Topology of critical points and Hawking-Page transition

Open Access

Topology of critical points and Hawking-Page transition

Pavan Kumar Yerra, Chandrasekhar Bhamidipati, and Sudipta Mukherji

Phys. Rev. D 106, 064059 – Published 30 September 2022

Abstract

Using the Bragg-Williams construction of an off-shell free energy, we calculate the topological charge of the Hawking-Page transition point for black holes in anti–de Sitter. A computation following from a related off-shell effective potential in the boundary gauge dual matches the value of topological charge obtained in the bulk. We also work out the topological charges of the equilibrium phases of these systems, which follow from the saddle points of the appropriate free energy. The locally stable and unstable phases turn out to have topological charges opposite to each other, with the total being zero, in agreement with the result obtained from a recent construction [S.-W. Wei, Y.-X. Liu, and R. B. Mann, arXiv:2208.01932].

Received 14 September 2022
Accepted 19 September 2022

DOI:https://doi.org/10.1103/PhysRevD.106.064059

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Alternative gravity theories Classical black holes Gauge-gravity dualities Gravity in dimensions other than four Quantum gravity Thermodynamics

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Pavan Kumar Yerra ^1,*, Chandrasekhar Bhamidipati ^2,†, and Sudipta Mukherji^1,‡

¹Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha, 751005 India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India
²School of Basic Sciences, Indian Institute of Technology Bhubaneswar Bhubaneswar, Odisha 752050, India

^*pk11@iitbbs.ac.in
^†chandrasekhar@iitbbs.ac.in
^‡mukherji@iopb.res.in

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Issue

Vol. 106, Iss. 6 — 15 September 2022

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Images

Figure 1
For Schwarzschild-AdS black holes: (a) Behavior of the Bragg-Williams free energy $\bar{f}$ , as a function of the order parameter $\bar{r}$ at different temperatures $\bar{T}$ . Temperature of the curves increases from top to bottom. Hawking-Page transition happens at the temperature ${\bar{T}}_{HP}$ (dashed red curve). For $\bar{T} < {\bar{T}}_{HP}$ , black holes are globally unstable, while for $\bar{T} > {\bar{T}}_{HP}$ , they are globally stable. (b) The coexistence curve ${\bar{T}}_{0}$ of the black hole phase and AdS phase, as a function of $\bar{r}$ , shows the HP transition point at its minima (red dot). Other colored dots correspond to respective colored curves in the left plot. Plots are displayed for $n = 2$ .
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Figure 2
For Schwarzschild-AdS black holes: (a) The vector field $n$ in the $θ - \bar{r}$ plane shows the presence of a Hawking-Page (HP) transition point (black dot, located at ${\bar{r}}_{HP} = 1$ ) at $ϕ = 0$ . Contour $C_{1}$ contains the HP transition point, while contour $C_{2}$ does not. (b) $Ω$ vs $ϑ$ for contours $C_{1}$ (blue curve) and $C_{2}$ (green curve). Plots are displayed for $n = 2$ , and the parametric coefficients of the contours are $(a, b, r_{0}) = (0.4, 0.5, 1)$ for $C_{1}$ and (0.4, 0.5, 2.8) for $C_{2}$ .
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Figure 3
For charged-AdS black holes: (a) The vector field $n$ in the $θ - \bar{r}$ plane, shows the presence of a Hawking-Page (HP) transition point (black dot, located at ${\bar{r}}_{HP} = \sqrt{1 - c^{2} {\bar{μ}}^{2}} = 0.866$ ) at $ϕ = 0$ . Contour $C_{1}$ contains the HP transition point, while contour $C_{2}$ does not. (b) $Ω$ vs $ϑ$ for contours $C_{1}$ (blue curve) and $C_{2}$ (green curve). Plots are displayed for $n = 2$ , $\bar{μ} = 0.5$ , and the parametric coefficients of the contours are $(a, b, r_{0}) = (0.4, 0.5, 0.86)$ for $C_{1}$ and (0.4, 0.5, 2.2) for $C_{2}$ .
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Figure 4
$T_{0}$ for the boundary gauge theory, whose minimum gives $T_{c}$ . Here, we set $n = 2$ , $l = 1$ , $μ = 0.5$ .
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Figure 5
For boundary gauge theory: (a) The vector field $ϕ$ vanishes at the confinement/deconfinement transition point (black dot, located at $Q_{c}$ ). Contour $C_{1}$ contains the transition point. (b) $Ω$ vs $ϑ$ for contour $C_{1}$ . Plots are displayed for $n = 2$ , $l = 1$ , $μ = 0.5$ .
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Figure 6
For Schwarzschild-AdS black hole solutions at various temperatures $\bar{T}$ : Left panel (i.e., plots (a), (c), (e) and (g)) shows the vanishing of vector field $ϕ$ at extremal points of $\bar{f}$ (black dots), i.e., at small black holes (SBH) and large black holes (LBH) and also at the black hole possessing the lowest temperature ${\bar{T}}_{\min}$ [see plot (a)]. Right panel (i.e., plots (b), (d), (f) and (h)) shows the behavior of deflection angle $Ω (ϑ)$ for corresponding black holes in the left panel. Plots are displayed for $n = 2$ .
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Physical Review D

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