φ must vanish at the hori-zon for A to be well-defined as a one-form, but I do notrequire it to vanish at infinity: thus the black hole cancarry charge under the U (1) gauge symmetry generatedby τ .The condensate w ( τ dx + τ dx ) breaks the U (1) ro-tation symmetry in the x - x plane as well as the U (1)gauge symmetry generated by τ , but it preserves a com-bination of the two. I will require w to be normalizablein the sense of making a finite contribution to the norm[16] || A || ≡ Z ∞ r H dr √− g g µν A aµ A aν , (7)where r H is the location of the horizon. Normalizabilityof w is what it will mean for the condensate to form“near” the horizon. It is an appropriate requirement inthe context of studying spontaneous symmetry breaking.In summary: the only conserved quantities associatedwith the black hole should be its mass density and itselectric τ charge density; and w , if it is non-zero, is acondensate whose presence spontaneously breaks U (1) ofspatial rotations times U (1) of τ into a diagonal sub-group. I want to know when this spontaneous symmetrybreaking occurs. II. SYMMETRY BREAKING SOLUTIONS
Plugging (6a) and (6b) into the equations of motion following from (2) results in four second order equations ofmotion: a ′′ + a ′ + 12 e − a w ′ + g e − a Φ w h = 0 (8a) h ′′ + 4 a ′ h ′ − e − a Φ ′ − g e − a Φ w h + e − a hw ′ − e − a hw ′ − g e − a w = 0 (8b)Φ ′′ + 2 a ′ Φ ′ − g e − a Φ w h = 0 (8c) w ′′ + (cid:18) a ′ + h ′ h (cid:19) w ′ + g e − a (cid:18) Φ wh − w h (cid:19) = 0 (8d)together with a zero-energy constraint,12 e a a ′ + 4 e a h ′ h a ′ − L h + Φ ′ h − w ′ − g e − a Φ w h + g e − a w h = 0 . (9) TABLE I: Scaling symmetries of the equations of motion (8), the constraint (9), of series coefficients appearing in (10) and(11), and of the thermodynamic variables appearing in (13). dt d~x dr e a e a h h H H Φ Φ p p w w W L g κ I ǫ s T ρ µ J
A 0 1 0 − − − − − − − − − − − − −
2B 1 1 − − − − − − − − − − − − − − − −
2C 1 1 1 0 0 0 − − − − − − − − −
2D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 − − − − − r = 0. This impliesseries expansions around r = 0 of the form a = a + a r + a r + . . .h = h r + h r + . . . Φ = Φ r + Φ r + . . .w = w + w r + w r + . . . . (10)It is also straightforward to obtain the asymptotic be-havior at large r : a = log rL + α + . . .h = H + H e − a + H e − a + . . . Φ = p + p e − a + . . .w = W e − a + . . . . (11)Once the six quantities a , h , Φ , w , g , and L arespecified, all the other coefficients in (10) may be com-puted, and one can then use the series expansions tospecify Cauchy data for the equations of motion (8) atsome radius r slightly outside the horizon. Of these sixparameters, only three are meaningful, because the equa- tions (8) and (9) have three scaling symmetries that actnon-trivially on the parameters. These symmetries aresummarized in the first three rows of table I, where as-signing a charge α to a quantity X means X → λ α X . (12)Recall that w is required to be normalizable at infinity.This requirement is imposed in the last equation of (11),but it is not the typical behavior of w : usually there isa constant term at large r , or else there is a singularityat finite r . The solutions with normalizable w form a co-dimension one locus in the three-dimensional space of pa-rameters modulo scaling symmetries. This locus includesas one branch the well-known Reissner-Nordstrom-anti-de Sitter solutions (hereafter RNAdS), which have theform (6) with w = 0: see for example [7].The energy density, entropy density, temperature,chemical potential, and charge density, as well as an or-der parameter, J , to be discussed further below, can beread off as follows: ǫ = − H κ LH s = 2 πκ e a µ = p L √ H T = 14 π e a h √ H ρ = − p κ √ H J = W Lκ . (13)Energy density, temperature, and chemical potential aremeasured in reference to the Killing time √ H t ratherthan t itself. The quantities in (13) can be regarded ascharacterizing a thermal state of a dual conformal fieldtheory, along the lines of the gauge-string duality [8, 9,10]. Their normalizations accord with the conventions of[7]. All dependence on κ can be removed by definingˆ ǫ = κ ǫ (2 π ) L ˆ s = κ s (2 π ) L ˆ ρ = κ ρ (2 π ) L ˆ J = κ J (2 π ) L , (14)where the factors of 2 π are included for later convenience.Any relation among the quantities in (13) and (14) mustbe expressible in terms of ratios which are invariant un-der the scaling symmetries summarized in table I. Forexample, the RNAdS solutions haveˆ ǫ ˆ s / = 1 + π ˆ ρ ˆ s . (15)When w = 0, one may ask what fraction q of the elec-tric charge density is carried by the non-abelian gaugebosons outside the horizon. The ratio of the flux of the τ electric field through the horizon to the flux at infinityis 1 − q . Therefore, q = 1 + Le a p H Φ p . (16) q by itself is invariant under the scaling symmetries.The conformal field theory dual to the AdS back-ground under consideration has currents J am satisfyingan SU (2) current algebra, where m runs over the t and x i directions. The symmetry breaking that arises fromnon-zero w corresponds to expectation values h J ai i ∝ Jδ ai , (17)where i runs over the values 1 ,
2. The tensor δ ai exhibitsthe locking of a spatial U (1) and a gauge U (1). (17)describes a form of long-range order which infrared fluc-tuations probably destroy; however, fluctuations are sup-pressed in the supergravity approximation, where κ ≪ L [17]. I will therefore ignore them. III. SUMMARY OF RESULTS
As explained following (12), there is a two-parameterfamily of black hole solutions with a non-vanishing, nor-malizable condensate, once all possible scaling symme-tries are used. A convenient choice of parameters is gL and T / √ ˆ ρ . Other scaling invariants include ˆ J/ ˆ ρ , q , and∆ ˆ f / ˆ ρ / , where ∆ ˆ f = κ ∆ f (2 π ) L . (18)Here ∆ f is the difference in the free energy density, ǫ − T s ,between a symmetry breaking solution and the RNAdSsolution with the same T and ρ . ∆ f < w everywhere positive. Solu-tions exist in which w has nodes, but they are probablyalways thermodynamically disfavored because spatial os-cillations in w increase energy density. A shooting strat-egy was employed to implement the constraint of nor-malizability on w . Solutions with non-zero, normalizable w were found only for T less than a critical temperature T c . This T c is, within numerical error, the temperatureat which the RNAdS solution admits a static linearizedperturbation, with w non-zero but infinitesimally small.The shooting algorithm was designed to work well near T c . In practice, it worked well for T > T c / T c goes to zero at g = g c ≈ . /L and does not ap-pear to have singular behavior there; however, the nu-merical problem becomes more difficult at small T c , sothe value of g c should be regarded as approximate. For g > ∼ g c , scaled thermodynamic quantities such as ˆ J/ ˆ ρ , q ,and ∆ ˆ f / ˆ ρ / exhibit nearly universal behavior as func-tions of T /T c from T = T c down at least to T c . Thisuniversality is related to a large g limit where the back-reaction of the gauge field on the metric can be neglected.For 0 . T c < T < T c , I found good fits to the followingscaling forms: q = q t ˆ J ˆ ρ = j / √ t , (19)where t = 1 − T /T c , and q and j / are positive anddepend on gL . More approximate fits can be found overthe same temperature range to the scaling form∆ ˆ f ˆ ρ / = − f t , (20)where f is