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Letters B 776 (2018) 236–241

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Physics Letters B

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Hairy AdS holes with a toroidal horizon in 4D Einstein-nonlinear σ -model system ∗ Marco Astorino a, Fabrizio Canfora b, Alex Giacomini c, , Marcello Ortaggio d,c a Universidad Adolfo Ibanez, Viña del Mar, Chile b Centro de Estudios Científicos (CECS), Casilla 1469, Valdivia, Chile c Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Edificio Emilio Pugin, cuarto piso, Campus Isla Teja, Valdivia, Chile d Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic a r t i c l e i n f o a b s t r a c t

Article history: An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear Received 19 October 2017 sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear SU(2) field is Received in revised form 16 November 2017 regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy– Accepted 21 November 2017 momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a Available online 24 November 2017 discrete parameter which has neither topological nor Noether charge associated with it and therefore Editor: M. Cveticˇ represents a hair. A U (1) gauge field interacting with Einstein can also be included. The is analyzed. Interestingly, the hairy black hole is always thermodynamically favoured with respect to the corresponding black hole with vanishing Pionic field. © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction modynamic properties of solitons (see [22]). These are necessary ingredients to build Pionic black holes with non-trivial thermody- The nonlinear sigma model is a useful theoretical tool with namics. applications ranging from quantum field theory to statistical me- Following the strategy devised in those references, we will con- chanics systems like quantum magnetism, the quantum Hall effect, struct a class of analytic black hole solutions of the Einstein-- 3 meson interactions, super fluid He, and theory [1]. The nonlinear sigma model system with intriguing geometrical prop- most relevant application of the SU(2) non-linear sigma model erties. This family of black holes possesses a flat horizon1 and a in physics is the description of the dynamics of Pions at discrete hairy parameter. The thermodynamics can be analysed ex- + low energy in 3 1 dimensions (see for instance [2]; for a detailed plicitly. The first law is satisfied and an interesting feature of these review [3]). Consequently, the analysis of the coupling of the non- black holes is that the hairy solution has always less free energy linear sigma model to is extremely important than the corresponding solution: thus, the present analysis both from the theoretical and from the phenomenological point suggests that the coupling of Pions with the gravitational field can of view. On the other hand, due to the complexity of the field act as a sort of catalysis for the Pions themselves. This family of equations (which usually reduce to a non-linear system of coupled black holes with flat horizons can also be generalized to the case in PDEs), the Einstein-nonlinear sigma model system has been ana- which there is a U (1) gauge field coupled to Einstein gravity. The lyzed mostly relying on numerical analyses (classic references are interesting thermodynamical features of these black holes remain [4–8]). However, the recent generalization of the boson ansatz to in this case as well. This is qualitatively similar (in a phenomeno- SU(2)-valued scalar fields (introduced in [9–19] and [20–22]) al- logical setting such as the Einstein–Pions system) to the recent lows to construct also non-trivial gravitating soliton solutions (see findings of [23] in which a family of asymptotically flat black holes in particular [10] and [16]) as well as to analyse explicitly ther- with non-Abelian hair has been presented which are thermody- namically favoured over the Reissner–Nordström solution.

