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.
We show that multi-critical points in which more than three phases coalesce are present in multiply rotating Kerr-AdS black holes in d-dimensions. We explicitly present a quadruple point for a triply rotating black hole in d = 8 and a quintuple point for a quadruply rotating black hole in d = 10. The maximal number of distinct phases n is one larger than the maximal number of independent rotations, and we outline a method for obtaining the associated n-tuple point. Situations also exist where more than three phases merge at sub-maximal multi-critical points. Our results show that multi-critical points in black hole thermodynamics are more common than previously thought, with systems potentially supporting many phases as long as a sufficient number of thermodynamic variables are present.
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