positive and depends on gL . A few solutionsclose to T c appeared to have ∆ f very slightly positive.But closer examination of some of these solutions usinghighly accurate numerics showed that in fact they have∆ f < numerics unreliableunbroken phase T = T c H g L L T = T c H g L L gL T (cid:144) Ρ` H A L phase diagram gL T (cid:144) T c H B L J ` (cid:144) Ρ` gL T (cid:144) T c H C L q gL T (cid:144) T c H D L D f ` (cid:144) Ρ` (cid:144) - - FIG. 1: (A) The phase diagram. Each point corresponds to a numerically constructed black hole solution. (B) ˆ J/ ˆ ρ as a functionof T /T c . (C) q as a function of T /T c . (D) ∆ ˆ f / ˆ ρ / as a function of T /T c . with simple critical exponents, from Reissner-Nordstrom-anti-de Sitter solutions to solutions with normalizablenon-abelian condensates. The critical temperature scalesas the square root of charge density and also dependson the gauge coupling. The solutions with condensatesbreak an abelian gauge symmetry, suggesting some linkor analogy to superconductivity. An analogy has beenalready suggested between charged anti-de Sitter blackholes without symmetry breaking condensates and thepseudogap state of high T c materials [11, 12, 13, 14]. Itwould be desirable to test the validity of this analogymore closely, and to extend it if possible. Given the wayspatial rotations enter into the description of the con-densate (17), it might be possible in some variant of theconstruction I have exhibited to find an analog of p -waveor d -wave superconductivity. Any such analog with high T c superconductivity should be viewed with caution if not skepticism, because the underlying degrees of free-dom of duals to AdS vacua are typically large N gaugetheories, seemingly distant from semi-realistic construc-tions like the Hubbard model. And yet, one might hopethat dynamics related to the global symmetries (usually U (2) or SO (4), and sometimes larger groups) of the Hub-bard model and its relatives can be mimicked, to someextent, by non-abelian gauge fields on the gravity side ofthe gauge-string duality. Acknowledgments
I thank C. Herzog for discussions. This work was sup-ported in part by the Department of Energy under GrantNo. DE-FG02-91ER40671 and by the NSF under awardnumber PHY-0652782.
APPENDIX A: STATIC PERTURBATIONS
The purpose of the appendices is to consider two limit-ing cases of the equations (8): the first, in this appendix,is the case where w is treated as a small perturbation toan RNAdS solution; and the second, in appendix B, is thecase where the entire non-abelian gauge field is treatedas a small perturbation to the AdS -Schwarzschild geom-etry.The RNAdS background is ds = − f dt + r L d~x + dr ff = − ǫL κ r + ρ κ L r + r L Φ = ρLκ (cid:18) r H − r (cid:19) , (A1)with w = 0. By definition, r H is the most positive zeroof f . If w is treated as a static linear perturbation, thenthe only equation of motion we must solve is (8d) withthe w term neglected: w ′′ + f ′ f w ′ + g Φ f w = 0 . (A2)Although (A2) does not appear to be solvable in termsof known functions, it is easy to make a series expansionnear the horizon: w = w + w ( r − r H ) + w ( r − r H ) + . . . . (A3)Once w is specified, all other coefficients in (A3) can becomputed. Because (A3) is linear, it suffices to considerthe case w = 1, even though our real interest is in so-lutions with very small w . In addition to (A3), there isalso a solution that has a logarithmic singularity at thehorizon, but I discard it based on the usual argumentthat the horizon is a non-singular location. Far from thehorizon, the typical behavior of w is for it to be asymptot-ically constant, but by adjusting r H /L one can arrangefor w to have 1 /r falloff, which