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Black hole heat engines, subregion complexity and bulk metric reconstruction

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Black hole heat engines, subregion complexity and bulk metric reconstruction

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Content BLACK HOLE HEAT ENGINES, SUBREGION COMPLEXITY AND BULK METRIC RECONSTRUCTION by Avik Chakraborty A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2021 Copyright 2021 Avik Chakraborty Dedication To Maa and Baba ... ii Acknowledgments I am grateful to my advisor Dr. Clifford V. Johnson for his guidance and patience and for devoting his time and energy to assist me during this long period of my PhD life. I want to thank him for sharing his vast knowledge of the subject and for his encouragement in my each little step taken towards my goal. In addi- tion to his technical guidance, his various interests and thoughtful advice in every occasion helped me shaping my career. I am indebted to him for all his guidance and support and I am going to miss his sense of humor and his homemade cake that we used to get during our weekly group meeting! I am also grateful to my dissertation committee members Dr. Stephan Haas, Dr. Krzysztof Pilch, Dr. Elena Pierpaoli and Dr. Aaron Lauda for their time and kind consideration. I would like to thank the faculty members of the high energy theory group at USC, Dr. Nicholas Warner, Dr. Krzysztof Pilch, Dr. Itzhak Bars, Dr. Hubert Saleur and Dr. Dennis Nemeschansky for their expertise and enthu- siasm for the discipline. I received constant help from the departmental staff members Betty Byers, Mary Beth Hicks, Lisa Moeller, Christina Tasulis Williams, Kimberly Burger and Allison Bryant in various issues during the program and I am really thankful to them. I wish to thank my friends and colleagues Dr. Robert Walker, Felipe Rosso, Ashton Lowenstein, Dr. Albin James, Dr. Ignacio Araya, Dr. Alexander Tyukov iii and all the other members of the group. I am specially indebted to Felipe, Ashton and Robert for many interesting discussions about physics and beyond. I also want to thank my friends Dr. Zoe González Izquierdo, Dr. Siavash Yasini, Dr. Arash Roshani and Nareg Mirzatuny for always being there. I am grateful and wish to thanktoallmyfriendswhohavemademefeellikehomeawayfromhome. Without my local and overseas friends, life in Los Angeles would have been very different. I must thank my old mentor Dr. Chethan Krishnan and my friend, philosopher and guide Bikash Chakraborty and Asis Sarkar for their guidance, support and advice throughout my whole career. Finally, I must express my respect and my gratitude to my parents, my sisters and other family members for their love and affection towards me. Without their mental support and encouragement throughout the program this thesis could have never been written. iv Contents Dedication ii Acknowledgments iii List of Tables viii List of Figures ix Abstract xi Overview 1 I Holographic Heat Engines 3 1 Brief Review of Black Hole Thermodynamics with Λ 4 1.1 Black Hole Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Extended Black Hole Thermodynamics . . . . . . . . . . . . . . . . 6 1.2.1 An example: Charged AdS Black Holes . . . . . . . . . . . . 9 2 Benchmarking Static Black Hole Heat Engines 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Heat Engines and Efficiency . . . . . . . . . . . . . . . . . . 13 2.1.2 Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Setting up the Circular Cycle . . . . . . . . . . . . . . . . . . . . . 17 2.3 The Ideal Gas Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Comparing different Heat Engines . . . . . . . . . . . . . . . . . . . 22 2.4.1 Einstein–Hilbert–Maxwell . . . . . . . . . . . . . . . . . . . 23 2.4.2 Gauss–Bonnet . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Born–Infeld . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.4 Comparison/Observations . . . . . . . . . . . . . . . . . . . 26 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 v 3 Benchmarking Rotating Black Hole Heat Engines 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The Generalized Benchmarking Cycle . . . . . . . . . . . . . . . . . 33 3.3 Benchmarking Kerr–AdS . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Background Thermodynamics . . . . . . . . . . . . . . . . . 35 3.3.2 Key Elements of the Tessellation . . . . . . . . . . . . . . . 37 3.3.3 An Exact Result . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 II Application of AdS/CFT: Holographic Superconduc- tor 44 4 AdS/CFT Basics 45 4.1 Anti-de Sitter Spacetime . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 AdS in Global Coordinates . . . . . . . . . . . . . . . . . . . 46 4.1.2 AdS in Static Coordinates . . . . . . . . . . . . . . . . . . . 47 4.1.3 AdS in Poincaré Coordinates . . . . . . . . . . . . . . . . . . 48 4.2 Conformal Field Theories . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.1 Conformal Group and Algebra . . . . . . . . . . . . . . . . . 49 4.3 AdS/CFT Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5 Entanglement Entropy 54 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 Entanglement Entropy in CFT . . . . . . . . . . . . . . . . . . . . . 58 5.4 Holographic Entanglement Entropy . . . . . . . . . . . . . . . . . . 59 5.4.1 Testing the Proposal: AdS 3 /CFT 2 . . . . . . . . . . . . . . . 60 5.4.2 HEE for a Straight Belt in CFT d+1 . . . . . . . . . . . . . . 61 6 Computational Complexity 63 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Holographic Subregion Complexity . . . . . . . . . . . . . . . . . . 66 6.2.1 HSC for a Straight Belt in CFT d . . . . . . . . . . . . . . . 66 6.2.2 Case of Pure AdS . . . . . . . . . . . . . . . . . . . . . . . . 68 7 Holographic Superconductor 69 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2 A Simple Model: Planar Schwarzschild–AdS Black Hole in D = 4 . 70 vi 8 The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 74 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.2 Dual Gravity Background of 2 + 1–dimensional Holographic Super- conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.2.1 RN–AdS Solution . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2.2 Hairy Black Hole Solution . . . . . . . . . . . . . . . . . . . 81 8.2.3 Zero Temperature Solution . . . . . . . . . . . . . . . . . . . 82 8.3 The Entanglement Entropy and The Subregion Complexity . . . . . 83 8.3.1 O 1 Superconductor . . . . . . . . . . . . . . . . . . . . . . . 86 8.3.2 O 2 Superconductor . . . . . . . . . . . . . . . . . . . . . . . 87 8.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 90 III Bulk Metric Reconstruction 93 9 Gravity Dual of ABJM from Holographic Entanglement Entropy 94 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.1.1 Spacetime from Entanglement . . . . . . . . . . . . . . . . . 96 9.2 ABJM and AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.3 Bulk metric reconstruction . . . . . . . . . . . . . . . . . . . . . . . 99 9.3.1 The bulk dual of ABJM in its ground state . . . . . . . . . . 102 9.3.2 Fixing the Last Metric Component . . . . . . . . . . . . . . 106 9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Bibliography 109 A Formulas for Benchmarking Kerr–AdS Black Hole Heat Engines129 A.1 Static Black Holes in D = 4 Dimensions . . . . . . . . . . . . . . . 129 A.1.1 Einstein–Hilbert–Maxwell . . . . . . . . . . . . . . . . . . . 129 A.1.2 Born–Infeld . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 A.2 Υ–function in Higher Dimensions . . . . . . . . . . . . . . . . . . . 131 vii List of Tables 4.1 The AdS/CFT dictionary . . . . . . . . . . . . . . . . . . . . . . . 53 viii List of Figures 1.1 Here are some sample isotherms. We have chosen q = 0.05 and T from 0.45 to 1.5 in steps of 0.15. The lower curves are at lower temperatures. . . . . . . . 10 2.1 A prototype engine (a) and the standard heat engine flows (b). . . . . . . . . 14 2.2 Adding cycles that share an edge. . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Example of tessellating the circular cycle for N = 10 (100 squares). Red lines are the tops of hot cells and blue lines are the bottoms of cold cells. As N increases, these lines converge to the boundary of the circle. The dashed black lines are sample isotherms. . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Efficiency of the ideal gas used in the benchmark cycle, computed as a function of circle radius L. The ideal gas (see text) equation of state was used in the algorithm for N = 500 with the circle origin at (110,20), and radius L = 10. The blue crosses plot the result forη. The red curve is a plot of the exact result from equation (2.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 The efficiency of our benchmarking cycle as a function of grid size, N. Here Einstein–Hilbert–Maxwell black holes are used as the working substance. Blue crosses represent the Carnot efficiency η C , while black squares represent η. For N = 500, η C and η converge to 0.6674942748 and 0.5653677678 respectively. . . 24 ix 2.6 The efficiency of our benchmark cycle as a function of grid size,N. Here Gauss– Bonnet black holes are used as the working substance. Blue crosses represent the Carnot efficiencyη C , while black squares represent η. ForN = 500,η C and η converge to 0.6674954523 and 0.5653678245 respectively. . . . . . . . . . . 24 2.7 The efficiency of our benchmark cycle as a function of grid size, N. Here Born– Infeld black holes are used as the working substance. Blue crosses represent the Carnot efficiency η C , while black squares represent η. For N = 500, η C and η converge to 0.6674942730 and 0.5653678967 respectively. . . . . . . . . . . . 25 2.8 The efficiencies of Einstein–Hilbert-Maxwell (lowest), Gauss–Bonnet (highest) and Born–Infeld (middle) black hole heat engines forN = 500 with circle origin at (110,20) and radiusL = 10. For additional comparison, the ideal gas case of section 2.3 has η' 0.56588 at N = 500, and is η = 2π/(π +8) exactly. . . . . 27 3.1 (a) Prototype engine for static black hole. (b) Prototype engine for rotating black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 A sample benchmarking circular cycle for N = 10, showing the tessellation along isobars and adiabats. The equation of state comes from D = 4 Kerr–AdS black holes. Red lines represent hot cells and blue lines represnt cold cells. As N increases, these lines converge to the boundary of the circle. Black dashed lines show a family of isotherms. We worked at fixed a here, with a = 0.04. . . 34 3.3 Results for the efficiency of the benchmarking circle for a range of rotation parameters. The results were obtained by direct integration of the heat to evaluate the efficiency. The blue dots are the numerical results, while the solid curve is the exact “ideal gas” result of equation (3.24). . . . . . . . . . . . . 40 x 3.4 The efficiency of our benchmarking cycle as a function of grid size, N. Here Kerr–AdS black holes inD = 4 are used as the working substance. Blue crosses represent the Carnot efficiency η C , while black squares represent η. For N = 100, η C and η begin to show convergence to approximately 0.295355525453494 and 0.234033334108944 respectively. We set a = 0.001. . . . . . . . . . . . . 40 3.5 TheefficienciesofKerr–AdS(lowest),Born–Infeld(middle)andEinstein–Hilbert– Maxwell (highest) black hole heat engines for N = 100 with circle origin at (50,30) and radius L = 5. (Since the static cases lie close to each other in the plot, an inset is included to resolve them.) . . . . . . . . . . . . . . . . . . 42 4.1 A cartoon of AdS spacetime where the conformal boundary is ar r→∞ and the Poincaré horizon is at r→ 0. This diagram is taken from [65]. . . . . . . 48 5.1 Thelinegeometryconsideredinthischapter. Herez denotestheradialdirection in the dual bulk geometry AdS 3 . The interval we consider isl on an infinite line of length L. z = z ∗ denotes the turning point of the minimal surface (in this case a semi circle of radius l/2) inside the bulk. . . . . . . . . . . . . . . . 60 5.2 Minimal surfaces in AdS d+2 : a) Straight belt A S and b) Circular disk A D . In this section we will compute the entanglement entropy for the straight belt. This figure is taken from refs. [73, 74]. . . . . . . . . . . . . . . . . . . . . 61 6.1 Penrose diagram of a double sided AdS black hole dual to a thermofield double state. The left and right CFTs live on the left and right boundaries. The dashed red line represents the singularity. The green line is a maximal codimension–one surface at t→∞. Note that it does not reach the singularity. Blue lines are few other maximal codimension–one surfaces. . . . . . . . . . . . . . . . . 64 6.2 A straight belt with width l and length L→∞ on the boundary z = 0. This diagram is taken from ref. [106]. . . . . . . . . . . . . . . . . . . . . . . 66 xi 7.1 A qualitative diagram of the condensate as a function of temperature for the O 2 superconductor. It goes to zero at T =T c . . . . . . . . . . . . . . . . . 72 8.1 The strip geometry considered in this chapter. Here z denotes the radial direc- tion in the dual bulk geometry AdS 4 . The strip width is l and the length is L which can be taken to infinity. The boundary is at z = 0 where the field theory lives and z =z ∗ denotes the turning point of the minimal surface inside the bulk. 83 8.2 The entanglement entropy (a) and the subregion complexity (b) as functions of thestrip–widthlfortheO 1 superconductorforafixedtemperature: R 1/2 (16πG4) 1/4 T √ ρ = 0.053. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. . . . . . . . . . . . . . . . . . . 86 8.3 The entanglement entropy (a) and the subregion complexity (b) as functions of thestrip–widthlfortheO 1 superconductorforafixedtemperature: R 1/2 (16πG4) 1/4 T √ ρ = 0. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. . . . . . . . . . . . . . . . . . . . . 86 8.4 The entanglement entropy (a) and the subregion complexity (b) as functions of thetemperatureT fortheO 1 superconductorforafixedl: √ ρ(16πG 4 ) 1/4 R −1/2 l/2 = 2.5. The red dashed(or dotted) curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. The black solid line denotes the transition temperature T c . Since the zero temperature solution is exactly known, we include our results for T = 0 in the plot. . . . . . . . . . . . . . 86 8.5 The entanglement entropy (a) and the subregion complexity (b) as functions of thestrip–widthlfortheO 2 superconductorforafixedtemperature: R 1/2 (16πG4) 1/4 100T √ ρ = 0.305. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. . . . . . . . . . . . . . . . . . . 88 xii 8.6 The entanglement entropy (a) and the subregion complexity (b) as functions of thestrip–widthlfortheO 2 superconductorforafixedtemperature: R 1/2 (16πG4) 1/4 100T √ ρ = 0. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. . . . . . . . . . . . . . . . . . . . . 88 8.7 (a) Behavior of f(z) for theO 1 superconductor. This has been shown for: R 1/2 (16πG4) 1/4 T √ ρ = 0.053. (b) Behavior of f(z) for theO 2 superconductor. This has been shown for: R 1/2 (16πG4) 1/4 100T √ ρ = 0.305. . . . . . . . . . . . . . . . . . 88 8.8 A comparison of the multivalued regions of the entanglement entropy (a) and the subregion complexity (b) as functions of the strip–width l for theO 2 super- conductor for a fixed temperature: R 1/2 (16πG4) 1/4 T √ ρ = 0. . . . . . . . . . . . . . 89 8.9 The entanglement entropy (a) and the subregion complexity (b) as functions of thetemperatureT fortheO 2 superconductorforafixedl: √ ρ(16πG 4 ) 1/4 R −1/2 l/2 = 2.5. The red dashed(or dotted) curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. The black solid line denotes the transition temperature T c . Since the zero temperature solution is exactly known, we include our results for T = 0 in the plot. . . . . . . . . . . . . . 89 9.1 A cartoon demonstrating how tuning the entanglement entropy to zero produces two disjoint spacetimes. The RT surface is shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.2 A sketch of an entangling region in a 3d CFT and its corresponding RT surface in the bulk. . . . . . . . . . . . . . . . . . . . . . . . . . 100 xiii Abstract This thesis examines three different areas in theoretical high energy physics: in the first part we talk about the extended black hole thermodynamics and con- sequently present a systematic way of computing and comparing the efficiencies of the holographic heat engines. In the second part, using the ideas derived from the AdS/CFT correspondence we compute the entanglement entropy and the sub- region complexity for a holographic superconductor and gain some novel physical insights that could be useful for our understanding of the strongly coupled field theories. Given the data of a boundary field theory, how to partially reconstruct the dual bulk geometry using general ideas from the AdS/CFT correspondence has been discussed in the third part. As we have just mentioned, in the first part of this thesis, we present the results of initiating a benchmarking scheme that allows for cross–comparison of the efficiencies of black holes used as working substances in heat engines. We use a circular cycle in the p−V plane as the benchmark engine. We test it on the Einstein–Maxwell, Gauss–Bonnet, and Born–Infeld black holes. Also, we derive a new and surprising exact result for the efficiency of a special “ideal gas” system to which all the black holes asymptote. Next we extend our benchmarking scheme to rotatingblackholesagainusingacircularcycleinthep−V planeasthebenchmark engine. We compare Kerr to Einstein–Maxwell and Born–Infeld black holes. As xiv in the static case, we derive an exact formula for the benchmark efficiency in an ideal–gas–like limit that may serve as an upper bound for (generic) rotating black hole heat engines in the thermodynamic ensemble we choose here. In the second part of this thesis, we present the results of our computation of the subregion complexity and also compare it with the entanglement entropy of a 2 + 1–dimensional holographic superconductor which has a fully backreacted gravity dual with a stable ground sate. We follow the “complexity equals volume” or the CV conjecture. We find that there is only a single divergence for a strip entangling surface and the complexity grows linearly with the large strip width. During the normal phase the complexity increases with decreasing temperature, butduringthesuperconductingphaseitbehavesdifferentlydependingontheorder of phase transition. We also show that the universal term is finite and the phase transition occurs at the same critical temperature as obtained previously from the free energy computation of the system. In one case, we observe multivaluedness in the complexity in the form of an “S” curve. Recent work has shown that the entanglement and the structure of spacetime areintimatelyrelated. Onewaytoinvestigatethisistobeginwithanentanglement entropyinaconformalfieldtheory(CFT)andusetheAdS/CFTcorrespondenceto calculate the bulk metric. In the final part, we perform this calculation for ABJM, a particular 3-dimensional supersymmetric CFT (SCFT), in its ground state. In particular we are able to reconstruct the pure AdS 4 metric from the holographic entanglemententropyoftheboundaryABJMtheoryinitsgroundstate. Moreover, we are able to predict the correct AdS radius purely from the entanglement. We also address the general philosophy of relating the entanglement and the spacetime through the Holographic Principle, as well as some of the philosophy behind our calculations. xv Overview This thesis contains three parts. The first part is named as “Holographic Heat Engines”. We will start with a brief review of black hole thermodynamics with Λ in chapter 1. We will discuss about the classic black hole thermodynamics and consequently the proposed extended black hole thermodynamics framework. Using a simple example of charged AdS black hole, we will show how this new idea works. In this new framework “holographic heat engine” is a natural concept. In chapter 2 we will present our benchmarking scheme for static black hole heat engines. After a short introduction to heat engines, we will set up our circular cycle as a benchmarking cycle and compute efficiencies of different heat engines and compare them. We will also present an exact result for the ideal–gas–like case. Then in chapter 3 we will extend our benchmarking scheme to include rotations as well. We will show how to modify our static–case algorithm to the non–static case and compute the efficiency of Kerr black hole heat engines. We will derive another ideal–gas–like exact result for this rotating case. We will conclude this chapter with a discussion section and some open problems. The second part is called “Application of AdS/CFT: Holographic Supercon- ductor”. Our aim is to compute and compare two interesting quantum information quantities for a holographic superconductor. In chapter 4 we will first review the basics of AdS/CFT correspondence, including the AdS geometry and the algebra 1 of CFT, as well as, the AdS/CFT dictionary. Then in chapter 5 we will review the entanglement entropy: its basic formalism, entanglement entropies in CFT and how to compute entanglement entropy holographically using a simple exam- ple of a 2d CFT on an infinite line. Chapter 6 will be devoted to the discussion of computational complexity and the proposed conjecture of computing holographic subregion complexity. We will introduce holographic superconductor with a simple example of planar Schwarzschild–AdS black hole in chapter 7. Finally in chapter 8, after reviewing the dual gravity background we will present our numerical results of the entanglement entropy and the subregion complexity for a 2 + 1–dimensional holographic superconductor. We will conclude this chapter with a short summary and possible areas for future research. Thenameofthethirdorthefinalpartis“BulkMetricReconstruction”. Mostof the key elements for this part are already discussed in second part of this thesis. So in chapter 9, after a very brief review of the holographic entanglement entropy and the spacetime, we will discuss about the ABJM theory that we will work with and the general philosophy for using the entanglement entropy to calculate the metric of the bulk spacetime. Finally We will consider ABJM theory, a particular 3-dim CFT in its ground state reduced to a strip in Minkowski space and present the techniques to reconstruct pure AdS 4 from the entanglement entropy of this CFT. At the end of this final chapter, we will discuss possible issues in our methodology and future directions to further develop and improve our techniques. In an appendix A we will write down the necessary formulae needed for the analysis in chapter 3. This thesis is heavily based on the content (and in some chapters a verbatim reproduction) of the work done by the author in refs. [1–4]. 2 Part I Holographic Heat Engines 3 Chapter 1 Brief Review of Black Hole Thermodynamics with Λ 1.1 Black Hole Mechanics The fact that the black holes behave as thermodynamic objects and can have an associated entropy has been known for years now. This surprising discovery has changed our understanding of the relationship between general relativity, ther- modynamics and quantum theory. Black holes are classical solutions to Einstein’s equations and classically they absorb everything and emit nothing. They have zero temperature since they can not radiate and they have zero entropy as well. So, there is no reason for them to exhibit thermodynamic behavior in the first place. The idea was first introduced by Bekenstein who observed the similarity of second law of thermodynamics and Hawking’s area theorem which says that the area of the event horizon of a black hole can never decrease [5–7]. Extending this analogy further, Bardeen, Carter and Hawking proposed the “four laws of black hole mechanics” with the assumption that the event horizon of a black hole is a killing horizon which is a null hypersurface generated by a corresponding killing vector field [8]. These laws show remarkable similarities with the laws of ordinary thermodynamics. They are stated below: • Zeroth Law: The surface gravity κ is constant over the event horizon of a sta- tionary black hole. This law is suggestive of a relationship between κ and the 4 temperature T of a black hole body in thermal equilibrium. The proof of this result is given in [8]. • First Law: For two stationary black holes differing only by small variations in the mass M, area A, angular momentum J and charge Q: δM = κ 8πG δA + ΩδJ + ΦδQ , (1.1) where Ω is its angular velocity, and Φ is its electric potential at the horizon. It looks like the first law of thermodynamics with κ being the temperature, A being the entropy and M playing the role of the energy. • Second Law: Hawking’s area theorem: δA≥ 0, i.e., the area A of a black hole’s event horizon can never decrease. This suggests the analogy between the horizon area A and the entropy S. • Third Law: It is impossible to reduce the surface gravity κ to zero in a finite number of processes. Soon after that, by taking quantum effects into account, Hawking showed that the black holes can thermally radiate and they have an associated temperature as well [9]: k B T = ~κ 2πc , (1.2) wherek B is the Boltzmann’s constant,c is the speed of light and~ is the Planck’s constant. With the discovery of Hawking’s radiation it was understood that the black holes are actual physical thermodynamic systems that have temperature and entropy. The First Law now reads: δM =TδS + ΩδJ + ΦδQ , (1.3) 5 with the following identification: M =U , T = κ 2π , S = A 4 , (1.4) where,U,T andS are respectively the energy, the temperature and the entropy of the black hole 1 . The thermodynamic variables in (1.3) satisfy the following Smarr relation (in D = 4 spacetime dimensions): M = 2(TS + ΩJ) + ΦQ , (1.5) thereby connecting the intensive (T, Ω, Φ) and extensive (M,J,Q) thermodynamic variables. 