PHASE TRANSITION
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Thesis Submitted For The Degree Of
[2.2ex] Doctor of Philosophy (Science)
[2ex] In
[2ex] Physics (Theoretical)
[2ex] By
[2ex] Dibakar Roychowdhury
University of Calcutta, India
[2ex] 2013
Acknowledgments
This thesis is the outcome of the cumulative efforts that has been paid for the last three and half years during my stay at S. N. Bose National Centre for Basic Sciences, Kolkata. I feel myself to be quite lucky in the sense that from the very first day of this journey I found many people around me who constantly assisted me in various occasions. I would like to convey my sincere thanks to all of them.
First of all I would like to convey my gratitude to Prof. Rabin Banerjee for giving me this nice opportunity to work under his supervision. The entire research work that I did for the last three and half years is based on the platform provided by Prof. Banerjee.
I am specially thankful to Dr. Amitabha Mukhopadhyay of North Bengal University and Prof. Subir Ghosh of ISI, Kolkata for their generous help and guidance during the early stages of my research career.
I am also thankful to my collaborators Dr. Bibhas Ranjan Majhi, Dr. Sujoy Kumar Modak, Dr. Kuldeep Kumar, Dr. Sunandan Gangopadhyay and Mr. Sudipta Das, who have actively collaborated with me in various occasions and helped me to accomplish various projects in different courses of time.
I am specially thankful to Mr. Srijit Bhattacharjee, with whom I have discussed lots of physics in various occasions.
I would also like to convey my gratitude to all of my friends with whom I have shared many colorful moments during these years. In this regard I would like to convey my sincere thanks to Mr. Ambika Prasad Jena, Mr. Soumyadipta Pal, Mr. Debmalya Mukhopadhyay and Dr. Sudhakar Upadhyay for their helpful assistance whenever it was required for me.
I am particularly thankful to my group mates Mr. Arindam Lala, Mr. Shirsendu Dey, Mr. Arpan Krishna Mitra and Ms. Poulami Chakraborty and for their active participation in various academic discussions.
I am specially thankful to Mr. Biswajit Paul with whom not only I have discussed physics in various occasions but also shared most of my memorable times in the S. N. Bose Centre.
I am also grateful to Mr. Debraj Roy for his generous help and assistance whenever it was required for me.
Finally, I would like to convey my sincere thanks to Ms. Debashree Chowdhury who has been always there for me to provide a constant mental support.
List of publications

“AdS/CFT superconductors with Power Maxwell electrodynamics: reminiscent of the Meissner effect”
Dibakar Roychowdhury
Published in Phys. Lett. B 718 (2013)10891094
ePrint: arXiv:1211.1612 [hepth]. 
“Effect of external magnetic field on holographic superconductors in presence of nonlinear corrections”
Dibakar Roychowdhury
Published in Phys.Rev. D 86, 106009 (2012)
ePrint: arXiv:1211.0904 [hepth]. 
“Analytic study of properties of holographic pwave superconductors”
Sunandan Gangopadhyay, Dibakar Roychowdhury
Published in JHEP 1208 (2012) 104
ePrint: arXiv:1207.5605 [hepth]. 
“Phase transition and scaling behavior of topological charged black holes in HoravaLifshitz gravity”
Bibhas Ranjan Majhi, Dibakar Roychowdhury
Published in Class. Quant. Grav 29 (2012) 245012
ePrint: arXiv:1205.0146 [grqc]. 
“Analytic study of GaussBonnet holographic superconductors in BornInfeld electrodynamics”
Sunandan Gangopadhyay, Dibakar Roychowdhury
Published in JHEP 1205 (2012) 156
ePrint: arXiv:1204.0673 [hepth]. 
“Critical behavior of Born Infeld AdS black holes in higher dimensions”
Rabin Banerjee, Dibakar Roychowdhury
Published in Phys.Rev. D85 (2012) 104043
ePrint: arXiv:1203.0118 [grqc]. 
“Analytic study of properties of holographic superconductors in BornInfeld electrodynamics”
Sunandan Gangopadhyay, Dibakar Roychowdhury
Published in JHEP 1205 (2012) 002
ePrint: arXiv:1201.6520 [hepth]. 
“Ehrenfest’s scheme and thermodynamic geometry in BornInfeld AdS black holes”
Arindam Lala, Dibakar Roychowdhury
Published in Phys.Rev. D86 (2012) 084027
ePrint: arXiv:1111.5991 [grqc]. 
“Critical phenomena in BornInfeld AdS black holes”
Rabin Banerjee, Dibakar Roychowdhury
Published in Phys.Rev. D85 (2012) 044040
ePrint: arXiv:1111.0147 [grqc]. 
“Thermodynamics of phase transition in higher dimensional AdS black holes”
Rabin Banerjee, Dibakar Roychowdhury
Published in JHEP 1111 (2011) 004
ePrint: arXiv:1109.2433 [grqc]. 
“A unified picture of phase transition: from liquidvapour systems to AdS black holes”
Rabin Banerjee, Sujoy Kumar Modak, Dibakar Roychowdhury
Published in JHEP 1210 (2012) 125
ePrint: arXiv:1106.3877 [grqc]. 
