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Subregion entropy for the doubly-holographic global black string

Karch, Andreas, Perez-Pardavila, Carlos, Riojas, Marcos, Youssef, Merna·2023·Journal of High Energy Physics
DOI 10.1007/jhep05(2023)195
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Abstract

We study the growth of entanglement entropy in a doubly holographic model of gravity for a spherical AdS black hole. Compared to previous work, which was limited to the case of planar black holes, this introduces an extra scale to the problem. This allows us to analyze the interplay between the reorganization of entanglement entropy due to island formation and the onset of the Hawking-Page phase transition and to find the appearance of a new critical black hole radius unrelated to the thermodynamics. We also find that the geometry of the Ryu-Takayanagi surface capturing the physics of islands exhibits drastically different behavior than in the planar case.

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Figure 1 The diameter of the black string is determined by the order parameter in the theory, u h . The bulk theory is dual to a thermal BCFT state on the surface of the sphere. When u h becomes larger than u HP = 1 2 1 / 3 we encounter the Hawking-Page transition and the string is no longer the global minimum of the free energy. The black string reaches minimum temperature at the spinodal point u s = √ 3 2 2 / 3 , where the specific heat changes sign from positive to negative. Also see figure 2.
Figure 1 The diameter of the black string is determined by the order parameter in the theory, u h . The bulk theory is dual to a thermal BCFT state on the surface of the sphere. When u h becomes larger than u HP = 1 2 1 / 3 we encounter the Hawking-Page transition and the string is no longer the global minimum of the free energy. The black string reaches minimum temperature at the spinodal point u s = √ 3 2 2 / 3 , where the specific heat changes sign from positive to negative. Also see figure 2.CC-BY · From the PDF
Figure 2 The temperature of the black hole scales non-monotonically with u h , but for large black holes, the relationship is approximately linear. Unlike in asymptotically flat space, where black holes always have negative specific heat, AdS black holes have positive specific heat when they are large enough. The canonical ensemble in AdS is sensible because a temperature can be defined in the dual theory. For β above the Hawking-Page phase transition, where the radius of the black hole is the AdS radius, the system will transition to a gas of thermal radiation. Such black holes cannot be in thermal equilibrium in the canonical ensemble. For u h past the spinodal point, black holes become hotter as they evaporate, so their specific heat is negative. Since AdS black holes do not exist above the value of β where the spinodal point occurs, they do not contribute at all to the canonical ensemble.
Figure 2 The temperature of the black hole scales non-monotonically with u h , but for large black holes, the relationship is approximately linear. Unlike in asymptotically flat space, where black holes always have negative specific heat, AdS black holes have positive specific heat when they are large enough. The canonical ensemble in AdS is sensible because a temperature can be defined in the dual theory. For β above the Hawking-Page phase transition, where the radius of the black hole is the AdS radius, the system will transition to a gas of thermal radiation. Such black holes cannot be in thermal equilibrium in the canonical ensemble. For u h past the spinodal point, black holes become hotter as they evaporate, so their specific heat is negative. Since AdS black holes do not exist above the value of β where the spinodal point occurs, they do not contribute at all to the canonical ensemble.CC-BY · From the PDF
Figure 3 This figure shows the critical anchor on the brane, as a function of brane angle, as a percentage of the horizon radius, for various choices of order parameter u h . The black string's diameter, and correspondingly, the black hole's size on the KR brane, decreases (increases) as u h is increased (decreased). The Hawking-Page transition lies at u h = 1 2 1 / 3 ≈ . 8 . We recover our result for a large black string, discussed in [21], by taking the limit u h → 0 . There is a 'hole' on the atoll when the critical anchor plot is not 1-to-1, which occurs for any non-zero value of u h . As u h →∞ , the black string's diameter shrinks to a point, and the plot approaches a square shape. There is only one critical anchor below the critical angle ( θ c ≈ . 98687 ), but there are two critical anchors between the critical angle and the saturation angle θ s .