* Corresponding author. E-mail addresses: [email protected] (M. Astorino), [email protected] (F. Canfora), [email protected] (A. Giacomini), [email protected] 1 Black holes with planar horizons have recently attracted a lot of attention due (M. Ortaggio). to their applications in , see, e.g., [24–26]. https://doi.org/10.1016/j.physletb.2017.11.051 0370-2693/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. M. Astorino et al. / Physics Letters B 776 (2018) 236–241 237

The paper is organized as follows. In section 2, the Einstein- Defining the quadratic combination Nonlinear σ -model is introduced and a convenient parametrization is described. The exact hairy black hole solutions are constructed S := i j =∇ 0∇ 0 +∇ i∇ i = ∇ i∇ j μν δijRμ Rν μY ν Y μY ν Y Gij(Y ) μY ν Y , in section 3. In section 4, the thermodynamic behaviour of the solutions is analyzed. In the last section 5, some conclusions are (11) drawn. with 2. The action Yi Y j Gij := δij + (12) 1 − Y k Y We consider the Einstein-non-linear sigma model system in k four dimensions, with a possible cosmological constant. This de- being the S3 metric of the target space,2 we obtain that the action scribes the low-energy dynamics of pions, whose degrees of free- (3) reads dom are encoded in an SU(2) group-valued scalar field U [3]. The √ action of the system is 4 1 i μ j SPions =−K d x −g Gij(∇μY )(∇ Y ) , (13) 2 S = SG + SPions, (1) while the energy–momentum tensor (6) takes the form where the gravitational action SG and the nonlinear sigma model action SPions are given by 1 Tμν = K Sμν − gμνS . (14) √ 2 1 4 SG = d x −g(R − 2), (2) 16π G The second of (9) means that Y I := (Y 0, Y i ) define a round unit √ K 4 μ 3-sphere in the internal space. A useful set of coordinates (H, A, G) SPions = d x −g Tr R Rμ . (3) 4 in the internal space is defined by Here we have defined Y 0 = cos H sin A, Y 1 = sin H cos G, = −1∇ ∈ Rμ U μU , U SU(2), (4) Y 3 = cos H cos A, Y 2 = sin H sin G, (15) while R is the Ricci scalar, G is Newton’s constant,  is the cos- where H ∈[0, π/2], while A ∈ [0, 2πk ], G ∈ [0, 2πk ] are both mological constant and the parameter K is positive. In our con- 1 2 periodic Killing coordinates of S3, with k and k positive integers ventions c = h¯ = 1, the space– signature is (−, +, +, +) and 1 2 (there is clearly some redundancy in this choice of the periodicities Greek indices run over space–time. In the case of the non-linear = = 3 sigma model on flat space–, the coupling constant K is fixed – the standard choice k1 k2 1already covers the whole S – experimentally. On the other hand, its true meaning in the context but this will become physically meaningful later on). In [16] and of AdS physics is to introduce a new scale. Indeed, it is natural to [22] it has been shown that similar parametrizations are extremely expect this new scale will be a relevant quantity in the thermody- useful both in curved and in flat spaces. The tensor Sμν (11) takes namics of the black holes which will be analyzed in the following the form sections. Our results confirm this expectation (we thank the anony- mous referee for this comment). 2 2 Sμν = (∇μ H)(∇ν H)+cos H(∇μ A)(∇ν A)+sin H(∇μG)(∇ν G). The resulting Einstein equations are (16) Gμν + gμν = 8π GTμν, (5) If one defines the following combinations of the Killing coordi- where Gμν is the and the energy–momentum ten- nates sor is = + = − K 1 α + G A, − G A, (17) Tμν =− Tr Rμ Rν − gμν R Rα . (6) 2 2 the field equations (7) can be written in the compact form3 For the nonlinear SU(2) sigma model, Tμν can be seen to satisfy 1 the dominant and strong energy conditions [27]. Finally, the H − sin(2H)∇ + ·∇ − = 0, (18) field equations are 2 sin(2H) + + 2∇ H · [∇ − + cos(2H)∇ +] = 0, (19) μ ∇ Rμ = 0. (7) sin(2H) − + 2∇ H · [∇ + + cos(2H)∇ −] = 0. (20) In the following, it will be useful to write Rμ as SU(2)-valued scalar fields may possess a non-trivial topolog- j ical charge which, mathematically, is a suitable