Thermodynamic phase structure and topological charge of Hayward-AdS black holes under phase space constraints

We investigate the thermodynamic behavior of the Hayward-AdS black hole and compare it with its singular counterpart from which it can be constructed through the imposition of an additional constraint. The singular black hole displays a rich phase structure, including reentrant phase transitions reminiscent of those observed in higher-dimensional Kerr-AdS spacetimes. After the constraint is imposed, the resulting Hayward-AdS black hole continues to exhibit Van der Waals-type $P-V$ criticality. However, its Gibbs free energy profile differs qualitatively from that of standard RN-AdS black holes. In addition, we extend the analysis by employing thermodynamic topology to characterize the global structure of the phase space. We find that the topological charge of the singular black hole is $-1$, whereas that of the Hayward-AdS black hole becomes $+1$. This change of topological charge indicates that the constraint not only regularizes the geometry but also induces a qualitative transformation in the thermodynamic configuration space.

💡 Research Summary

The paper investigates the thermodynamic behavior of a singular black hole sourced by nonlinear electrodynamics in Einstein‑gravity with a negative cosmological constant, and then constructs the regular Hayward‑AdS black hole by imposing an additional algebraic constraint that relates the mass, charge and the nonlinear coupling. The authors first solve the field equations for the singular solution, obtaining a metric function
(f(r)=1-\frac{2M}{r}-\frac{\Lambda r^{2}}{3}+ \frac{64\sqrt{2},\pi,\alpha Q^{3}}{r^{4}}+\frac{(2Q^{2}\alpha)^{3/4}}{r^{3}}).
From this they derive the Hawking temperature, the pressure (P=-\Lambda/8\pi) and the equation of state (P=P(T,r_{+},Q,\alpha)). By solving (\partial_{r_{+}}P=0) and (\partial_{r_{+}}^{2}P=0) numerically (for (Q=0.01,;\alpha=2)) they locate a Van‑der‑Waals‑type critical point ((T_{c}\simeq0.365,;r_{c}\simeq0.336,;P_{c}\simeq0.022)). The (P!-!r_{+}) curve shows an extra branch in the small‑radius region, reminiscent of re‑entrant phase transitions found in higher‑dimensional rotating AdS black holes.

The heat capacity at fixed ((Q,P,\alpha)) exhibits three divergences when (P<P_{c}), separating two stable branches (small and large black holes) from two unstable ones (intermediate and very small black holes). For pressures below a special value (P_{0}\approx0.3) a pronounced peak appears, analogous to a Schottky anomaly, suggesting discrete micro‑states.

The Gibbs free energy (G=M-TS) displays a rich structure: for (P<P_{0}) the large‑black‑hole phase dominates at low temperature; for (P_{0}<P<P_{z}) ((P_{z}\approx0.018)) a zeroth‑order transition occurs; for (P_{z}<P<P_{c}) a swallow‑tail appears, indicating a first‑order transition that smoothly connects to the zeroth‑order one. This phase diagram is more intricate than that of the standard RN‑AdS black hole.

To obtain the regular Hayward‑AdS black hole the authors impose the constraint
(M=\frac{(16\pi)^{3/4}}{2^{3/4}},Q^{3/2},\alpha^{1/4})
(equivalently Eq. (3.2)), which eliminates one degree of freedom in the thermodynamic phase space. The resulting metric function simplifies to a form that reduces to Hayward’s regular black hole when (\Lambda=0). Because the constraint ties the parameters, the usual first law (dM=TdS+\Phi dQ+A d\alpha+V dP) no longer holds directly; the authors therefore recompute temperature and pressure from the constrained metric, obtaining a new equation of state and a critical point ((T_{c}\simeq0.189,;r_{c}\simeq0.53,;P_{c}\simeq0.067)) for (\alpha=1.5).

The heat capacity of the constrained Hayward‑AdS black hole remains positive for all admissible pressures, showing no divergences; thus the system is thermodynamically stable throughout. The Gibbs free energy, however, exhibits multiple branches when (P<P_{c}). The (G!-!T) curves form an “8‑shaped” loop for low pressures, which morphs into a swallow‑tail‑like profile as pressure increases, and eventually collapses to a single branch for (P\ge P_{c}). At a particular pressure (P\approx0.043) the lower loop disappears, giving rise to a “0‑shaped” curve; further increase leads to a “C‑shaped” structure and a zeroth‑order transition between small and large black holes. These topologically distinct free‑energy profiles are absent in RN‑AdS black holes.

The final part of the work applies Duan’s (\phi)-mapping topological current method. By defining a vector field from derivatives of the Helmholtz free energy (F=M-TS) with respect to the order parameters ((r_{+},\tau=1/T)), the zero points of this field are treated as topological defects. The winding number (Hopf index) multiplied by the Brouwer degree yields a topological invariant (topological charge). For the singular black hole the calculation gives (Q_{\text{top}}=-1); after imposing the constraint, the regular Hayward‑AdS black hole acquires (Q_{\text{top}}=+1). Hence the regularization not only smooths the spacetime curvature but also flips the topological charge of the thermodynamic phase space, indicating a qualitative change in the global configuration of equilibrium states.

In summary, the paper demonstrates that (i) imposing a constraint to regularize a singular black hole reduces the dimensionality of the thermodynamic phase space and modifies the first law; (ii) the constrained system retains Van‑der‑Waals‑type criticality but displays novel Gibbs‑free‑energy topologies (8‑shape, C‑shape, zeroth‑order transitions) not seen in standard charged AdS black holes; and (iii) the topological charge associated with the thermodynamic vector field changes sign from –1 to +1, providing a clear topological signature of the regularization process. These results highlight the deep interplay between spacetime regularity, thermodynamic phase structure, and topological aspects of black‑hole physics.