Figure 3 This figure shows the critical anchor on the brane, as a function of brane angle, as a percentage of the horizon radius, for various choices of order parameter u h . The black string's diameter, and correspondingly, the black hole's size on the KR brane, decreases (increases) as u h is increased (decreased). The Hawking-Page transition lies at u h = 1 2 1 / 3 ≈ . 8 . We recover our result for a large black string, discussed in [21], by taking the limit u h → 0 . There is a 'hole' on the atoll when the critical anchor plot is not 1-to-1, which occurs for any non-zero value of u h . As u h →∞ , the black string's diameter shrinks to a point, and the plot approaches a square shape. There is only one critical anchor below the critical angle ( θ c ≈ . 98687 ), but there are two critical anchors between the critical angle and the saturation angle θ s .CC-BY · From the PDF
Figure 4 This cartoon illustrates typical RT surfaces, above the critical angle θ c , for an eternal black string dual to a BCFT on the surface of a sphere. When the KR brane lies above the critical angle, RT surfaces can travel from the KR brane to the defect from either the upper or lower critical anchor. RT surfaces can reach the boundary only from the atoll, and the region between the critical anchors is always a hole in the atoll. Increasing the brane angle decreases the size of the hole; at the saturation angle, the upper and lower critical anchors annihilate at the saturation anchor. Note that similar observations apply to anchors within a finite distance of the defect on the non-gravitating bath.
Figure 4 This cartoon illustrates typical RT surfaces, above the critical angle θ c , for an eternal black string dual to a BCFT on the surface of a sphere. When the KR brane lies above the critical angle, RT surfaces can travel from the KR brane to the defect from either the upper or lower critical anchor. RT surfaces can reach the boundary only from the atoll, and the region between the critical anchors is always a hole in the atoll. Increasing the brane angle decreases the size of the hole; at the saturation angle, the upper and lower critical anchors annihilate at the saturation anchor. Note that similar observations apply to anchors within a finite distance of the defect on the non-gravitating bath.CC-BY · From the PDF
Figure 5 This figure shows the saturation angle for the brane on the left, and the saturation anchor for the brane on the right, both as a function of u h . For a given value of u h , the saturation angle is the largest brane angle with a hole on the atoll. As the brane angle increases, the hole shrinks to a point that we call the saturation anchor. Our numerics are consistent with result 2, in the appendix, which states that the saturation anchor always lies inside the photon sphere.
Figure 5 This figure shows the saturation angle for the brane on the left, and the saturation anchor for the brane on the right, both as a function of u h . For a given value of u h , the saturation angle is the largest brane angle with a hole on the atoll. As the brane angle increases, the hole shrinks to a point that we call the saturation anchor. Our numerics are consistent with result 2, in the appendix, which states that the saturation anchor always lies inside the photon sphere.CC-BY · From the PDF
Image 6
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Image 7
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Figure 8 Here we present our phase diagram for the doubly-holographic global black string model. Since the thermal BCFT state lives on the surface of a sphere, there is a scale dependence in the model that depends on u h -see figure 1 and figure 2. Each point on this plot gives the brane angle for which the area difference vanishes, which we call the Page angle θ p . The Page angle was found to increase with u h . The island surfaces are dominant when the area difference is negative. It was shown in previous work [21] that the tiny island phase is dominant above the critical angle because the area difference diverges to -∞ . In contrast, finite-size islands dominate below the critical angle when the brane lies above the Page angle. When u h > u crit h , which is ≈ . 62777 , the island surfaces are dominant only when the brane lies above the critical angle.
Figure 8 Here we present our phase diagram for the doubly-holographic global black string model. Since the thermal BCFT state lives on the surface of a sphere, there is a scale dependence in the model that depends on u h -see figure 1 and figure 2. Each point on this plot gives the brane angle for which the area difference vanishes, which we call the Page angle θ p . The Page angle was found to increase with u h . The island surfaces are dominant when the area difference is negative. It was shown in previous work [21] that the tiny island phase is dominant above the critical angle because the area difference diverges to -∞ . In contrast, finite-size islands dominate below the critical angle when the brane lies above the Page angle. When u h > u crit h , which is ≈ . 62777 , the island surfaces are dominant only when the brane lies above the critical angle.CC-BY · From the PDF

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Metadata

Authors
Karch, A., Perez-Pardavila, C., Riojas, M., & Youssef, M.
ORCID
Youssef, Merna
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2023-05-25
Volume
2023
Issue
5
ISSN
1029-8479
DOI
10.1007/jhep05(2023)195
Cited by
7
References
2
Language
en
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