homotopy class Rμ = iRμσ j, (8) or winding number W . Its explicit expression as an integral over where σ j are the Pauli matrices. Furthermore, we adopt the stan- a suitable three-dimensional hypersurface can be found, e.g., dard parametrization of the SU(2)-valued scalar U (xμ) in [1]. However, in the present paper we will consider only con- figurations with W = 0. ±1 μ 0 μ j μ 0 2 i U (x ) = Y (x )I ± iY (x )σ j , Y + Y Yi = 1 , (9) × where I is the 2 2identity matrix. From (8) one thus finds 2 For the special field configuration Y 0 = 0, eq. (12) should be replaced by := Gij δij. This case will not be considered in this paper. i = ijk ∇ + 0∇ i − i∇ 0 3 μν =∇ ·∇ +  Rμ ε Y j μYk Y μY Y μY . (10) It is useful to recall the simple identity g hf,ν ;μ h f h f . 238 M. Astorino et al. / Physics Letters B 776 (2018) 236–241 dr2 3. Toroidal black hole solutions ds2 =−f (r) dt2 + + r2 k2dx2 + k2dy2 , (29) f (r) 1 2 3.1. Uncharged black hole μ r2 f (r) =−b2 − + . (30) r 2 We now analyze configuration which are a natural non- topological generalization of the ansatz of [16,22]. Note that b2 is not an integration constant, but is fixed by the We will consider a static with a flat base manifold coupling constants of the theory. and the following diagonal metric Let us mention a possible alternative parametrization of this solution. Rescaling ds2 =−f (r) dt2 + h (r) dr2 + r2 dθ 2 + dϕ2 . (21) ˜ ˜ 3 ˜ k1 = k1, k2 = k2, μ = μ˜ / , r = r˜/, t = t, In this geometry, for the matter field (15) we choose the “adapted” configuration (31) π one obtains H = , A = θ, G = ϕ, (22) 4 ˜2 2 2 dr 2 ˜2 2 ˜2 2 ds =−f r˜ dt˜ + + r˜ k dx + k dy , (32) which is consistent with (15) if we set the periodicity of the angu- f r˜ 1 2 lar spacetime coordinates as μ˜ r˜2 f (r˜) =−2b2 − + . (33) ˜ 2 θ ∈[0, 2πk1], ϕ ∈[0, 2πk2], (23) r This allows one to set  = 0, if desired, thus recovering the known making the base manifold in (21) a (flat) 2-torus.4 It is easy to toroidal vacuum solution [28–32] – this limit is however only “for- see that, thanks to (21) and (22), the field equations (18)–(20) are mal”, since the Pionic field (22), (26) degenerates discontinuously satisfied identically. In addition, by construction we have W = 0 to a single point in the internal space. (since the field (22) is 2-dimensional), as we required. We further note that this Pionic configuration has also zero Noether charge From now on we stick to the parametrization (29), (30). The associated with the global isospin symmetry of the model. Indeed, base space is a flat 2-torus with Teichmüller parameter ik1/k2 the Noether charge is proportional to the spatial integral of the (cf., e.g., [32]). Metric (29) is similar to the well-known vacuum time-component of the Nother current associated with the global topological black holes, for which the base manifold can be spher- Isospin symmetry of the model. In the present case, due to the fact ical, hyperbolic or toroidal [28–32]. However, while in the absence that the SU(2)-valued field does not depend on time, its Noether of Pions the constant term in the lapse function is fixed by the charge vanishes identically. curvature of the base manifold (i.e., +1, −1, 0in the spherical, hy- One still needs to solve Einstein’s equations (5) with (14), perbolic and toroidal case, respectively), in (30) it takes a negative which here becomes value despite the fact that the base is flat. As a consequence, the base space area K t t r r Tμν = ( f δ δ − hδ δ ). (24) 2r2 μ ν μ ν σ = k1k2, (34) Eq. (5) is thus solved by (21) (up to a constant rescaling of t) with plays the role of an extra (discrete) parameter that cannot be 1 μ  rescaled away by a redefinition of μ (as opposed to toroidal black = f (r) =−4π GK − − r2, (25) h(r) r 3 holes in vacuum [28–32]). Recalling that the present Pionic con- figuration has neither topological nor Noether charge, σ can be where μ is an integration constant. considered as an integer Pionic hair of the