Multicritical points of gravitational solitons and a black hole in four dimensions

We present the first realization of multicritical points in four-dimensional general relativity, specifically within the context of Pleba\'nski nonlinear electrodynamics, using a polynomial structural function denoted as $\mathcal{H}(P)$. We show that this construction provides a systematic mechanism to engineer multicritical behavior in gravitational systems. By establishing an explicit mapping between matter theories expressed as power series in the Maxwell invariant $F$ and the Pleba\'nski formulation, we construct new families of electrically charged asymptotically anti-de Sitter black holes and magnetically charged solitons. In the grand-canonical ensemble, we analyze their thermodynamic properties and uncover a rich phase structure. We demonstrate that the soliton sector develops multiple swallowtail structures, signaling first-order phase transitions and allowing the coexistence of several magnetically charged solitons with a single electrically charged black hole. These configurations define multicritical points that generalize previously known triple points. We further show that the number of coexisting phases is controlled by the degree of the polynomial structural function, providing a direct link between the nonlinear electrodynamics couplings and the thermodynamic phase structure. In contrast, the black hole branch does not display swallowtail behavior, and it does not allow multiple electrically charged black holes to coexist with a magnetically charged soliton.