Microscopic entropy of higher-dimensional nonminimally dressed Lifshitz black holes

In arbitrary dimension, we consider a theory described by the most general quadratic curvature corrections of Einstein gravity together with a self-interacting nonminimally coupled scalar field. This theory is shown to admit five different families of Lifshitz black holes dressed with a nontrivial scalar field. The entropy of these configurations is microscopically computed by means of a higher-dimensional anisotropic Cardy-like formula where the role of the ground state is played by the soliton obtained through a double analytic continuation. This involves to calculate the correct expressions for the masses of the higher-dimensional Lifshitz black hole as well as their corresponding soliton. The robustness of this Cardy-like formula is checked by showing that the microscopic entropy is in perfect agreement with the gravitational Wald entropy. Consequently, the calculated global charges are compatible with the first law of thermodynamics. We also verify that all the configurations satisfy an anisotropic higher-dimensional version of the Smarr formula.


I. INTRODUCTION
Gauge/gravity duality can be extended to nonrelativistic systems by using anisotropic spacetimes. In this context, the archetypal example is the Lifshitz spacetime [1] whose main feature is the isometry allowing time and space scale with different exponents. Here, z is the dynamical critical exponent responsible of the anisotropic scaling characterizing nonrelativistic systems. As it was preliminarily emphasized in [1], standard vacuum Einstein gravity cannot allows Lifshitz spacetimes, except in the isotropic case z = 1 where they turn out to be antide Sitter (AdS) spaces. Nevertheless, this problem can be circumvented by considering instead higher-order corrections to gravity theories or by introducing specific matter sources. It has then becomes important to find specific gravity models that can accommodate the Lifshitz spacetimes together with their black hole extensions recovering the anisotropic scaling asymptotically. These so-called Lifshitz black holes are supposed to holographically capture the finite-temperature behavior of their strongly correlated nonrelativistic dual systems. New Massive Gravity [2] was one of the first gravity models that was shown to admit an analytic Lifshitz black hole as part of its vacua [3]. A property that later resulted to be generic for higher-order pure gravity theories in higher dimensions [4]. In presence of specific matter sources, Lifshitz solutions have also been investigated, see e.g. [5][6][7][8]. Also charged Lifshitz solutions can be engineering through a Maxwell-Proca model [9] or its nonlinear generalization [10] or in the presence of dilaton scalar fields [11].
The relevance of Lifshitz black holes lies in the hope that strongly coupled condensed matter systems can be better understood at finite temperature from an holographic point of view. But because of their unconventional asymptotic behavior, these black holes present interesting features which deserve more profound investigations. For example, their thermodynamic properties are usually quite different from those of the isotropic AdS black holes and in particular if the solutions are charged, see e.g. [12,13]. On the other hand, threedimensional configurations are usually excellent laboratories to investigate important conceptual questions about the gauge/gravity duality. For example, it has been shown that the semiclassical entropy of three-dimensional black holes with Lifshitz asymptotics can be recovered through a Cardy-like formula where the mass of their corresponding Lifshitz solitons explicitly appears, giving a prominent role to these regular configurations [14]. The solitons are obtained from the black holes by means of a double Wick rotation that involves inverting the dynamical critical exponent, as result they enjoy the same sort of uniqueness than the black holes [15]. The robustness of this formula has been successfully tested in a system exhibiting a wide spectrum of Lifshitz configurations as is the case of self-interacting scalar fields nonminimally coupled to New Massive Gravity [16]. Recently, this Cardylike formula has been extended to higher-dimensional anisotropic black holes [17]. In the present work, we pretend to test the validity of this higher-dimensional Cardy-like formula by considering again self-interacting scalar fields, but nonminimally coupled now to the most general quadratic curvature corrections of Einstein gravity in higher dimensions. We hope this study will contribute to highlight the importance of the role played by the soliton in the description of the thermal properties of black holes, and particulary for those with unconventional asymptotic behaviors.
The paper is organized as follows. In the next section, we will present the theory, the field equations as well as a specific ansatz which permits to obtain particular Lifshitz black hole solutions. Using this ansatz unspecifically, the thermodynamics quantities of interest as the entropy, the temperature and the mass of the black holes will be generically computed together with the mass of the soliton. In section IV, we will explicitly present four concrete classes of Lifshitz black holes fitting our ansatz. For these solutions, we will check that their gravitational entropy, calculated with the standard Wald formula, can be correctly reproduced by means of the Cardy-like formula. In the last section, we show the existence of a fifth class of Lifshitz black hole that is slightly different of our working ansatz. For this specific solution, we also test the robustness of the Cardy-like formula. In all cases we verify the fulfillment of the first law of black hole thermodynamics, which point the correctness of the quasilocal off-shell extension of the ADT formalism that we use to compute global charges. Finally, we provide an Appendix for reporting some of the involved expressions for the (coupling) constants of the specifics solutions.

