Maxwell construction and multi-criticality in uncharged generalized quasi-topological black holes

we demonstrate the existence of N-tuple critical points of uncharged AdS black holes in generalized quasi-topological (GQT) theories. The criticality is shown to have a geometrical interpretation described by the Maxwell’s equal area rule. We present a compact reformulation of the area rule and identify a criterion for the emergence such points. Using this criterion, we construct several multi-critical points with genuine GQT densities, including a quadruple and a quintuple points.


Introduction
It is well known that black holes behave like thermodynamic systems that radiate a thermal flux of particles [1] and whose equilibrium states are governed by the four laws of black hole mechanics [2], reinterpreted as the four laws of thermodynamics.For asymptotically AdS black holes, the negative cosmological constant can be identified as a thermodynamic pressure that validates the extended first law of thermodynamics [3][4][5][6].In this context black hole mass is interpreted as enthapy, and thus the mechanics of black holes can be understood in terms of chemical thermodynamics [7,8].Over the past decade, the perspective of black hole chemistry has led to the discovery of a number of rich properties, including Van der Waals phase transitions [9], re-entrant phase transitions [10,11], superfluid-like phase transitions [12][13][14], and triple points [11,[15][16][17].
A recent interesting discovery in black hole chemistry was that of multi-critical points.These were first seen in Einstein gravity coupled to non-linear electrodynamics [18], but were shortly afterward found to be present in multiply rotating Kerr-AdS black holes [19], and in Lovelock gravity [20].In the latter case multi-critical behaviour can even occur for asymptotically flat black holes [21].An N -th order multi-critical point occurs when N distinct phases merge at a single value of pressure and temperature, generalizing the notion of a triple point (with N = 3).Explicit examples of quadruple and quintuple points have been found in all cases examined so far.
In this paper, we demonstrate an alternative method for finding multi-critical points in black hole phase transtions.Previous methods exploited the fact that each extremum of the temperature (regarded as a function of horizon radius) corresponded to a cusp in the Gibbs free energy [18,20].2N − 2 distinct extrema were thus required to obtain N distinct phases, each with its own swallowtail in the Gibbs free energy diagram.The intersection points of corresponding swallowtails will merge if two adjacent inflections of T (r + ) occur at the same temperature.Extending this to all the inflections will then yield an N -tuple critical point where all such intersections merge.However, the number of parameters used in this approach is more than needed -in fact only approximately half of the horizon radii of the extrema are needed.This redundancy results in extremely low efficiency in computing the (finely tuned) parameters need to obtain a multicritical point, as iteratively tuning the various parameters such that all inflections occur at the same temperature becomes very time-consuming.This problem is particularly acute when dealing with more complicated higher-curvature theories.Indeed the situation deteriorates when considering black holes with large numbers of thermodynamic degrees of freedom since considerably high precision is required because of the finely tuned nature of multi-criticality.
The new method overcomes these difficulties.Inspired by Maxwell's equal area rule [22], we provide an equivalent but more compact description of multi-criticality, which can be perfectly adapted to the construction without the redundancy introduced in previous methods.Instead of iteratively manipulating input parameters, our approach directly indicates whether a given set of non-redundant parameters can or cannot have multi-critical points.If the latter holds, we obtain their accurate values in thermodynamic phase space with notably less computation.
To illustrate our method we consider black holes in higher curvature theories of gravity.Specifically we consider Generalized quasi-topological (GQT) gravity.The reasons for this are as follows.
For the past 2 decades there has been a revival of interest in higher curvature gravity in the theoretical physics community.Such theories have proven to be significant in a variety of contexts in physics, including string theory, holography, the AdS/CFT correspondence, tests of general relativity, and black hole thermodynamics.The gravitational action becomes renormalizable when supplemented with higher-curvature terms [23], making such theories candidates for a quantum theory of gravity.String-theoretic versions of quantum gravity motivate the possibility of higher dimensional spacetimes, and the addition of higher-curvature corrections allows for broader generalizations of the Einstein-Hilbert action to dimensions larger than four.These higher-curvature theories provide toy models for studying the AdS/CFT correspondence and allow for holographic study of Conformal Field Theories (CFTs).
