Multi-critical Points in Black Hole Phase Transitions

We present the first examples in black hole thermodynamics of multicritical phase transitions, in which more than three distinct black hole phases merge at a critical point. Working in the context of non-linear electrodynamics, we explicitly present examples of black hole quadruple and quintuple points, and demonstrate how $n$-tuple critical points can be obtained. Our results indicate that black holes can have multiple phases beyond the three types observed so far, resembling the behaviour of multicomponent chemical systems. We discuss the interpretation of our results in the context of the Gibbs Phase Rule.

The importance of black hole thermodynamics in providing clues about the nature of quantum gravity cannot be underestimated. Much has been learned from studying asymptotically anti de Sitter (AdS) black holes, beginning with the pioneering work of Hawking and Page who demonstrated the existence of a phase transition between thermal radiation and a large AdS black hole [1]. Such black holes can be in thermal equilibrium with their Hawking radiation, and the Hawking-Page phase transition corresponds to the confinement/deconfinement of the dual quark gluon plasma [2].
More generally, the cosmological constant can be regarded as a thermodynamic variable corresponding to pressure in the first law [3][4][5] (arising, e.g. as a (d − 1)form gauge field [6]). Within this context, the black hole mass takes on the interpretation of enthalpy and a rich variety of thermodynamic phase behaviour has been shown to emerge. The Hawking-Page transition can be understood as a solid-liquid transition [7], and black holes have been shown to exhibit Van der Waals [8], re-entrant [9], superfluid [10], and polymer-type phase transitions [11], snapping transitions [12] for accelerating black holes [13], and universal scaling behaviour of the Ruppeiner curvature [14]. This panoply of behaviour has suggested a molecular interpretation of the underlying constituent degrees of freedom [15]. For these reasons this subdiscipline has come to be called Black Hole Chemistry [16].
Triple points have been observed for quite some time [17][18][19], and a recent interpretation of the microstructure of black holes at such points was recently proposed [20]. However multicritical points -in which many phases merge together at a single critical point -have never been observed.
We present here the first examples of black hole multicritical points in the context of non-linear electrodynamics in 4-dimensional Einstein gravity. We find that regions exist within the parameter space of such theories in which many distinct black hole phases exist for a range of sufficiently large pressure. Transitions between each such phase are of 1st-order, and the coexistence lines for each terminate in distinct 2nd-order critical points. As the pressure is lowered to a certain critical value, all these phases merge into a single multicritical point. Below this critical pressure only two distinct phases exist, separated by a 1st-order phase transition between the largest and the smallest possible black holes allowed.
We explicitly illustrate this for a black hole quadruple point. At high pressures only a single black hole phase exists. As the pressure is lowered, first two, then three, and finally four distinct black hole phases emerge, each distinguished by their size. As the pressure is further lowered, all fours phases merge at a single quatro-critical point (a quadruple point). For pressures below this point there are only two distinct phases -the largest and smallest possible black holes -separated by a first-order phase transition.
Our results provide the first indication that black holes can behave like multicomponent systems in nature that exhibit similar multicritical behaviour [21]. We find that an n-tuple critical point for a charged black hole requires 2n − 1 conjugate pairs in Power-Maxwell theory. The Gibbs Phase Rule then implies that there are n degrees of freedom at this multicritical point, whose implications for the microstructure of black holes has yet to be clarified.
In the extended thermodynamic phase space of black hole chemistry [4,16], the relation is posited between a (negative) cosmological constant Λ and the thermodynamic pressure P , with l is the radius of the D-dimensional AdS space and G the (dimensionful) Newton gravitational constant; we set = c = 1. The black hole mass M is interpreted as thermodynamic enthalpy rather than internal energy. The first law of thermodynamics and corresponding Smarr relation are for a black hole of charge Q, surface gravity κ, angular momentum J, and area A in D-dimensional Einstein arXiv:2207.03505v1 [hep-th] 7 Jul 2022 gravity, where entropy S and temperature T are with φ, Ω the respective conjugates to Q, J, and where is the thermodynamic volume conjugate to P [4,5,27]. We consider a general form of non-linear electrodynamics minimally coupled to D = 4 Einstein gravity [22]. The action of this theory is with ) and A µ is the U(1) Maxwell field. We recover Einstein-Maxwell theory when α 1 = 1 and by varying the action with respect to the metric. The generalized Maxwell equations follow by varying the action with respect to the field A µ . It has been shown [22] that the following ansatz admits asymptotically flat black holes with multiple horizons provided the coupling constants α i are appropriately chosen. This result is straightforwardly generalized to the asymptotically AdS case by writing where the field equations imply with Q an integration constant corresponding to the electric charge of the black hole, and the prime denoting a derivative with respect to r.
Setting α 1 = 1 without loss of generality, we find ... = ......, with M corresponding to the ADM mass of the black hole. If all α i for (i > 1) are set to zero, we recover which is the usual Reissner-Nordstrom AdS-black hole having two horizons. For appropriate choices of nonvanishing α i with i ≤ n − 1, n distinct horizons can be obtained. Appropriate choices of the α i can also yield multicritical thermodynamic behaviour. We begin by illustrating the existence of a quadruple point, which can be obtained for α i = 0 with i ≤ 7, sufficient to support up to eight black hole horizons [22]. Explicitly The thermodynamic quantities are given by and Gibbs free energy G = M − T S. By carefully choosing the locations of the local extrema in T (r + ), it is possible to produce three separate swallowtails in the Gibbs free energy, characteristic of three stable first order phase transitions (Fig. 1). This in turn determines the charge Q, pressure P , and coupling constants α i . The parameter space admitting four phases is small, but is not a set of measure zero.