II. SET-UP OF THE PROBLEM
In arbitrary dimension D, we consider a gravity action given by the most general quadratic-curvature corrections of the Einstein-Hilbert action sourced by a selfinteracting nonminimally coupled scalar field with The cosmological constant λ, the coupling constants β i , the self-interacting potential U (Φ) and eventually the nonminimal coupling parameter ξ will depend explicitly on the concrete solutions presented in the sections that follow. The field equations obtained by varying the action with respect to the metric and the scalar field read where we have defined and the energy-momentum tensor is given by In order to look for Lifshitz black holes, we will opt for the following ansatz where the structural metric function satisfies that lim r→∞ f (r) = 1, condition ensuring the metric to reproduce the Lifshitz asymptotic (1). As shown below, four of the five classes of Lifshitz solutions that will be presented can be generically parameterized as where χ is a nonnegative decay exponent modulating the Lifshitz asymptotics, r h denotes the location of the horizon and Φ 0 characterizes the positive strength of the field. The positivity of the scalar field can be explained from the fact that for these four classes of solutions, the discrete transformation Φ → −Φ will be a symmetry of the problem. The ansatz (4)(5) is also motivated by the fact that for Φ 0 = 0 most of the vacuum Lifshitz black hole solutions known for the theory have precisely this form [4], which also occurs for their charged extensions [12]. The remaining solution belongs to a different class where the structural metric function involves two different radial powers and will be presented in Sec. V. One the main aims of this work is to confirm the importance of the role played by the gravitational soliton for the thermal properties of the Lifshitz black holes. In order to achieve this task correctly, we will need the Lifshitz soliton counterparts of the black holes (4)(5). The solitons will be generically described by the following metric The solitons are obtained from the black holes by means of a double Wick rotationt = −ix 1 andx 1 = −it supplemented by a re-definition of the horizon location which ensures the correct identification.
The configurations that will be described below are fully determined in term of three parameters, namely the decay exponent χ, the strength Φ 0 and the dynamical exponent z. In order to simplify the discussion, we start by evaluating first the formulas of interest as the entropy, temperature and mass of the black hole configurations as well as the mass of the solitons for generic values of these constants, i.e. not for those that actually satisfy all the system constraints. The precise thermodynamic quantities are given later for each genuine solution with the help of these formulae.

III. PRELIMINARY THERMODYNAMIC QUANTITIES
First of all, the Wald entropy formula [18] for black holes that eventually extreme action (2) by fitting ansatz (4-5) generically reads where P abcd ≡ ∂(L g + L s )/∂R abcd and Ω D−2 represents the finite volume of the (D − 2)-dimensional planar base manifold. On the other hand, their temperature is given by In order to compute the masses of the black hole and soliton configurations defined in Eqs. (4-7), we will opt for the quasilocal formalism as defined in [19,20]. Notice that this formalism has proved to be well suited for correctly computing the masses of black holes of higherorder gravity theories with rather unconventional asymptotic, see e.g. [16]. The quasilocal formalism is based on an off-shell prescription [19] for the ADT potential [21] which allows the following concise expression for the conserved charge associated to a Killing vector field k where s is a parameter interpolating between the solution of interest at s = 1 and the asymptotic one at s = 0, the difference between their off-shell Noether potentials is denoted by ∆N µν (k) ≡ N µν s=1 (k) − N µν s=0 (k) and Θ ν is the surface term arising after varying the action. In the present case, these tensors are given by For a timelike Killing vector field, ∂ t = k µ ∂ µ , the evaluation of the mass formula for action (2) in the black hole ansatz (4)(5) gives rise to the expression where Ψ 1 and Ψ 2 are two dimensionless linear combinations of the squared corrections coupling constants reported in App. A. For the soliton ansatz (4-7) with timelike Killing vector field ∂t = k µ ∂ µ the mass formula reads where the dimensionless coupling constants combinations Ξ 1 and Ξ 2 are also defined in App. A. For actual solutions the mass expressions (11) and (12) cannot depend on the radial coordinates r andr, respectively. Interestingly, this imposes constraints on the constants z, χ and Φ 0 giving indications on the possible solutions within the ansatz; concretely, only two families of exponents χ are possible since they are the only giving rise to a nontrivial global charge mass.
In what follows, we will report four different classes of Lifshitz black hole solutions fitting our ansatz (4)(5). For each solution, we will check that its gravitational Wald entropy (8) is correctly reproduced by means of a higherdimensional anisotropic Cardy-like formula [17] given by This expression is the higher-dimensional extension of the one obtained for two-dimensional Lifshitz field theory [14]. Here we have used the notation S C for the microscopic entropy in order to reflect that the anisotropic Cardy-like expression is a priori different from the gravitational Wald entropy (8). Nevertheless, as shown below, both entropies will coincide for the different classes of solutions reported. For completeness, we will also verify that the first law of black hole thermodynamics consistently holds for each Lifshitz black hole solution.