But the inclusion of higher order curvature terms comes at a price -it can yield equations of motion containing higher derivatives in the metric that give rise to instabilities and negative energy modes [24,25].Notably, this inconsistency with general relativity (GR) is absent in some classes of higher-curvature theories [26][27][28][29] in which only a massless spin-2 graviton can propagate to infinity .This subclass of higher-curvature gravity theories is considerably more promising than the others and thus warrants further investigation.
Generalized quasi-topological (GQT) gravities, a class of recently proposed higherderivative theories, satisfies the requirements noted above.Theories in the GQT class characterize generalizations of GR in any dimension and to any order in curvature insofar as they contain non-hairy black hole solutions and second-order-differential equations for the metric in any linearized maximally symmetric background [30][31][32].Generally, the bulk part of their action can be written as where Λ is the cosmological constant, S (k) n are independent densities constructed from different constructions of n Riemann tensors and the metric, α n,k is the associated kth higher-curvature coupling, and the Newtonian constant G is set to 1 for simplicity.Quite remarkably, in the context of gravitational effective field theory, any higher-curvature theory can be mapped into a subset of GQT theories via field redefinition [30].Furthermore, there exists a subset in the parameter space of higher-curvature couplings where these theories only allow a massless spin-2 graviton propagating at the linearized level.
In general, the field equations of this class contain metric derivatives up to fourth order.However, for a static spherically symmetric (SSS) ansatz (1.2), where dΩ 2 d−2,κ describes the (d − 2)-dimensional transverse surface of constant curvature normalized to κ = +1, 0, −1 denoting spherical, flat and hyperbolic topologies, respectively.The metric function f (r) is fully determined by the vanishing of a total derivative of a second order differential or algebraic expression, with the constant of integration related to the ADM mass.Theories with an algebraic equation of motion for f are identified as quasi-topological (QT) gravities [33][34][35], and they likely satisfy a Birkhoff theorem [35][36][37][38].However, theories in the QT class only exist in d ≥ 5. Another interesting GQT subclass is that of Lovelock theories [26,27], which are the most direct generalizations of GR in that the field equations are always second order differential equations for any metric.Similar to the QT class, for the ansatz (1.2), the metric function f is fully characterized by a single algebraic equation that only differs from that of the QT theories by an overall constant.Thus Lovelock gravity corresponds to a subclass of QT theories.
However, Lovelock theories seem too restrictive -Einstein gravity is the only possible Lovelock theory in d = 4, and a Lovelock curvature density of order n yields non-trivial dynamics only if d > 2n + 1.So the first non-trivial Lovelock theory appears in d = 5, corresponding to Gauss-Bonnet gravity with n = 2, whereas non-trivial GQT gravity theories exist for any order n ≥ 3 in d ≥ 4 [30,31].In addition, as far as (1.2) is concerned, GQT theories allow multiple inequivalent densities at a given order in d ≥ 5, but QT (Lovelock) theories have one unique density at any order [31].Thus GQTs constitute a much broader class of higher-curvature theories than have already been specified.
We therefore choose to illustrate our approach in GQT gravity theories.The thermodynamics of these theories have not been explored to the same extent that Lovelock theories have.Previous studies have been carried out in limited contexts, with only a few low order couplings in dimensions not much larger than four [39][40][41][42][43][44].Inspired by the multi-critical behaviour found for a broad range of black holes in different contexts [18][19][20][21], our interests lie both in demonstrating our method and in understanding multi-criticality in GQT gravity.
Our paper is organized as follows.In sections 2 and A we review some properties of GQT black hole solutions.In section 3, we provide an interpretation of multi-criticality and introduce the K-rule, obtained by the reformulation of the Maxwell area rule.We develop in section 4 a method based on the K-rule, carry out a discussion on its feasibility, and construct quadruple and quintuple points based on this approach.