black hole. Geometric features of the spacetime are described more trans- Because of the form of the lapse function (30), one can easily ∈[ ] parently if we introduce two “normalized” coordinates x, y 0, 1 adapt to the present context the discussion of the causal struc- such that ture and horizons of hyperbolic vacuum black holes given in [31, 32] (to which we refer for more details). First, since r = 0is a θ = 2 k x, = 2 k y , (26) π 1 ϕ π 2 curvature singularity, we restrict ourselves to the range r > 0. The = and additionally rescale discriminant of f (r) 0reads (up to an overall positive factor)  = 4 2b6 − 27μ2, while the three roots satisfy r r r = μ 2 > 0 r μ 1 2 3 t → 2πt, r → , μ → . (27) and r1 + r2 + r3 = 0. It follows that there cannot be three real roots 2 2 3 π ( π) with the same sign, and that there are no real positive roots (only) Defining the convenient parameters5 when  and μ are both negative. To be more precise, let us define the (positive) critical value of μ (such that  = 0for μ =±μc , 3 b2 = 16π 3GK , 2 =− , (28) cf. [31,32])  √ 2 3 3 the final form of the solution is given by the field (22), (26) in the μc = b . (35) spacetime 9 There exists a unique, positive simple root r+ for μ ≥ 0, two dis- tinct positive roots r+ and r− (with r+ > r−) for −μc < μ < 0, 4 Here the integers k1 and k2 are used to define the identifications of points in a double positive root for μ =−μc , and no positive roots for the physical spacetime and are therefore not redundant anymore (as opposed to μ < −μc , the latter case thus describing a . The (15)). Also note that the cylindrical [and planar] is included in the formal =− limit k →+∞ [and k →+∞]. critical value√ μ μc gives rise to a degenerate Killing horizon at 1 2 = ≡ 5 Here we are interested in solutions in AdS, but in (25)  can take any sign or r re b/ 3, however this spacetime does not describe a black vanish. hole [31,32]. In order to have a black hole solution, one thus needs M. Astorino et al. / Physics Letters B 776 (2018) 236–241 239 ⎛  ⎞ − to take μ > μc . Therefore r+ is a monotonically increasing func- 2 3b2 = ⎝ + 2 2 + ⎠ tion of μ and r+ > r . r+ 2π T 4π T , (42) e 3 2 3.2. therefore no bifurcations of the free energy similar to the Hawking– Page [33] transition are possible here. The generalization of the above model to the presence of a U (1) The Euclidean action (which we denote by I to distinguish it gauge field minimally coupled to General Relativity is direct. The main motivation for this extension consists in analyzing whether from its Lorentzian counterpart) is a sum of three contributions, or not the thermodynamic behaviour disclosed in the Einstein- namely nonlinear sigma model system (discussed in section 4) resists after = + + the inclusion of further reasonable matter fields. I Ibulk Isurf Ict, (43) More precisely, we add to the action (1) the electromagnetic with (recall (11)–(13)) term √ 1 √ 1 4 6 4 μν Ibulk =− d x g R + SMax =− d x −g Fμν F , (36) 2 16π G 16π G M where F = ∂ A − ∂ A , which adds to the energy–momentum √ μν μ ν ν μ K 4 μν i j + d x gGijg ∂μY ∂ν Y , (44) tensor (14) the usual Maxwell term 2 M 1 ρ − 1 ρσ √ Fμρ Fν gμν Fρσ F . (37) 1 3 4π G 4 Isurf =− d x hKB , (45) 8π G We will focus on static electric fields defined by the potential ∂M q where hab is the metric induced on the boundary ∂M at r = rB , Aμ = − , 0, 0, 0 . (38) M r and K B is the trace of the extrinsic curvature of ∂ as embedded in M. For r →+∞, both I and I are divergent when eval- This form of A has the property of being compatible with the B bulk surf μ uated on our solutions, nevertheless (43) can be made finite by an metric (29), i.e., the Maxwell equations appropriate definition of the counterterm action I . The latter con- √ ct μν sists of boundary integrals involving scalars constructed only from ∂μ( −gF ) = 0 boundary quantities. At this purpose, taking inspiration from [26], are automatically satisfied. Solving Einstein’s equations