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

1. Problem Statement

The paper addresses the thermodynamic behavior and phase structure of black holes, specifically comparing a singular black hole (derived from nonlinear electrodynamics) with its regular counterpart, the Hayward-AdS black hole.

• Context: While singularities are topological defects in General Relativity, regular black holes (like the Hayward solution) avoid them. However, deriving the thermodynamics of regular black holes directly often leads to inconsistencies with the First Law of Thermodynamics.
• Core Issue: The authors investigate whether imposing a specific constraint on the singular black hole's parameters can generate the Hayward-AdS solution and how this constraint alters the thermodynamic phase space, criticality, and topological properties compared to the singular case.

2. Methodology

The authors employ a multi-step analytical and numerical approach:

• Construction of Solutions:
  • They first derive a singular black hole solution using Einstein gravity coupled with a specific nonlinear electrodynamics (NED) source and a cosmological constant ($\Lambda$).
  • They then impose an algebraic constraint ($M = 16 \times 2^{3/4} \pi Q^{3/2} \alpha^{1/4}$) on the singular solution's parameters. This constraint effectively "regularizes" the geometry, yielding the Hayward-AdS black hole.
• Extended Phase Space Thermodynamics:
  • The cosmological constant is treated as thermodynamic pressure ($P = -\Lambda/8\pi$).
  • They analyze the $P-V$ criticality by calculating the equation of state $P(T, r_+)$ and finding critical points where $\partial P/\partial r_+ = 0$ and $\partial^2 P/\partial r_+^2 = 0$.
  • Heat Capacity ($C$) and Gibbs Free Energy ($G$) are calculated to determine local stability and phase transition orders.
  • Crucial Adjustment: For the Hayward-AdS black hole, the authors note that the standard First Law ($dM = TdS + \dots$) becomes invalid due to the constraint reducing the dimensionality of the phase space. Consequently, they derive $T$, $C$, and $G$ by applying the constraint directly to the singular black hole's expressions rather than using standard Legendre transformations.
• Thermodynamic Topology:
  • They utilize Duan Yishi's $\phi$-mapping topological current theory.
  • A vector field $\phi$ is constructed from the derivative of the Helmholtz free energy ($F = M - S/\tau$).
  • The topological charge (winding number) is calculated by analyzing the zero points of this vector field using the Hopf index and Brouwer degree. This characterizes the global stability of the phase space.

3. Key Contributions

• Constraint-Induced Regularization: Demonstrated that the Hayward-AdS black hole can be rigorously derived from a singular NED black hole via a specific mass-charge constraint, linking the two solutions mathematically.
• Thermodynamic Inconsistency Resolution: Highlighted that for constrained regular black holes, standard thermodynamic definitions (like $C = \partial M/\partial T$) fail. The paper provides a corrected methodology for calculating thermodynamic quantities under such constraints.
• Novel Phase Structures: Identified unique phase transition behaviors in the Hayward-AdS black hole that differ significantly from standard RN-AdS or singular counterparts, specifically the emergence of "8-shaped" and "C-shaped" Gibbs free energy curves.
• Topological Charge Flip: Discovered a fundamental change in the topological charge of the phase space: the singular black hole has a net charge of 0 (unstable + stable branches cancel out), while the regular Hayward-AdS black hole has a net charge of +1.

4. Key Results

A. The Singular Black Hole

• Phase Structure: Exhibits rich phase behavior including reentrant phase transitions and Van der Waals-like $P-V$ criticality.
• Transitions: Shows both first-order and zeroth-order phase transitions. The Gibbs free energy ($G-T$) displays a "swallowtail" structure for certain pressures, indicating a first-order transition, and a zeroth-order transition at lower pressures.
• Heat Capacity: Shows divergent points corresponding to temperature extrema. It exhibits a Schottky-like anomaly (a peak) at specific pressures, suggesting discrete energy levels in the microstructure.
• Topology: The total topological charge is 0. This is composed of a stable branch (winding number +1) and an unstable branch (winding number -1), which cancel each other out.

B. The Hayward-AdS (Regular) Black Hole

• Criticality: While it retains Van der Waals-like $P-V$ criticality, its $G-T$ profile is qualitatively different from standard RN-AdS black holes.
• Unique Phase Transitions:
  • Instead of a standard swallowtail, the $G-T$ curve exhibits an "8-shaped" knot at low pressures.
  • As pressure increases, this evolves into a "0-shaped" single loop, and finally a "C-shaped" structure.
  • The "C-shaped" structure indicates a zeroth-order phase transition between small and large black hole phases.
  • No phase transitions are observed for $P > P_c$.
• Stability: The heat capacity is consistently positive in the positive temperature region for $P \ge P_c$, indicating stability, but the phase structure is more complex than the singular case.
• Topology: The total topological charge is +1. The constraint eliminates the unstable branch (or alters the vector field such that only the stable branch contributes), resulting in a non-zero net topological charge.

5. Significance

• Theoretical Insight: The paper provides a mechanism to understand how "regularizing" a black hole (removing the singularity) fundamentally alters its thermodynamic topology. The shift from a topological charge of 0 to +1 suggests that the constraint does not just smooth the geometry but induces a qualitative transformation in the thermodynamic configuration space.
• Methodological Advancement: It offers a robust framework for studying the thermodynamics of regular black holes where the standard First Law is compromised by parameter constraints, ensuring consistent calculation of $T$, $C$, and $G$.
• New Phase Phenomena: The discovery of "8-shaped" and "C-shaped" phase structures in the Gibbs free energy expands the known landscape of black hole phase transitions, suggesting new types of zeroth-order transitions that are distinct from those found in higher-dimensional Kerr-AdS or Gauss-Bonnet-AdS spacetimes.
• Topological Classification: The use of thermodynamic topology to distinguish between singular and regular solutions provides a new invariant (winding number) for classifying black hole thermodynamic stability and global structure.