2 1.2 Extended Black Hole Thermodynamics One interesting feature of the first law is the missing usual pδV term. There is no concept of pressure or volume of a black hole unlike our ordinary thermody- namics where we have well defined pressure and volume. In recent years there have been many efforts to incorporate this notion into the first law. The idea is that the pressurecanbeassociatedwithanegativecosmologicalconstant Λ, whichisaform of energy whose (positive) pressure is equal in magnitude to its (negative) energy density. One can think of this as a black hole “immersed” in the environment of a 1 We have set G,c,~,k B to unity. 2 For a selection of review papers on standard black hole thermodynamics, see refs. [10–14]. 6 negative cosmological constant. Λ< 0 gives rise to asymptotically Anti–de Sitter black holes which are the solutions of Einstein’s equations: R ab − 1 2 g ab R + Λg ab =T ab , (1.6) where T ab is the stress-energy tensor and Λ is parametrized by the AdS radius l via the following equation: Λ =− (D− 1)(D− 2) 2l 2 < 0 , (1.7) where D is the number of spacetime dimensions. The notion of Λ being a dynamical variable was first proposed by Teitelboim andBrown[15, 16]byconsideringathree–indextensor coupled tothegravitational field. In the black hole solution the cosmological constant Λ then appears as a constant of integration (hairy black hole). In fact, in many theories Newton’s constant, Yukawa coupling, gauge coupling and other physical constants can arise asvacuumexpectation values andhence they canvary. Anothermotivationbehind this idea was the problem of the observed vanishing value of Λ. Consequently, this idea was explored by many and a generalized first law was written down [17–20]: dM =TdS +Vdp + ΩdJ + ΦdQ , (1.8) where, p =−Λ/8π = (D− 1)(D− 2)/16πl 2 , (1.9) 7 is interpreted as the thermodynamic pressure and the conjugate thermodynamic volume is given by [21, 22]: V≡ (∂M/∂p) S,Q,J . (1.10) Withthisidentification, theblackholemassisnowrelatedtotheenthalpyH ofthe system instead of the internal energy U [20]: M =H≡U +pV. This is the total energy required to create a black hole and put it in the AdS space. In general, the thermodynamic volume does not necessarily be same as the “geometric” volume of the black holes, but for static black holes, this indeed turns out to be true. In the static case, V is simply the “geometric” volume, namely, the volume of the ball of radius r + . The generalized Smarr relation now becomes: D− 3 D− 2 M =TS− 2 D− 2 pV + ΩJ + D− 3 D− 2 ΦQ . (1.11) This new framework is known as the extended black hole thermodynamics and it has shed new lights on our understanding of concepts like Van der Waals flu- ids, solid/liquid phase transitions, reentrant phase transitions, heat engines, triple points etc. from the gravitational perspective. See refs. [13, 23] for a detail review of the extended thermodynamics (sometimes called “black hole chemistry”) and many different aspects of this idea. 8 1.2.1 An example: Charged AdS Black Holes Let us now discuss a simple example of charged AdS black holes in four space- time dimensions and see how all of these come into play. The bulk action for the Einstein–Hilbert–Maxwell system in D = 4 dimensions is: I =− 1 16π Z d 4 x √ −g R− 2Λ−F 2 , (1.12) where Λ = −3/l 2 , and hence using the above-mentioned prescription, p = −Λ/8π = 3/8πl 2 . The solution of the equations of motion coming from this action gives rise to Reissner–Nordström black hole with the following metric: ds 2 =−f(r)dt 2 + dr 2 f(r) +r 2 (dθ 2 + sin 2 θdφ 2 ) , (1.13) where, f(r) = 1− 2M r + q 2 r 2 + r 2 l 2 . (1.14) M and q are the black hole mass and the charge respectively. The horizon is the zero locus of f(r) and is located at r = r + . The gauge potential is given by: A t = q(r−r + )/rr + and it vanishes at the horizon. By Wick-rotating the metric (1.13) to Euclidean signature and imposing regularity at the horizon, the temperature T can be computed as: T = |f 0 | 4π | r=r + = 3r 4 + +l 2 r 2 + −q 2 l 2 4πl 2 r 3 + . (1.15) The entropy S and the thermodynamic volume V are given by: S =πr 2 + and V = 4 3 πr 3 + . (1.16) 9 So the equation of state for a given charge q becomes [24]: p = 1 8π 4π 3 4 3 3T V 1 3 − 3 4π 2 3 1 V 2 3 + q 2 V 4 3 , (1.17) where we have replaced r + by V using the previous equation. Since we are using a canonical ensemble (fixed charge) the Φdq term vanishes. The enthalpy H(S,p) can be computed as: H(S,p) = 1 2 s S π 1 + πq 2 S + 8Sp 3 . (1.18) One can check that the Maxwell’s relations are also satisfied: ∂T ∂p S = 3V 4π 1 3 = 2 S π 1 2 = ∂V ∂S p . (1.19) Note that we can write the temperature T in terms of S and p as follows: T = 1 4 √ πS 1− πq 2 S + 8pS . (1.20) So the specific heat at constant volume becomes [24]: C V = T ∂S ∂T V = 1− 2S 1 2 √ π ∂p ∂T V 2S 8pS 2 +S−πq 2 8pS 2 −S + 3πq 2 . (1.21) But this quantity vanishes since (∂p/∂T ) V =π 1/2 /2S 1/2 . This fact will be of great importance and we will exploit it when we will discuss the static black hole heat engines in the next chapter. Note that the equation of state (1.17) can be recast as [13]: p = T v − 1 2πv 2 + 2q 2 πv 4 , v = 2r + = 2 3V 4π 1 3 , (1.22) 10 0.2 0.4 0.6 0.8 v -1 0 1 2 3 4 5 p Figure 1.1: Here are some sample isotherms. We have chosen q = 0.05 and T from 0.45 to 1.5 in steps of 0.15. The lower curves are at lower temperatures. which tantalizingly looks like the Van der Waals equation: p + a v 2 (v−b) =T , (1.23) where the parameters a and b correspond to the attraction between the particles and the volume of the fluid particles respectively. The Gibbs free energy in this case is given by: G =H−TS = l 2 r 2 + −r 4 + + 3q 2 l 2 4r + l 2 , (1.24) which exhibits a swallowtail behavior for pressures less than the critical value p c [13]. This is indicative of a first order phase transition. This behavior ends at the following critical point where the phase transition becomes second order: p c = 1 96πq 2 , v c = 2 √ 6q , T c = √ 6 18πq . (1.25) Notice that the ratio: p c v c /T c = 3/8 is exactly the same as we find for a Van der Waals fluid! In figure 1.1 we have shown few isotherms in the p−v plane 11 using (1.22). The jumps in the first order small-black-hole/large-black-hole phase transition removes the negative pressure region of the diagram. See ref. [13] for further details in this context. In the next chapter we will show how to define black hole heat engines in this extended thermodynamics framework and present a benchmarking scheme to compare different heat engines. 12 Chapter 2 Benchmarking Static Black Hole Heat Engines 2.1 Introduction Let us first summarize the key concepts discussed in the previous chapter. The classic black hole thermodynamics [5, 9, 25, 26] relates the massM, surface gravity κ, and outer horizon area A of a black hole solution to the energy, temperature, and entropy (U, T, and S, resp.) according to 1 : M =U , T = κ 2π , S = A 4 . (2.1) The formalism has been extended 2 by allowing the cosmological constant of the theory to be dynamical, supplying a pressure via p = −Λ/8π, along with its conjugate volumeV. Now, the black hole mass is related to the enthalpyH of the system instead of the energyU [20]: M =H≡U +pV. The First Law now reads as: dM =TdS +Vdp + Φ i dq i + Ω i dJ i . (2.2) 1 Here we are using geometrical units where G,c,~,k B have been set to unity. 2 For a selection of references, see refs. [18, 20–22, 27–31], including the reviews in refs. [13, 23, 32]. See also the early work in refs. [15, 33, 34]. 13 The temperature and the entropy remain related to the surface gravity and area of the black hole as in equations (2.1). The q i are gauge charges, and J i are angular momenta, while Φ i and Ω i are their conjugate potentials and angular velocities, respectively. The black holes may have other parameters and they enter additively with their conjugates to the First Law (2.2) in theusual way. This formalism works inmultipledimensions. Interestingly, forthestaticblackholes, thethermodynamic volume V is just the naive “geometric” volume of the black holes: the volume of the ball of radius r + (our notation for the horizon radius in this paper) 3 . 2.1.1 Heat Engines and Efficiency In this extended black hole thermodynamics, since the pressure and volume are now in play, alongside temperature and entropy (2.2), it is natural to study devices whichcan extract useful mechanical work fromheat energy, i.e., traditional heat engines [24]. These devices were named “holographic heat engines”, since for negative cosmological constant (i.e. with positive pressure, sincep =−Λ/8π) such cycles represent a journey through a family of holographically dual [36–40] non– gravitational field theories (at largeN c ) defined in one dimension fewer. Although we have holographic applications in mind for some of this work, for this thesis our focus will be on the black hole side of the story, an interesting context in its own right. So for the purposes of the gravitational theory, the working substance of the heat engine is a particular black hole solution of the gravity system. It supplies an equationofstatethroughtherelationbetweenitstemperatureT andtheblackhole parameters defined in the usual way (we will give examples below). The precise form of all these relations depends on the type of black hole, and the parent theory 3 This coincides with the definition of the volume of a static black hole proposed in ref. [35]. 14 of gravity under discussion. One may extract mechanical work from such an engine [24]viathepdV termintheFirstLawofthermodynamicsintheclassicway: Define a closed cycle in state space during which there is a net input heat flowQ H , a net output heat flow Q C , and a net output work W. So Q H = W +Q C . A central p V 1 2 3 4 (a) Q H Q C W (b) Figure 2.1: A prototype engine (a) and the standard heat engine flows (b). quantity, the efficiency for the cycle, is defined as η = W/Q H = 1−Q C /Q H . Its value is sensitive to the details of the equation of state of the system and also to the choice of cycle in state space. Consider the cycle given in figure 2.1a. In refs. [24, 41, 42] it is explained why this is a natural choice for static black holes 4 . For such holes, the entropy and the volume are not independent, being both simple functions ofr + , the horizon radius. So isochores are adiabats, and so the only heat flows are along the top and bottom lines. Computing the efficiency boils down to evaluating R C p dT along those isobars, where C p is the specific heat at constant pressure. Figure 2.1b shows the standard logic of the heat energy flows during one cycle of the heat engine. In general, calculation of efficiency is a difficult task 4 Refs. [43–50] have presented further studies of such heat engines. 15 to perform exactly using this approach, and high temperature or high pressure computations are used to get approximate results [41, 42]. Recently, however, ref. [51] showed a much simpler way to evaluate the effi- ciency. The First Law is: dH =TdS +Vdp , (2.3) and along the isobars, dp = 0. Therefore the total heat flow along an isobar is simply the enthalpy change. Normally, that might not be a useful rewriting, but in extended gravitational thermodynamics, a precise expression for the enthalpy is readilyavailablesinceitisjusttheblackholemassM. Thisresultsinaremarkably simple exact formula: η = 1− M 3 −M 4 M 2 −M 1 , (2.4) where the black hole mass is evaluated at each corner of the rectangle, with the labelling given in figure 2.1a. M is usually written as a function of r + and p. In the examples of this chapter, since V is a simple function of r + , we will be easily able to write down M as a function of p and V. It was also shown in ref. [51] that the result (2.4) can be used as the basis for an algorithm for computing the efficiency of a cycle of arbitrary shape to any desired accuracy. Any closed shape on the state space can be approximated by tiling with a regular lattice of rectangles. This is possible because cycles are additive (see Figure 2.2: Adding cycles that share an edge. figure 2.2). Consequently, only the cells at the edge contribute. Any mismatch 16 between the edge of the cycle’s contour and the tiling’s edge can be reduced by simply shrinking the size of the unit cell. Edge cells are called hot cells if they have their upper edges open, and cold cells if they have their lower edges open. Summing all the hot cell mass differences (evaluated at the top edges) will give Q H and summing all the cold cell mass differences (evaluated at the bottom edges) will yield Q C . So the efficiency is: η = 1− Q C Q H , Q H = X ith hot cell (M (i) 2 −M (i) 1 ), Q C = X ith cold cell (M (i) 3 −M (i) 4 ) .(2.5) where we have labelled all cells’ corners in the same way as the prototype cycle in figure 2.1a. An example with a triangular cycle was given in [51] to show the algorithm in action supporting the previous argument. 2.1.2 Benchmarking As already stated, a given black hole, thought of as a working substance for a heat engine, supplies a particular equation of state. The efficiency will depend upon this choice. Moreover, the efficiency will also depend upon the details of the choice of cycle. For maximizingη, certain choices of cycle will be better adapted to a particular working substance (choice of black hole) than others. (For example, for the same cycle of figure 2.1a, a non-static black hole will generically have a larger Q H due to non–zero heat flows along the isochores, and therefore a smaller η.) So a natural question arises: How does one compare the efficiency of different working substances? We have in mind a comparison that depends as little as possi- ble on special choices of cycle. In other words, in comparing working substances for making a heat engine, we should not choose a special cycle that favours one black 17 hole’s particular properties over another. Notice that this requirement requires us to make a choice that is in opposition to what is normally done: Cycles are usually chosen in a way that is naturally adapted to the equation of state in order to simplify computation. So we are asking that a more difficult choice of cycle be made, by necessity. This is where the exact formula and algorithm reviewed above come in. We can pick a benchmark cycle of whatever shape seems appropriate and implement the algorithm to compute η to any desired accuracy. This freedom allows us to make the following choice of benchmark: We choose thecycletobeacircleinthep−V plane[1]. Thelogicofthischoiceisthatthecircle is a simply parametrised shape which is also unlikely to favour any species of black hole (working substance) whatsoever. No thermodynamic variable is unchanged on any segment of the cycle, so it is, in some sense, a difficult cycle for all black holes. All that needs be specified is the origin of the circle and its radius. These properties make it an excellent choice of benchmark. The outline of this chapter is as follows. In section 2.2, we set up the circu- lar cycle as our benchmarking tool, and explain our implementation of the exact formula and algorithm of ref. [51] for calculating the efficiency. We then discuss, in section 2.3, a very special case of working substance: an “ideal gas"–like sys- tem. It allows us to derive some exact results that help test our implementation, and which also set a new benchmark standard for later use. In section 2.4, we compare three examples of black holes as working substances for heat engines: Charged (Reissner–Nordstrom–like) black holes, Gauss–Bonnet black holes, and Born–Infeld black holes. We conclude in section 2.5 with a brief discussion of future applications of our benchmarking procedure. 18 2.2 Setting up the Circular Cycle For our circle, we implemented the algorithm and exact formula of ref. [51], with the aid of a computer, as follows [1]: Imagine that we have chosen the origin and radius,L, of the circle in thep−V plane. We next overlaid it onto theN×N regular lattice of squares of total side length 2L. For simplicity, we used evenN so that there are same number of squares both in the upper half and in the lower half of the circle. Next we computed the pressure and volume at each corner of all the squares. Using simple geometry, we determined which squares intersect the circle. We checked for cases where two squares share a common isobar and both intersect the circle. Then, if we are in the upper part of the circle, we remove the one below and keep only the upper square. We did this check in the lower half of the circle in a similar fashion. This allowed us to identify all the hot cells and cold cells of the approximation, and their (p,V ) coordinates. The black hole mass is a function of pressurep and volumeV only (with some parameters that we have already fixed), so we can compute its value at each corner. Then we use the formula (2.5) to give us the approximateη for that level of granularity. Increasing the value ofN makes the size of the unit cell smaller, making the path traced by the hot and cold cells a better fit to our circle, reducing the error in η. Indeed, we found that just as for the triangle prototype of ref. [51], the efficiency converges nicely for large N. (See the examples in section 2.4.) We can even do more. Since temperature is also a function of p and V, we can compute it at each corner. Then while we run over all the cells to compute η, we can keep track of the maximum and the minimum temperatures (T H and T C ) achieved in the entire cycle. Hence, we can compute the Carnot efficiency η C = 1−T C /T H for this engine. This will be a check of our results because no cycle can have a greater efficiency than a Carnot cycle. 19 Figure 2.3: Example of tessellating the circular cycle for N = 10 (100 squares). Red lines are the tops of hot cells and blue lines are the bottoms of cold cells. As N increases, these lines converge to the boundary of the circle. The dashed black lines are sample isotherms. Figure 2.3 shows an example forN = 10. The green crosses show the points of the square lattice. The circle is our circular cycle. Red segments are the tops of the hot cells and blue segments are the bottoms of cold cells. The black dashed lines show a few sample isotherms determined from the underlying equation of state of the system in question. (This example is a snapshot of the Einstein–Hilbert– Maxwell case more fully explored in subsection 2.4.1). In choosing our benchmark cycle to compare different black holes, we should fix the circle origin and radiusL. Generically, the choices don’t matter, as long as they are the same across the comparison. We chose (p = 20,V = 110) here, and in the following sections, purely arbitrarily, except for making sure that we avoided any regions where the equations of state of the black holes under comparison had any multi–valuedness that would signal non–trivial phase transitions [52–54]. Such 20 regimes require a separate, more careful study in this heat engine context that are beyond the scope of this thesis. One might worry that since the circular cycle is presumably not even close to a cycle for which one has an analytically computable result, if there was an error, it might not be noticed. The Carnot test above is useful, but it is a rather weak upper bound on the efficiency. We derive some complementary tests, and a stronger (exact) bound, in the next section. 2.3 The Ideal Gas Case Before we proceed to study some black hole examples we briefly pause to study a simple but instructive case. It is in fact a limiting form of all of the black hole solutions we’ll discuss shortly. As discussed in ref. [42] it deserves to be called an ideal gas case, and as such, sets an additional standard by which we might assess other working substances. In dimension D, the leading large horizon radius (r + ) limit of all the asymptotically anti–de Sitter black holes we will discuss is rather simple, with dependence for the mass and temperature as follows: M =ω D−2 r D−1 + D− 1 p +··· , T = 4r + D− 2 p··· , (2.6) where ω D−2 is the volume (i.e., surface area) of the unit round S D−2 sphere. The exact thermodynamic volume for all of the static black holes under study is: V = ω D−2 (D− 1) r D−1 + , (2.7) 21 and so we have the familiar “ideal gas" behaviour in this large r + limit: pV 1/(D−1) ∼T , (2.8) a family of hyperbolae in the p−v plane where v = V 1/(D−1) . This ideal gas can be obtained as a limit for any of our black holes (in later sections) as either a large r + limit or as a high temperature limit. Before moving on to those cases, we can study this in its own right, taking the above as the equation of state everywhere in the p−V plane. Notice first that the efficiency of any cell such as the prototype of figure 2.1a simplifies nicely in this case. This is because the mass is simply M = pV, and hencep factors out in each mass difference, leaving only a volume difference. So η for figure 2.1a is just [24] η = 1−p 4 /p 1 . Turning to the efficiency of the circle, the factorization into sums of volume differences means that there is no dependence of the result on the volume coor- dinate of the circle’s origin: Any shift in the origin will cancel out everywhere. We can say even more in this case however. In fact, the terms in the sums in the algorithm (2.5) are actually entirely geometrical in interpretation! For example, for a hot cell a term is of the form p(V 2 −V 1 ). This is simply the area of the rectangular strip that starts on theV axis and is bounded above by the top of the cell. This is a clue to writing an exact formula for the efficiency in the case of our ideal gas. The simplest way to do it is to rewrite η as the ratio of work to heat flowing in,W/Q H . NowW =πL 2 while from our observation above,Q H is, in the 22 large N limit, exactly the area underneath the upper semi–circle of the circular path: Q H =πL 2 /2 + 2Lp, so our result is [1]: η = 2π π + 4p/L , (ideal gas) (2.9) where p is the pressure at the centre of the circle. This exact formula is rather surprising. Notably, in addition to being indepen- dent of V it is also independent of spacetime dimension, but the real surprise is that the algorithm assembled itself into a purely geometric result that yielded an exact formula for what is, on the face of it, a difficult shape of cycle. In fact, this exact geometrical result will work for any cycle shape. Perhaps there can be other surprises of this sort for other systems besides this special ideal gas case. The formula is also a rather useful check on our methods for a number of reasons. The first is that the p and L dependence are non–trivial predictions, and so we were obliged to check to see if our discrete algorithm reproduces such dependence, and indeed it did. For example, figure 2.4 shows, for N = 500, some example points computed by inserting the ideal gas into our algorithm. The red curve is the exact result of equation (2.9). The second reason this is a strong check is that it presents a lower upper bound on our results than the upper bound given by Carnot (discussed in the previous section). Our black holes, in the regions where we study them, can be thought of as perturbations of this ideal gas case, and so we should expect that the efficiencies we obtain approach (but do not exceed) the ideal gas result. We have, for the comparisons to come, the circle’s origin at p = 20, V = 110, and its radius as L = 10, for which the ideal gas efficiency is (to six significant figures) η = 2π/(π+8)' 0.56394. Itisworthnotingthatusingthediscretisationalgorithm 23 Figure 2.4: Efficiency of the ideal gas used in the benchmark cycle, computed as a function of circle radiusL. The ideal gas (see text) equation of state was used in the algorithm for N = 500 with the circle origin at (110,20), and radiusL = 10. The blue crosses plot the result for η. The red curve is a plot of the exact result from equation (2.9). to compute the ideal gas case gives η' 0.56588 at N = 500 and η' 0.56493 at N = 1000. (Moving significantly beyond N = 1000 to see further convergence proved beyond the numerical capabilities of the system we were using.) 2.4 Comparing different Heat Engines We now apply our benchmark cycle to a sampling of different black holes acting as working substances. We will only briefly introduce the black holes since they are well known in the literature. They were used in heat engines in refs. [24, 41, 42], with some analysis and comparison presented there, but now we have a clearer, more systematic benchmarking procedure. We will work in D = 5 for definiteness (it is trivial to insert the formulae for other dimensions into our algorithm; we saw no compelling reason to present the results for other dimensions here), and our benchmark circle will be centred at p = 20,V = 110, with radiusL = 10. In each case we list the bulk action inD = 5 24 dimensions and the mass and temperature of the black hole. For static black holes thevolumeV issimply: V =π 2 r 4 + /2. Also, recallthatthecosmologicalconstant Λ is related to pressurep viap =−Λ/8π, and inD dimensions Λ sets a length scalel through Λ =−(D− 1)(D− 2)/(2l 2 ). So in D = 5 dimensions, p = 3/(4πl 2 ). The mass and temperature formulae we present will have had l eliminated in favour of p. Note that in presenting our results for the efficiency, the engine’s actual effi- ciency will be denoted by η (without a subscript; the surrounding text will make it clear which case is being discussed) and the associated Carnot efficiency will be denotedη C (again with context making it clear as to which case is being discussed). This will help us avoid a proliferation of subscripts. 2.4.1 Einstein–Hilbert–Maxwell The bulk action for the Einstein–Hilbert–Maxwell system in D = 5 is 5 : I = 1 16π Z d 5 x √ −g R− 2Λ−F 2 . (2.10) We can now write the mass and the temperature of the Einstein–Hilbert–Maxwell (i.e., Reissner–Nordstrom–like) black hole solution, parametrized by a charge q (which we will later choose as q = 0.1): M = 3π 8 r 2 + + q 2 r 2 + + 4πp 3 r 4 + , and T = 1 4π 16πp 3 r + + 2 r + − 2q 2 r 5 + , (2.11) 5 We’re using the conventions of ref. [52]. 25 Figure 2.5: The efficiency of our benchmarking cycle as a function of grid size, N. Here Einstein–Hilbert–Maxwell black holes are used as the working substance. Blue crosses represent the Carnot efficiency η C , while black squares represent η. For N = 500, η C and η converge to 0.6674942748 and 0.5653677678 respectively. and we can write them entirely in terms ofp andV, usingr 4 + = 2V/π 2 . Figure 2.5 shows the results of the algorithm for computingη C andη for the benchmark circle in this case. 2.4.2 Gauss–Bonnet In the presence of a Gauss–Bonnet sector, the action becomes 6 : I = 1 16π Z d 5 x √ −g R− 2Λ +α GB (R γδμν R γδμν − 4R μν R μν +R 2 )−F 2 ,(2.12) whereα GB istheGauss–Bonnetparameterwhichhasdimensionsof (length) 2 . Ifwe setα GB = 0 in (2.12) we go back to the previous case of Einstein–Hilbert–Maxwell system (2.10). 6 We are using the conventions of ref. [55], with a slight modification of the Maxwell sector. 26 Figure 2.6: The efficiency of our benchmark cycle as a function of grid size, N. Here Gauss– Bonnetblackholesareusedastheworkingsubstance. BluecrossesrepresenttheCarnotefficiency η C , while black squares represent η. For N = 500, η C and η converge to 0.6674954523 and 0.5653678245 respectively. The mass and temperature of the black hole, parametrized by q and α are: M = 3π 8 α +r 2 + + q 2 r 2 + + 4πp 3 r 4 + , and T = 1 4π(1 + 2α r 2 + ) 16πp 3 r + + 2 r + − 2q 2 r 5 + , (2.13) where α = 2α GB . We will again work with q = 0.1 and we choose a sample value of the coupling as α = 0.001. See figure 2.6 for η C and η from the benchmark analysis. 27 2.4.3 Born–Infeld The so–called 7 Born–Infeld action [56–58] is a non-linear generalization of the Maxwell action, controlled by the parameter β : L(F ) = 4β 2 1− s 1 + F μν F μν 2β 2 (2.14) If we take the limitβ→∞ in (2.14) we recover old Maxwell action. The Einstein– Hilbert–Born–Infeld bulk action in D = 5 is obtained by replacing the Maxwell sector in equation (2.10) with this action. The exact results for the Born–Infeld Figure 2.7: The efficiency of our benchmark cycle as a function of grid size, N. Here Born– Infeld black holes are used as the working substance. Blue crosses represent the Carnot efficiency η C , while black squares represent η. For N = 500, η C and η converge to 0.6674942730 and 0.5653678967 respectively. 7 See e.g. the remarks in ref. [42] about the terminology 28 black hole’s mass and temperature are known 8 , but for our purposes, it is enough to expand them in 1/β, keeping only leading non–trivial terms. For the mass: M = 3π 8 r 2 + + 4πp 3 r 4 + + q 2 r 2 + (1− 9q 2 16β 2 r 6 + ) +O 1 β 4 , (2.15) and the temperature: T = 1 4π 16πp 3 r + + 2 r + − 2q 2 r 5 + (1− 3q 2 4β 2 r 6 + ) +O 1 β 4 . (2.16) The exact formulae are computationally intensive, and for any significantN, there are far too many computations to allow computation of the efficiency in a reason- able amount of time (especially inD = 5) and so we chose to make this truncation at the outset. We worked with q = 0.1 andβ = 0.1 in our benchmark studies, the results of which are shown in figure 2.7 for η C and η. 2.4.4 Comparison/Observations In figure 2.8 we gather all the efficiencies computed using the benchmark cycle together. The Gauss–Bonnet and Born–Infeld cases, thought of as perturbations of the Einstein–Maxwell case, have higher efficiencies, although it is interesting that the differences begin to show only in the 8th significant figure, for the parameter values chosen forα andβ. We explored other parameter values (while making sure to stay in the physical range allowed by reality of the mass for the Gauss–Bonnet case) and found a very weak dependence of the efficiency as they varied. (This all matches observations made in refs. [41, 42] in the high temperature limit.) They 8 See refs. [59–61] for further details. 29 all in turn have significantly lower efficiency than the ideal gas case listed at the end of section 2.3. Figure 2.8: The efficiencies of Einstein–Hilbert-Maxwell (lowest), Gauss–Bonnet (highest) and Born–Infeld (middle) black hole heat engines for N = 500 with circle origin at (110,20) and radius L = 10. For additional comparison, the ideal gas case of section 2.3 has η' 0.56588 at N = 500, and is η = 2π/(π +8) exactly. 2.5 Discussion We’ve defined a new way of comparing different black holes [1], given meaning in the context of defining black hole heat engines [24] in extended thermodynamics. Our benchmarking allowed us to compare four important cases against each other, and we found results consistent with earlier studies reported in refs. [41, 42], but herewe’veestablishedamorerobustframeworkforcomparison(astandardcircular cycle) facilitated by the exact formula and algorithm of ref. [51]. Along the way, we foundafascinatingcasewherethealgorithmitselfcollapsestoanotherexactresult, this time the exact efficiency of an “ideal gas" example. It would be fascinating to see if other exact results of this kind can be obtained for other non–trivial systems. It would be interesting to study other black holes using this same benchmarking scheme in order to compare more properties of heat engine working substances. 30 Extending all these to non–static cases would be particularly worthwhile, which is the topic of the next chapter [2]. Finally, the possible applications of all of this to holographically dual strongly coupled field theories is worth exploring. In fact, in ref. [62] the authors have shown a connection between the d–dimensional CFTs in flat spacetime and the extended thermodynamics of hyperbolic black holes in (d + 1)–dimensional AdS spacetime. When the CFTs are reduced on a sphere, the engine efficiencies become simple functions of the ratio of the number of degrees of freedomoftwoCFTs. Seealsoref.[63]foranoveldescriptionofcertainholographic heat engines as models of quantum heat engines in a particular regime. 31 Chapter 3 Benchmarking Rotating Black Hole Heat Engines 3.1 Introduction As mentioned in chapter 2, holographic heat engines [24] are a natural concept in the extended black hole thermodynamics framework 1 . The engine does mechan- ical work via the pdV term in the First Law of thermodynamics, and is defined using a closed cycle in state space during which there is a net input heatQ H flow, a net output heat flowQ C , and a net output workW such thatQ H =W +Q C . The efficiency of the cycle,η =W/Q H = 1−Q C /Q H , is determined by the equation of state of the system and the choice of cycle in the state space. The cycle given in figure 3.1(a) is a natural choice for static black holes. This is because, in that case entropy and volume, both purely powers of r + , are not independent, and hence the isochores are adiabats, (i.e.,C V = 0 [23]), making all the heat flows take place on the isobars, simplifying the analysis[41, 42] of the efficiency 2 . In general, the calculation of efficiency is a difficult task to perform exactly and one adopts various approximation schemes (such as high temperature limits) 1 See e.g., refs. [20, 27–29], the early work in refs. [15, 33, 34], and the reviews in refs. [13, 23]. 2 Refs. [43–50] have since done further studies of such heat engines. 32 p V 1 2 3 4 (a) (b) Figure 3.1: (a) Prototype engine for static black hole. (b) Prototype engine for rotating black hole. to proceed, but ref. [51] showed that for this cycle, a simple exact formula can be derived for it: η = 1− M 3 −M 4 M 2 −M 1 , (3.1) whereM i is the mass of the black hole (which is also the enthalpy [20]) evaluated at theith corner of the cycle. Moreover, since two such cycles can be placed together (sharing a common edge) to give a larger cycle, the simple formula can be used in an algorithm for computing the efficiency of an engine defined by an arbitrary cycle shape in the (p,V ) plane, by tessellating the plane using the basic cycle as a unit cell. Only the edges of the cells that intersect the cycle path will contribute to the efficiency, and since the heat flows are on the isobars only, there are only two types of cell that contribute: “hot” cells that are on upward–facing parts of the cycle (therefore contributing to Q H ) and “cold" cells otherwise (contributing to Q C ), and hence: η = 1− P j (M (j) 3 −M (j) 4 ) P k (M (k) 2 −M (k) 1 ) , (3.2) where the additional labels j,k are for cold and hot cells respectively. As the number of cells used in this prescription increases (or equivalently, the cell size decreases), the accuracy of the algorithm increases. 33 In the previous chapter this algorithm was used in a prototype benchmarking schemeforcomparingefficienciesofdifferentkindsofheatengines[1]. Theideawas that since certain special shapes of cycle might be more advantageous for certain kinds of black hole (e.g. having isochores favours static black holes since C V = 0), it makes sense to use a benchmark shape that is equally disadvantageous to all equations of state. A circle was the natural choice 3 : ((p−p c ) 2 + (V−V c ) 2 =L 2 , i.e., centered at (p c ,V c ) and of radiusL) and the algorithm gave a means by which η for each case could be computed and compared. Proceeding in this way produced another useful outcome: In the “ideal gas" limit, where black hole equations of state take the form pV 1/(D−1) ∼T, the mass becomesM =pV, and the entire scheme becomes a simple exactly solvable geom- etry problem with the result: η = 2π π + 4p c /L , (3.3) where L is the circle’s radius and p c (≤ L) is the pressure at the centre of the circle. This suggests a universal (dimension independent) bound on the efficiency in this benchmarking scheme, coming from the ideal gas sector. (It is stronger than the bound presented by the second law of thermodynamics through the Carnot efficiency.) It is natural to wonder how all of this generalizes to cases for whichC V 6= 0. In other words, what happens if the constant volume lines are not adiabatic curves. Rotatingblackholesfallintothiscategory, asnon–zerorotationparameteraenters the formula for the thermodynamic volume and entropy in such a way as to make them independent from each other. In this chapter we will present the details 3 It’s a circle in our choice of units for p and V, which is part of the choice of benchmarking scheme if making comparisons. 34 of an interesting generalization of our previous results. While this project was underway, a different generalization from the approach we took appeared in the literature [64]. That paper also looks at theC V 6= 0 case, but instead opts to keep the prototype cycle in figure 3.1(a) made of isobars and isochores. A generalization of the exact efficiency formula 3.1 can be readily written down, and they study the Kerr (and other) black holes using it. In the benchmarking circle, the paper revisits the C V = 0 case and derives an analytic expression for a lower bound on η. While not as universal (and dimension independent) as our upper bound 3.3, it may be useful in further studies of benchmarking. Their approach using the rectangular cycle for C V 6= 0 does not extend to a simple algorithm for solving arbitrary shapes, however, and so they study C V 6= 0 benchmarking by direct numerical evaluation of the heat around the circle. Since our interest was primarily in generalizations of the tessellation algorithm for the purposes of computing the efficiency of the benchmarking circle, we took a different approach, using two isobars and two adiabats to form our basic engine, giving a new prototype cycle that looks like that shown in figure 3.1(b). With this choice, the heat flows are again just along the top and bottom paths of the cycle, and the simple algorithm that involves tessellating the (p,V ) plane with them in order to determine the efficiency for an arbitrary shape can be implemented again [2]. The dependence of S on the various parameters can be used, in com- bination with the equation of state, to determine the equations for the adiabatic curves at each step in the algorithm in such a way as to avoid needing to solve for the adiabatic curves in closed form (which is in general hard to do). We describe this generalized scheme in section 3.2 and then apply it to the example of the Kerr black hole in section 3.3. 35 As with our previous work [1], it is possible to ignore the tessellation algorithm and do a direct numerical integration of the heat in order to compute the efficiency of the circle cycle. However, proceeding in that way would have meant missing the opportunity to derive the exact formula (3.3). Indeed, pursuing the generalization of the tessellation method for the rotating case, we found a generalization of that exact formula by again taking an ideal gas like limit. This formula again suggests itselfasanupperboundongenericrotatingcases. Thisispresentedinsection3.3.3. 3.2 The Generalized Benchmarking Cycle To generate the tessellation of the benchmarking cycle with unit cells of the form shown in figure 3.1(b), the procedures of ref. [1] need some modification. The benchmarking path is still a circle with center at (V c ,p c ) and radius L, of course, but the core tessellating grid will now be N× 2N, to accommodate the bending of the adiabats. Here N is again chosen to be even, since it gives equal numbers of cells in the upper and lower halves of the circle. Having chosen the origin and the radius (denoted L) of the circle in the (p,V ) plane, the lattice is constructed as follows. We started from the top left corner of the grid. The vertical decrement for pressure will be Δp = 2L/N, and we have N rows as before, resulting in an exact fit of the circle between the top most and bottom most isobars, along the vertical direction. In the horizontal direction, pressure is constant, and we have generated 2N columns in stead ofN, i.e., we have extended the length of the grid along this direction. A similar increment is used for ΔV to locate the corners of the tops of the cells in the top row. Since we have 2N columns now, the total length of the top isobar is 4L as opposed to our static case where it was 2L. Next, for the adiabatic segments, setting dS = 0 gives an equation that can be used 36 to determine the required ΔV =F (V,p)Δp where F (V,p) may also depend upon other parameters (such asa orJ in the rotating black hole case). This determines all the steps needed to move vertically and horizontally, and the grid can be readily generated. For black hole solutions, it is often most useful to write quantities in terms of the horizon radius r + , so we computed r + at each corner of the tops of the top cells in the top row using the explicit equation for V in terms of r + and p since its (p,V ) coordinates are known, with other parameters being held fixed. To generate the corners of the bottoms of the cells in the top row, we can now use ΔV because ΔV is a function of V, p and r + with all the variables known. Corners of the tops of the cells in the next row are same as those of the corners of the bottoms of the cells in the top row. So we can start from those corners and compute ΔV using the same trick as before which gives us the corners of the bottoms of the cells in the second row and this process continues until we reach theN-th row. At the end of the process, r + at each corner is known (along with p and V), and can be readily used to evaluate mass M, temperature T and other quantities at each corner. Identifying hot and cold cells is done using the same method as in [1] and the efficiency η (for that value of N) is readily computed. As described above, the circle fits exactly on the core N×N tessellating grid only when the adiabats are vertical (the static black hole case). Since the adiabats bend away from the vertical, the core grid was extended to N× 2N, by inspection, in order to ensure that the circle fits on it. As N increases Δp and ΔV get smaller, resulting in more accurate fits to the exactadiabats, andalsothehotcellsandcoldcellsmakeabetterfittothecircle, so η becomes more accurate for largeN. Since the algorithm identifies all the hot cells and cold cells, it is easy to determine the maximum and minimum temperatures 37 on the benchmarking cycle in order to compute the Carnot efficiency (η C ) of the engine as a check our results for η. Figure 3.2: A sample benchmarking circular cycle for N = 10, showing the tessellation along isobars and adiabats. The equation of state comes from D = 4 Kerr–AdS black holes. Red lines represent hot cells and blue lines represnt cold cells. As N increases, these lines converge to the boundary of the circle. Black dashed lines show a family of isotherms. We worked at fixed a here, with a = 0.04. Figure 3.2 shows an example for the D = 4 Kerr black hole for N = 10 where we worked at fixed rotation parameter a = 0.04. (See the next section.) The benchmarking circular cycle is shown, with (V = 50,p = 30) as the origin and L = 5 as the radius. The crosses (green) show the points of the tessellation. The black dashed lines show a few sample isotherms. 38 3.3 Benchmarking Kerr–AdS 3.3.1 Background Thermodynamics We use the D = 4 Einstein–Hilbert action: I = 1 16π Z d 4 x √ −g R− 2Λ , (3.4) where the cosmological constant Λ =−3/l 2 sets a length scale l. The Kerr–AdS solution in Boyer-Lindquist coordinates is given by: ds 2 =− Δ r ρ 2 dt− a sin 2 θ Ξ dφ 2 + ρ 2 Δ r dr 2 + ρ 2 Δ θ dθ 2 + Δ θ sin 2 θ ρ 2 adt− r 2 +a 2 Ξ dφ 2 , (3.5) where, ρ 2 =r 2 +a 2 cos 2 θ , Ξ = 1− a 2 l 2 , (3.6) Δ r = (r 2 +a 2 ) 1 + r 2 l 2 − 2mr , Δ θ = 1− a 2 l 2 cos 2 θ . (3.7) Hereaandmarerotationandmassparameterrespectively. Notethatthissolution is valid only fora<l and becomes divergent in the limita→l. The event horizon is at the largest real root, r + , of Δ r = 0. The horizon area A and entropy S are given by: A = 4π(r 2 + +a 2 ) Ξ , S = A 4 . (3.8) Analytically continuing the metric by t→ iτ and a→ ia gives the Euclidean section and if we identify τ∼τ +β and φ∼φ +iβΩ H , where Ω H is the angular velocity of the event horizon Ω H = a Ξ r 2 + +a 2 . (3.9) 39 and β is the inverse temperature: β −1 =T = 1 2πr +     (a 2 + 3r 2 + ) r 2 + l 2 + 1 2(a 2 +r 2 + ) − 1     , (3.10) then the geometry is free of conical singularities. Note that the temperature van- ishes at the following extremal value of a: a 2 ext = r 2 + 1 + 3r 2 + l 2 1− r 2 + l 2 . (3.11) The angular velocity measured at spatial infinity is what enters the first law [22] (see below) and is: Ω = Ω H + a l 2 = a 1 + r 2 + l 2 r 2 + +a 2 . (3.12) The mass, angular momentum and thermodynamic volume are [22]: M = m Ξ 2 = 1 2r + Ξ 2 (r 2 + +a 2 ) 1+ r 2 + l 2 , J = ma Ξ 2 , and V = r + A 3     1 + a 2 1 + r 2 + l 2 2r 2 + Ξ     . (3.13) The first law of black hole thermodynamics for a rotating black hole is [22]: dH =TdS +Vdp + ΩdJ , (3.14) where Ω is the angular velocity and J is the angular momentum. The enthalpy H is simply the mass of the black hole, usually written as a function of r + and p, parametrized by a, where, the rotation parameter a =J/M (in D = 4 ). 40 3.3.2 Key Elements of the Tessellation It is possible to explicitly rewrite all the thermodynamic quantities in terms of the angular momentum J, and, as a conserved quantity, it is indeed natural thermodynamic variable to hold fixed. On the other hand for the purposes of exploring the generalized tessellation we (for simplicity) choose is to work with fixed a (amounting to fixing J’s conjugate, the angular velocity at infinity). Fixing a gives dJ =adM. So along the isobars (dp = 0) the total heat flow is R TdS = R dM− R aΩdM. It turns out that, perhaps surprisingly, for Kerr–AdS black holes, aΩdM is exactly integrable over r + . For D = 4, it is [2]: aΩdM = a 2 1 + r 2 + l 2 3r 4 + + (a 2 +l 2 )r 2 + −a 2 l 2 2(a 2 +r 2 + )r 2 + l 2 Ξ 2 dr + . (3.15) Then integrating over r + keeping l fixed results in: Υ(r + ,l,a) = arctan r + a a + r + a 2 l 2 Ξ " 1 + l 2 2r 2 + Ξ 1 + 2r 2 + l 2 + r 4 + l 4 # . (3.16) Appendix A.2 shows how this generalized to higher dimensions. This means that the efficiency formula extends to: η = 1− (M 2 −M 1 )− (Υ 2 − Υ 1 ) (M 3 −M 4 )− (Υ 3 − Υ 4 ) , (3.17) where M i and Υ i are the value of those functions at each corner of the cycle. 41 This extended result for a cell of type shown in figure 3.1(b) is what is used in the tesselation algorithm for computing the efficiency of the benchmark circle, with η = 1−Q C /Q H , where: Q H = X ith hot cell (M (i) 2 −M (i) 1 )− (Υ (i) 2 − Υ (i) 1 ) , (3.18) Q C = X ith cold cell (M (i) 3 −M (i) 4 )− (Υ (i) 3 − Υ (i) 4 ) . (3.19) Our benchmark circle will be centred at p c = 30, V c = 50, with radius L = 5. For generating the moves along adiabats (as discussed in the previous section), we used equation (3.8), setting dS = 0 giving: dr + =− 4πa 2 (r 2 + +a 2 ) 3r + 1− 8πpa 2 3 dp . (3.20) Then using equation (3.13) for V and equation (3.20) for dr + , one gets: dV = 8π 2 a 4 9r 3 + (r 2 + +a 2 ) 2 1 + 8πpr 2 + 3 1− 8πpa 2 3 3 dp . (3.21) Before showing our results of implementing the algorithm, we pause to note an exact result that will be a useful guide. 3.3.3 An Exact Result For static black holes, there is an “ideal gas” limit in which the equation of state inD dimensions becomes formpV 1/(D−1) ∼T. It is essentially a high temperature or large volume limit, and can be obtained by keeping the leadingr + behaviour of the various expressions for T, M, etc. In this limit, the mass/enthalpy becomes 42 M =pV withaninterestingconsequenceforthetessellationalgorithm. Recallthat the input heat, Q H = P k (M (k) 2 −M (k) 1 ) is summed along the tops of a stepwise discretization of the top of the benchmarking circle. But in this case it is Q H = P p(V )ΔV along the circle’s top half. Therefore in the limit of small step size this becomes the area under the curve, a pleasing geometrical result which allows the formula (3.3) to be written down. It is natural to wonder if a similar limit might apply here, allowing for another exact formula. The answer is yes. The larger + limit of the mass is againM =pV, whiletheangularvelocitybecomes Ω =a/l 2 . Thereforeisobaricheatflowbecomes: Z TdS = Z 1− a 2 l 2 ! dM−→ Z 1− 8πa 2 3 p(V ) ! p(V )dV . (3.22) So again the result is geometrical, with a weighting factor in the integral. So for a circle centered at p =p c and V =V c of radius L, Q H = Z Vc+L Vc−L 1− 8πa 2 3 p(V ) ! p(V )dV , where (p(V )−p c ) 2 + (V−V c ) 2 =L 2 . (3.23) The first (square root) term is the original result (the sum of the areas of the upper semi–circle and the rectangle), easily solved by the substitutionV =L sinθ. Meanwhile the second term is an elementary polynomial. Recalling thatW =πL 2 , the final result is [2]: η = 2π h π + 4pc L − 16π 3 a 2 4 3 L +p c π + 2p 2 c L i . (3.24) As with the a = 0 version, since it comes from an ideal gas limit, it probably expresses a limit on the efficiency achievable by a rotating black hole, at least in generic situations. Our results in the next section are consistent with this. 43 Figure 3.3: Results for the efficiency of the benchmarking circle for a range of rotation param- eters. The results were obtained by direct integration of the heat to evaluate the efficiency. The blue dots are the numerical results, while the solid curve is the exact “ideal gas” result of equation (3.24). 3.3.4 Numerical Results Our benchmarking circle has origin at (50, 30) and radius 5. Figure 3.4 shows the results of the algorithm for computing η C and η. We worked with a = 0.001 as a sample value of the rotation parameter 4 . As N grows, the tessellation scheme, while successful, becomes numerically labour intensive as compared to the static case ofa = 0. That difficult grows with larger a. Being able to do large enough N for a range of values of a, to ensure reliable convergence, eventually became beyond the capabilities of our computers. Instead, for a study of the a–dependence of the system, we relied on a direct 4 Recall that the metric is valid only fora<l which sets an upper limit ona. With our chosen cycle, this means a has to be less than 0.06. We also have another restriction on a coming from the extremality condition in equation (3.11). For our choices of cycle a ext is always greater than 0.06. Notice that we also avoided the regions of the phase diagram where values of a at which multi–valuedness associated with phase transitions would develop. 