with the where similar matter field was studied, we naturally generalise the ansatz (21), (22) and (38) shows that the contribution of the elec- counterterm action to the more general collection of scalar fields, tric field consists just in an upgrading of the lapse function (30) we are considering, in the following way to 1 √ 4 2 2 I = d3x h R + μ r q ct B 2 f (r) =−b2 − + + . (39) 8π G 2 r 2 r2 ∂M √ Of course, the presence of an electric field modifies the structure K 3 ab i j − d x h Gijh ∂aY ∂b Y , (46) of the black hole horizons. In general in this charged case we have 2 M both an inner horizon r− and an outer horizon r+, see [31] for a ∂ related discussion. where RB is the Ricci scalar of the boundary metric. When the sigma model coupling of the scalar fields is trivialised to become 4. Thermodynamic behaviour of the asymptotically locally AdS a kinetic interaction, i.e. Gij = δij, the axion counterterm of [26] black hole solution is, indeed, recovered from (46). The AdS gravitational part of the above counterterms is well-known [35] (see also [36,37]). Taking 4.1. Uncharged black hole the limit rB →+∞ gives the renormalized action Let us begin by analyzing the electrically uncharged case. We βσ r+ 2 2 2 I =− r+ + b . (47) employ the Euclidean approach [34] and thus replace t → it in 16π G 2 (29). The is given by the inverse of the Euclidean time It is easy to see that I ∼ T 2 for a large T (similarly as for hyper- period, i.e., bolic vacuum black holes [32]). The energy E =−∂β log Z, the S = (1 − β∂β ) log Z and −1 1 1 μ 2r+ T ≡ β = f (r+) = + , (40) the free energy F =−T log Z of the solutions in the semiclassical 4π 4π r2 2 + approximation log Z ≈−I thus read = 2 2 − 2 or, expressing μ r+(r+/ b ) as a function of r+, by 2 2 σ r+ r+ σμ σ r+ E = − b2 = , S = , 1 −b2 3r+ 8π G 2 8π G 4G T = + . (41) 4π r+ 2 σ r+ F =− r2 + b2 2 . (48) 2 + This is a monotonically increasing function of r+ (and thus of μ), 16π G with T ∈[0, +∞) vanishing for the extremal solution with μ = The area law is thus clearly obeyed. By defining the as −μc . Inverting this relation admits only one positive root for r+, which expresses the horizon radius as a function of T M = E, (49) 240 M. Astorino et al. / Physics Letters B 776 (2018) 236–241 it is easy to check with (41) that the first law of black hole ther- 4.2. Charged black hole modynamics is satisfied, i.e., The extension to the electrically charged case is straightforward. 2 σ 2 3r+ Now the lapse function is given by (39). The reads T δS = −b + δr+ = δM . (50) 2 = qσ 8π G Q 4π and the mass is the same as in the uncharged case. Ex- pressing M = M(r+, Q ) as a function of the horizon radius and of The local thermodynamic stability with respect to thermal fluc- the electric charge, one readily finds that the first law of thermo- tuations is given by the positivity of the function C, dynamics i.e., −1 2 2 2 2 δM = T δS + δ Q (56) ∂ S ∂ S ∂ T r+σ (3r+ − b ) C := T = T = . (51) 2 ∂ T ∂r+ ∂r+ 2G(b2 2 + 3r+) is again fulfilled. The Coulomb electric potential at the horizon is defined, as usual, as b Since black hole solutions satisfy r+ > √ (section 3.1), they are 3 μ μ q stable under temperature fluctuation (this is also true in the ab- = χ Aμ − χ Aμ = , (57) sence of the Pionic matter, cf. [31,32]). In other words, M is a r∞ r+ r+ monotonically increasing function of T (see also (52) below). μ where χ represents the Killing vector ∂t , while the temperature We further note that (49) gives a partly negative mass spec- of the charged solution is trum (similarly as obtained in [36] for hyperbolic vacuum black holes – cf. [31,32] for a different approach). Note also that M and 2 4 1 μ 2r+ q 1 3r+ 2 2 2 S depend not only on the radial length r+ (i.e., on the integra- T = + − = − b r+ − q . 