[Paper Review] Topology of Born-Infeld-AdS Black Hole Phase Transitions: Bulk and CFT Sides

Md Sabir Ali, H. El Moumni+2 more|arXiv (Cornell University)|Jun 20, 2023

Black Holes and Theoretical Physics122 references4 citations

TL;DR

This paper investigates the topological structure of phase transitions in Born-Infeld-AdS black holes using Duan’s topological current formalism and off-shell free energy analysis. It identifies critical points as vortex/anti-vortex pairs, revealing a topological breakdown in certain parameter regimes and uncovering a novel phase transition between unstable black hole phases, with results analogous to BKT transitions and swallowtail-like thermodynamic behavior.

ABSTRACT

The thermodynamic criticality of the AdS black holes serves as an important structure during the thermal phase transition. This paper discusses about the critical points and their topology during thermal phase transitions of the Born-Infeld AdS black holes. We make such investigations using two different topological approaches, namely, using Duan's topological current $ϕ$-mapping theory, and the off-shell free energy. Within Duan's formalism, we observe that for a given value of the Born-Infeld parameter $b$, there exists an associated electric charge parameter $Q$, which is highly sensitive to the topological phase transitions. This way we examine the connections of the first-order phase transition and the topological nature of the critical points. We find that the topological nature has a possible breakdown in certain parametric ranges. In effect, we determine the unconventional and the conventional phase critical points as the creation (topologically vortex) and annihilation (topologically anti-vortex) points (pairs). As the second approach, we call the off-shell free energy to determine the topological classes: of which one corresponds to the AdS-Schwarzschild black hole phases, while the other provides a possible topological phase transition. Here we also reveal a novel phase transition between two unstable phases, namely, the unstable small black hole and the intermediate black holes. For a certain parametric values of the Born-Infeld parameter and the pressure, we also study the different topological descriptions that inevitably correspond to the AdS-Reissner-Nordstr$\Ddot{o}$m black hole phases. As a consistency check of the critical points during the topological phase transitions, we study the vortex/anti-vortex annihilation thermodynamics from local as well as global thermodynamic viewpoint.

Motivation & Objective

• To understand the topological nature of critical points in Born-Infeld-AdS black hole phase transitions.
• To explore how the Born-Infeld parameter and electric charge influence topological phase transitions.
• To identify and classify topological phases using Duan’s $φ$-mapping theory and off-shell free energy.
• To analyze the thermodynamic stability and topology of small, intermediate, and large black hole phases.
• To establish a correspondence between vortex/anti-vortex annihilation and thermodynamic phase transitions.

Proposed method

• Application of Duan’s topological current $φ$-mapping theory to map critical points in Born-Infeld-AdS black holes to topological defects.
• Construction of a vector field from the Gibbs free energy to analyze topological charges and classify phases.
• Use of off-shell free energy to determine topological classes for S-type and RN-type black holes under varying $Q$ and $T$.
• Plotting vortex/anti-vortex creation and annihilation points in the $(1/Q, T)$ plane to model thermodynamic swallowtail behavior.
• Analysis of local and global thermodynamic stability to validate topological phase transition signatures.
• Comparison of results with known phase transitions, including BKT-like transitions and RN-AdS behavior.

Experimental results

Research questions

• RQ1 How do critical points in Born-Infeld-AdS black holes manifest topologically, and what is their nature as vortex or anti-vortex pairs?
• RQ2 What role does the Born-Infeld parameter $b$ play in the breakdown of topological structure during phase transitions?
• RQ3 Can the off-shell free energy be used to classify distinct topological phases in S-type and RN-type Born-Infeld-AdS black holes?
• RQ4 What is the thermodynamic significance of vortex/anti-vortex pair creation and annihilation in the context of black hole phase transitions?
• RQ5 How does the system's behavior change when transitioning between unstable black hole phases, and what topological features emerge?

Key findings

• Critical points in Born-Infeld-AdS black holes are topologically classified as vortex (creation) and anti-vortex (annihilation) pairs, with topological charge $\mathcal{Q} = -1$ at $Q = Q_m$.
• The topological phase transition breaks down for $b = 8\pi Q_m$, beyond which the system behaves like an RN-AdS black hole with no unstable phases.
• A novel phase transition is identified between the unstable small black hole (SSBH) and intermediate black hole (IBH) phases, not observed in standard RN-AdS systems.
• The $(1/Q, T)$ plane exhibits a swallowtail-like structure analogous to first-order phase transitions, with distinct thermodynamic processes for LBH/SSBH annihilation and SSBH creation.
• Local and global thermodynamic analyses confirm the consistency of vortex/anti-vortex annihilation with thermodynamic stability and free energy behavior.
• For $Q < Q_m$, the Gibbs free energy shows a cusp-like structure and a Hawking-Page-like transition, while for $Q > Q_m$, no such cusp appears, indicating topological and thermodynamic differences.