This paper studies when and how ordinary black holes can switch to a different state that carries a surrounding scalar field. The authors work in a modified gravity model called Einstein–scalar–Gauss–Bonnet (EsGB) gravity. In that model a scalar field is coupled to a particular curvature combination (the Gauss–Bonnet invariant). Depending on the form of that coupling, the ordinary Schwarzschild black hole of general relativity can become unstable and give rise to a new family of “scalarized” black holes. The paper asks whether such a change is a thermodynamic phase transition, and if so whether it is continuous or abrupt.

The team restricted the study to static, spherically symmetric black holes and solved the field equations numerically. They tested three representative classes of coupling functions between the scalar field and the Gauss–Bonnet term. One is a simple quadratic coupling that triggers spontaneous scalarization in a linear analysis. A second is an exponential-type coupling that can be tuned by a parameter. A third coupling removes the linear trigger and can produce scalarized solutions only through nonlinear effects. The authors read off mass, scalar charge, Hawking temperature and the Wald entropy from the numerical solutions and computed the free energy F = M − T_H S to compare configurations at a given temperature.

Their results show that the phase behaviour depends strongly on the coupling. For the simplest quadratic coupling the scalarized black holes exist but are thermodynamically disfavored compared with Schwarzschild black holes, so no phase transition occurs. For the exponential couplings the outcome depends on the coupling parameter: there can be no transition, a continuous second‑order transition, or a discontinuous first‑order transition. For the couplings that only allow nonlinear scalarization the authors find either a first‑order transition or no transition. To classify these cases they used a Landau-style expansion of the free energy in powers of the scalar charge (the order parameter). The sign and size of the expansion coefficients indicate whether the switch between phases is smooth or abrupt.