Further adjusting the locations of the extrema, the three inflections can be made to occur at the same temperature, resulting in the quadruple point shown in Fig. 2. For high pressures and temperatures, only one black hole phase is observed. As the temperature and pressure are lowered, new phases emerge at distinct critical points, separated by first order phase transitions. In the pressure range P ∈ (6.90 × 10 −5 , 7.83 × 10 −5 ), four distinct black hole phases exist, characterised by their sizes. As the pressure is lowered to 6.90 × 10 −5 , the four phases merge at a critical point, and only the largest and smallest phases exist at lower pressures.
Multi-critical points of higher order can likewise be achieved by introducing more coupling constants. In general, two additional coupling constants are needed for each new n-tuple point; for example two coupling constants (α 2 and α 3 ) are needed to obtain a triple point. for P, Q, α i , where the prime denotes the derivative with respect to r + , as each pair of local minima and maxima produce a swallowtail in the Gibbs free energy. Once a sufficient number of swallowtails are obtained, it is possible to make additional adjustments to the locations of the extrema so that the inflections of T (r + ) occur at the same temperature, forming n − 1 swallowtails that overlap in the Gibbs free energy. Fig. 3 shows a quintuple point realized using this method.
An alternate method for finding multi-critical points is to choose values of r + where a line of constant temperature T * intersects T (r + ). An n-tuple point has 2n − 1 such intersects, obtained by solving for T * , P, Q, α i . This yields n − 1 swallowtails in the Gibbs free energy that only require slight adjustments to merge. This approach can more easily generate multicritical points using the minimum number of coupling constants; using it we can obtain a quadruple point with α i = 0 for i ≤ 5, two fewer couplings than for the quadruple point shown in Fig. 2. By generalizing the scaling arguments used to obtain the Smarr relation [4], we can regard each α i as a thermodynamic variable. Noting that dimensionally both hold for the solutions we obtain, where is thermodynamically conjugate to the α i coupling. These quantities play a role similar to polarization susceptiblities in non-linear optics, generalizing the notion of vacuum polarization in Born-Infeld Electrodynamics [23].
Our results indicate that black holes can behave like multicomponent chemical systems seen elsewhere in nature [21,24,25]. In such systems the generalized Gibbs phase rule [26] relates the number of coexistent phases P to the number of thermodynamic conjugate pairs W via where setting W = C + 1 recovers the Gibbs phase rule for simple systems, with C the number of constituents in a multicomponent chemical system. Modern functional materials are not simple, having additional degrees of freedom that can do thermodynamic work represented by the more general quantity W [26]. The number of degrees of freedom F is the number of independent intensive parameters. n-tuple points are marked by simplices connecting n phases on the internal energy surface in terms of the extensive thermodynamic variables U (S, V, ...), and F is determined by the nullity of the set of points in R n defining the simplex [26]. The maximum number of coexistent phases is P = W + 1, attained for F = 0, as a set of W + 1 affinely independent points in R W +1 has 0 nullity. The situation is somewhat different in Black Hole Chemistry. A charged AdS black hole has only two phases (P = 2), large and small, respectively analogous to the high-entropy liquid and low-entropy gas phases of a Van der Waals fluid [8]. The analogue of the lowestentropy solid phase is thermal AdS [7], which cannot be attained due to charge conservation. If charge vanishes, this phase exhibits a first order phase transition with a higher-entropy large black hole [1] analogous to a solidliquid transition [7] but is missing the analogous highest entropy gaseous phase, again giving P = 2. Consequently neither the Schwarzschild-AdS nor Reissner-Nordstrom-AdS black holes can exhibit triple points and F = 2.
More generally 2n − 1 thermodynamic conjugate pairs allow for n-tuple phase transitions. Triple points have been observed in black hole systems in D = 4 but with at least five thermodynamic conjugate pairs [28]. In higher dimensions, triple points can exist with fewer conjugate pairs [17][18][19], but still within bounds of the generalized Gibbs' phase rule. By regarding the coupling constants in Power-Maxwell theory as extensive thermodynamic variables, we find that each new phase requires two additional thermodynamic conjugate pairs. The generalized Gibbs phase rule indicates that an n-tuple point in non-linear electrodynamics has F = n degrees of freedom. The implications of this for the microstructure of black holes [14,15,20] remains to be understood.
Schwarzschild-AdS black holes, however, can exhibit radiation-small-large triple points with only three conjugate pairs in higher curvature gravity [29], fully analogous to the triple point of water, provided their horizon geometry does not have constant curvature. This suggests that the Gibbs Phase rule (31) can be satisfied with F = 0 for an n-tuple critical point for at least some gravitational systems. Elucidation of the necessary and sufficient conditions to attain a black hole multicritical point remains an interesting open question.