44 Figure 3.4: The efficiency of our benchmarking cycle as a function of grid size, N. Here Kerr– AdS black holes in D = 4 are used as the working substance. Blue crosses represent the Carnot efficiencyη C , while black squares representη. ForN = 100,η C andη begin to show convergence to approximately 0.295355525453494 and 0.234033334108944 respectively. We set a = 0.001. numericalintegrationoftheheataroundthecycleinordertoevaluatetheefficiency at different a. The results are shown in figure 3.3 as a series of points indicated by circles. The solid (red) line is the exact result (3.24) coming from the ideal gas case and again it seems to be consistent with our expectation that it is an upper bound. To compare the efficiencies of Kerr–AdS black hole heat engines with that of the other black holes, we use our new choice of benchmarking, i.e., a circle with origin at (50, 30) and radius 5, to the algorithm for Einstein-Hilbert–Maxwell and Born–Infeld cases (see refs. [1] for the details). In D = 4 dimensions, Gauss– Bonnet black holes are purely topological and so we choose not to include that for the present chapter. We performed our computation for N = 100 and the result is shown in figure 3.5. As expected, inclusion of rotation results in lower efficiency than the static cases. In D = 4, Born–Infeld case has lower efficiency than the 45 Einstein–Hilbert–Maxwell case (as opposed to theD = 5 case), and Kerr–AdS has the lowest efficiency (a was set to 0.001), and of course all of them have efficiencies lower than their corresponding Carnot efficiencies. Note that in [64] they have found that in D = 4 Born–Infeld black hole heat engines have higher efficiencies than the Einstein–Hilbert–Maxwell heat engines for the intermediate ranges of the charge Q. But this difference is consistent with the fact that we are working in a different ensemble than theirs. This also implies that the engine efficiency depends on the choice of thermodynamic ensemble we make, different ensemble produces different results. Figure 3.5: The efficiencies of Kerr–AdS (lowest), Born–Infeld (middle) and Einstein–Hilbert– Maxwell (highest) black hole heat engines for N = 100 with circle origin at (50,30) and radius L = 5. (Since the static cases lie close to each other in the plot, an inset is included to resolve them.) 46 Anotherinterestingobservationofournumericalresultsisthatforsmallenough value of the charge parameter q, both Einstein–Hilbert–Maxwell and Born–Infeld heat engines have same efficiencies (up to the decimal places our machine can gen- erate). As we increase the value of q, Einstein–Hilbert–Maxwell engine efficiency increases, as well as Born–Infeld engine efficiency but at a slower rate than the Einstein–Hilbert–Maxwell case. Ultimately, Born–Infeld engine efficiency reaches itspeakwhenq≈ 0.5 (Einstein–Hilbert–Maxwellstill having higherefficiencythan the Born–Infeld). If we further increaseq, then Born–Infeld engine efficiency starts decreasing, while Einstein–Hilbert–Maxwell engine efficiency still keeps increasing. So, as far as we have checked the ordering of the efficiencies does not depend on the parameter q, which is in contrast with the result in [64] 3.4 Conclusion We have shown that the circular cycle can be used as a standard benchmark to compute efficiency of rotating black hole heat engines. Specifically, we did it for four dimensional Kerr–AdS black hole used as a working substance for the engines. We used our modified algorithm and exact efficiency formula (3.17) to break the circle into a regular lattice of cycles of the form in figure 3.1b and using the exact formula for this particular case at least we computed the efficiency. We also computed efficiencies of Einstein–Hilbert–Maxwell and Born–Infeld black hole heat engines in D = 4 and compared those with the rotating case. We have already shown in refs. [1] that perturbations of Einstein–Maxwell cases yield higher efficiencies inD = 5-dimensions. In this chapter we observed that inD = 4 this trend changes (now Born–Infeld has lower efficiency than Einstein–Hilbert– Maxwell case) and more importantly, inclusion of rotation results in efficiency 47 lower than that of both the static cases discussed here. It would be interesting to implementouralgorithmforotherthermodynamicensembles(specificallyforfixed- J case) and then compare the results we already have. Another possible direction would be to extend all this to other black holes, e.g., multiply spinning, super- entropic, lovelock etc. And finally, how all of this can be applied to holographically dual strongly coupled field theories would be interesting to explore (see e.g. [62]). 48 Part II Application of AdS/CFT: Holographic Superconductor 49 Chapter 4 AdS/CFT Basics The basic statement of AdS/CFT correspondence is that any conformal field theory on d–dimensional spacetime is equivalent to a quantum theory of gravity in asymptotically AdS d+1 ×M spacetime where M is some compact manifold. The most famous example of this duality was first discovered by Maldacena [36]: N = 4 Super Yang–Mills theory in 3 + 1–dimensions is equivalent to type IIB superstring theory on AdS 5 ×S 5 . We will start with a brief review of both sides of this correspondence. 4.1 Anti-de Sitter Spacetime Anti-de Sitter spacetime, or AdS spacetime for short, is the maximally sym- metric spacetime with constant negative curvature. AdS spacetime in (d + 1)–dimension, denoted by AdS d+1 , can be obtained from (d + 2)–dimensional Minkowski spacetime in (2,d) signature, with metric ¯ η = diag(−, +, +,..., +,−), i.e., ds 2 =−(dX 0 ) 2 + (dX 1 ) 2 +... + (dX d ) 2 − (dX d+1 ) 2 . (4.1) Then AdS d+1 is given by the following hypersurface or submanifold: − (X 0 ) 2 + d X i=1 (X i ) 2 − (X d+1 ) 2 =−l 2 , (4.2) where l is the radius of curvature of the AdS spacetime. 50 Apart from being maximally symmetric, AdS space has several other important properties. First of all, it is a solution of Einstein’s field equation with negative cosmological constant withT μν = 0. Simple calculations lead to the following nice results for the AdS spacetime: R μνρσ = R d(d + 1) (g μρ g νσ −g νρ g μσ ) , (4.3) R μν = R (d + 1) g μν , (4.4) R =− d(d− 1) l 2 , (4.5) and we can compute cosmological constant Λ as: Λ = − d(d−1) 2l 2 . Secondly, it has a large isometry group given by the Lorentz group of (d + 2)–dimensional Minkowski spacetime: SO(d, 2). It is a homogeneous and locally isotropic space. Using this isometry group SO(d, 2), AdS can be written as the coset space: SO(d, 2)/SO(d, 1). 4.1.1 AdS in Global Coordinates Let us consider the following parametrisation: X 0 =l coshρ coshτ , (4.6) X d+1 =l coshρ sinτ , (4.7) X i =lΩ i sinhρ , for i = 1,...,d , (4.8) where Ω i ’s are angular coordinates of unit sphereS d−1 with normalization P i Ω 2 i = 1 and the other two coordinates take the ranges ρ∈ [0,∞) and τ∈ [0, 2π]. The 51 coordinates (ρ,τ, Ω i ) are called Global coordinates of AdS d+1 . The resulting metric is: ds 2 =l 2 (cosh 2 ρdτ 2 +dρ 2 + sinh 2 ρdΩ 2 d−1 ) . (4.9) This metric has closed timelike curve since τ is periodic in 2π. To solve this problem we should consider universal covering of AdS space by unwrapping the timelike circle, i.e., τ∈R. To investigate the causal structure of AdS d+1 , let us consider the transformation: tanθ = sinhρ. Then the metric becomes: ds 2 = l 2 cos 2 θ (−dτ 2 +dθ 2 + sin 2 θdΩ 2 d−1 ) . (4.10) This is half of the Einstein static universeR×S d since 0≤θ<π/2. Ignoring the Weyl factor in the metric and adding the point θ =π/2 at spatial infinity, we get the following compactified spacetime: ds 2 =−dτ 2 +dθ 2 + sin 2 θdΩ 2 d−1 , (4.11) where 0≤θ≤π/2 and 0≤τ < 2π. The asymptotic boundary is at θ =π/2 and it has the topologyR×S d−1 . 4.1.2 AdS in Static Coordinates Let us now choose a convenient set of radial and time coordinates as follows: r =l sinhρ , t =lτ , (4.12) 52 where r∈ [0,∞) and t∈ (−∞,∞). Then from equation (4.9) we get AdS d+1 in static coordinates: ds 2 =− 1 + r 2 l 2 ! dt 2 + 1 + r 2 l 2 ! −1 dr 2 +r 2 dΩ 2 d−1 . (4.13) 4.1.3 AdS in Poincaré Coordinates Figure 4.1: A cartoon of AdS spacetime where the conformal boundary is ar r→∞ and the Poincaré horizon is at r→ 0. This diagram is taken from [65]. Another useful parametrisation is given by local coordinates or Poincaré patch coordinates: X 0 = l 2 2r 1 + r 2 l 4 ( − → x 2 −t 2 +l 2 ) ! , (4.14) X i = rx i l fori = 1,...,d− 1 , (4.15) X d = l 2 2r 1 + r 2 l 4 ( − → x 2 −t 2 −l 2 ) ! , (4.16) X d+1 = rt l , (4.17) 53 where t∈R, − → x = (x 1 ,...,x d−1 )∈R d−1 and r∈R + . The metric in these coordi- nates reads: ds 2 = l 2 r 2 dr 2 + r 2 l 2 (−dt 2 +d − → x 2 ) . (4.18) This metric covers only half of the AdS d+1 spacetime sincer> 0. For a fixed value ofr thetransversespacetimeissimplythed–dimensionalflatMinkowskispacetime. In these coordinates the conformal boundary is at r→∞ (see fig. 4.1). Another important form of this metric which is widely used in AdS/CFT correspondence can be obtained by the transformation z = l 2 /r. In this case the horizon is at z→∞ and the conformal boundary is at z = 0. The metric takes the following form: ds 2 = l 2 z 2 (−dt 2 +dz 2 +d − → x 2 ) . (4.19) 4.2 Conformal Field Theories A conformal field theory (CFT) is a quantum field theory that is invariant under the conformal transformations. We all are familiar with the usual Poincaré transformations in flat spacetime. In addition to these symmetries, CFTs have extra spacetime symmetries. An interesting addition to the Poincaré symmetry is thescale invariancesymmetrywhichlinks physics atdifferentlengthscales, e.g., 4d Yang-Mills theory is scale invariant. Field theories tend to exhibit an RG flow from one scale invariant UV fixed point to another scale invariant IR fixed point. Apart from string theory, CFTs have important applications in statistical mechanics and condensed matter systems as well. Hence, studying these scale invariant theories is interesting in its own right. 54 4.2.1 Conformal Group and Algebra The conformal group can be thought of as a group of transformations that leave the metric invariant up to an arbitrary but positive spacetime dependent scale factor: g μν (x)→ Ω 2 (x)g μν (x). Thus, conformal transformations preserve angles locally while distorting the lengths. A manifold which obeys this property is called conformally flat. The conformal group includes usual Poincaré transformations, i.e., translations and Lorentz transformations, along with the scale transformation: x μ →λx μ , (4.20) and the special conformal transformations: x μ → x μ +a μ x 2 1 + 2x ν a ν +a 2 x 2 . (4.21) It can be easily checked that there are (d + 2)(d + 1)/2 number of generators of the conformal group. Hence, in Lorentzian signature the conformal algebra is isomorphic to the algebra of SO(d, 2). Let us now denote the generators of these transformations as follows: P μ for translations, J μν for Lorentz transformations, D for for dilatation andK μ for special conformal transformation. Then they obey the conformal algebra: [J μν ,K ρ ] =i(η μρ K ν −η νρ K μ ) , (4.22) [D,P μ ] =iP μ , [D,K μ ] =−iK μ , [D,J μν ] = 0 , (4.23) [K μ ,K ρ ] = 0 , [K μ ,P ν ] =−2i(η μν D−J μν ) , (4.24) [J μν ,J ρσ ] =−iη μρ J νσ ±permutations. (4.25) 55 As we have already mentioned, the conformal group ind–dimensions is isomorphic to the groupSO(d, 2) and the fields transform in irreducible representations of this group algebra. Under dilatations x→x 0 =λx, a field φ transforms as: φ(x)→φ 0 (x 0 ) =λ −4 φ(x) . (4.26) The field φ has a fixed scaling dimension4 and hence it is an eigenstate of the dilatation operator D with eigenvalue−i4: [D,φ(0)] =−i4φ(0) . (4.27) Fields (or operators) which satisfy the commutation relation: [K μ ,φ(0)] = 0 , (4.28) are called primary fields. The primary fields are fields of lowest scaling dimension determined by this equation. All other fields, called descendants are obtained by acting withP μ on the primary fields. Though we have written down this relation at x = 0, using the momentum vector P μ one can shift the argument to an arbitrary point x to derive the general transformation rule. Conformal algebra in d = 2–dimensions is special. They have infinite number of generators and hence the conformal group is infinite dimensional in this case. The generators L m ’s satisfy the Virasoro algebra: [L m ,L n ] = (m−n)L m+n + c 12 m(m 2 − 1)δ m+n,0 , (4.29) 56 wherec is known as the central charge or the central extension or central anomaly. Though this group is infinite dimensional, there is a finite dimensional subgroup. The finite form of the group is identified as SL(2,C)/Z 2 =SO(3, 1). 4.3 AdS/CFT Dictionary We started this chapter with the statement of the AdS/CFT correspondence. A couple of questions immediately arise from the statement: i) what is the map between observables on one side to that of the other side? ii) The other ques- tion is that which CFTs define the semi-classical gravity theories where gravity is approximately described by Einstein’s equations? The answer is called the dictionary. Note that we have seen earlier that both the symmetric group of asymptotically AdS d+1 and the conformal group of d– dimensional CFT are isomorphic to the group SO(d, 2). This is a hint that there could be some connection between these two. It leads to the fact that there is an isomorphism between the Hilbert spaces: φ :H AdS →H CFT , (4.30) and the unitary operators which implements these symmetries are related by φ: φ◦U AdS =U CFT ◦φ . (4.31) This suggests that the spectrum of the Hamiltonian are same on both sides and the states decompose into the same irreducible representation of SO(d, 2). To answer the second question, let us consider a local bulk effective action S eff [φ i , Λ] where φ i ’s are a finite set of bulk fields including the metric and Λl 57 CFT d AdS d+1 Scalar operator,O Scalar field, φ Fermionic operator, Ψ Dirac field, ψ Global current, J μ Maxwell field, A a Enery-momentum tensor, T μν Metric tensor, g ab Global symmetry Local Symmetry Scaling dimension of operator Mass of the corresponding field Temperature, T Hawking temperature, T H Phase transition Gravitational instability Table 4.1: The AdS/CFT dictionary is a cutoff. Also consider that there exists a finite set of CFT primary operators O i . Then a d–dimensional CFT has a semi-classical dual near the vacuum if: Z Dφ i e iS eff [φ i ,Λ] O i1 (t 1 , Ω 1 )...O in (t n , Ω n )≈hO i1 (t 1 , Ω 1 )...O in (t n , Ω n )i CFT , (4.32) lim r→∞ r 4 φ i (r,t, Ω) =O i (t, Ω) , (4.33) where4 i is the scaling dimension ofO i . The second equation is known as the “extrapolate dictionary” which shows a nice relationship between the boundary primary operators and the bulk fields. We have listed the (incomplete) AdS/CFT dictionary in table 4.1. Based on the ideas in this chapter, now we will move on to discuss holographic entanglement entropy, holographic subregion complexity and holographic superconductor in the next few chapters. Finally, we will apply all these techniques which are derived using the AdS/CFT correspondence for a 2 + 1–dimensional holographic superconductor in chapter 8. 58 Chapter 5 Entanglement Entropy 5.1 Introduction The idea that black holes exhibit thermodynamic behavior was first introduced by Bekenstein who observed the similarity of second law of thermodynamics and Hawking’s area theorem which says that the area of the event horizon of a black hole can never decrease [5–7]. By taking quantum effects into account, Hawking showed that black holes thermally radiate and they have an associated temper- ature as well [9]. This work confirmed the idea of entropy of an black hole and is called Bekenstein–Hawking entropy. One remarkable feature of Bekenstein– Hawking entropy is that it is proportional to the area of the black hole horizon which is in sharp contrast with our usual knowledge of entropy: if you consider a thermal gas in a box, then the entropy is proportional to the volume of the box. However, the origin of this entropy and its interesting scaling behavior couldn’t be explained. In 1985, ’t Hooft computed the entropy of the thermal gas of Hawk- ing particles that propagate just outside the horizon [66]. It was shown that the entropy indeed is proportional to the area of the horizon and one needs to place a brick wall, a boundary placed at a very small distance from the actual horizon to regulate the theory. In 1986, Bombelli et al. computed Bekenstein–Hawking entropy from reduced density matrix by tracing over the degrees of freedom of a quantum field that are inside the horizon [67]. The idea was that the Bekenstein– Hawkingentropy,S BH couldhaveaspecialquantummechanicalorigin, specifically, 59 it could arise from the quantum entanglement between the degrees of freedom inside and outside of the horizon. Srednicki used a similar approach in 1993 [68] in four dimensional flat spacetime. This method seems very natural for black holes since the horizon (or, the boundary of the sphere) behaves like a causal boundary which does not allow anything outside the horizon to have access to the interior. Quantum entanglement is one of the key concepts in modern physics and below we will briefly review basic ideas, definitions and properties of the entanglement entropy. 5.2 Density Matrix Formalism Given a QFT, consider the system in a pure ground state Ψ and divide the system (mentally) into two subsystems A and B = ¯ A. A bipartite system is a system with Hilbert space equal to the direct product of two subsystems:H tot = H A ⊗H B and the total density matrix is given by: ρ tot =|ΨihΨ|. Let us now take the partial trace over the system B and define the reduced density matrix ρ A for the system A: ρ A ≡ Tr B (ρ tot )≡ X i B hi|ρ tot |ii B . (5.1) Consequently, we define a measure of the entanglement, the entanglement entropy of the subsystem A by the von Neumann entropy of the reduced density matrix ρ A : S A =−Tr A (ρ A logρ A ) . (5.2) Let us again consider the pure ground state Ψ in a general form: |Ψi = X i,j c ij |ii A ⊗|μi B , (5.3) 60 where, |ii A and|ji B are orthonormal bases forH A = {|ii A ,i = 1,...d A } and H B ={|ji B ,j = 1,...d B } respectively and the coefficient c ij is a d A ×d B matrix with complex entries. • The ground state Ψ is called a pure state or a product state or a separable state if c ij factorizes (c ij = c A i c B j ) and Ψ can be written as: |Ψi =|Ψ A i⊗|Ψ B i, where |Ψ A i≡ P i c A i |ii A and|Ψ B i≡ P j c B j |ji B . It follows that in this case the reduced density matrix also becomes pure and hence has vanishing entropy: ρ A =|Ψ A ihΨ A | , S A = 0 . (5.4) • The ground state is called an entangled state if c ij 6=c A i c B j , i.e., the state is not separable. Using Schmidt decomposition we can then write: Ψ = min(d A ,d B ) X k=1 √ p k |ψ k i A ⊗|ψ k i B , (5.5) where, p k are positive real numbers obeying P k p k = 1 and |ψ k i A,B are new orthonormal bases for the subsystems A and B. Using this decomposition it is easy to write the reduced density matrix in the following form: ρ A = min(d A ,d B ) X k=1 p k |ψ k i AA hψ k | . (5.6) Note that even if the total system is in pure ground state, the reduced density matrix of a subsystem can still be mixed. The entanglement entropy is given by the Shannon entropy: S A =− min(d A ,d B ) X k=1 p k logp k . (5.7) Using the condition P k p k = 1, we get the entropy of maximally entangled state : S A | max = log min(d A ,d B ) at p k = 1/ min(d A ,d B ) for any k. 61 In short, the entanglement entropy measures how much a given state differs from a pure state. It reaches the maximum value when the given state is a super- position of all possible quantum states with an equal weight. Below we will write down couple of useful properties: • Given a pure ground state wave function, the entanglement entropy of a subsys- tem A and its complement B are same: S A =S B . • Subadditivity: Given two disjoint subsystems A and B, the entanglement entropies satisfy: S A∪B ≤S A +S B . • We will conclude this section with a simple example. Let us consider a sys- tem of two particles A and B each with spin 1/2. Each particle can have two possible spin states: up or down. Hence the corresponding Hilbert spaces are spanned by: H A,B ={|0i A,B ,|1i A,B } where we have chosen orthonormal bases: A,B hi|ji A,B =δ i,j for i,j = 0, 1. The complete Hilbert space is the tensor product of these two subsystems with bases: H tot =H A ⊗H B ={|00i,|01i,|10i,|11i} where|iji≡|ii A ⊗|ji B . Let us now consider one of the four Bell states: Ψ = 1 √ 2 |01i−|10i . (5.8) Taking partial trace of the total density matrix overB gives us the reduced density matrix for A: ρ A = 1 2 |0i AA h0| +|1i AA h1| . (5.9) In matrix form: ρ A =    1/2 0 0 1/2    . (5.10) This means that ρ A is not pure and the corresponding entanglement entropy is: S A = log 2. This is a maximally entangled state with d A =d B = 2. 62 5.3 Entanglement Entropy in CFT For a QFT on ad–dimensional manifold, the subsystemA and its complement B are defined by (d− 1)–dimensional submanifolds at a fixed time. Then the entanglement entropy S A is defined by the above mentioned von Neumann for- mula (5.2). Since it depends on the geometry of the submanifoldA, sometimes its called geometric entropy. The entanglement entropy in this case looks very similar to Bekenstein–Hawking entropy S BH and can be expressed as [67, 68]: S A =γ Area(∂A) a d−2 + subleading terms + universal term , (5.11) wherea is a UV cutoff or lattice spacing,∂A is the boundary ofA,γ is a constant that depends on the system. It turns out that this simple area law does not hold for generic quantum field theories (for example 2d CFTs) and is also violated in the presence of Fermi surface [69]. As mentioned earlier, conformal field theories are relativistic quantum field the- ories that are invariant under dilations and special conformal transformations in addition to the usual Poincaré symmetry. 2d CFTs are particularly special because they have infinite number of symmetry generators and they are characterized by a single number c which is know as the central charge of the theory. In general computingvonNeumannentropydirectlyinCFTisextremelyhardandoneimple- ments certain “replica trick” to overcome this problem [70–72]. The first step is to compute Tr A ρ n A , then differentiate it with respect to n and finally take the limit n→ 1: S A = lim n→1 Tr A ρ n A − 1 1−n =− ∂ ∂n Tr A ρ n A | n=1 . (5.12) 63 Then the computation boils down to performing an euclideanized path integral which is an integration over field configurations with periodic boundary condi- tions. This gives us the reduced density matrix ρ A . To obtain Tr A ρ n A , we take n copies of the original manifold and glue them together by the fields with a certain identification. Then we take the limitn→ 1. A very simple and exact result using this replica trick can be found in 2d CFTs. For a 2d CFT on an infinite line the entanglement entropy is given by [70]: S A = c 3 log l a ! , (5.13) where c = 3R 2G N is the central charge, l is the length of the sub-system A and a is the UV cut-off or lattice spacing. But using replica trick in higher dimension is quite complicated and not very successful yet for computing entanglement entropy in d> 2. 5.4 Holographic Entanglement Entropy As mentioned above, for CFTs in d > 2, it is extremely hard to compute the entanglement entropy even for free field theories. Motivated by this, the Bekenstein–Hawking area formula and the fact that the entanglement entropy of a subregion A can be obtained using reduced density matrix by smearing out the inaccessible subregion B, Ryu and Takayanagi proposed a simple formula from the AdS/CFT point of view. It tells us how to compute the entanglement entropy holographically [73, 74]. To be more specific, it says that the holographic entangle- ment entropy (HEE) of a subregionA with its complementB in thed–dimensional boundary is given by: S A = Area(γ A ) 4G d+1 , (5.14) 64 where γ A is the extremal surface in the bulk, that extends from the boundary of the regionA and G d+1 is the Newton’s constant in the d + 1–dimensional bulk. 5.4.1 Testing the Proposal: AdS 3 /CFT 2 With the holographic proposal in hand, let us now compute the entanglement entropy of a 2d CFT on an infinite line and compare that with what we got using the replica trick in section 5.3. We will consider AdS 3 in Poincaré coordinates: ds 2 = R 2 z 2 (−dt 2 +dx 2 +dz 2 ) , (5.15) whereR is the AdS radius andz is the radial direction in the dual bulk geometry. The subsystemA is defined by:− l 2 ≤x≤ l 2 in a constant time slice and we will Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Chapter 7. The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 52 l L y z B BA x z= z ⇤ Figure 7.1: The strip geometry considered in this paper. Here z denotes the radial direction in the dual bulk geometry AdS 4 .Thestripwidthis l and the length is L which can be taken to infinity. The boundary is at z=0 where the field theory lives and z = z ⇤ denotes the turning point of the minimal surface inside the bulk. Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [102] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [73] the authors have further discussed the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area( A )= L Z l/2 l/2 dx R 2 z 2 1+ z 0 (x) 2 f(z) ! 1/2 . (7.22) The minimization problem leads to the Hamiltonian which is independent of x and hence this is a constant of motion: 1 z 2 ⇤ = 1 z 2 1 q 1+ z 0 (x) 2 f(z) , (7.23) where z = z ⇤ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface A : dz dx = p (z 4 ⇤ z 4 )f(z) z 2 . (7.24) Since z = z ⇤ is the turning point of the minimal surface in the bulk we will require that: l(z ⇤ ) 2 = Z z ⇤ 0 dz z 2 p (z 4 ⇤ z 4 )f(z) . (7.25) Figure 5.1: The line geometry considered in this chapter. Here z denotes the radial direction in the dual bulk geometry AdS 3 . The interval we consider is l on an infinite line of length L. z =z ∗ denotes the turning point of the minimal surface (in this case a semi circle of radius l/2) inside the bulk. 65 use the following parametrization: z = z(x). The 1–dimensional minimal surface is given by minimizing the area functional: Area(γ A ) =R Z l/2 −l/2 dx q 1 +z 0 (x) 2 z . (5.16) From the equation of motion of the function z = z(x) we get the following equation which determines the minimal surface: dz dx = q (z 2 ∗ −z 2 )f(z) z , (5.17) where z =z ∗ is the turning point of the minimal surface in the bulk. Integrating this we get the relation: l = 2z ∗ . So the entanglement entropy becomes: S A = 2R 4G 3 Z z∗ a dz z ∗ z 1 q (z 2 ∗ −z 2 ) = c 3 log l a , (5.18) where a is the UV cut-off. To get the final expression we have used l = 2z ∗ and the Brown–Henneaux relation: c = 3R/2G 3 . Note that this is exactly the same result (5.13) we found in the previous section using the replica trick! 