Fundamentals of thermodynamics of GQT black holes
Evaluated on a static spherically symmetric metric of the form (1.2), the GQT class of curvature order n ≥ 2 has exactly n − 1 inequivalent densities in d ≥ 5 [31].As required by the integrability of the field equation for f (r), on-shell GQT Lagrangian densities should be total derivatives of the form [31] where k labels one of the n − 1 inequivalent densities, S (n,j) is given by n,j are some constrained coefficients such that S n is induced by a real off-shell GQT density.Then the constraints n,j j(j − 1) = 0.
(2.3) are required to obtain the most general Lagrangian density [31].The first constraint ensures that all densities contribute with a power of r d−1 ; the second ensures that the field equations are of 2nd order when linearized about constant curvature backgrounds.Note that any linear combination of on-shell densities still satisfies (2.1), (2.2) and (2.3); therefore any density can be decomposed into n − 1 independent densities.In other words, it is sufficient to study one particular choice of λ (k) n,j .Incorporating the constraints (2.3), the following choice n,j form a family of GQT densities, with the remaining coefficients identical to 0. This simple choice is employed in our analysis but the results in the remaining part of this section hold generally.
Upon integrating the equations of motion, we obtain [30] where 2 ) , M is the ADM mass [45][46][47][48], and where prime is defined as taking derivatives with respect to r.
For j = 0, 1, F (n,j) becomes an algebraic quantity (having no derivative terms), which implies the density defined by (2.4) with k = 1 is purely QT (Lovelock).To make a consistent notation with common definitions, we separate QT densities from GQT ones (2.4) and define as QT densities.The remainder, with k ≥ 2, are genuine GQT densities.Despite the complexity of (2.5), the thermodynamics of GQT black holes is determined by two simple equations: f (r + ) = 0, which defines the outermost black hole horizon at r = r + , and f (r + ) = 4πT , which defines the temperature of the black hole.These two relations are given by [31] ) where the couplings of the lowest two orders are set to be α 0,1 ≡ −(d − 1)(d − 2)/(2 2 ), and α 1,1 ≡ 1/2 for consistency with Einstein gravity.The parameter here is the AdS length, and so α 0,1 is identical to the cosmological constant Λ.In the context of black hole chemistry, all couplings except for α 1,1 are identified as thermodynamic variables, with as their corresponding conjugate potentials.As a well-known consequence in GR, the first law of thermodynamics and the Smarr relation hold as well in the extended phase space: where the pressure P is a redefinition of α 0,1 and V is its corresponding conjugate volume The thermodynamic quantity S is the Wald entropy [49], which reads [31] This quantity is not always positive, and in such situations it has been common to simply discard solutions for which this is the case.However ambiguities exist in the definition of the black hole entropy.For example, adding to the Lagrangian a term proportional to the induced metric on the horizon will, without having an effect on the other properties of the solution, shift the entropy by an arbitrary constant.One example is that of adding an Euler density to the action [44].We shall therefore retain solutions with S < 0 in considerations, appropriately indicating in our figures where this occurs.
Henceforth we shall consider the Gibbs free energy G = M −T S for investigating phase transitions.The global minimum of G yields the most stable thermodynamic phase at any given temperature.
Before continuing, we note that physical theories should only propagate one type of massless spin-2 graviton on constant curvature backgrounds.This in turn implies the effective Newtonian constant must have the same sign as the one in general relativity, which means that for the class of metrics (1.2) having asymptotically AdS solutions of the form we shall only consider black holes with f ∞ > 0, h (f ∞ ) < 0 and γ 2 > 01 as satisfying the requisite physical criteria.We discuss these issues in appendix A.