IV. FOUR CLASSES OF LIFSHITZ BLACK HOLES
Here, we will present the four classes of solutions that fit within ansatz (4)(5) and compute their definitive thermodynamic quantities through the preliminary formulas derived in the previous section. For each solution, we will corroborate that their Wald entropy can be reproduced from the anisotropic Cardy-like formula (13).
A. Class with arbitrary dynamical exponent and arbitrary nonminimal coupling parameter The first family of solutions is obtained for the standard potential where a mass term is supplemented by a quartic interaction and exists for arbitrary values of the dynamical exponent z and of the nonminimal coupling parameter ξ. Because of cumbersome formulae, the concrete form of the potential, and the parameterizations obeying the different coupling constants as well as the cosmological constant are reported in App. B. Its line element and the nontrivial scalar field are given by where the polynomials P n and the remaining details of the solution are defined in App. B. This solution is obtained from the proposed ansatz (5) by using one of the only two decay exponents allowing a well-defined Lifshitz mass, namely χ = (z + D − 2)/2. The result is the higher-dimensional lifting from D = 3 of the black hole family with Lifshitz decay (z + 1)/2 originally derived in Ref. [8] for New Massive Gravity, whose thermodynamics was studied in Ref. [16]. It is interesting to notice that the above higher-dimensional line element has been previously obtained also as a vacuum solution in [4], but for a more restrictive election of the coupling constant. The vacuum limit of [4] is easily recovered by fixing the coupling constant β 3 in solution (15) in order to obtain a vanishing scalar strength. In this sense the present solution is a generalization of the one of [4] allowing the same black hole to be dressed by a self-interacting nonminimally coupled scalar field.
The thermodynamic properties of the lifted configuration follow from the following expressions, first, the preliminary formula (8) gives the Wald entropy where the dimensionless coefficient Υ 1 , depending on the free coupling constants, is mutual to all the extensive thermodynamic quantities associated with the solution and is defined in App. D. It also measures the depar-ture of the theory from the behavior of standard gravity, whose areal interpretation of black holes entropy forces Υ 1 = 1. The Hawking temperature (9) in this case reads The formulae (11) and (12) lead to the Lifshitz black hole and soliton masses, respectively Now, it is straightforward to verify that the Cardy-like formula (13) for the entropy correctly reproduces the gravitational Wald entropy, that is S W = S C . Another interesting feature of these thermodynamic quantities is that they obey the following anisotropic higher-dimensional version of the Smarr formula [24] which in fact is not unexpected since they consequently respect the first law (14). For the other decay exponent compatible with a welldefined mass, χ = z + D − 2, it happens that one obtains a vanishing mass, as will be exhibited in Subsec. IV C. However, there is an exception for the critical exponent z = D, this is the solution we present below.

B. Solution with a fixed value of the dynamical exponent z = D
The second family of Lifshitz black hole solutions exists for a dynamical exponent z = D and for a nonminimal coupling parameter ξ < (D − 1)/(5D − 2), In this case, the self-interacting potential is also given by a mass term plus a Φ 4 −interaction and the coupling constants β 1 and β 3 are arbitrary This solution consistently fits the working ansatz (5) within the other admisible family of decay exponents χ = z +D−2 = 2(D−1) when the critical exponent takes the value z = D. Its Hawking temperature becomes while the entropy together with the masses of the black hole and its soliton counterpart are given by where the mutual extensive coefficient is again defined in App. D. As before, one can check the validity of the first law and the Cardy-like formula (13), as well as of the Smarr formula (20).