5.4.2 HEE for a Straight Belt in CFT d+1 Let us first denote the subsystem as A and consider a straight belt A S with width l (see fig. 5.2). The area functional is then given by (we have set x 1 =x): Area =R d L d−1 Z l/2 −l/2 dx q 1 + ( dz dx ) 2 z d , (5.19) 66 Figure 5.2: Minimal surfaces in AdS d+2 : a) Straight belt A S and b) Circular disk A D . In this section we will compute the entanglement entropy for the straight belt. This figure is taken from refs. [73, 74]. where we are using the parametrization z = z(x). Then by minimizing this area functional we get the following equation which determines thed–dimensional min- imal surface γ A : dz dx = q z 2d ∗ −z 2d z d , (5.20) where z ∗ is the turning point of the minimal surface. Then the belt width as a function of z ∗ becomes: l 2 = Z z∗ a dz z d q z 2d ∗ −z 2d = √ πΓ( d+1 2d ) Γ( 1 2d ) z ∗ , (5.21) where we have introduced a small cutoffa. Finally, the area, and hence the entan- glement entropy becomes: S A S = 1 4G (d+2) N   2R d d− 1 L a d−1 − 2 d π d/2 R d d− 1 Γ( d+1 2d ) Γ( 1 2d ) ! d L l d−1   . (5.22) Note that the first term is divergent and it is proportional to the area, i.e.,L d−1 as is expected from the area law. The second term is finite and does not depend on the cutoff, hence this is the universal term. In the following chapters we will use 67 this method again and again in various cases to compute entanglement entropy and complexity, as well as to reconstruct the dual bulk metric given the entanglement entropy of the field theory. 68 Chapter 6 Computational Complexity 6.1 Introduction Inthepreviouschapterwediscussedoneimportantquantuminformationquan- tity: theentanglemententropy. Wealsobrieflyreviewedtheholographicderivation of the entanglement entropy. It turns out that there is another quantum informa- tion quantity which is reflected in the bulk geometry: the quantum complexity. The computational complexity of a quantum state can be roughly interpreted as the minimum number of gates required to implement a certain unitary operator to prepare this state from a given reference state. In other words, this quantity has to do with the difficulty of converting one quantum state to another quantum state. Although the entropy is connected to the counting of the degrees of freedom in the dual field theory description of a black hole, in the interior of the black hole it is not quite the right quantity that can describe the time evolution of the Einstein-Rosen bridge (the wormhole). When thermal equilibrium of the black hole is achieved and the entanglement entropy is saturated, the wormhole still continues to grow with time at late times [75–78]. For a double sided AdS black hole geometry which is dual to a thermo-field double state, the left and right CFTs are entangled through the black hole interior by a wormhole which grows linearly with time (see figure 6.1). Computational complexity turns out to be the viable candidate on the dual field theory to describe such behavior. Let us consider a 69 simple example to demonstrate our argument: • A generic state as a superposition of K qubits with complex coefficients:|ψi = P 2 K i=1 c i |ii. Note that we need 2 K complex numbers to specify this state. • A simple reference state|0...0i which is a product state. • Choose a universal gate set of 2-qubit simple unitary operations. The maximum entropy is given by the logarithm of the number of microstates (2 K in this case) and hence: S max = K log 2. But the maximal complexity is exponential in K: C∼ e K . Initially the quantum comlexity grows linearly in t for a long time which is much much larger than the thermalization time, and it saturates at a constant valueC max after an exponentially large time t∼e K . t=0 Figure6.1: PenrosediagramofadoublesidedAdSblackholedualtoathermofielddoublestate. The left and right CFTs live on the left and right boundaries. The dashed red line represents the singularity. The green line is a maximal codimension–one surface at t→∞. Note that it does not reach the singularity. Blue lines are few other maximal codimension–one surfaces. A precise definition of the complexity in the boundary CFT remains an open problem and recently there have been many attempts to address this question. The 70 main issues are the choice of the reference states, the allowed set of elementary quantum gates etc. But there are some nice work/explanations regarding these issues lately. Studies of circuit complexity of Gaussian states in the free field theories using Nielsen’s geometric approach and Fubini-Study metric have been investigated in refs. [79–84]. A path-integral optimization procedure to define the computational complexity is explored in refs. [85–88]. In [89, 90] the (subregion) complexity has been explained as the purification complexity using tensor network model. Sofartherearetwoconjecturestocomputethequantumcomplexityholograph- ically [75–78]. The first conjecture is known as the “complexity equals volume” or CV. It says that for an eternal black hole the complexity is proportional to the spatial volume of the Einstein-Rosen bridge connecting two boundaries: C = V Gl , (6.1) wherel is some length scale which can be the AdS radius or the radius of the black hole under consideration and needs to be chosen by hand case by case. Because of this ad hoc choice the second conjecture came up as a more refined version. This second one is known as the “complexity equals action” or CA which relates the complexity to the action on a Wheeler-DeWitt (WDW) patch 1 : C = A π~ . (6.2) In this case one does not require to choose the length scale l by hand. But the computation of the action on the WDW patch is more complicated and include 1 For a selection of references see [91–103] where the authors have developed and further extended these ideas. 71 surface terms from the null boundaries, corner terms where these boundaries meet and a surface term from the singularity. It turns out that the rate of change of the WDW action at late times is [77, 78]: dA dt = 2M , (6.3) which leads to the following conjecture (here, E is the energy of the system): dC dt ≤ 2E π~ , (6.4) i.e., black holes are the fastest computers in nature [77, 78, 104]. 6.2 Holographic Subregion Complexity Based on the CV conjecture discussed above, Alishahiha in [105] has proposed that the holographic complexity of a subregionA is equal to the codimension–one maximal volume of the bulk enclosed by the entangling region and the RT surface that appears in the holographic entanglement entropy computation: C = Volume(γ A ) 8πRG d+1 , (6.5) where R is the AdS radius. This quantity is known as the holographic subre- gion complexity. In this part we will be interested to compute subregion volumes, i.e., the complexity of a holographic superconductor and discuss several physical properties of that system. Note that we will be working in a time-independent geometry. A covariant description of how to compute the volume associated with 72 a subregion on the boundary (which is also applicable ti time-dependent geome- tries) is given in ref. [91]. See also ref. [106] where the authors have further studied codimension–one volumes for different strip and spherical entangling surfaces and uncovered some interesting properties such as non–monotonic behavior of the com- plexity with respect to the temperature and so on. 6.2.1 HSC for a Straight Belt in CFT d Figure 6.2: A straight belt with width l and length L→∞ on the boundary z = 0. This diagram is taken from ref. [106]. Inthissubsectionwewillpresentagenericformulaforthesubregioncomplexity for a strip region in a d–dimensional CFT which is dual to a d + 1–dimensional AdS spacetime. Let us then consider a general asymptotically AdS d+1 metric: ds 2 = 1 z 2 [f 0 (z)dt 2 +f 1 (z)dx 2 μ +f 2 (z)dz 2 ] , (6.6) where f 0 (z), f 1 (z) and f 2 (z) are some arbitrary functions of z with f i (z = 0) = 1. We choose the entangling region as a belt of width l and length L→∞ (see fig 6.2). It turns out that for a strip entangling region, the subregion volumes 73 include a finite term and a single divergent term which is proportional to the width l of the subsystem [106]: V =c d−1 lL d−2 d−1 +c finite (6.7) The entanglement entropy turns out to be the following: S(z ∗ ) = 2L d−2 4G N Z z∗ dz z d−1 q f 2 f d−1 1 r 1− f d−1 1 (z∗)z 2d−2 f d−1 1 (z)z 2d−2 ∗ , (6.8) and the subregion complexity is given by: C(z ∗ ) = 2L d−2 8πGR Z z∗ dz q f d−1 1 f 2 z d x(z) , (6.9) where, x(z) = Z z∗ z dZ r f 2 (Z) f 1 (Z) r f d−1 1 (Z)z 2d−2 ∗ f d−1 1 (z∗)Z 2d−2 − 1 . (6.10) As usual, z =z ∗ is the turning point of the minimal surface inside the bulk and is a small UV cut-off. The strip width l is: l(z ∗ ) 2 = Z z∗ dz r f 2 (z) f 1 (z) r f d−1 1 (z)z 2d−2 ∗ f d−1 1 (z∗)z 2d−2 − 1 . (6.11) If we choose: f 0 =−R 2 f(z)e −χ(z) , f 1 =R 2 , f 2 = R 2 f(z) and d = 3, we arrive at our results (8.25-8.29). 74 6.2.2 Case of Pure AdS For pure or empty AdS spacetime, f i (z) = 1. Then the entanglement entropy is given by: S(l) = 1 2(d− 2) L a d−2 − 2 d−3 π d−1 2 d− 2 Γ( d 2d−2 ) Γ( 1 d−2 ) ! d−1 L l d−2 , (6.12) and the volume is: V (l) = c 1 a d−1 − c 0 l d−2 , (6.13) where c 1 and c 0 are as follows: c 1 = lL d−2 d− 1 , c 0 = (2L) d−2 π d−1 2 (d− 1) 2 Γ( d 2d−2 ) Γ( 1 d−2 ) ! d−3 . (6.14) Notice that the finite part of both the entanglement entropy and the complex- ity has the same l dependence. Next, let us briefly talk about the holographic superconductor and then apply all these ideas in chapter 8. 75 Chapter 7 Holographic Superconductor 7.1 Introduction It has been known for a long time that the electrical resistivity of certain metals drops suddenly to zero when the temperature is lowered below a critical temper- ature. These metals are called superconductors. Below the critical temperature, these materials also expel the magnetic field and this behavior is known as the Meissner effect. The famous Landau–Ginzburg model explains superconductivity as a second order phase transition where the order parameter is a complex scalar field. For temperatures below the critical value, the minimum of the free energy is at some non-zero value of this scalar field. This is similar to the usual Higgs Mechanism which is related with spontaneous breaking of a U(1) symmetry. A more rigorous treatment of superconductivity is given by the BCS theory in terms of Cooper pair which is a charged boson. Below the critical temperature these bosons condense producing a second order phase transition and the DC conduc- tivity becomes infinite thereby giving rise to a superconductor. Many aspects of the condensed matter physics have been explored using the AdS/CFT correspondence (or strong/weak duality) for the last two decades. So a natural question arises: is there a holographic description of the superconductors? The answer is yes. Holographic superconductors are strongly coupled field theories which undergo a superconducting phase transition below a critical temperature T c and which have a gravity dual that is mathematically more tractable model. 76 To construct a gravitational dual of a superconductor there are certain minimal ingredients that are required [107–111]. The first is a notion of the temperature. We introduce it by adding a black hole in AdS. The AdS/CFT correspondence identifies the Hawing temperature of the black hole with the temperature of the dual field theory. Next we need a condensate. In the bulk, this role is played by a charged scalar field coupled to gravity. Hence, the scalar fields are naturally complex. So to describe a superconductor phase diagram we require a black hole that has no hair at high temperatures but admits non-vanishing scalar profile at low temperatures. 7.2 A Simple Model: Planar Schwarzschild–AdS Black Hole in D = 4 Let us consider the simple action in [107]: S = Z d 4 x √ −g R + 6 L 2 − 1 4 F μν F μν −|∇Ψ−iqAΨ| 2 −m 2 |Ψ| 2 . (7.1) This is just gravity coupled to a Maxwell field and a complex scalar field with mass mandchargeq inanegativecosmologicalbackground. Notethattheeffectivemass of Ψ is: m 2 eff =m 2 +q 2 g tt A 2 t . Butg tt is negative outside the horizon and near the horizon it goes to−∞. So there is chance that near the horizon m 2 eff can become sufficiently negative inducing the instability of the scalar field. Let us now start with the planar Schwarzschild–AdS black hole in D = 4 spacetime dimensions: ds 2 =−f(r)dt 2 + dr 2 f(r) +r 2 (dx 2 +dy 2 ) , (7.2) 77 where, f(r) = r 2 L 2 1− r 3 H r 3 . (7.3) Here L is the AdS radius and r H is the Schwarzschild radius. We also assume the following plane symmetric ansatz: Ψ =ψ(r) , A t =φ(r) . (7.4) The Hawking temperature of the black hole is: T = 3r H 4πL 2 . (7.5) We will be working in the probe limit where the Maxwell field and the scalar field do not back react on the metric. The Maxwell equations allow us to set the phase of the scalar field to be zero, soψ can be taken to be real. The scalar field and the Maxwell field equations become: ψ 00 + f 0 f + 2 r ψ 0 + φ 2 f 2 ψ− m 2 f ψ = 0 , (7.6) φ 00 + 2 r φ 0 − 2ψ 2 f φ = 0 . (7.7) At the horizon r = r H , φ must vanish so that φdt has finite norm. This implies ψ and ψ 0 are not independent. So we have a two parameter family of solutions which is regular at the horizon. Asymptotically these solutions have the following fall-offs: ψ = ψ (1) r + ψ (2) r 2 +... , (7.8) φ =μ− ρ r +... . (7.9) 78 For ψ, one can impose the boundary condition that either of the ψ (i) ’s vanishes. With that imposed we get a one parameter family of solutions. The asymptotic behavior ofφ defines the chemical potentialμ and the charge densityρ of the field theory. The scalar fieldψ defines the condensate of the charged scalar operatorO in the dual field theory: hO i i = √ 2ψ (i) , i = 1, 2 , (7.10) withtheboundarycondition ij ψ (j) = 0. It’shardtofindanalyticsolutionsofthese 1.0 0.0 0 8 4 0.5 the AdS radius is L=1. Recallthat T has mass dimension one, and ρ has mass dimension two so!O i "/T i and ρ/T 2 are dimensionless quantities. An exact solution to eqs (6,7) is clearly Ψ=0and Φ = μ− ρ/r.Itappearsdifficult to find other analytic solutions to these nonlinear equations. However, it is straightforward to solve them numerically. We find that solutions exist with all values of the condensate !O".However,asshowninfigure1,inorderfortheoperatortocondense, a minimal ratio of charge density over temperature squared is required. 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 T T c !O 1 " T c 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 T T c !O 2 " T c Figure 1: The condensate as a function of temperature for the two operatorsO 1 andO 2 . The condensate goes to zero at T = T c ∝ ρ 1/2 . The right hand curve in figure 1 is qualitatively similar to that obtained in BCS theory, and observed in many materials, where the condensate goes toaconstantatzerotempera- ture. The left hand curve starts similarly, but at low temperature the condensate appears to diverge as T −1/6 .However,whenthecondensatebecomesverylarge,thebackreaction on the bulk metric can no longer be neglected. At extremely lowtemperatures,wewill eventually be outside the region of validity of our approximation. By fitting these curves, we see that for small condensate thereisasquarerootbehaviour that is typical of second order phase transitions. Specifically, for one boundary condition we find !O 1 "≈ 9.3T c (1−T/T c ) 1/2 , as T → T c , (11) where the critical temperature is T c ≈ 0.226ρ 1/2 .Fortheotherboundarycondition !O 2 "≈ 144T 2 c (1−T/T c ) 1/2 , as T → T c , (12) where now T c ≈ 0.118ρ 1/2 .Thecontinuityofthetransitioncanbecheckedbycomputing the free energy. Finite temperature continuous symmetry breaking phase transitions are 4 the AdS radius is L=1. Recallthat T has mass dimension one, and ρ has mass dimension two so!O i "/T i and ρ/T 2 are dimensionless quantities. An exact solution to eqs (6,7) is clearly Ψ=0and Φ = μ− ρ/r.Itappearsdifficult to find other analytic solutions to these nonlinear equations. However, it is straightforward to solve them numerically. We find that solutions exist with all values of the condensate !O".However,asshowninfigure1,inorderfortheoperatortocondense, a minimal ratio of charge density over temperature squared is required. 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 T T c !O 1 " T c 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 T T c !O 2 " T c Figure 1: The condensate as a function of temperature for the two operatorsO 1 andO 2 . The condensate goes to zero at T = T c ∝ ρ 1/2 . The right hand curve in figure 1 is qualitatively similar to that obtained in BCS theory, and observed in many materials, where the condensate goes toaconstantatzerotempera- ture. The left hand curve starts similarly, but at low temperature the condensate appears to diverge as T −1/6 .However,whenthecondensatebecomesverylarge,thebackreaction on the bulk metric can no longer be neglected. At extremely lowtemperatures,wewill eventually be outside the region of validity of our approximation. By fitting these curves, we see that for small condensate thereisasquarerootbehaviour that is typical of second order phase transitions. Specifically, for one boundary condition we find !O 1 "≈ 9.3T c (1−T/T c ) 1/2 , as T → T c , (11) where the critical temperature is T c ≈ 0.226ρ 1/2 .Fortheotherboundarycondition !O 2 "≈ 144T 2 c (1−T/T c ) 1/2 , as T → T c , (12) where now T c ≈ 0.118ρ 1/2 .Thecontinuityofthetransitioncanbecheckedbycomputing the free energy. Finite temperature continuous symmetry breaking phase transitions are 4 Figure 7.1: A qualitative diagram of the condensate as a function of temperature for theO 2 superconductor. It goes to zero at T =T c . equations, but one can solve them numerically. It turns out that for all values of the condensate there exists solutions. Interestingly, near the critical temperature T c , there is a square root behavior:hO i i∼T i c (1−T/T c ) 1/2 , which is typical of the second order phase transitions predicted by the Landau-Ginzburg theory. 79 A nonzero condensate implies that the black hole has developed scalar hair. To decide whether the hairy black hole solution is thermodynamically preferred over the Schwarzschild–AdS black hole, we need to compute the free energy using euclidean action. It turns out that the free energy is always less for hairy solution and the difference in free energy of these two solutions goes to zero asT→T c . We can also compute the critical temperatureT c from the free energy vs. temperature plot. It is determined by where the order parameter vanishes for the second order phase transition. Moreover, we can determine the conductivity using the linear response theory. It turns out that above T c the conductivity is constant. But as we lower the temperature below the critical temperature, a gap opens up in the real part of the conductivity at low frequency. There is also a delta function at the frequency ω = 0, i.e., the DC conductivity becomes infinite for all T < T c . In the next chapter we will first review a similar but more complicated model obtained from String theory. We will present the full solution with the backre- action. Then using the prescriptions discussed previously in this part, we will compute the entanglement entropy and the subregion complexity of this system holographically. 80 Chapter 8 The Entanglement Entropy and The Subregion Complexity of a Holographic Superconductor 8.1 Introduction Let us start with summarizing the main ideas and formulae that we discussed in the last four chapters and implement those for a 2 + 1–dimensional holographic superconductor. As we reviewed above, the AdS/CFT correspondence provides us a dual description of the d–dimensional strongly interacting field theories on the boundary and the d + 1–dimensional weakly coupled gravity theories in the bulk [36, 112]. Two quantities on the boundary field theory play important roles in the quantum information theory: the entanglement entropy and the computational complexity, and both these quantities are reflected in the bulk geometry. The entanglement entropy is a measure of the quantum correlation of a quan- tumstateandisextremelyusefulforstudyingblackholephysics, condensedmatter systems etc. Given a quantum mechanical system, let us divide it into two sub- systems:A and its complementB = ¯ A. On one hand, the Ryu-Takayanagi (RT) conjecture tells us how to compute the entanglement entropy holographically [73, 81 74]. To be more specific, it tells that the holographic entanglement entropy of a subregionA with its complement in the d–dimensional boundary is given by: S = Area(γ A ) 4G d+1 , (8.1) where γ A is the extremal surface in the bulk, that extends from the boundary of the regionA and G d+1 is the Newton’s constant in the d + 1–dimensional bulk. On the other hand, the computational complexity of a quantum state can be roughly interpreted as the minimum number of gates required to implement a certain unitary operator to prepare this state from a given reference state. Based on the “complexity=volume” conjecture, Alishahiha in [105] has proposed that the holographic complexity of a subregionA is equal to the codimension–one maximal volume of the bulk enclosed by the entangling region and the RT surface that appears in the holographic entanglement entropy computation above: C = Volume(γ A ) 8πRG d+1 , (8.2) where R is the AdS radius. This quantity is known as the holographic subregion complexity. In the AdS/CFT framework, one important object which is widely investigated is the holographic superconductor [108–111, 113, 114] (see the previous chapter for a very short introduction to the holographic superconductor). In recent years there has been numerous work carried out to model such a holographic superconductor dual to gravity theories coupled to a Maxwell field plus a scalar. In general the task is highly non-trivial and finding a stable vacuum is difficult. Recently, ref. [115] has developed one such model in the context of AdS 4 /CFT 3 . Their solution is fully back reacted and the ground state is stable. Ref. [116] has further discussed about 82 the phase transition of this system by studying the entanglement entropy using RT prescription. The action in ref. [115] arises as an SO(3)×SO(3) invariant trun- cation of four–dimensionalN = 8 gauged supergravity. One needs to numerically solve the equations of motion coming from this action. At high temperature, the solution is simply the RN–AdS solution. The scalar has zero value and hence there is no condensate. At low enough temperature, below some critical value a new type of solution exists with a non–zero charged scalar hair. This solution is ther- modynamically preferred over the RN–AdS solution. Depending on the boundary conditions, there are two types of solutions: one gives rise to a second order phase transition and the other one first order phase transition. At zero temperature, the solution is an RG flow between two AdS spaces. Because the complexity measures the difficulty of turning one quantum state into another, it is expected that the subregion complexity should capture behavior of the phase transition of such a model and can provide useful information as well. In refs. [117–121] the authors have discussed the subregion complexity for different types of holographic super- conductors which have backreaction included and we will compare our results with theirs in section 8.3 and 8.4. In this chapter we will present our results of the subregion complexity of the 2 + 1–dimensional holographic superconductor system mentioned above [3]. We will see that there is a single divergence and no other divergences during the phase transition, the complexity remains finite during both the first order and the second order phase transitions. The complexity curves go linearly with l for large strip width l. Moreover, the temperature where normal phase turns into a superconducting phase is exactly same in both the entanglement entropy and the subregion complexity computation and is equal to the transition temperature T c obtained from the free energy calculation in [116]. Another interesting fact that we 83 will describe in section 8.3 is that of the multivaluedness captured in different ways for both the cases. We will also see the discontinuous but finite jump behavior of the complexity for the first order phase transition. The outline of this chapter is as follows. In the next section we will review the dual gravity background of our 2 + 1–dimensional superconductor system, along with the solutions in different temperature regime. In section 8.3 we will present our results for both the entanglement entropy and the subregion complexity of this system. We will conclude this chapter with a short summary and discussions. 8.2 Dual Gravity Background of 2 + 1– dimensional Holographic Superconductor In this section we will briefly review the dual gravity background of our 2 + 1– dimensional holographic superconductor system. The Lagrangian that gives rise to this superconductor is [115, 116]: e −1 L = 1 16πG 4 R− 1 4 F μν F μν −2∂ μ λ∂ μ λ− sinh 2 (2λ) 2 ∂ μ ϕ− g 2 A μ ∂ μ ϕ− g 2 A μ −P , (8.3) where the potentialP is given by: P =−g 2 6 cosh 4 λ− 8 cosh 2 λ sinh 2 λ + 3 2 sinh 4 λ . (8.4) The action in (8.3) is the geometry of a black hole coupled with complex scalar fields λ and ϕ with a non-trivial potential, A μ being the gauge field and G 4 is the 84 3 + 1-dimensional gravitational constant. We choose the ansatz for the metric, the gauge field and the scalar field as follows: ds 2 =− R 2 z 2 f(z)e −χ(z) dt 2 + R 2 z 2 (dx 2 1 +dx 2 2 ) + R 2 z 2 dz 2 f(z) , A t = Ψ(z) , λ =λ(z) , (8.5) wherewehavesetthescalarϕ tozerousingthe equations ofmotionandsymmetry. Next, we define a dimensionless coordinate: z =R˜ z. Then substituting this ansatz into the equations of motion arising from the Lagrangian (8.3) we get the following system of ordinary differential equations: χ 0 − 2˜ z(λ 0 ) 2 − ˜ ze χ sinh 2 (2λ)Ψ 2 8f 2 = 0 , (8.6) (λ 0 ) 2 −   f 0 ˜ zf   + ˜ z 2 e χ (Ψ 0 ) 2 4f + R 2 P 2˜ z 2 f + 3 ˜ z 2 + e χ sinh 2 (2λ)Ψ 2 16f 2 = 0 , (8.7) Ψ 00 +   χ 0 2   Ψ 0 − sinh 2 (2λ)Ψ 4˜ z 2 f = 0 , (8.8) λ 00 +   − χ 0 2 + f 0 f − 2 ˜ z   λ 0 − R 2 4˜ z 2 f dP dλ + e χ sinh(4λ)Ψ 2 16f 2 = 0 . (8.9) The horizon is the zero locus of f(˜ z). We will assume that it occurs at ˜ z = ˜ z H . There are three scaling symmetries of the equations of motion [116]: t→γ −1 1 t , χ→χ− 2 lnγ 1 , Ψ→γ 1 Ψ , t→γ −1 2 t , z→γ −1 2 z , R→γ −1 2 R , x μ →γ −1 x μ , f(z)→f(z) , Ψ(z)→γΨ(z) , λ(z)→λ(z) , χ(z)→χ(z) .