is normalizable in thesense described around (7). The discretely many allowedvalues of r H /L depend on ˆ ǫ , ˆ ρ , g , and L , but because ofscaling symmetries (essentially, the ones in table I), di-mensionless quantities such as T / √ ˆ ρ depend only on gL .Thus one may convert the allowed values of r H /L intoallowed values of T / √ ˆ ρ at some specified gL . The largestallowed value of T / √ ˆ ρ corresponds to a solution where w has no zeroes, and in this case the temperature is pre-cisely the T c discussed in the paragraph following (18).The next-to-largest allowed value of T / √ ˆ ρ correspondsto a solution with one node; the next-to-next-to-largestto two nodes; and so forth. See figure 2.The last term in (A3) shows that the condensate devel-ops for essentially the same reason as in the scalar case[2]: w acquires a negative effective mass squared nearthe horizon. When the w term in (8d) becomes signifi- cant, it decreases or limits the tendency of w to condense.An analogous situation in the lagrangian studied in [2]would be to have a positive | ψ | term in the scalar poten-tial, similarly limiting the tendency of the charged scalar ψ to condense. APPENDIX B: STRONG COUPLING LIMIT
The large gL limit is simple because the gauge fielddoesn’t back-react significantly on the metric [18]. Tosee this, note that upon scaling A → A/g and F → F/g ,the action for the gauge fields is just g ( F aµν ) . Thestress tensor acquires a similar factor of 1 /g , and it isthis factor that suppresses the back-reaction. Approxi-mate solutions to the full equations of motion (8) andthe constraint (9) can therefore be constructed by start-ing with the AdS -Schwarzschild line element, ds = r L (cid:2) − (1 − r H /r ) dt + d~x (cid:3) + L r dr − r H /r , (B1)and then solving the Yang-Mills equations in that back-ground. Let us set r H = 1. Then (B1) has the form (6a)with a = log rL h = 1 − r . (B2)Let us also define˜Φ = gL Φ ˜ w = gL w . (B3)Then the Yang-Mills equations (8c) and (8d) become˜Φ ′′ + 2 r ˜Φ ′ − r ( r −
1) ˜ w ˜Φ = 0˜ w ′′ + 1 + 2 r r ( r −
1) ˜ w + r ( r − ˜Φ ˜ w − r ( r −
1) ˜ w = 0 . (B4)A charming feature of (B4) is that there are no free pa-rameters in the equations. Factors of g are absent be-cause the definitions (B3) include the aforementionedscaling A → A/g . It is perhaps surprising that one alsoneeds explicit factors of L in (B3) to avoid them in (B4).This is because of the choice of radial variable in (B1): ifI had used z = L /r , such factors would not be needed,because then L enters into the line element only as anoverall factor, and overall factors don’t effect conformallyinvariant equations like the classical Yang-Mills equationsof motion.The equations (B4) cannot be solved analytically, butthey are straightforward to solve numerically, startingwith expansions˜Φ = ˜Φ ( r −
1) + ˜Φ ( r − + . . . ˜ w = ˜ w + ˜ w ( r −
1) + . . . (B5) T = T c H g L L o n e n o d e t w o n o d e s n o b a c k - r e a c ti o n gL T (cid:144) Ρ` Static perturbations at gL = r (cid:144) r H - - w FIG. 2: Left: Temperatures where static linear perturbations of the RNAdS solution arise. The curve labeled “no back-reaction”is given analytically by (B8). It comes from neglecting the back-reaction of Φ as well as of w on the metric. As discussed insection B, neglecting back-reaction is a good approximation when gL is large. Right: Static normalizable perturbations, scaledso that w = 1 at the horizon. The solution that is everywhere positive corresponds to the dot at gL = 6 on the T = T c curvein the left-hand plot, and the other solutions correspond to the dots on the other curves. èèèèèèèèèèèèè è è è è è è è è è è è è è è