C. Lifshitz black holes with vanishing mass
There exist two other families fitting our ansatz (5) which are defined for an arbitrary value of the dynamical exponent. These solutions present the peculiarity of having a vanishing mass and a zero Wald entropy, and hence the first law of thermodynamics is trivially satisfied. These zero mass solutions are also an interesting ground to test the validity of the Cardy-like formula since their solitons counterpart must also have a vanishing mass which in turn trivially implies that S W = 0 = S C . Here, we just report for completeness these Lifshitz black hole configurations of vanishing mass.
The first solution is obtained by choosing the other decay exponent compatible with a well-defined Lifshitz mass χ = z + D − 2, this gives where the second grade polynomial in the critical exponent at the denominator of the scalar strength is defined as This solution is the higher-dimensional lifting of the family with Lifshitz decay z + 1 obtained in Ref. [8]. The self-interaction potential supporting the solution and the values of the coupling constants are extended as while the cosmological constant λ takes the same expression given in App. B. It is straightforward to check by means of formulas (11)(12) that the masses of the black hole and its soliton counterpart are zero. A property also shared by the Wald entropy (8).
The other zero-mass solution is obtained for the decay exponent χ = 2(z − 1) which does not gives in general a global charge. However, this exponent works just because it exactly cancels the coefficients in front of the decaying powers preventing the mass formula to become a conserved charge, causing at the same time its vanishing. The resulting solution is where the polynomial in the critical exponent is given bỹ and the specific parameterizations of the coupling constants together with the cosmological one are presented in App. C. It is interesting to emphasize that this line element corresponds to other of the vacuum solutions previously obtained in [4] for the same theory, but with a more restrictive choice of the coupling constants. Notice that if we fix the coupling constant β 3 by demanding the vanishing of the scalar strength we recover the black hole of [4] without scalar field. In other words, this solution generalize the other vacuum example of higher-dimensional Lifshitz black hole by dressing it with a self-interacting nonminimally coupled scalar field.

V. LAST CLASS OF LIFSHITZ BLACK HOLE
There exists a fifth class of Lifshitz black holes that does not fit within our ansatz (5). This solution has a fixed value of the dynamical exponent z = D and is valid also for a precise value of the nonminimal coupling parameter .

The configuration in question reads
where α is a coupling constant appearing in the potential and P 2 (D) = 6D 2 − 16D − 1. Indeed, this solution exists provided that the potential and the coupling constants are given by The event horizon of this black hole is located at the radius and in terms of this radius the quantities of interest to corroborate the first law (14), the validity of the Cardylike (13) and Smarr (20) formulas are given by where the extensive coefficient is expressed as all the previous ones in App. D.

VI. CONCLUSION
Here, we have extended the work done in three dimensions in the case of a scalar field nonminimally coupled to New Massive Gravity [16]. Indeed, we have considered a gravity theory given by the most general quadratic corrections to Einstein gravity supplemented by a source action describing a self-interacting nonminimally coupled scalar field. For this theory, we have presented five different classes of Lifshitz black hole solutions. Each solution is specified with a particular self-interacting potential and for a certain parametrization of the coupling constants β i accompanying the different gravity invariants. Interestingly, some of the obtained solutions describe Lifshitz black hole that were known previously as part of the vacuum of the studied theories [4], but for more restrictive elections of the coupling constants. Hence, they constitute generalizations of these vacuum higher-dimensional Lifshitz black holes that turn to be dressed by self-interacting nonminimally coupled scalar fields. It must be emphasized that, in contrast with the three-dimensional case, none of the different choices of the coupling constants β i corresponds to the recently discussed critical gravity points [22,23]. We would like to stress that our work constitute a new example putting in light the importance played by the gravitational solitons in order to describe the thermal properties of black holes with (un)usual asymptotics. In this spirit, it will be desirable to keep exploring this issue from the holographic point of view. In particular, a promising work to be done will consist in identifying or interpreting the role of the soliton in the field theory side. Another interesting aspect that has to do with these solutions concerns the Smarr formula [24]. Indeed, since all the solutions reported here verify the higher-dimensional anisotropic Cardy-like formula (13) as well as the first law of thermodynamics, they will also satisfy an anisotropic higherdimensional version of the Smarr formula (20). This last formula is in perfect accordance with the one obtained in Ref. [25] for different theories admitting Lifshitz black holes. It is evident that the emergence of the solutions presented here is essentially due to the higher-order nature of the gravity theory together with the nonminimal coupling of the scalar field to these gravity through the term RΦ 2 . One eventually can pursue the exploration on this issue by studying other Lifshitz black hole solutions that may arise from other nonminimal couplings as those recently put in spotlight through the Horndeski Lagrangian [26]. The dimensionless combinations of coupling constants appearing in the black hole mass formula (11) are defined by The corresponding dimensionless combinations ap-pearing in the soliton mass formula (12) are Appendix B: Parameters associated to the first class of solutions (15) The potential associated to the first class of solutions of Subsec. IV A reads where for simplicity we have defined where the polinomial P 5 (z; ξ) was previously defined in App. B. The dimensionless extensive coefficient related to the second class of solutions of Subsec. IV B is written as The two classes of solution with vanishing mass of Subsec. IV C consequently have vanishing extensive coefficients. Finally, the dimensionless extensive coefficient of the last class in Sec. V is .