(8.10) 85 Usingthesescalingsymmetrieswecanchoosearbitraryvaluesofthepositionofthe event horizon, the coupling constant of gauged supergravity,g, and the asymptotic value of the field χ(z). We choose the following: ˜ z H = 1 , g = 1 , lim z→0 χ = 0 . (8.11) To solve the equations of motion we need to know the IR and the UV behavior of various fields. Near the IR, i.e. the horizon, the fields have an expansion: λ(˜ z) =λ (0) +λ (1)   1− ˜ z ˜ z H   +... , χ(˜ z) =χ (0) +χ (1)   1− ˜ z ˜ z H   +... , f(˜ z) =f (1)   1− ˜ z ˜ z H   +... , Ψ(˜ z) = Ψ (1)   1− ˜ z ˜ z H   + Ψ (2)   1− ˜ z ˜ z H   2 +... . (8.12) Plugging this into the equations of motion (8.6-8.9) leaves us with three inde- pendent parameters which are our initial conditions for the numerical shooting method. We choose the following parameters: λ (0) , χ (0) , Ψ (1) . (8.13) 86 In the UV, i.e. near the AdS boundary ˜ z = 0, the fields have the following expansion: λ(˜ z) =λ 1 ˜ z +λ 2 ˜ z 2 +... , χ(˜ z) =χ 0 +λ 2 0 ˜ z 2 +... , f(˜ z) = 1 +λ 2 0 ˜ z 2 +f 3 ˜ z 3 +... , Ψ(˜ z) = Ψ 0 + Ψ 1 ˜ z +... . (8.14) Using our initial conditions we first fix λ (0) and χ (0) . We then tune Ψ (1) so that either λ 1 = 0 or λ 2 = 0. In general this will generate some non-zero value for χ 0 which we shift using the scaling symmetry to χ 0 = 0. The UV asymptotic values of the field λ define the vacuum expectation value of the charged operators in the theory and they are defined as: O 1 = 2λ 1 √ 16πG 4 , O 2 = 2λ 2 √ 16πG 4 R . (8.15) Usingtheholographicdictionary, theUVasymptoticsofthegaugefield Ψ(z)define a chemical potential μ and the charge density ρ given by: μ = e χ 0 /2 √ 16πG 4 Ψ 0 , ρ =− e χ 0 /2 R √ 16πG 4 Ψ 1 . (8.16) The temperature can be computed in the usual way by Wick-rotating the metric (8.5) to Euclidean signature and then imposing regularity at the horizon [115]: T = 1 4πR˜ z H e −(χ (0) −χ 0 )/2 32   61 + 36 cosh 2λ (0) − cosh 4λ (0) − 8˜ z 2 H e χ (0) Ψ (1) 2   . (8.17) 87 AthightemperaturethesolutionisthefamiliarRN–AdSblackhole. Atlowenough temperature, below some critical value, there exists a new type of solution which has scalar hair. By computing the free energy in both cases it has been shown that the hairy black hole solution is thermodynamically preferred to that of the RN–AdS solution and the transition temperature T c has been obtained as well. See [115, 116] for further details. 8.2.1 RN–AdS Solution The high temperature RN–AdS solution is obtained by setting λ(z) = 0 and χ(z) = 0. That means both the operatorsO 1 andO 2 vanish and there is no condensate. The metric and the gauge field are given by: Ψ(z) = 2QR z H   1− z z H   , f(z) = 1− (1 +Q 2 ) z 3 z 3 H +Q 2 z 4 z 4 H . (8.18) Using equations (8.16-8.17) the temperature, the chemical potential and the charge density become: T = 1 4πz H (3−Q 2 ) , μ = R √ 16πG 4 2Q z H , ρ = R √ 16πG 4 2Q z 2 H . (8.19) 8.2.2 Hairy Black Hole Solution As mentioned before, at low enough temperatures there exists a new branch of solutions which is a black hole with a charged scalar hair. There is no analytic solution available. So we employ a numerical shooting technique to obtain the solution. We impose the initial conditions in the IR and set χ (0) = 1. Then by tuning Ψ (1) we set either λ 1 = 0 or λ 2 = 0, meaning, eitherO 2 orO 1 being 88 turned on respectively. Below the critical temperatureT c this new type of solution represents the superconducting phase with non–zero condensate. 8.2.3 Zero Temperature Solution It is argued in [122] that the zero temperature solution is an RG flow between two AdS 4 spaces. We will again use the numerical shooting technique and impose the initial conditions in the IR. Since there is no black hole horizon at zero tem- perature, the IR is now at ˜ z→∞. In the IR, the fields have an expansion of the form [115]: λ(˜ z) = log(2 + √ 5) +λ 1 ˜ z −α +... , Ψ(˜ z) =ψ 1 ˜ z −β +... , f(˜ z) = 7 3 +... , χ(˜ z) =χ 0 +... , (8.20) where, α = s 303 28 − 3 2 , β = s 247 28 − 1 2 . (8.21) As before, using the scaling symmetries (8.10) of the equations of motion, we can fix the values of Ψ 1 and χ 0 and then tune the free parameter λ 1 to either have λ 1 = 0 or λ 2 = 0 in the UV. We set Ψ 1 = 1 and χ 0 = 4. 89 8.3 The Entanglement Entropy and The Subre- gion Complexity It is instructive to reproduce the results of the entanglement entropy reported in [116] for completeness and also so that we can compare that with our subregion complexity results in the next section. To that end, let us choose a strip region A with width l and length L→∞ in a constant time slice (figure 8.1). Now l L y z B B A x z= z ∗ Figure 8.1: The strip geometry considered in this chapter. Here z denotes the radial direction in the dual bulk geometry AdS 4 . The strip width is l and the length is L which can be taken to infinity. The boundary is at z = 0 where the field theory lives and z = z ∗ denotes the turning point of the minimal surface inside the bulk. following the RT proposal, we need to find the minimal surface γ A bounded by the perimeter ofA and that extends into the bulk of the geometry. Then the area of this minimal surface will give us the entanglement entropy of the subregion A using equation (8.1) and the volume enclosed by this minimal surface and the strip regionA will give us the subregion complexity using equation (8.2). Since our background is static we can compute this volume by slicing the bulk with planes of constant z. See [106] for a review on the computation of volumes of subregions in various AdS black hole geometries. In ref. [91] the authors have further discussed 90 the divergence structure of the volume and a covariant generalization of computing the volume which includes time-dependent geometries as well. The 2-dimensional minimal surface is given by minimizing the area functional: Area(γ A ) =L Z l/2 −l/2 dx R 2 z 2   1 + z 0 (x) 2 f(z)   1/2 . (8.22) The minimization problem leads to the Hamiltonian which is independent ofx and hence this is a constant of motion: 1 z 2 ∗ = 1 z 2 1 r 1 + z 0 (x) 2 f(z) , (8.23) wherez =z ∗ is the turning point of the minimal surface in the bulk. This equation determines the minimal surface γ A : dz dx = q (z 4 ∗ −z 4 )f(z) z 2 . (8.24) Sincez =z ∗ is the turning point of the minimal surface in the bulk we will require that: l(z ∗ ) 2 = Z z∗ 0 dz z 2 q (z 4 ∗ −z 4 )f(z) . (8.25) Then using (8.1) and (8.23) or (8.24), the entanglement entropy becomes: 4G 4 S = 2LR 2 Z z∗ dz z 2 ∗ z 2 1 q (z 4 ∗ −z 4 )f(z) = 2LR 2   s + 1   , (8.26) where s is the finite part and has dimension of inverse length [116]. Note that we have introduced a small cut-off to regularize the area integral. To compute the subregion complexity, we need to find out the volume enclosed 91 by the minimal surface γ A and the strip regionA. This can easily be done by integrating the inside of the minimal surface: V (z ∗ ) = 2LR 3 Z z∗ dz 1 z 3 q f(z) x(z) , (8.27) where, x(z) = Z z∗ z du u 2 q (z 4 ∗ −u 4 )f(u) . (8.28) Then using (8.2) and (8.27) the subregion complexity becomes: 8πGRC = 2LR 3   c fin + x(0) 2 2   , (8.29) where, c fin is the finite part of the subregion complexity and has dimension of inverse length. Note that, x(0) = Z z∗ 0 dz z 2 q (z 4 ∗ −z 4 )f(z) ≡ l(z ∗ ) 2 . (8.30) Sincethedivergingtermhasz ∗ dependenceweshoulddividethequantityinthebig parenthesis in equation (8.29) byx(0) and then plot this re-scaled finite complexity c as a function of the strip width l or the temperature T. This ensures that we subtract the same diverging term for each z ∗ . Finally, symmetries allow us to use the following dimensionless quantities to analyze our system: T √ ρ , O 1 √ ρ , O 2 ρ , √ ρ l , s √ ρ , c ρ . (8.31) 92 0.5 1.0 1.5 2.0 2.5 3.0 ρ l 2 (16πG4) 1/4 R 1/2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 s ρ R 1/2 (16πG4) 1/4 (a) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ρ l 2 (16πG 4) 1/ 4 R 1/2 -6 -4 -2 0 2 c ρ R (16πG4) 1/2 (b) Figure 8.2: The entanglement entropy (a) and the subregion complexity (b) as functions of the strip–widthl for theO 1 superconductor for a fixed temperature: R 1/2 (16πG4) 1/4 T √ ρ = 0.053. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. 8.3.1 O 1 Superconductor Figure 8.2 shows our results for the entanglement entropy s and the subregion complexitycasfunctionsofthestripwidthlforafixedtemperature: R 1/2 (16πG 4 ) 1/4 T √ ρ = 0.053, which is below the transition temperature T c . In both cases we see the expected linear growth behavior for large l (for the entanglement entropy this is known as the “area law”) and we find that the entropy and the complexity are lower in the superconducting phase than that of the normal phase. The RN– AdS case having higher entropy than the superconducting case represents the fact that the degrees of freedom have condensed in the latter case. As we decrease the temperature all the curves still go linearly for large l though the slopes of the curves in superconducting cases are smaller. In figure 8.3 we plot our results for T = 0. Remember that the zero temperature solution is an RG flow between two AdS vacua [115, 122]. Now, large l probes more deeply into the IR and for empty AdS since there is no horizon, the IR is at ˜ z→∞ where f(˜ z) is constant. Hence, 93 0.5 1.0 1.5 2.0 2.5 3.0 ρ l 2 (16πG4) 1/4 R 1/2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 s ρ R 1/2 (16πG4) 1/4 (a) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ρ l 2 (16πG 4) 1/ 4 R 1/2 -5 -4 -3 -2 -1 0 1 2 c ρ R (16πG4) 1/2 (b) Figure 8.3: The entanglement entropy (a) and the subregion complexity (b) as functions of the strip–width l for theO 1 superconductor for a fixed temperature: R 1/2 (16πG4) 1/4 T √ ρ = 0. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. 0.02 0.04 0.06 0.08 0.10 0.12 0.14 T ρ R 1 /2 (16πG4) 1 /4 -1.0 -0.5 0.0 0.5 1.0 s ρ R 1 /2 (16πG4) 1 /4 (a) 0.02 0.04 0.06 0.08 0.10 0.12 T ρ R 1 /2 (16πG4) 1 /4 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c ρ R (16πG4) 1 /2 (b) Figure 8.4: The entanglement entropy (a) and the subregion complexity (b) as functions of the temperature T for theO 1 superconductor for a fixed l: √ ρ(16πG 4 ) 1/4 R −1/2 l/2 = 2.5. The red dashed(or dotted) curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. The black solid line denotes the transition temperature T c . Since the zero temperature solution is exactly known, we include our results for T = 0 in the plot. the entropy and the complexity approaching different constant values for large l during the superconducting phase is not surprising. In figure 8.4 we present how s and c change with the temperature while the strip width l is kept fixed. For the entropy plot, the physical curve is deter- mined by choosing the point of lowest entropy for a given temperature [116]. As 94 we lower the temperature the entropy decreases in both the phases and there is a discontinuity in the slope at the transition temperature T c . On the contrary, as we lower the temperature the complexity increases during the normal phase but decreases during the superconducting phase. At some low temperature, the complexity in the superconducting phase rises slightly and then drops to a finite minimum value at zero temperature. Note that we do not plot all the supercon- ductor results due to lack of numerical control in our shooting technique. Similar to the entropy plot, there is again a discontinuity in the slope at the transition temperature T c ≈ 0.1199 √ ρ(16πG 4 ) 1/4 R 1/2 . Also note that both plots lead to the same transition temperature T c . 8.3.2 O 2 Superconductor 2.522.542.562.582.60 0.420 0.425 0.430 0.435 0.440 0.445 0.450 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ρ l 2 (16πG4) 1/4 R 1/2 -1.0 -0.5 0.5 1.0 1.5 s ρ R 1/2 (16πG4) 1/4 (a) 2.48 2.50 2.52 2.54 2.56 2.58 2.60 0.25 0.30 0.35 0.40 1 2 3 4 ρ l 2 (16πG4) 1/4 R 1/2 -2 -1 0 1 c ρ R (16πG4) 1/2 (b) Figure 8.5: The entanglement entropy (a) and the subregion complexity (b) as functions of the strip–widthl for theO 2 superconductor for a fixed temperature: R 1/2 (16πG4) 1/4 100T √ ρ = 0.305. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. Figure 8.5 again shows our results for the entanglement entropy s and the subregion complexity c as functions of the strip width l for a fixed temperature: R 1/2 (16πG 4 ) 1/4 100T √ ρ = 0.305. We choose the fixed temperature to be below the transition 95 2.15 2.20 2.25 2.30 2.35 2.40 2.45 0.28 0.30 0.32 0.34 0.36 0.38 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ρ l 2 (16πG4) 1/4 R 1/2 -1.0 -0.5 0.0 0.5 1.0 1.5 s ρ R 1/2 (16πG4) 1/4 (a) 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 0.1 0.2 0.3 0.4 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ρ l 2 (16πG4) 1/4 R 1/2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 c ρ R (16πG4) 1/2 (b) Figure 8.6: The entanglement entropy (a) and the subregion complexity (b) as functions of the strip–width l for theO 2 superconductor for a fixed temperature: R 1/2 (16πG4) 1/4 100T √ ρ = 0. The red dashed curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. temperatureT c as before. ForO 2 superconductor the critical temperature is given by: T c ≈ 0.3638 √ ρ(16πG 4 ) 1/4 R 1/2 . In [116], the origin of this multivaluedness has been argued to be the non–monotonic behavior of f(z), which applies to the results of the complexity as well. In figure 8.6 we plot our results forT = 0. In general both the plots behave similar to that of theO 1 case for largel. But there are few crucial differences here now. For the temperature below the critical temperature T c both the quantities now show multivaluedness for a given range of l. For the entropy this fact is reflected in the form of a swallowtail [116, 123]. In the complexity plot this is captured as an “S” curve. Also note that the superconducting phase now has higher complexity than the normal phase in the region of multivaluedness and for largel as well which is in contrast with the entropy plot. It has been observed that in theO 2 case,f(z) can develop a minimum and a maximum (at low temper- atures). When the turning point of the minimal RT surface in the bulk, z ∗ lies in the neighborhood of the minimum off(z), the entropy and the complexity become multivalued. See [116] for further details and a clear demonstration of how it hap- pens using the domain wall analysis. We show the behavior of f(z) in figure 8.7. 96 0.0 0.2 0.4 0.6 0.8 1.0 z 0.0 0.2 0.4 0.6 0.8 1.0 1.2 f(z) (a) 0.0 0.2 0.4 0.6 0.8 1.0 z 0.0 0.2 0.4 0.6 0.8 1.0 f(z) (b) Figure 8.7: (a) Behavior of f(z) for the O 1 superconductor. This has been shown for: R 1/2 (16πG4) 1/4 T √ ρ = 0.053. (b) Behavior off(z) for theO 2 superconductor. This has been shown for: R 1/2 (16πG4) 1/4 100T √ ρ = 0.305. 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 ρ l 2 (16πG4 ) 1/4 R 1/2 0.28 0.30 0.32 0.34 0.36 0.38 s ρ R 1/2 (16πG4) 1/4 (a) 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 ρ l 2 (16πG4 ) 1/4 R 1/2 0.1 0.2 0.3 0.4 0.5 c ρ R (16πG4) 1/2 (b) Figure 8.8: A comparison of the multivalued regions of the entanglement entropy (a) and the subregion complexity (b) as functions of the strip–width l for theO 2 superconductor for a fixed temperature: R 1/2 (16πG4) 1/4 T √ ρ = 0. In figure 8.8 we compare different parts of the entanglement entropy and the sub- region complexity curves for zero temperature by mapping out the corrsponding features of the multivalued regions. From the entropy plot (figure 8.8a) we see that the physical curve must follow the sequence 6→ 2→ 1 as explained in [116]. Accordingly, there is a finite jump towards a lower value in the complexity plot (figure 8.8b) from 6→ 2→ 1. Wepresentinfigure8.9howsandcchangewiththetemperaturewhilethestrip width l is kept fixed. There is again a discontinuity in the slope at the transition 97 0.1 0.2 0.3 0.4 0.5 100T ρ R 1 /2 (16πG4) 1 /4 0.35 0.40 0.45 0.50 0.55 0.60 s ρ R 1 /2 (16πG4) 1 /4 (a) 0.1 0.2 0.3 0.4 100T ρ R 1 /2 (16πG4) 1 /4 0.0 0.1 0.2 0.3 0.4 0.5 c ρ R (16πG4) 1 /2 (b) Figure 8.9: The entanglement entropy (a) and the subregion complexity (b) as functions of the temperature T for theO 2 superconductor for a fixed l: √ ρ(16πG 4 ) 1/4 R −1/2 l/2 = 2.5. The red dashed(or dotted) curve is the Reissner–Nordstrom solution and the solid blue curve is the superconductor solution. The black solid line denotes the transition temperature T c . Since the zero temperature solution is exactly known, we include our results for T = 0 in the plot. temperature T c in both the cases. Moreover, as we lower the temperature, the value of the entropy drops discontinuously whereas the value of the complexity rises discontinuously. There is another special feature in this case. For our chosen value ofl, there is an additional discontinuity in the slope of the decreasing entropy at some lower temperature and the entropy curve is a combination of two types of curve joined by this new discontinuity. This special feature is due to a new length scale in the theory as argued in [116]. In the complexity plot, correspondingly we see a discontinuous but finite jump exactly at the temperature where the above- mentioned new discontinuity shows up in the entropy plot. 8.4 Summary and Outlook In summary, we have performed a numerical shooting technique to compute the subregion complexity for a 2+1–dimensional holographic superconductor using the “Complexity equals Volume” or the CV conjecture [3]. Our analysis reveals that 98 apart from the universal divergent term, there are no other divergences in the complexity as far as the phase transition is concerned. The subregion complexity grows linearly withl for large strip widthl. From our computation it is clear that the complexity captures phase transition as well and the transition temperature T c is exactly same as computed from the free energy analysis of the system. Same T c has been read off from the entropy vs. temperature plot as well. This is not surprising since the gravity background is dual to a field theory with a single tran- sition temperature and hence our result is a nice confirmation of holography. 1 Apart from that, the complexity actually behaves differently. Specially, for the first order phase transition, the complexity of the superconducting phase is higher than the normal phase, and it also increases with decreasing temperature. The second order phase transition rather shows a similar behavior to that of the entropy analysis, though, at very low temperature the complexity behavior is quite strange and the physics behind it is yet to be investigated. We have shown the zero temperature solution in all these cases as well. We have also observed the multivaluedness similar to the entropy plot except the fact that in this case the multivaluedness is of different form, namely in the form of an “S” curve. Finally, the behavior of the complexity suggests that it may be used as another independent probe to the physics of the phase transition. Our results are in agreement with that reported in [118] and in [119] where the authors have studied 1 + 1–dimensionsals-wave andp-wave holographic supercon- ductor respectively: the complexity remains finite during the phase transition 2 and the subregion complexity plot leads to the same transition temperatureas observed 1 We thank the anonymous referee for pointing this out. 2 Notice that our results do not match with the results reported in [117]. They have found that during the phase transition the complexity becomes infinite for a 1+1-dimensional s-wave superconductor. 99 from the entropy plot. Ref. [119] has also observed multivaluedness and discon- tinuous but finite jump in the complexity during the first order phase transition. While this project was near completion two new articles appeared in the litera- ture: In [120] the authors have studied the time dependent complexity and how the complexity (of formation) scales with the temperature where the scaling factor is a function of the superconductor model parameters by using the CV conjecture in an asymptotically AdS d+1 geometry. In [121] the authors have investigated the subregion complexity for the Stückelberg superconductor which is very similar to our set up in this paper. Apart from the second order phase transition result, all other main results reported here agree with theirs as well. During the second order phase transition the complexity of the superconducting phase is opposite to what they have found. 34 There are multiple questions which need to be answered. It has been reported that the complexity decreases with increasing temperature during the second order phasetransition, butinthispaperweactuallyfoundthatitbehavesquitesimilarto that of the entropy for the superconducting phase. It seems like the temperature dependence of the complexity is not universal. It will be nice to see if there is any deeper physics behind it. Also, the behavior of the complexity of theO 1 superconductor at low temperature needs some careful explanation. Some other questions that we might ask is what happens if we compute the complexity using theCAconjectureandhowthecomplexityevolvesafterathermalquench? Finally, 3 See also [124] for analytic expressions of the complexity in the high and the low temperature regime for the Schwarzschild–AdS and the RN–AdS black holes. 4 In refs. [125] the authors have calculated holographic entanglement entropy, subregion complexity and fisher information metric for a class of nonsupersymmetric D3 branes and also reviewed the same for the AdS black brane as well. 100 what all of these mean in the the dual field theory is worth exploring. We will try to answer some of these in future work. 101 Part III Bulk Metric Reconstruction 102 Chapter 9 Gravity Dual of ABJM from Holographic Entanglement Entropy 9.1 Introduction ThediscoveryoftheAdS/CFTcorrespondence[36,37,40,126]hasproventobe one of the most important and powerful toolkit in modern theoretical physics. The correspondenceprovidesarealizationoftheHolographicPrinciple[127, 128], which statesthatthedegreesoffreedomofa‘bulk’theorycanbedescribedbythedegrees of freedom on its boundary. Another attractive feature of the correspondence is its connection between strongly coupled field theory and weakly coupled gravity. In its most basic form, the correspondence relates quantities computable in a theory of gravity, specifically a spacetime which is asymptotically AdS, to ones calculated in a ‘dual’ quantum field theory through what’s called a ‘holographic dictionary.’ For example, the behavior of scalar fields in the bulk gravity theory can be determined by studying correlation functions of operators in the boundary field theory [37]. The metric of the bulk gravity theory is also determined in part by the stress tensor of the dual conformal field theory (CFT) [129]. In [73, 74] Ryu and Takayanagi proposed a relationship between the entanglement entropy in the dual field theory and the area of an extremal surface in the bulk gravity theory. 103 Inparticular, supposethefield theory isdefinedonamanifoldM whichhasthe topologyR× Σ, where Σ is a spacelike hypersurface. Let ρ be a global pure state in the field theory andA⊂ Σ a constant-time subregion ofM. The boundary∂A is called the entangling surface. The reduced density matrix ρ A is defined as the partialtraceofρoverdegreesoffreedominthecomplementA C . Theentanglement entropy of the state ρ A is then defined as the von Neumann entropy of ρ A : S A ≡S[ρ A ] =−Tr A (ρ A logρ A ) . (9.1) The dual spacetime geometry will be an asymptotically AdS manifold whose conformal boundary isM. Let Γ be the constant-time surface which is anchored along ∂A on the boundary with the extremal areaA[Γ] (calculated with the bulk metric). Then the Ryu-Takayanagi (RT) formula states that: S A = A[Γ] 4G N . (9.2) In practice it is easiest to use this formula to compute the entanglement entropy for a CFT in some global state (usually the vacuum or a thermal state) reduced to a region A. That is, one gains information about the boundary field theory by performing a calculation using geometric information in the bulk. Recent stud- ies of the entanglement entropy in the context of AdS/CFT indicate that the entanglement is intimately related to the structure of spacetime and gravity the- ories [130–138]. From this perspective, one might ask how much can be learned about the bulk geometry from the entanglement in the CFT. This question has been explored in several ways, including the ER = EPR proposal, entanglement wedge reconstruction with and without using the Petz map, reconstructing bulk geodesics, andreconstructingthebulkmetric. In[139, 140]theauthorshaveshown 104 thattheRyu-Takayanagi(RT)andtheHubeny-Rangamani-Takayanagi(HRT)for- mulae are sufficient for boundary regions of disk topology (and a generalization thereafter) to uniquely fix the bulk metric in some portions of the bulk. Their construction is covariant and generally requires no symmetry assumptions. Ref. [141] proposed a new method of metric reconstruction based on the entanglement entropy, specifically, on its reformulation in terms of bit threads. For more gen- eral states other than the vacuum, their method seems to be more efficient than those based on RT surfaces because for many excited states, the RT surfaces that compute the entanglement of the boundary regions do not generally foliate the full bulk spacetime, while the bit threads associated with them do reach these “shadow” regions. Somehow similar to our set up, the authors in [142] have used the chain rule and imprecise and discrete boundary data to reconstruct a generic bulk metric from the derivatives of the entanglement entropy. Their method also allows to predict the two-point functions of heavy operators or Wilson loops at different energy scales given the same external control parameters. 