Geometric interpretation of interphase equilibrium
We seek to obtain the conditions under which three or more phases merge at a particular temperature and pressure.The Gibbs free energy provides a diagnostic for this.Its global minimum as a function of the temperature T determines the thermodynamically stable state of the system for a given fixed choice of the other thermodynamic parameters.The presence of swallowtails in the Gibbs free energy indicates multiple phases, with first order phase transitions between two distinct phases taking place at the intersection point of the swallowtail.There must be N − 1 swallowtails in order to have N distinct phases.Whenever the intersection points of j different swallowtails coincide, then there is a j-th order multicritical point, where j ≤ N .Previous methods for finding multiple phases and N -tuple critical points exploited the fact that, regarding temperature as a function of horizon radius, each extremum of T (r + ) corresponds to a cusp in the Gibbs free energy [18,20].Hence N distinct phases require 2N −2 distinct extrema.If two adjacent inflections of T (r + ) occur at the same temperature, then the intersection points of corresponding swallowtails will merge.If this takes place for all the inflections, then all such intersection points will merge, correseponding to an N -tuple critical point.These critical points can be found by finely tuning the other thermodynamic parameters.
Here we demonstrate an alternate method that is considerably more efficient.We start with a brief review of the Maxwell construction [22].It is well-known that the multiplicity of the Gibbs free energy G(P, T ) corresponds to the non-monotonic behavior of the pressure P (V, T ).As illustrated in figure 1, a full oscillation AaBbC in the pressure at a fixed temperature T * leads to a swallowtail on the Gibbs phase diagram.With some abuse of notation, integrating dG along the loop A → b → a → C yields The second equality holds because the temperature is fixed, the fourth one comes from integration by parts, and the last expression follows from P (V A ) = P (V C ) = P * , where P * characterizes the swallowtail intersection point in the Gibbs phase plot.The geometric interpretation of (3.1) is obvious: P * corresponds to a pressure that partitions the oscillatory parts of the P − V diagram into equal areas.
It is useful to define the function K(V, V i ) and its derivative K (V, V i ) as follows: It is obvious that K(V A , V A ) = 0 and that the last expression of (3.1) can be rewritten as However for any two points in thermodynamic phase space whose volumes satisfy (3.3), the relation (3.3) alone doesn't imply that their difference in free energy is zero.It is also necessary to ensure that P (V C ) = P (V A ) so that (3.1) holds.In a plot of G vs. P , this requirement is equivalent to the condition that A and C are the same point.This in turn implies that where the first condition ensures that P (V A , T * ) = P * .Geometrically, (3.3) and (3.4) ensure that A and C are the same point in the Gibbs energy diagram, and the continuity of K between A and C guarantees that this point is on some closed loop.A true self-intersection point (or double point) therefore emerges.We note that if the second derivative vanishes at some point then K will no longer be an extremum there.This is illustrated for point C in the rightmost diagram of figure 1 by the red curve.The pressure will then be an extremum at this point (as shown in the leftmost diagram in figure 1), and the corresponding part of the curve in the free energy diagram will get reflected through P * , as shown by the red curve in the middle diagram in figure 1.By convention, we still regard this as a double point.These considerations can be easily generalized to any N -tuple point.We say that an N -tuple point exists at (P * , T * ) if and only if the K-rule is satisfied: namely that the function K(V, V 0 ) has N real zero points {V n } for some fixed V 0 and K vanishes for all those roots, namely are satisfied by exactly N different values of {V n }, including V 0 itself.Since the above argument about multicriticality is quite general, we would expect these discussions apply to any thermodynamic system for any conjugate pair of thermodynamic quantities, such as the temperature and the entropy.

Multiple phases and N -tuple critical points
We shall now construct multiple phases and N -tuple points for GQT black holes based on the K-rule introduced in section 3. The procedure is simple.
1. Write the function K as where P is given from (2.9) (which can be regarded as the the equation of state) with2 replaced by (2.13), V is identified as the thermodynamic volume defined in (2.13), and {α n,k } is the set of undetermined couplings.
2. Apply the K-rule to N positive distinct values of r + (where r 0 is taken to be any one of these values), then solve for P * , T * , {α n,k } from the 2N − 1 independent equations 2 (3.6).This implies that a minimum number of 2N − 3 non-zero higher-curvature couplings are required.