ŸŸŸŸŸŸŸŸŸŸŸŸŸŸŸŸŸ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ éééééééééééé é é é é é é é é é é é éãããããããããã ã ã ã ã ã ã ã ã ã ã ã ãøøøøøøøøøø ø ø ø ø ø ø ø ø ø t J ` (cid:144) Ρ` èèèèèèèèèèèè è è è è è è è è è è è è è è è ŸŸŸŸŸŸŸŸŸŸŸŸŸŸŸ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ ééééééééééé é é é é é é é é é é é é éãããããããããã ã ã ã ã ã ã ã ã ã ã ã ãøøøøøøøøøø ø ø ø ø ø ø ø ø ø t q gL =¥ ø gL = á gL = ç gL = à gL = æ gL = FIG. 3: Plots of ˆ J/ ˆ ρ and q as a function of t = 1 − T /T c for several values of gL . The gL = ∞ curve is obtained from solutionsto (B4), where back-reaction of the gauge fields on the metric is neglected. near the horizon and˜Φ = ˜ p + ˜ p r + . . . ˜ w = ˜ W r + . . . (B6)far from it. The only free parameters are ˜Φ and ˜ w : allhigher coefficients of (B5) can be determined in terms ofthem. The normalizable form of ˜ w shown in (B6) is notthe typical behavior: instead, ˜ w usually asymptotes toa non-zero constant at large r . Requiring normalizable w amounts to a condition on ˜Φ and ˜ w , and one windsup with a one-parameter family of allowed solutions to(B4). Again I restrict attention to solutions with ˜ w > w has nodes are never thermodynamically preferred. Starting from (B3) and (14), it is easy to show that T √ ˆ ρ = 3 s − πgL p q = 1 + ˜Φ ˜ p ˆ J ˆ ρ = − ˜ W ˜ p . (B7)I found that ˜ p is never greater than about 3 .
65, andas this value is approached, ˜ w →
0. Using (B7), thistranslates into a critical temperature T c ≈ . p gL ˆ ρ . (B8)(See figure 2.) I constructed solutions to (B4) corre-sponding to a range of temperatures T c < T < T c .A summary of their thermodynamic behaviors is shownin figure 3.At lower temperatures, the numerical problem be-comes more difficult, partly because a larger range of r must be integrated over. Also, for smaller values of T ,extra solutions exist where ˜ w has nodes. This is problem-atic because shooting algorithms generally rely on numer-ically finding zeroes of a merit function, and when there are multiple zeroes, one cannot easily control which ofthem one will find. [1] S. S. Gubser, Class. Quant. Grav. , 5121 (2005), hep-th/0505189.[2] S. S. Gubser (2008), arXiv:0801.2977 [hep-th].[3] R. Bartnik and J. McKinnon, Phys. Rev. Lett. , 141(1988).[4] P. Bizon, Phys. Rev. Lett. , 2844 (1990).[5] E. Winstanley (2008), arXiv:0801.0527 [gr-qc].[6] M. S. Volkov and D. V. Gal’tsov, Phys. Rept. , 1(1999), hep-th/9810070.[7] S. A. Hartnoll and P. Kovtun, Phys. Rev. D76 , 066001(2007), arXiv:0704.1160 [hep-th].[8] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998),hep-th/9711200.[9] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys.Lett. B428 , 105 (1998), hep-th/9802109.[10] E. Witten, Adv. Theor. Math. Phys. , 253 (1998), hep-th/9802150.[11] C. P. Herzog, P. Kovtun, S. Sachdev, and D. T. Son,Phys. Rev. D75 , 085020 (2007), hep-th/0701036.[12] S. A. Hartnoll, P. K. Kovtun, M. Muller, and S. Sachdev,Phys. Rev.
B76 , 144502 (2007), arXiv:0706.3215 [cond- mat.str-el].[13] S. A. Hartnoll and C. P. Herzog, Phys. Rev.
D76 , 106012(2007), arXiv:0706.3228 [hep-th].[14] S. A. Hartnoll and C. P. Herzog (2008), arXiv:0801.1693[hep-th].[15] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz (2008),arXiv:0803.3295 [hep-th].[16] The norm (7) can be made positive definite by passingto a gauge where A a = 0, but then one has the problemthat the norm for a static electric field grows linearly withtime. Such issues do not affect the question of whether w makes a finite contribution to || A || .[17] Infrared fluctuations can also be regulated using finitevolume constructions, such as toroidal compactificationin the x - x directions, or a generalization to sphericallysymmetric black holes in global AdS4