9.1.1 Spacetime from Entanglement To philosophically motivate the calculations later in the chapter, we can begin by summarizing an argument in [131]. As a thought experiment, picture a field theory defined on a sphere S 2 , and split this sphere into two regions A,B. The extremal surface anchored along the boundary of A is a sheet that splits the bulk ballintotwodistinctregions ˜ A, ˜ B. Nowsupposewecancalculatetheentanglement entropy of some stateρ reduced toA, and suppose that the entanglement entropy is tuneable with some parameter. That is, if we increase the parameter the entan- glement goes up, and vice versa. Imagine we tune the entanglement entropy down to 0. By the RT formula, the area of of the bulk extremal surface will also shrink 105 to 0, meaning the regions ˜ A, ˜ B become totally disjoint. The effect of having no entanglement entropy between the two regions in the CFT is to have two separate spacetimes. In this sense, the bulk spacetime is held together by entanglement entropy. A cartoon of this process in one less dimension is presented in fig. 9.1. Figure 9.1: A cartoon demonstrating how tuning the entanglement entropy to zero produces two disjoint spacetimes. The RT surface is shown in red. The rest of the chapter is organized as follows. In section 9.2 we review the basics of ABJM [143] and present results for the entanglement entropy of sev- eral entangling surfaces. We argue that when the entangling surface is a finite length rectangular strip the entanglement entropy should match the RT result. In section 9.3 we introduce the framework for computing the bulk metric from the entanglement entropy and perform the calculation for ABJM in Minkowski space. We successfully reconstruct the pureAdS 4 metric with the correct AdS radius from the entanglement entropy of the ABJM vacuum reduced to a strip in Minkowski space. Examples of this methodology for 2-dimensional CFTs in various states and N = 4 SYM in the vacuum were explored in [144]. We will conclude this chapter with a discussion section and future prospective 9.4. 106 9.2 ABJM and AdS/CFT The ABJM theory [143] is a three dimensional supersymmetric Chern-Simons- matter theory with gauge group U(N) k ×U(N) −k where (k,−k) are the Chern- Simons levels of the action. The theory is superconformal, generally withN = 6 supersymmetry, and is weakly coupled in the limitkN. The supersymmetry is enhanced toN = 8 when N = 2, as well as when k = 1, 2. The theory can be constructed by considering a coincident stack of M2-branes in 11-dimensional supergravity. When the stack of N branes is probing aC 4 /Z k singularity, the field theory description of the branes is anN = 3 predecessor of ABJM, namely a supersymmetric Chern-Simons-matter theory with massive dynamical gauge fields. One can integrate out the gauge fields in the low energy limit to recover theN = 6 ABJM theory. In the limit of large N this brane construction can be thought of as M-theory on AdS 4 /Z k . From this perspective, the Chern-Simons level k is related to the circle one would dimensionally reduce M-theory on to get type IIA string theory. Thus this pairing of AdS 4 and ABJM gives a realization of the AdS/CFT correspondence. There has been a great deal of success calculating the partition function for ABJM on (squashed and branched) 3-spheres using localization techniques (for a review, see [145]). The partition function of ABJM on S 3 can then be used to calculate the entanglement entropy for ABJM in a different context. Using conformal transformations, it can be shown that [146] the entanglement entropy of the vacuum reduced to a disk is given by: S disk = logZ S 3 , (9.3) 107 where Z S 3 is the S 3 partition function. To leading order in N the universal part of the result is [147]: logZ S 3 = √ 2π 3 k 1/2 N 3/2 . (9.4) Including the divergent area term and choosing the UV regulator appropriately, the entanglement entropy becomes [73]: S disk = √ 2π 3 k 1/2 N 3/2 ` a − 1 ! , (9.5) where ` is the radius of the disk and a is the UV regulator. This is expected for a 3-dimensional CFT (see e.g. [148] and references therein). The construction of Casini et al. [146] which make the entanglement entropy calculation tractable on the field theory side relies heavily on the spherical sym- metry of the entangling surface. However, using a spherical entangling surface renders the bulk metric reconstruction calculation very difficult. It will be much easier to use a rectangular strip as our entangling surface. In fact, it is possible to calculate the entanglement entropy of the vacuum reduced to a rectangular strip using holography [73]. If the strip has width ` and length L (understood to be very large), the result is: S strip = √ 2 3 k 1/2 N 3/2 L a − 4π 3 L Γ(1/4) 4 ` ! . (9.6) 9.3 Bulk metric reconstruction The general philosophy for using the entanglement entropy to calculate the metric of the bulk spacetime is as follows. Suppose we start with just the idea of a holographic dictionary. In particular, putting the boundary CFT in its vacuum 108 state should be dual to a relatively ‘calm’ bulk geometry. This is exemplified by Fefferman-Grahamcoordinates[149, 150], inthesensethatthegeometryisaffected by the expectation value of the stress tensor of the field theory. Further, we want the conformal boundary of our bulk spacetime to match the spacetime the CFT is defined on. Along with this, we are free to use diffeomorphism invariance of the bulk gravity theory to choose coordinates which are convenient. The final part of the holographic dictionary that we need is the RT formula (9.2). Using these principles we write down an ansatz for the bulk metric. For the methodology of the calculation we follow [136, 137]. Consider a CFT defined on a d-dimensional Minkowski space. Suppose we know the entanglement entropy of this theory’s vacuum reduced to a rectangular strip. Schematically, we expect that since the bulk geometry is not too perverse (i.e. calm), the RT surface should share some of the symmetries of the entangling surface. As such, we choose the metric ansatz: ds 2 = R 2 z 2 −h(z)dt 2 +f(z) 2 dz 2 + d−2 X i=1 dx 2 i ! . (9.7) Arrange the entangling region to be: A : x i |−`/2<x 1 <`/2, 0<x 2 ,x 3 ,...,x d−2 <L . (9.8) An example is shown in fig. 9.2. The symmetry between x 2 ,...,x d−2 informs us to assume the RT surface can be parametrized in the bulk by z = z(x 1 ). The induced metric on the RT surface is: ds 2 = R 2 z 2 z 02 f 2 + 1 dx 2 1 + d−2 X i=2 dx 2 i ! , (9.9) 109 l L y z B B A x z= z ∗ Figure 9.2: A sketch of an entangling region in a 3d CFT and its corresponding RT surface in the bulk. where z 0 ≡dz/dx. Call the RT surface γ. The volume of γ is: A[γ] =R d−1 L d−2 Z `/2 `/2 dx q 1 +z 0 (x) 2 f(z(x)) 2 z(x) d−1 , (9.10) where we have integrated out the extra coordinates. This volume is a functional of the embedding z(x) and the metric function f(z). AccordingtotheRTprescription, wemustextremize(9.10)withrespecttof,z. We can use the calculus of variations to solve the problem. Call the integrand the Lagrangian: L(z,z 0 ;x) = q 1 + (z 0 f(z)) 2 z d−1 , (9.11) where z,z 0 are the generalized coordinates and x is the ‘time’. SinceL does not depend explicitly on x, its Legendre transform (the Hamiltonian) is conserved: H =z 0 ∂L ∂z 0 −L, dH dx = 0 . (9.12) 110 The Hamiltonian is given by: H =− 1 z d−1 q 1 + (z 0 f) 2 . (9.13) Because the Hamiltonian is conserved, we can set it equal to something that does not depend on x. In particular, we choose to relate it to the turning point of the RT surface, called z ∗ : H =− 1 z d−1 ∗ . (9.14) Thenextsteps, whichwillbedisplayedspecificallyforthecaseofinterestbelow, are to rewrite the volume functional using z ∗ , relate the field theory entanglement entropy to the area functional using the RT formula, and then solve the resulting integral equation for the metric function f(z). Note that this procedure does not give us the time component of the metric, h(z). 9.3.1 The bulk dual of ABJM in its ground state Write the metric ansatz: ds 2 = R 2 z 2 −h(z)dt 2 +f(z) 2 dz 2 +dx 2 +dy 2 . (9.15) Assume the RT surface is given by z =z(x) with x∈ (−`/2,`/2) and y∈ (0,L). Since the RT surface γ is 2-dimensional, we will start referring toA as the area. The area functional is: A = 2LR 2 Z `/2 0 dx q 1 + (z 0 f(z)) 2 z 2 . (9.16) 111 In the calculus of variations problem the Lagrangian is: L(z,z 0 ;x) = q 1 + (z 0 f(z)) 2 z 2 , (9.17) and the conserved Hamiltonian is: H =− 1 z 2 q 1 + (z 0 f) 2 . (9.18) Let z ∗ be the turning point of the surface, so that dz dx z=z∗ = 0. Then z 0 = q z 4 ∗ −z 4 z 2 f(z) , (9.19) which implies ` = 2 Z z∗ a dz z 2 f(z) q z 4 ∗ −z 4 , (9.20) where a is a UV cutoff. In terms of z ∗ the Lagrangian isL =z 2 ∗ /z 4 , and our area functional becomes: A = 2LR 2 Z z∗ a dz z 2 ∗ f(z) z 2 q z 4 ∗ −z 4 . (9.21) Next, we view the entanglement entropy in eqn. (9.6) and the area functional in eqn. (9.21) as functions of the parameter `. Their ` derivatives are: dS strip d` = √ 2 3 k 1/2 N 3/2 4π 3 L Γ(1/4) 4 ` 2 , (9.22) dA d` = dA dz ∗ dz ∗ d` . (9.23) 112 BecauseA does not explicitly depend on ` we must use the chain rule. Write: A = 2LR 2 Z z∗ a F (z ∗ ,z)dz & ` = 2 Z z∗ a z 4 z 2 ∗ F (z ∗ ,z)dz , (9.24) where F (z ∗ ,z) = z 2 ∗ f(z) z 2 q z 4 ∗ −z 4 . (9.25) By the Leibnitz rule for differentiation under an integral sign we get: dA dz ∗ = 2LR 2 " lim z→z∗ F (z ∗ ,z)− 2 Z z∗ a z ∗ z 2 f(z) (z 4 ∗ −z 4 ) 3/2 dz # d` dz ∗ = 2z 2 ∗ " lim z→z∗ F (z ∗ ,z)− 2 Z z∗ a z ∗ z 2 f(z) (z 4 ∗ −z 4 ) 3/2 dz # . Hence dA d` = LR 2 z 2 ∗ . (9.26) Upon differentiating the RT formula with respect to` and solving for` in terms of z ∗ , the entanglement entropy can be rewritten as: S strip (z ∗ ) = √ 2 3 k 1/2 N 3/2 L a − LRπ 3/2 k 1/4 N 3/4 2 1/4 q 3G (4) N Γ(1/4) 2 z ∗ . (9.27) The RT formula S strip (z ∗ ) =A(z ∗ )/4G (4) N then gives us the integral equation for the function f(z) which we now begin to solve. First we write: F(z ∗ ) = 4G (4) N S EE (z ∗ ) 2LR 2 z 2 ∗ , p(z) = f(z) z 2 , g(z) =z 4 , (9.28) so that F(z ∗ ) = Z z∗ a dz p(z) q g(z ∗ )−g(z) . (9.29) 113 This integral equation is called Volterra equation of the first kind and it has the solution: p(z) = 1 π d dz Z z a dz ∗ F(z ∗ )g 0 (z ∗ ) q g(z)−g(z ∗ ) , (9.30) where g 0 (z) = dg dz > 0. So, we find: f(z) = 8G (4) N z 2 πLR 2 d dz Z z a dz ∗ z ∗ S EE (z ∗ ) q z 4 −z 4 ∗ . (9.31) Lets us now define: C 1 = √ 2 3 k 1/2 N 3/2 L a , (9.32) C 2 = LRπ 3/2 k 1/4 N 3/4 2 1/4 q 3G (4) N Γ(1/4) 2 z ∗ . (9.33) Thus the radial function f(z) becomes: f(z) = 8G (4) N z 2 πLR 2 d dz Z z a dz ∗ C 1 z ∗ −C 2 q z 4 −z 4 ∗ ≡ 8G (4) N z 2 πLR 2 dM dz , (9.34) where M(z) is the integral part in the above expression. One then finds: M(z) =C 1 " π 4 − 1 2 tan −1 a 2 √ z 4 −a 4 !# −C 2   √ π Γ 5 4 zΓ 3 4 − a 2 F 1 , 1 4 , 5 4 , a 4 z 4 z 2   , (9.35) where 2 F 1 is a hypergeometric function. Therefore: z 2 dM dz = C 1 a 2 z q 1− a 4 z 4 + C 2 √ πΓ 5 4 Γ 3 4 − C 2 a z   1 q 1− a 4 z 4 + 2 F 1 1 4 ,, 5 4 , a 4 z 4 !   . (9.36) 114 We are only interested in the regime a/z 1 because it corresponds to the UV cutoff becoming small and the metric not probing too close to the boundary at z = 0. In this limit a/z→ 0 we have: lim a/z→0 8G (4) N z 2 πLR 2 dM dz = C 2 √ πΓ 5 4 Γ 3 4 = const. (9.37) Using the expression of C 2 and the fact that Γ 5 4 = Γ 1 4 4 , (9.38) we end up with: lim a/z→0 f(z) 2 = G (4) N k 1/2 N 3/2 2 √ 2 3R 2 . (9.39) After rescaling z, the metric is: ds 2 = ˜ R 2 ˜ z 2 −h(z)dt 2 +dz 2 +dx 2 +dy 2 , (9.40) where the AdS radius is: ˜ R =   G (4) N k 1/2 N 3/2 2 √ 2 3   1/2 . (9.41) As a sanity check, notice that from [74] we find: G (4) N = 48π 3 ` 9 p (2R AdS 4 ) 7 , 2R AdS 4 =` p (32π 2 k 1/3 N) 1 6 . (9.42) Solving these two gives us: R 2 AdS 4 = G (4) N k 1/2 N 3/2 2 √ 2 3 , (9.43) 115 which is exactly what we have found! So, we have been successfully reconstructed the radial function and the AdS radius using this technique. The one part left is to fix the time component. 9.3.2 Fixing the Last Metric Component The time-independent RT prescription is only enough to determine one of the unknown functions in the metric ansatz. With our assumptions about the holo- graphic dictionary at the beginning of section 4 and some mild assumptions about the entanglement entropy in ABJM, it is safe to assume (see for example [151]) that the bulk metric satisfies Einstein’s equations. Using Λ =−3/ ˜ R 2 , the three unique Einstein’s equations are: 12h z 2 + 2h 00 − (h 0 ) 2 h − 6h 0 z 2 = 12 z 2 , (9.44) z 2h h 0 −2h 00 )+z(h 0 ) 2 h 2 − 12 4z 2 =− 3 z 2 , (9.45) zh 0 h − 6 2z 2 =− 3 z 2 . (9.46) The final one implies h = const. while the first two are consistent only if h = 1. Thus we have recovered the metric of pure AdS 4 : ds 2 = ˜ R 2 z 2 (−dt 2 +dz 2 +dx 2 +dy 2 ) . (9.47) with ˜ R given above. 116 9.4 Discussion In this chapter we have demonstrated again the relevance of the entanglement entropy to the structure of spacetime. In particular, we have used general ideas from holography to reconstruct the metric of the bulk spacetime dual to ABJM in its ground state in flat space. This result enlarges the list of AdS/CFT pairs that have been investigated through this specific methodology. It is worth noting an apparent tension between our philosophy and the actual source of the entanglement entropy in eqn. (9.6). This result was derived in the original paper by Ryu and Takayanagi [73]. That is to say, they predicted the entanglement entropies for a theory dual to AdS 4 ×S 7 by doing a calculation in the bulk (they did not know that they were computing the entanglement for ABJM withk = 1 because the RT paper predated the original ABJM paper [143]). There appears to be a circularity in using a bulk calculation to reconstruct the bulk metric. However, there is a good reason to overlook that. It was shown by direct computation on the field theory side [147, 148] that the RT prediction for a spherical entangling surface can be trusted. Moreover, the authors of [144] used a result from RT [73] to reconstruct the bulk metric dual toN = 4 SYM in flat space. It would be interesting to test the methodology of this chapter with a time dependent entanglement entropy in the CFT. We suspect that the covariant gen- eralization of the RT prescription [152] would be necessary. In the static examples considered here and in [144], the time component of the metric is determined by demanding that the Einsten’s equations be satisfied, i.e. not using the entangle- ment entropy. It would further probe the connection between entanglement and spacetime by trying to fully reconstruct a dynamic bulk metric using a dynamic entanglement entropy. 117 Bibliography [1] Avik Chakraborty and Clifford V. Johnson. “Benchmarking black hole heat engines, I”. In: Int. J. Mod. Phys. D 27.16 (2018), p. 1950012. doi: 10. 1142/S0218271819500123. arXiv: 1612.09272 [hep-th]. [2] AvikChakrabortyandCliffordV.Johnson.“BenchmarkingBlackHoleHeat Engines, II”. In: Int. J. Mod. Phys. D 27.16 (2018), p. 1950006. doi: 10. 1142/S0218271819500068. arXiv: 1709.00088 [hep-th]. [3] Avik Chakraborty. “On the complexity of a 2 + 1-dimensional holographic superconductor”. In: Class. Quant. Grav. 37.6 (2020), p. 065021. doi: 10. 1088/1361-6382/ab6d09. arXiv: 1903.00613 [hep-th]. [4] Ashton Lowenstein and Avik Chakraborty. “Reconstructing the Bulk Dual of ABJM from Holographic Entanglement Entropy”. In: (Jan. 2021). arXiv: 2101.04624 [hep-th]. [5] Jacob D. Bekenstein. “Black holes and entropy”. In: Phys. Rev. D7 (1973), pp. 2333–2346. doi: 10.1103/PhysRevD.7.2333. [6] J. D. Bekenstein. “Black holes and the second law”. In: Lett. Nuovo Cim. 4 (1972), pp. 737–740. doi: 10.1007/BF02757029. [7] S. W. Hawking. “Black holes in general relativity”. In: Commun. Math. Phys. 25 (1972), pp. 152–166. doi: 10.1007/BF01877517. 118 [8] James M. Bardeen, B. Carter, and S. W. Hawking. “The Four laws of black hole mechanics”. In: Commun. Math. Phys. 31 (1973), pp. 161–170. doi: 10.1007/BF01645742. [9] S. W. Hawking. “Particle Creation by Black Holes”. In: Commun. Math. Phys. 43 (1975). [,167(1975)], pp. 199–220. doi: 10.1007/BF02345020,10. 1007/BF01608497. [10] Robert M. Wald. “The thermodynamics of black holes”. In: Living Rev. Rel. 4 (2001), p. 6. doi: 10.12942/lrr-2001-6. arXiv: gr-qc/9912119. [11] Jennie H. Traschen. “An Introduction to black hole evaporation”. In: 1999 Londrona Winter School on Mathematical Methods in Physics. Aug. 1999. arXiv: gr-qc/0010055. [12] Daniel Grumiller, Robert McNees, and Jakob Salzer. Black holes and ther- modynamics - The first half century. 2014. arXiv: 1402.5127 [gr-qc]. [13] David Kubiznak, Robert B. Mann, and Mae Teo. “Black hole chem- istry: thermodynamics with Lambda”. In: Class. Quant. Grav. 34.6 (2017), p. 063001. doi: 10 . 1088 / 1361 - 6382 / aa5c69. arXiv: 1608 . 06147 [hep-th]. [14] S. Carlip. “Black Hole Thermodynamics”. In: Int. J. Mod. Phys. D 23 (2014), p. 1430023. doi: 10.1142/S0218271814300237. arXiv: 1410.1486 [gr-qc]. [15] C. Teitelboim. “The Cosmological Constant as a Thermodynamic Black Hole Parameter”. In: Phys.Lett. B158 (1985), pp. 293–297. doi: 10.1016/ 0370-2693(85)91186-4. 119 [16] J. David Brown and C. Teitelboim. “Neutralization of the Cosmological Constant by Membrane Creation”. In: Nucl. Phys. B297 (1988), pp. 787– 836. doi: 10.1016/0550-3213(88)90559-7. [17] Jolien D. E. Creighton and Robert B. Mann. “Quasilocal thermodynam- ics of dilaton gravity coupled to gauge fields”. In: Phys. Rev. D52 (1995), pp. 4569–4587. doi: 10.1103/PhysRevD.52.4569. arXiv: gr-qc/9505007 [gr-qc]. [18] Marco M. Caldarelli, Guido Cognola, and Dietmar Klemm. “Thermody- namics of Kerr-Newman-AdS black holes and conformal field theories”. In: Class. Quant. Grav. 17 (2000), pp. 399–420. doi: 10.1088/0264-9381/17/ 2/310. arXiv: hep-th/9908022 [hep-th]. [19] T. Padmanabhan. “Classical and quantum thermodynamics of horizons in spherically symmetric space-times”. In: Class. Quant. Grav. 19 (2002), pp. 5387–5408. doi: 10.1088/0264-9381/19/21/306. arXiv: gr-qc/ 0204019 [gr-qc]. [20] David Kastor, Sourya Ray, and Jennie Traschen. “Enthalpy and the Mechanics of AdS Black Holes”. In: Class.Quant.Grav. 26 (2009), p. 195011. doi: 10.1088/0264-9381/26/19/195011. arXiv: 0904.2765 [hep-th]. [21] Brian P. Dolan. “The cosmological constant and the black hole equation of state”. In: Class.Quant.Grav. 28 (2011), p. 125020. doi: 10.1088/0264- 9381/28/12/125020. arXiv: 1008.5023 [gr-qc]. [22] M. Cvetic, G.W. Gibbons, D. Kubiznak, and C.N. Pope. “Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume”. In: Phys.Rev. D84 (2011), p. 024037. doi: 10.1103/PhysRevD.84.024037. arXiv: 1012.2888 [hep-th]. 120 [23] Brian P. Dolan. “Where is the PdV term in the fist law of black hole ther- modynamics?” In: (2012). arXiv: 1209.1272 [gr-qc]. [24] Clifford V. Johnson. “Holographic Heat Engines”. In: Class. Quant. Grav. 31 (2014), p. 205002. doi: 10.1088/0264-9381/31/20/205002. arXiv: 1404.5982 [hep-th]. [25] Jacob D. Bekenstein. “Generalized second law of thermodynamics in black hole physics”. In: Phys. Rev. D9 (1974), pp. 3292–3300. doi: 10.1103/ PhysRevD.9.3292. [26] S.W. Hawking. “Black Holes and Thermodynamics”. In: Phys.Rev. D13 (1976), pp. 191–197. doi: 10.1103/PhysRevD.13.191. [27] Shuang Wang, Shuang-Qing Wu, Fei Xie, and Lin Dan. “The First laws of thermodynamics of the (2+1)-dimensional BTZ black holes and Kerr- de Sitter spacetimes”. In: Chin.Phys.Lett. 23 (2006), pp. 1096–1098. doi: 10.1088/0256-307X/23/5/009. arXiv: hep-th/0601147 [hep-th]. [28] Yuichi Sekiwa. “Thermodynamics of de Sitter black holes: Thermal cos- mological constant”. In: Phys.Rev. D73 (2006), p. 084009. doi: 10.1103/ PhysRevD.73.084009. arXiv: hep-th/0602269 [hep-th]. [29] Eduard Alexis Larranaga Rubio. “Stringy Generalization of the First Law ofThermodynamicsforRotatingBTZBlackHolewithaCosmologicalCon- stant as State Parameter”. In: (2007). arXiv: 0711.0012 [gr-qc]. [30] BrianP.Dolan.“Compressibilityofrotatingblackholes”.In: Phys.Rev.D84 (2011), p. 127503. doi: 10.1103/PhysRevD.84.127503. arXiv: 1109.0198 [gr-qc]. 121 [31] Brian P. Dolan. “Pressure and volume in the first law of black hole ther- modynamics”. In: Class.Quant.Grav. 28 (2011), p. 235017. doi: 10.1088/ 0264-9381/28/23/235017. arXiv: 1106.6260 [gr-qc]. [32] Natacha Altamirano, David Kubiznak, Robert B. Mann, and Zeinab Sherkatghanad. “Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume”. In: Galaxies 2 (2014), pp. 89–159. doi: 10.3390/galaxies2010089. arXiv: 1401.2586 [hep-th]. [33] M. Henneaux and C. Teitelboim. “The Cosmological Constant as a Canoni- cal Variable”. In: Phys.Lett. B143 (1984), pp. 415–420.doi:10.1016/0370- 2693(84)91493-X. [34] M. Henneaux and C. Teitelboim. “The Cosmological Constant and General Covariance”. In: Phys.Lett. B222 (1989), pp. 195–199. doi: 10.1016/0370- 2693(89)91251-3. [35] Maulik K. Parikh. “The Volume of black holes”. In: Phys.Rev. D73 (2006), p. 124021. doi: 10.1103/PhysRevD.73.124021. arXiv: hep-th/0508108 [hep-th]. [36] JuanMartinMaldacena.“TheLargeNlimitofsuperconformalfieldtheories and supergravity”. In: Int. J. Theor. Phys. 38 (1999), pp. 1113–1133. doi: 10.1023/A:1026654312961. arXiv: hep-th/9711200. [37] Edward Witten. “Anti-de Sitter space and holography”. In: Adv. Theor. Math. Phys. 2 (1998), pp. 253–291. doi: 10.4310/ATMP.1998.v2.n2.a2. arXiv: hep-th/9802150. [38] S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov. “Gauge theory correlators from non-critical string theory”. In: Phys. Lett. B428 (1998), pp. 105–114. eprint: hep-th/9802109. 122 [39] Edward Witten. “Anti-de Sitter space, thermal phase transition, and con- finement in gauge theories”. In: Adv. Theor. Math. Phys. 2 (1998), pp. 505– 532. eprint: hep-th/9803131. [40] Ofer Aharony, Steven S. Gubser, Juan Martin Maldacena, Hirosi Ooguri, and Yaron Oz. “Large N field theories, string theory and gravity”. In: Phys. Rept. 323 (2000), pp. 183–386. doi: 10.1016/S0370-1573(99)00083-6. arXiv: hep-th/9905111. [41] Clifford V. Johnson. “Gauss-Bonnet Black Holes and Holographic Heat Engines Beyond Large N”. In: (2015). arXiv: 1511.08782 [hep-th]. [42] Clifford V. Johnson. “Born-Infeld AdS Black Holes as Heat Engines”. In: (2015). arXiv: 1512.01746 [hep-th]. [43] A. Belhaj, M. Chabab, H. El Moumni, K. Masmar, M. B. Sedra, and A. Segui. “On Heat Properties of AdS Black Holes in Higher Dimensions”. In: JHEP 05 (2015), p. 149. doi: 10.1007/JHEP05(2015)149. arXiv: 1503. 07308 [hep-th]. [44] J. Sadeghi and Kh. Jafarzade. “Heat Engine of black holes”. In: (2015). arXiv: 1504.07744 [hep-th]. [45] Elena Caceres, Phuc H. Nguyen, and Juan F. Pedraza. “Holographic entan- glement entropy and the extended phase structure of STU black holes”. In: JHEP 09 (2015), p. 184. doi: 10.1007/JHEP09(2015)184. arXiv: 1507.06069 [hep-th]. [46] M. R. Setare and H. Adami. “Polytropic black hole as a heat engine”. In: Gen. Rel. Grav. 47.11 (2015), p. 133. doi: 10.1007/s10714-015-1979-0. 123 [47] Ming Zhang and Wen-Biao Liu. “f(R) Black Holes as Heat Engines”. In: Int. J. Theor. Phys. 55.12 (2016), pp. 5136–5145. doi: 10.1007/s10773- 016-3134-4. [48] Chandrasekhar Bhamidipati and Pavan Kumar Yerra. “Heat Engines for Dilatonic Born-Infeld Black Holes”. In: (2016). arXiv: 1606.03223 [hep-th]. [49] Shao-WenWeiandYu-XiaoLiu.“Implementingblackholeasefficientpower plant”. In: (2016). arXiv: 1605.04629 [gr-qc]. [50] J. Sadeghi and Kh. Jafarzade. “The modified Horava-Lifshitz black hole from holographic engine”. In: (2016). arXiv: 1604.02973 [hep-th]. [51] Clifford V. Johnson. “An Exact Efficiency Formula for Holographic Heat Engines”. In: Entropy 18 (2016), p. 120. doi: 10.3390/e18040120. arXiv: 1602.02838 [hep-th]. [52] Andrew Chamblin, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. “Charged AdS black holes and catastrophic holography”. In: Phys. Rev. D60 (1999), p. 064018. eprint: hep-th/9902170. [53] Andrew Chamblin, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. “Holography, thermodynamics and fluctuations of charged AdS blackholes”.In:Phys. Rev.D60(1999),p.104026.eprint:hep-th/9904197. [54] DavidKubiznakandRobertB.Mann.“P-VcriticalityofchargedAdSblack holes”. In: JHEP 1207 (2012), p. 033. doi: 10.1007/JHEP07(2012)033. arXiv: 1205.0559 [hep-th]. [55] Rong-Gen Cai, Li-Ming Cao, Li Li, and Run-Qiu Yang. “P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space”. In: 124 JHEP 09 (2013), p. 005. doi: 10.1007/JHEP09(2013)005. arXiv: 1306. 6233 [gr-qc]. [56] M. Born. “Modified Field Equations with a Finite Radius of the Electron”. In: Nature 132 (1933), pp. 282–282. doi: 10.1038/132282a0. [57] M. Born. “Quantum theory of the electromagnetic field”. In: Proc. Roy. Soc. Lond. A143 (1934), pp. 410–437. doi: 10.1098/rspa.1934.0010. [58] M. Born and L. Infeld. “Foundations of the new field theory”. In: Proc. Roy. Soc. Lond. A144 (1934), pp. 425–451. doi: 10.1098/rspa.1934.0059. [59] Sharmanthie Fernando and Don Krug. “Charged black hole solutions in Einstein-Born-Infeld gravity with a cosmological constant”. In: Gen. Rel. Grav. 35 (2003), pp. 129–137. doi: 10.1023/A:1021315214180. arXiv: hep-th/0306120 [hep-th]. [60] Rong-Gen Cai, Da-Wei Pang, and Anzhong Wang. “Born-Infeld black holes in (A)dS spaces”. In: Phys. Rev. D70 (2004), p. 124034. doi: 10.1103/ PhysRevD.70.124034. arXiv: hep-th/0410158 [hep-th]. [61] Tanay Kr. Dey. “Born-Infeld black holes in the presence of a cosmological constant”. In: Phys. Lett. B595 (2004), pp. 484–490. doi: 10.1016/j. physletb.2004.06.047. arXiv: hep-th/0406169 [hep-th]. [62] Clifford V. Johnson and Felipe Rosso. “Holographic Heat Engines, Entan- glement Entropy, and Renormalization Group Flow”. In: Class. Quant. Grav. 36.1 (2019), p. 015019. doi: 10.1088/1361-6382/aaf1f1. arXiv: 1806.05170 [hep-th]. [63] Clifford V. Johnson. “Holographic Heat Engines as Quantum Heat Engines”. In: Class. Quant. Grav. 37.3 (2020), p. 034001. doi: 10.1088/ 1361-6382/ab5ba9. arXiv: 1905.09399 [hep-th]. 