4π 2 2 3 3 2 tion constant μ), but also on the angular periods via σ . It can thus r+ r+ 4πr+ be seen that, for a given M, larger are attained for larger (58) (discrete) values of σ . In addition, for a given T (i.e., for a fixed r+) there exists a discrete infinity of black holes parametrized by σ , for In order to check the thermodynamic stability of this solution, we which the functions (48) take different values, namely compare the free energy of the electrically charged black hole with Pionic field with the electrically charged black hole without Pionic σ M = 2π T + 4π 2 T 2 2 + 3b2 field. The free energy in the grand canonical ensemble, i.e. when 108π G both the temperature T and the Coulomb potential are consid- × 4π 2 T 2 2 + 2π T 4π 2 T 2 2 + 3b2 − 3b2 , (52) ered fixed, for the charged pionic solution is given by the Gibbs ⎛  ⎞ free energy 4σ 3b2 3b2 S = ⎝2π 2 T 2 + π T 4π 2 T 2 + + ⎠ , (53) F = M − TS− Q . (59) 9G 2 4 2 When μ is expressed in terms of r+, the free energy becomes σ =− + 2 2 2 + 2 F 2π T 4π T 3b 4 2 2 2 2 108π G r+ + (q + b r+) F (r+, q) =− . (60) 2 σ × 3b2 + 2π 2 T 2 2 + π T 4π 2 T 2 2 + 3b2 . (54) 16πr+ In the electrically charged case, it is convenient to write the free At high , M ∼ T 3, S ∼ T 2 and F ∼− T 3. σ σ σ energy as a function of the intensive parameters such as the tem- It is interesting to compare this black hole solution dressed perature T and the Coulomb potential . This can be done invert- with a matter field with the corresponding vacuum solution (i.e., ing eqs. (57) and (58). Again, there is only one positive root of the with b = 0). The entropy of the toroidal vacuum black hole is radial position of the horizon as a function of the tempera- [31,32] ture r+(T ) l4 = σ 2 2 2 S0 4π T . (55) r+(T ) = π 2 T + 4π 2 T 2 + 3(b2 + 2). (61) 9G 3 3 Clearly S < S, i.e., at equal temperature the vacuum black hole 0 Finally the Gibbs free energy reads as follows has always lower entropy than the dressed solution. In order to see which black hole is thermodynamically favoured at a given −σ F (T , ) = 2π T + 4π 2 2 T 2 + 3b2 + 3 2 temperature, it is necessary to compare the respective free ener- 108π gies. From (54), it is obvious that the free energy is more negative 2 for b2 = 0, i.e., the solution with Pionic matter fields is thermody- × 3b + π T 2π T + 4π 2 2 T 2 + 3b2 + 3 2 . namically favoured at any temperature. (62) This result is remarkable, since usually black holes dressed with matter fields in asymptotically locally AdS are thermo- This expression for the free energy is the generalisation of the dynamically favoured over their matter-free counterparts only for uncharged one, given in eq. (54), which can be straightforwardly low temperatures (see, e.g., [38,39]). Recently, results qualitatively recovered in the vanishing electric potential limit. similar to ours have been found in [23]. However, this reference Again, the free energy of the dressed black hole turns out to obtained numerically solutions of a higher order version of Yang– be more negative at the same temperature and electric potential, Mills theory minimally coupled to General Relativity which are than the one of the black hole without Pionic matter field, as can asymptotically flat – here we have constructed analytic results for be seen from eq. (62) and in Fig. 1. Therefore the charged pionic Pions minimally coupled with AdS gravity. solution is always preferred. M. Astorino et al. / Physics Letters B 776 (2018) 236–241 241

Fig. 1. Gibbs free energy F (T , ) comparison between the electrovacuum (yellow) and the pionic (blue and green) black holes as function of the temperature T and electric potential for some fixed numerical values of the cosmological constant and the base manifold area = π, σ = 1. The picture does not change qualitatively for different values of the chosen pionic parameter b (chosen as 29 and 41 respectively in blue and the green surfaces): the hairy configuration is always preferred at a given temperature and electric potential since its free energy is lower. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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