3. Check if the solution P * , T * , {α n,k } provides a real N -tuple point in a sense that exactly N roots solve (3.6) as desired.If not, change the choice of as many horizon radii as needed until a real N -tuple point occurs.
We pause to make a few supplementary comments regarding the feasibility of the method.First of all, K is constructed from P and V instead of T and S because we want K to be a simple function such that the equations in step 2 are solvable: the definition (4.1) fulfills this requirement since K is in fact a polynomial in r + .For convenience, K is defined as a function of the radius rather than the volume.It should be pointed out that the density S (2) n≥3 is quasi-topological and becomes trivial in d = 2n.Therefore, for even dimensions, we exclude α d/2,2 from our considerations.We shall also restrict ourselves to spherical black holes with κ = 1 for simplicity.Since the pressure becomes a polynomial in r + (with the temperature T considered as a non-dynamical parameter), then Descartes' rule of signs can be applied, which relates the largest number of oscillations in the region r + > 0 to the number of sign changes in the sequence of a polynomial's coefficients.Thus we discuss the feasibility of our method through studying the possibility for the occurrence of N − 1 oscillations in P by manipulating signs of couplings in the next paragraph.
The feasibility of step 2 can be seen by induction.As indicated by (2.9) or table 1, switching on an arbitrary genuine GQT coupling (k ≥ 2) always introduces three independent terms proportional to different powers of r + in the expression for the pressure in addition to which is the expression with all higher-curvature couplings set to zero.Since P 0 has one sign change, then turning on any particular coupling can introduce another sign change.
Respecting the fact that temperature and the coupling are free, it is possible to make P have a full oscillation, which means that (3.6) has real solutions and a double point can be obtained.Similarly, for critical points involving more phases, we can obtain additional oscillations by including two additional couplings per oscillation, as long as they switch on at least two monomials in r + that differ from those already present.Hence not any choice of 2N − 3 higher-order densities yields an equilibrium state with four phases or more.For example, as highlighted in table 1, switching on {α 7,2 , α 8,4 , α 9,6 , α 10,8 , α 11,10 } and keeping other couplings zero only introduces 1/r 11 + , 1/r 12 + , 1/r 13 + into the pressure.Together with P 0 , a total of five monomials are present in the expression for the pressure, which is only enough to construct a triple point.In this example we must therefore avoid turning on more than three couplings that contribute to the same three powers of r + .
The number 2N −3 should be considered as the minimum number of couplings required by our method.This may not be the smallest number of couplings needed for the emergence of an N -tuple point in general, since it is possible to have another sign change internally between the three monomials corresponding to the α n,k .The explicit form of new terms that are activated by switching on α n,k reads We claim that there can be at most one sign change between these three terms.If the sign switches twice, the first two coefficients must have a negative product, namely, Since we restrict ourselves to genuine GQT densities (k ≥ 2) in d ≥ 5 only, the above can be simplified to Note that the dimension d cannot be any of 2n − k, 2n − k + 1, 2n − k + 2; otherwise one of the terms would vanish and it would be no longer possible to have two sign changes between two terms.Therefore the first two factors in (4.4) must be positive, which leads to where k ≥ 2 is applied to determine the directions of inequalities.Meanwhile, we require the last two terms produce a sign change as well.Through a similar analysis, we arrive at Our claim is thus proved, since (4.6) contradicts (4.5).Even if there is an extra internal sign change, a half oscillation in the pressure does not necessarily occur, since these variables are not only integers but also constrained relative to each other in a complicated way.However, notice that the range of n in (4.6) is proportional to d, implying that we have a rather large parameter space for n and k.Hence we can still expect that there exists some choices of d, k, n that can give rise to an N -tuple point with less than 2N − 3 couplings.As a consequence, we would also expect N − 1 to be a lower bound on the number of couplings needed for the occurrence of an N -tuple point.Because of these considerations, step 3 is added to guarantee that exactly N phases are obtained.