125 [64] Robie A. Hennigar, Fiona McCarthy, Alvaro Ballon, and Robert B. Mann. “Holographicheatengines:generalconsiderationsandrotatingblackholes”. In: (2017). arXiv: 1704.02314 [hep-th]. [65] M. Ammon and J. Erdmenger. “Gauge/Gravity Duality: Foundations and Applications”. In: 2015. [66] Gerard ’t Hooft. “On the Quantum Structure of a Black Hole”. In: Nucl. Phys. B256 (1985), pp. 727–745. doi: 10.1016/0550-3213(85)90418-3. [67] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin. “A Quantum Source of Entropy for Black Holes”. In: Phys. Rev. D34 (1986), pp. 373–383. doi: 10.1103/PhysRevD.34.373. [68] MarkSrednicki.“Entropyandarea”.In:Phys. Rev. Lett.71(1993),pp.666– 669. doi: 10.1103/PhysRevLett.71.666. arXiv: hep-th/9303048. [69] MichaelM.Wolf.“ViolationoftheentropicarealawforFermions”.In:Phys. Rev. Lett. 96 (2006), p. 010404. doi: 10.1103/PhysRevLett.96.010404. arXiv: quant-ph/0503219. [70] ChristophHolzhey,FinnLarsen,andFrankWilczek.“Geometricandrenor- malized entropy in conformal field theory”. In: Nucl. Phys. B 424 (1994), pp. 443–467. doi: 10.1016/0550-3213(94)90402-2. arXiv: hep-th/ 9403108. [71] Pasquale Calabrese and John L. Cardy. “Entanglement entropy and quan- tum field theory”. In: J. Stat. Mech. 0406 (2004), P06002. doi: 10.1088/ 1742-5468/2004/06/P06002. arXiv: hep-th/0405152. [72] Pasquale Calabrese and John Cardy. “Entanglement entropy and conformal field theory”. In: J. Phys. A 42 (2009), p. 504005. doi: 10.1088/1751- 8113/42/50/504005. arXiv: 0905.4013 [cond-mat.stat-mech]. 126 [73] Shinsei Ryu and Tadashi Takayanagi. “Holographic derivation of entangle- ment entropy from AdS/CFT”. In: Phys. Rev. Lett. 96 (2006), p. 181602. doi: 10.1103/PhysRevLett.96.181602. arXiv: hep-th/0603001. [74] Shinsei Ryu and Tadashi Takayanagi. “Aspects of Holographic Entangle- ment Entropy”. In: JHEP 08 (2006), p. 045. doi: 10.1088/1126-6708/ 2006/08/045. arXiv: hep-th/0605073. [75] Leonard Susskind. “Entanglement is not enough”. In: Fortsch. Phys. 64 (2016), pp. 49–71. doi: 10.1002/prop.201500095. arXiv: 1411.0690 [hep-th]. [76] Leonard Susskind. “Computational Complexity and Black Hole Horizons”. In: Fortsch. Phys. 64 (2016). [Fortsch. Phys.64,24(2016)], pp. 44–48. doi: 10.1002/prop.201500093,10.1002/prop.201500092. arXiv: 1403.5695 [hep-th]. [77] Adam R. Brown, Daniel A. Roberts, Leonard Susskind, Brian Swingle, and Ying Zhao. “Holographic Complexity Equals Bulk Action?” In: Phys. Rev. Lett. 116.19 (2016), p. 191301. doi: 10.1103/PhysRevLett.116.191301. arXiv: 1509.07876 [hep-th]. [78] Adam R. Brown, Daniel A. Roberts, Leonard Susskind, Brian Swingle, and Ying Zhao. “Complexity, action, and black holes”. In: Phys. Rev. D93.8 (2016), p. 086006.doi: 10.1103/PhysRevD.93.086006. arXiv: 1512.04993 [hep-th]. [79] Shira Chapman, Michal P. Heller, Hugo Marrochio, and Fernando Pastawski. “Toward a Definition of Complexity for Quantum Field The- ory States”. In: Phys. Rev. Lett. 120.12 (2018), p. 121602. doi: 10.1103/ PhysRevLett.120.121602. arXiv: 1707.08582 [hep-th]. 127 [80] Ro Jefferson and Robert C. Myers. “Circuit complexity in quantum field theory”. In: JHEP 10 (2017), p. 107. doi: 10.1007/JHEP10(2017)107. arXiv: 1707.08570 [hep-th]. [81] Rifath Khan, Chethan Krishnan, and Sanchita Sharma. “Circuit Complex- ity in Fermionic Field Theory”. In: Phys. Rev. D98.12 (2018), p. 126001. doi: 10.1103/PhysRevD.98.126001. arXiv: 1801.07620 [hep-th]. [82] Shira Chapman, Jens Eisert, Lucas Hackl, Michal P. Heller, Ro Jefferson, Hugo Marrochio, and Robert C. Myers. “Complexity and entanglement for thermofield double states”. In: (2018). arXiv: 1810.05151 [hep-th]. [83] Tibra Ali, Arpan Bhattacharyya, S. Shajidul Haque, Eugene H. Kim, and Nathan Moynihan. “Time Evolution of Complexity: A Critique of Three Methods”. In: JHEP 04 (2019), p. 087. doi: 10.1007/JHEP04(2019)087. arXiv: 1810.02734 [hep-th]. [84] Tibra Ali, Arpan Bhattacharyya, S. Shajidul Haque, Eugene H. Kim, and Nathan Moynihan. “Post-Quench Evolution of Distance and Uncertainty in a Topological System: Complexity, Entanglement and Revivals”. In: (2018). arXiv: 1811.05985 [hep-th]. [85] Pawel Caputa, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi, and KentoWatanabe.“Anti-deSitterSpacefromOptimizationofPathIntegrals in Conformal Field Theories”. In: Phys. Rev. Lett. 119.7 (2017), p. 071602. doi: 10.1103/PhysRevLett.119.071602. arXiv: 1703.00456 [hep-th]. [86] Pawel Caputa, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi, and Kento Watanabe. “Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT”. In: JHEP 11 (2017), p. 097. doi: 10.1007/JHEP11(2017)097. arXiv: 1706.07056 [hep-th]. 128 [87] Arpan Bhattacharyya, Pawel Caputa, Sumit R. Das, Nilay Kundu, Masamichi Miyaji, and Tadashi Takayanagi. “Path-Integral Complexity for PerturbedCFTs”.In:JHEP 07(2018),p.086.doi:10.1007/JHEP07(2018) 086. arXiv: 1804.01999 [hep-th]. [88] Tadashi Takayanagi. “Holographic Spacetimes as Quantum Circuits of Path-Integrations”. In: JHEP 12 (2018), p. 048. doi: 10 . 1007 / JHEP12(2018)048. arXiv: 1808.09072 [hep-th]. [89] Cesar A. Agón, Matthew Headrick, and Brian Swingle. “Subsystem Com- plexity and Holography”. In: (2018). arXiv: 1804.01561 [hep-th]. [90] Elena Cáceres, Josiah Couch, Stefan Eccles, and Willy Fischler. “Holo- graphic Purification Complexity”. In: (2018). arXiv: 1811.10650[hep-th]. [91] Dean Carmi, Robert C. Myers, and Pratik Rath. “Comments on Holo- graphic Complexity”. In: JHEP 03 (2017), p. 118. doi: 10 . 1007 / JHEP03(2017)118. arXiv: 1612.00433 [hep-th]. [92] Josiah Couch, Willy Fischler, and Phuc H. Nguyen. “Noether charge, black hole volume, and complexity”. In: JHEP 03 (2017), p. 119. doi: 10.1007/ JHEP03(2017)119. arXiv: 1610.02038 [hep-th]. [93] Hyat Huang, Xing-Hui Feng, and H. Lu. “Holographic Complexity and Two Identities of Action Growth”. In: Phys. Lett. B769 (2017), pp. 357–361.doi: 10.1016/j.physletb.2017.04.011. arXiv: 1611.02321 [hep-th]. [94] Rong-Gen Cai, Shan-Ming Ruan, Shao-Jiang Wang, Run-Qiu Yang, and Rong-Hui Peng. “Action growth for AdS black holes”. In: JHEP 09 (2016), p. 161. doi: 10.1007/JHEP09(2016)161. arXiv: 1606.08307 [gr-qc]. 129 [95] Rong-Gen Cai, Misao Sasaki, and Shao-Jiang Wang. “Action growth of charged black holes with a single horizon”. In: Phys. Rev. D95.12 (2017), p. 124002. doi: 10.1103/PhysRevD.95.124002. arXiv: 1702.06766 [gr-qc]. [96] Zicao Fu, Alexander Maloney, Donald Marolf, Henry Maxfield, and Zhencheng Wang. “Holographic complexity is nonlocal”. In: JHEP 02 (2018), p. 072. doi: 10.1007/JHEP02(2018)072. arXiv: 1801.01137 [hep-th]. [97] Mohsen Alishahiha, Amin Faraji Astaneh, Ali Naseh, and Mohammad Has- san Vahidinia. “On complexity for F(R) and critical gravity”. In: JHEP 05 (2017), p. 009. doi: 10.1007/JHEP05(2017)009. arXiv: 1702.06796 [hep-th]. [98] Davood Momeni, Mir Faizal, Aizhan Myrzakul, and Ratbay Myrzakulov. “Fidelity susceptibility for Lifshitz geometries via Lifshitz Holography”. In: Int. J. Mod. Phys. A33.17 (2018), p. 1850099. doi: 10 . 1142 / S0217751X18500999. arXiv: 1701.08660 [quant-ph]. [99] Mohsen Alishahiha, Amin Faraji Astaneh, M. Reza Mohammadi Mozaf- far, and Ali Mollabashi. “Complexity Growth with Lifshitz Scaling and Hyperscaling Violation”. In: JHEP 07 (2018), p. 042. doi: 10.1007/ JHEP07(2018)042. arXiv: 1802.06740 [hep-th]. [100] Seyed Ali Hosseini Mansoori, Viktor Jahnke, Mohammad M. Qaemmaqami, andYaithdD.Olivas.“Holographiccomplexityofanisotropicblackbranes”. In: (2018). arXiv: 1808.00067 [hep-th]. 130 [101] Mohsen Alishahiha, Komeil Babaei Velni, and M. Reza Mohammadi Mozaf- far. “Subregion Action and Complexity”. In: (2018). arXiv: 1809.06031 [hep-th]. [102] Mahdis Ghodrati. “Complexity growth in massive gravity theories, the effects of chirality, and more”. In: Phys. Rev. D96.10 (2017), p. 106020. doi: 10.1103/PhysRevD.96.106020. arXiv: 1708.07981 [hep-th]. [103] Mahdis Ghodrati. “Complexity growth rate during phase transitions”. In: Phys. Rev. D98.10 (2018), p. 106011. doi: 10.1103/PhysRevD.98.106011. arXiv: 1808.08164 [hep-th]. [104] Seth Lloyd. “Ultimate physical limits to computation”. In: Nature 406.6799 (2000), 1047?1054. issn: 1476-4687. doi: 10.1038/35023282. url: http: //dx.doi.org/10.1038/35023282. [105] Mohsen Alishahiha. “Holographic Complexity”. In: Phys. Rev. D92.12 (2015), p. 126009.doi: 10.1103/PhysRevD.92.126009. arXiv: 1509.06614 [hep-th]. [106] Omer Ben-Ami and Dean Carmi. “On Volumes of Subregions in Holography and Complexity”. In: JHEP 11 (2016), p. 129.doi:10.1007/JHEP11(2016) 129. arXiv: 1609.02514 [hep-th]. [107] Steven S. Gubser. “Breaking an Abelian gauge symmetry near a black hole horizon”. In: Phys. Rev. D 78 (2008), p. 065034. doi: 10.1103/PhysRevD. 78.065034. arXiv: 0801.2977 [hep-th]. [108] Sean A. Hartnoll, Christopher P. Herzog, and Gary T. Horowitz. “Building a Holographic Superconductor”. In: Phys. Rev. Lett. 101 (2008), p. 031601. doi: 10.1103/PhysRevLett.101.031601. arXiv: 0803.3295 [hep-th]. 131 [109] Sean A. Hartnoll, Christopher P. Herzog, and Gary T. Horowitz. “Holo- graphicSuperconductors”.In:JHEP 12(2008),p.015.doi:10.1088/1126- 6708/2008/12/015. arXiv: 0810.1563 [hep-th]. [110] Steven S. Gubser, Christopher P. Herzog, Silviu S. Pufu, and Tiberiu Tesileanu. “Superconductors from Superstrings”. In: (2009). arXiv: 0907. 3510 [hep-th]. [111] Gary T. Horowitz. “Introduction to Holographic Superconductors”. In: (2010). arXiv: 1002.1722 [hep-th]. [112] Ofer Aharony, Steven S. Gubser, Juan M. Maldacena, Hirosi Ooguri, and Yaron Oz. “Large N field theories, string theory and gravity”. In: Phys. Rept. 323 (2000), pp. 183–386. eprint: hep-th/9905111. [113] Francesco Aprile, Diederik Roest, and Jorge G. Russo. “Holographic Super- conductors from Gauged Supergravity”. In: JHEP 06 (2011), p. 040. doi: 10.1007/JHEP06(2011)040. arXiv: 1104.4473 [hep-th]. [114] Alberto Salvio. “Holographic Superfluids and Superconductors in Dilaton- Gravity”. In: JHEP 09 (2012), p. 134. doi: 10.1007/JHEP09(2012)134. arXiv: 1207.3800 [hep-th]. [115] Nikolay Bobev, Arnab Kundu, Krzysztof Pilch, and Nicholas P. Warner. “Minimal Holographic Superconductors from Maximal Supergravity”. In: JHEP 1203 (2012), p. 064. doi: 10.1007/JHEP03(2012)064. arXiv: 1110. 3454 [hep-th]. [116] Tameem Albash and Clifford V. Johnson. “Holographic Studies of Entan- glement Entropy in Superconductors”. In: JHEP 1205 (2012), p. 079. doi: 10.1007/JHEP05(2012)079. arXiv: 1202.2605 [hep-th]. 132 [117] Davood Momeni, Seyed Ali Hosseini Mansoori, and Ratbay Myrzakulov. “Holographic Complexity in Gauge/String Superconductors”. In: Phys. Lett. B756 (2016), pp. 354–357. doi: 10.1016/j.physletb.2016.03.031. arXiv: 1601.03011 [hep-th]. [118] Mahdi Kord Zangeneh, Yen Chin Ong, and Bin Wang. “Entanglement Entropy and Complexity for One-Dimensional Holographic Superconduc- tors”. In: Phys. Lett. B771 (2017), pp. 235–241.doi:10.1016/j.physletb. 2017.05.051. arXiv: 1704.00557 [hep-th]. [119] Mitsutoshi Fujita. “Holographic subregion complexity of a 1+1 dimensional p-wave superconductor”. In: (2018). arXiv: 1810.09659 [hep-th]. [120] Runqiu Yang, Hyun-Sik Jeong, Chao Niu, and Keun-Young Kim. “Com- plexity of Holographic Superconductors”. In: (2019). arXiv: 1902.07586 [hep-th]. [121] HongGuo,Xiao-MeiKuang,andBinWang.“Noteonholographicentangle- ment entropy and complexity in St¨ uckelberg superconductor”. In: (2019). arXiv: 1902.07945 [hep-th]. [122] Steven S. Gubser and Fabio D. Rocha. “The gravity dual to a quantum critical point with spontaneous symmetry breaking”. In: Phys. Rev. Lett. 102 (2009), p. 061601. doi: 10.1103/PhysRevLett.102.061601. arXiv: 0807.1737 [hep-th]. [123] Tameem Albash and Clifford V. Johnson. “Evolution of Holographic Entan- glement Entropy after Thermal and Electromagnetic Quenches”. In: New J. Phys. 13 (2011), p. 045017. doi: 10.1088/1367-2630/13/4/045017. arXiv: 1008.3027 [hep-th]. 133 [124] Pratim Roy and Tapobrata Sarkar. “Note on subregion holographic com- plexity”. In: Phys. Rev. D96.2 (2017), p. 026022. doi: 10.1103/PhysRevD. 96.026022. arXiv: 1701.05489 [hep-th]. [125] Aranya Bhattacharya and Shibaji Roy. “Holographic Entanglement Entropy, Subregion Complexity and Fisher Information metric of ’black’ Non-SUSY D3 Brane”. In: (2018). arXiv: 1807.06361 [hep-th]. [126] GaryT.HorowitzandJosephPolchinski.“Gauge/gravityduality”.In:(Feb. 2006), pp. 169–186. arXiv: gr-qc/0602037. [127] LeonardSusskind.“TheWorldasahologram”.In:J. Math. Phys.36(1995), pp. 6377–6396. doi: 10.1063/1.531249. arXiv: hep-th/9409089. [128] Gerard ’t Hooft. “Dimensional reduction in quantum gravity”. In: Conf. Proc. C 930308 (1993), pp. 284–296. arXiv: gr-qc/9310026. [129] Sebastian de Haro, Sergey N. Solodukhin, and Kostas Skenderis. “Holo- graphic reconstruction of space-time and renormalization in the AdS / CFT correspondence”. In: Commun. Math. Phys. 217 (2001), pp. 595–622. doi: 10.1007/s002200100381. arXiv: hep-th/0002230. [130] Juan Maldacena and Leonard Susskind. “Cool horizons for entangled black holes”. In: Fortsch. Phys. 61 (2013), pp. 781–811. doi: 10.1002/prop. 201300020. arXiv: 1306.0533 [hep-th]. [131] Mark Van Raamsdonk. “Building up spacetime with quantum entangle- ment”. In: Gen. Rel. Grav. 42 (2010), pp. 2323–2329. doi: 10.1142/ S0218271810018529. arXiv: 1005.3035 [hep-th]. [132] Xi Dong, Daniel Harlow, and Aron C. Wall. “Reconstruction of Bulk Oper- ators within the Entanglement Wedge in Gauge-Gravity Duality”. In: Phys. 134 Rev. Lett. 117.2 (2016), p. 021601. doi: 10.1103/PhysRevLett.117. 021601. arXiv: 1601.05416 [hep-th]. [133] Jordan Cotler, Patrick Hayden, Geoffrey Penington, Grant Salton, Brian Swingle, and Michael Walter. “Entanglement Wedge Reconstruction via Universal Recovery Channels”. In: Phys. Rev. X 9.3 (2019), p. 031011.doi: 10.1103/PhysRevX.9.031011. arXiv: 1704.05839 [hep-th]. [134] Geoffrey Penington. “Entanglement Wedge Reconstruction and the Infor- mation Paradox”. In: (May 2019). arXiv: 1905.08255 [hep-th]. [135] Chi-Fang Chen, Geoffrey Penington, and Grant Salton. “Entanglement Wedge Reconstruction using the Petz Map”. In: JHEP 01 (2020), p. 168. doi: 10.1007/JHEP01(2020)168. arXiv: 1902.02844 [hep-th]. [136] Samuel Bilson. “Extracting spacetimes using the AdS/CFT conjecture”. In: JHEP 08 (2008), p. 073. doi: 10.1088/1126-6708/2008/08/073. arXiv: 0807.3695 [hep-th]. [137] Samuel Bilson. “Extracting Spacetimes using the AdS/CFT Conjecture: Part II”. In: JHEP 02 (2011), p. 050. doi: 10.1007/JHEP02(2011)050. arXiv: 1012.1812 [hep-th]. [138] Thomas Faulkner. “Bulk Emergence and the RG Flow of Entanglement Entropy”. In: JHEP 05 (2015), p. 033. doi: 10.1007/JHEP05(2015)033. arXiv: 1412.5648 [hep-th]. [139] NingBao,ChunJunCao,SebastianFischetti,andCynthiaKeeler.“Towards Bulk Metric Reconstruction from Extremal Area Variations”. In: Class. Quant. Grav. 36.18 (2019), p. 185002. doi: 10.1088/1361-6382/ab377f. arXiv: 1904.04834 [hep-th]. 135 [140] Ning Bao, Chunjun Cao, Sebastian Fischetti, Jason Pollack, and Yibo Zhong. “More of the Bulk from Extremal Area Variations”. In: Class. Quant. Grav. 38.4 (2021), p. 047001. doi: 10.1088/1361-6382/abcfd0. arXiv: 2009.07850 [hep-th]. [141] CesarA.Agón,ElenaCáceres,andJuanF.Pedraza.“Bitthreads,Einstein’s equations and bulk locality”. In: JHEP 01 (2021), p. 193. doi: 10.1007/ JHEP01(2021)193. arXiv: 2007.07907 [hep-th]. [142] Niko Jokela and Arttu Pönni. “Towards precision holography”. In: Phys. Rev. D 103.2 (2021), p. 026010. doi: 10.1103/PhysRevD.103.026010. arXiv: 2007.00010 [hep-th]. [143] Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, and Juan Maldacena. “N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals”. In: JHEP 10 (2008), p. 091. doi: 10.1088/1126-6708/ 2008/10/091. arXiv: 0806.1218 [hep-th]. [144] Ashis Saha, Sourav Karar, and Sunandan Gangopadhyay. “Bulk geometry from entanglement entropy of CFT”. In: Eur. Phys. J. Plus 135.2 (2020), p. 132. doi: 10.1140/epjp/s13360-020-00110-7. arXiv: 1807.04646 [hep-th]. [145] Marcos Marino. “Lectures on localization and matrix models in supersym- metric Chern-Simons-matter theories”. In: J. Phys. A 44 (2011), p. 463001. doi: 10.1088/1751-8113/44/46/463001. arXiv: 1104.0783 [hep-th]. [146] HoracioCasini,MarinaHuerta,andRobertC.Myers.“Towardsaderivation of holographic entanglement entropy”. In: JHEP 05 (2011), p. 036. doi: 10.1007/JHEP05(2011)036. arXiv: 1102.0440 [hep-th]. 136 [147] Nadav Drukker, Marcos Marino, and Pavel Putrov. “From weak to strong coupling in ABJM theory”. In: Commun. Math. Phys. 306 (2011), pp. 511– 563. doi: 10.1007/s00220-011-1253-6. arXiv: 1007.3837 [hep-th]. [148] Igor R. Klebanov, Silviu S. Pufu, Subir Sachdev, and Benjamin R. Safdi. “Entanglement Entropy of 3-d Conformal Gauge Theories with Many Fla- vors”. In: JHEP 05 (2012), p. 036. doi: 10.1007/JHEP05(2012)036. arXiv: 1112.5342 [hep-th]. [149] Charles Fefferman and C. Robin Graham. “Conformal invariants”. en. In: Élie Cartan et les mathématiques d’aujourd’hui - Lyon, 25-29 juin 1984. Astérisque S131. Société mathématique de France, 1985, pp. 95–116. url: http://www.numdam.org/item/AST_1985__S131__95_0. [150] C. Fefferman and C. R. Graham. “The Ambient Metric”. In: (). eprint: 0710.0919. [151] Thomas Faulkner, Felix M. Haehl, Eliot Hijano, Onkar Parrikar, Charles Rabideau, and Mark Van Raamsdonk. “Nonlinear Gravity from Entan- glement in Conformal Field Theories”. In: JHEP 08 (2017), p. 057. doi: 10.1007/JHEP08(2017)057. arXiv: 1705.03026 [hep-th]. [152] Veronika E. Hubeny, Mukund Rangamani, and Tadashi Takayanagi. “A Covariant holographic entanglement entropy proposal”. In: JHEP 07 (2007), p. 062. doi: 10.1088/1126-6708/2007/07/062. arXiv: 0705.0016 [hep-th]. 137 Appendix A Formulas for Benchmarking Kerr–AdS Black Hole Heat Engines A.1 Static Black Holes in D = 4 Dimensions Here we write down the mass and the temperature of Einstein-Hilbert-Maxwell andBorn-Infeldblackholes, whichwerepreviouslyusedinourbenchmarkingstudy in D = 5 and with a different choice of benchmarking circle for static black holes [1]. A.1.1 Einstein–Hilbert–Maxwell The bulk action for the Einstein–Hilbert–Maxwell system in D = 4 is 1 : I = 1 16π Z d 4 x √ −g R− 2Λ−F 2 . (A.1) We can now write the mass and the temperature of the Einstein–Hilbert– Maxwell (i.e., Reissner–Nordstrom–like) black hole solution, parametrized by a charge q (which we will later choose as q = 0.1): 1 We’re using the conventions of ref. [52]. 138 M = 1 2 r + + q 2 r + + 8πp 3 r 3 + , and T = 1 4π 8πpr + + 1 r + − q 2 r 3 + , (A.2) and we can write them entirely in terms of p and V, using r + = (3V/4π) 1/3 . A.1.2 Born–Infeld The so–called 2 Born–Infeld action [56–58] is a non-linear generalization of the Maxwell action, controlled by the parameter β: L(F ) = 4β 2 1− s 1 + F μν F μν 2β 2 . (A.3) If we take the limitβ→∞ in (A.3) we recover old Maxwell action. The Einstein– Hilbert–Born–Infeld bulk action in D = 4 is obtained by replacing the Maxwell sector in equation (A.1) with this action. The exact results for the Born–Infeld black hole’s mass and temperature are known 3 , but for our purposes, it is enough to expand them in 1/β, keeping only leading non–trivial terms. For the mass: M = 1 2 r + + 8πp 3 r 3 + + q 2 r + (1− 2q 2 15β 2 r 4 + ) +O 1 β 4 , (A.4) and the temperature: T = 1 4π 1 r + + 8πpr + − q 2 r 3 + (1− q 2 4β 2 r 4 + ) +O 1 β 4 . (A.5) We worked with q = 0.1 and β = 0.1 in our benchmarking scheme. 2 See e.g. the remarks in ref. [42] about the terminology 3 See refs. [59–61] for further details. 139 A.2 Υ–function in Higher Dimensions The Υ–function that we have written down explicitly in section 3.3 was derived for D = 4. In this appendix we will show that it can also be done for higher dimensional singly spinning Kerr-AdS black holes. A singly spinning Kerr-AdS black hole in general D-dimensions can be described by one non-zero rotation parameter a and the metric takes the form: ds 2 = − Δ r ρ 2 dt− a sin 2 θ Ξ dφ 2 + ρ 2 Δ r dr 2 + ρ 2 Δ θ dθ 2 + Δ θ sin 2 θ ρ 2 adt− r 2 +a 2 Ξ dφ 2 +r 2 cos 2 θdΩ 2 D−4 , (A.6) where, ρ 2 =r 2 +a 2 cos 2 θ , Ξ = 1− a 2 l 2 , (A.7) Δ r = (r 2 +a 2 ) 1 + r 2 l 2 − 2mr 5−D , Δ θ = 1− a 2 l 2 cos 2 θ . (A.8) Other useful thermodynamic quantities are: M = mω D−2 4πΞ 2 1 + (D− 4)Ξ 2 , J = maω D−2 4πΞ 2 , (A.9) V = r + A D− 1 1 + a 2 (1 + r 2 + l 2 ) (D− 2)r 2 + Ξ , Ω = a 1 + r 2 + l 2 r 2 + +a 2 . (A.10) 140 The area and the entropy are given by: A = ω D−2 (r 2 + +a 2 )r D−4 + Ξ , and S = A 4 , (A.11) and ω D−2 = 2π ( D−1 2 ) /Γ( D−1 2 ) is the usual volume of the unit (D− 2)-sphere. As before, p =− Λ 8π , where Λ is the cosmological constant and related to l by Λ = − (D−1)(D−2) 2l 2 . The statement that r + is the largest root of Δ r gives us mass M in term of the horizon radius, r + : M = ω D−2 4πΞ 2 1 + (D− 4)Ξ 2 (r 2 + +a 2 )(1 + r 2 + l 2 ) 2r 5−D + . (A.12) Now we can easily compute dJ from (A.9) using equation (A.12) (while keeping p, i.e., l constant). Next, we multiply dJ by Ω using (A.10). This quantity, ΩdJ, is integrable exactly along the isobars in any D. For general D, ΩdJ takes the following form: a 2 ω D−2 (1 + r 2 + l 2 ) (D− 5)(a 2 +r 2 + )(1 + r 2 + l 2 )r D−6 + + 2(1 + 2 r 2 + l 2 + a 2 l 2 )r D−4 + 8π(a 2 +r 2 + )Ξ 2 dr + . (A.13) The result upon integrating (A.13) is what we called Υ (forD = 4) in section 3.3. We present two such results here. In D = 5 , Υ takes the form: Υ 5 = a 2 ω 3 8π ln(a 2 +r 2 + ) + r 2 + l 2 Ξ 1 + 2 Ξ + r 2 + l 2 Ξ , (A.14) and in D = 6 , the result is: Υ 6 = a 2 ω 4 4π r + l 2 Ξ r 2 + ( 1 3 + 1 Ξ ) + (l 2 −a 2 ) + (l 4 +r 4 + ) 2l 2 Ξ − arctan( r + a )a . (A.15) 141 One can derive exact results for higher dimensions too using (A.13). 142

Abstract (if available)

Abstract This thesis examines three different areas in theoretical high energy physics: in the first part we talk about the extended black hole thermodynamics and consequently present a systematic way of computing and comparing the efficiencies of the holographic heat engines. In the second part, using the ideas derived from the AdS/CFT correspondence we compute the entanglement entropy and the subregion complexity for a holographic superconductor and gain some novel physical insights that could be useful for our understanding of the strongly coupled field theories. Given the data of a boundary field theory, how to partially reconstruct the dual bulk geometry using general ideas from the AdS/CFT correspondence has been discussed in the third part. ❧ As we have just mentioned, in the first part of this thesis, we present the results of initiating a benchmarking scheme that allows for cross-comparison of the efficiencies of black holes used as working substances in heat engines. We use a circular cycle in the p-V plane as the benchmark engine. We test it on the Einstein-Maxwell, Gauss-Bonnet, and Born-Infeld black holes. Also, we derive a new and surprising exact result for the efficiency of a special ""ideal gas'' system to which all the black holes asymptote. Next we extend our benchmarking scheme to rotating black holes again using a circular cycle in the p-V plane as the benchmark engine. We compare Kerr to Einstein-Maxwell and Born-Infeld black holes. As in the static case, we derive an exact formula for the benchmark efficiency in an ideal-gas-like limit that may serve as an upper bound for (generic) rotating black hole heat engines in the thermodynamic ensemble we choose here. ❧ In the second part of this thesis, we present the results of our computation of the subregion complexity and also compare it with the entanglement entropy of a 2 + 1-dimensional holographic superconductor which has a fully backreacted gravity dual with a stable ground sate. We follow the ""complexity equals volume'' or the CV conjecture. We find that there is only a single divergence for a strip entangling surface and the complexity grows linearly with the large strip width. During the normal phase the complexity increases with decreasing temperature, but during the superconducting phase it behaves differently depending on the order of phase transition. We also show that the universal term is finite and the phase transition occurs at the same critical temperature as obtained previously from the free energy computation of the system. In one case, we observe multivaluedness in the complexity in the form of an ""S'' curve. ❧ Recent work has shown that the entanglement and the structure of spacetime are intimately related. One way to investigate this is to begin with an entanglement entropy in a conformal field theory (CFT) and use the AdS/CFT correspondence to calculate the bulk metric. In the final part, we perform this calculation for ABJM, a particular 3-dimensional supersymmetric CFT (SCFT), in its ground state. In particular we are able to reconstruct the pure AdS₄ metric from the holographic entanglement entropy of the boundary ABJM theory in its ground state. Moreover, we are able to predict the correct AdS radius purely from the entanglement. We also address the general philosophy of relating the entanglement and the spacetime through the Holographic Principle, as well as some of the philosophy behind our calculations.

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Creator Chakraborty, Avik (author)

Core Title Black hole heat engines, subregion complexity and bulk metric reconstruction

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 04/14/2021

Defense Date 03/09/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag ABJM,AdS/CFT,bulk metric reconstruction,computational complexity,entanglement entropy,extended black hole thermodynamics,holographic heat engines,holographic superconductor,OAI-PMH Harvest,phase transition

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Johnson, Clifford (committee chair), Haas, Stephan (committee member), Lauda, Aaron (committee member), Pierpaoli, Elena (committee member), Pilch, Krzysztof (committee member)

Creator Email avikchak@usc.edu,avikchak88@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-443468

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Legacy Identifier etd-Chakrabort-9455.pdf

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