From the previous discussion, we can see that to figure out a general Gibbs phase rule is challenging.For neutral multi-rotating black holes [19], because the phase structure is invariant under the exchange of any two angular momenta, turning on any two additional couplings always creates a new phase and vice versa.However in the GQT scenario, this symmetry is broken between any two couplings, and theories with distinct values of k (even if they have the same n) differ a lot in their phase structures.A multi-critical point may not occur even if infinitely many couplings are turned on.This implies that the problem is not as simple as it is in Lovelock gravity where k = 1 and each density (2.7) similarly contributes to the thermodynamics [20].We leave this question for future investigation.
In order to obtain multiple phases and multi-critical points, the physical constraints and ensuring P > 0 everywhere must also be considered.Since it is difficult to find a case with positive γ 2 everywhere, we only impose γ 2 > 0 in a neighbourhood of each critical point.In practice, we keep manipulating r + 's until a critical point satisfying all constraints occurs.Under these considerations, we explicitly obtain a quadruple point (figures 2 and 3) and a quintuple point (figures 4 and 5) for two different spherical GQT black holes.Note that to see the merging of multiple swallowtails requires high precision in computations due to the finely tuned nature of multi-critical points.Both multi-critical points have negative Gibbs free energies, implying stable phase transitions.For the quintuple point, an extra coupling α 3,2 is fixed to be 1 before running the procedure in order to make physical cases easier to emerge.
Compared with the previous methods, the advantages of our procedure are quite significant.First, the application of the Maxwell construction turns the problem into algebra which enables our method to produce critical points with arbitrarily high precision such that very tiny phase structures can be discovered easily.More importantly, the method is more efficient in the sense that it does not need any fine-tuning procedure such as those previously employed [18][19][20][21] where N -tuple points were obtained by manipulating thermodynamic variables so that a common point of inflection occurred between multiple maxima and minima of the temperature as a function of r + .However, due to the additional physical constraint γ 2 > 0 induced from the non-algebraic nature of equation of motions in genuine GQTGs, finding a physical multicritical point with a large N is still time-consuming.

Conclusions
We have exploited Maxwell's equal area law to find an interphase equilibrium for black holes with multiple phases, reformulating it into what we call the K-rule.Utilizing the K-rule, we developed a novel approach for constructing N -tuple points in the phase space of black holes.
We applied our results to GQT theories with 2N − 3 genuine couplings.Our analysis suggests that the minimum number of couplings required for the formation of an N -tuple point is likely confined between N − 1 and 2N − 3. We presented quadruple and quintuple points to illustrate the effectiveness of the method.
Future work would involve applying the K-rule to other kinds of black hole holes, particularly those whose horizon structures are not spherically symmetric.These include accelerating black holes in non-linear electrodynamics [50], multiply rotating black holes [21], and various black hole solutions in supergravity theories [51].where the superscripts (±) indicate the sign of γ 2 .In order that the homogenous part is subdominant at large r, we require γ 2 > 0 and A = 0.
Apart from the correct asymptote, physical theories should only propagate one type of massless spin-2 graviton on constant curvature backgrounds.This implies the effective Newtonian constant must have the same sign as the one in general relativity, which means the third term in (2.15) should become negative for positive mass, that is h (f ∞ ) < 0 [42].

Figure 1 .
Figure 1.The Maxwell equal-area construction implies P = P * divides AaBbC into two regions AaB and BbC with equal areas.The red curve indicates the trajectories of the plots if K = 0.

Figure 5 .
Figure 5.The figure shows further magnifications about the quintuple point A in the figure 4. We can clearly see the intersection of 5 curves appears at A.

Table 1 .
The table shows a general pattern that the pressure follows when some genuine GQT densities (k ≥ 2) are turned on.The top element of each column indicates the power of 1/r + , and each row contains all possible couplings with a constant sum of subscripts.The table tells what powers of 1/r + in the expression of pressure are influenced by which couplings.For example, if α