“Symmetries of Snyderde Sitter space and relativistic particle dynamics”
Rabin Banerjee, Kuldeep Kumar, Dibakar Roychowdhury
Published in JHEP 1103 (2011) 060
ePrint: arXiv:1101.2021 [hepth]. 
“Corrected area law and Komar energy for noncommutative inspired ReissnerNordstroem black hole”
Sunandan Gangopadhyay, Dibakar Roychowdhury
Published in Int.J.Mod.Phys. A27 (2012) 1250041
ePrint: arXiv:1012.4611 [hepth]. 
“New type of phase transition in Reissner Nordstrom  AdS black hole and its thermodynamic geometry”
Rabin Banerjee, Sumit Ghosh, Dibakar Roychowdhury
Published in Phys.Lett. B696 (2011) 156162
ePrint: arXiv:1008.2644 [grqc]. 
“Thermodynamics of Photon Gas with an Invariant Energy Scale”
Sudipta Das, Dibakar Roychowdhury
Published in Phys.Rev. D81 (2010) 085039
ePrint: arXiv:1002.0192 [hepth]. 
“Relativistic Thermodynamics with an Invariant Energy Scale”
Sudipta Das, Subir Ghosh, Dibakar Roychowdhury
Published in Phys.Rev. D80 (2009) 125036
ePrint: arXiv:0908.0413 [hepth].
This thesis is based on the papers numbered by [ 2, 4, 5, 6, 7, 9, 10, 11] whose reprints are attached at the end of the thesis.
PHASE TRANSITION
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Chapter 1 Introduction
1.1 Overview
Gravity is the weakest among the all four fundamental interactions in nature and could be clearly distinguished from the other three forces through its distinguished property known as the principle of equivalence which states that the trajectories of all freely moving particles get affected in an identical way in presence of an external gravitational field. Mathematically this equivalence could be understood as saying that gravity is always coupled to the energy momentum tensor of some matter field. In spite of such a remarkable feature, gravity is still believed to be one of the mysterious forces of nature. The reason for this is quite clear : In spite of numerous attempts for the past several decades till date we do not have a fully satisfactory and consistent theory of Quantum Gravity. The classical General Relativity is still considered to be the most successful theory describing gravity.
In the classical description of gravity we have a classical metric that describes our space time and which may be obtained as a solution of Einstein equations of General Relativity namely^{1}^{1}1 is the gravitational constant and is the speed of light.,
(1.1) 
where is the cosmological constant and is the energy momentum tensor of some classical matter field. For the past several decades, many people have tried to develop various theories with gravity that basically treat the metric classically while, on the other hand, consider the matter fields quantum mechanically. Presumably such a scheme may be considered as a good approximation of the full quantum theory of gravity.
Since its birth in 1916, the theory of General Relativity has gained renewed attention due to its several remarkable predictions about nature, among which the existence of black holes may be regarded as the most significant one. According to General Relativity, a sufficiently high concentration of mass in certain region of space time could produce a strong gravitational field so that even light can not escape from that region. Such a region of space time is known as black hole.
Black holes are perhaps the most tantalizing objects of Einstein’s General Relativity. For space time containing black holes one must encounter a space time singularity provided General Relativity is correct and the energy momentum tensor satisfies certain positive definite inequality. This singularity, which is known as black hole singularity, may be regarded as the place where all the known laws of physics formulated on a classical back ground break down. In a spacetime containing black hole, there exists a boundary known as the event horizon, which behaves as a trapped surface such that all future directed null geodesics orthogonal to it are converging. Interestingly black hole singularity is not visible to an observer sitting outside the event horizon. Literally speaking, the event horizon acts as a one way membrane such that an infalling observer can easily pass through the horizon from out side to inside, whereas on the other hand, nothing can escape out of it. Due to this spatial behavior classical black holes do not radiate and therefore they possess absolute zero temperature [1].
The discovery of a close connection between the laws of black hole physics to that with the laws of ordinary thermodynamics is considered to be one of the remarkable achievements of theoretical physics to have occurred during the last forty years. The existence of such a deep analogy has been found to be playing the central role towards our understanding of the theory of Quantum Gravity. On top of it, this also illuminates the role of black holes in order to develop a consistent quantum theory of gravity as well as some fundamental aspects regarding the nature of ordinary thermodynamics itself^{2}^{2}2For excellent reviews on this subject see [2].
A possible identification of black holes as thermodynamic objects was first realized by Bardeen, Carter and Hawking [3] during their remarkable derivation of the four laws of black hole mechanics. Based on the notion of the geometry of space time, in their calculations the authors have found that the change in mass () of a rotating black hole could be related to the change in horizon area () and the change in angular momentum () by the following relation,
(1.2) 
which is known as the first law of black hole mechanics. Here is the surface gravity which is constant over the event horizon of the stationary black hole and is the angular velocity at the event horizon. There could be some additional terms present on the R.H.S of (1.2) when matter fields are present. The relation (1.2) exactly looks like the first law of ordinary thermodynamics which states that the difference in energy (), entropy () and the other state variables of two nearby equilibrium states may be related as,
(1.3) 
Thus in order to fit the laws of black hole mechanics to that with the laws of ordinary thermodynamics we have to have the following analogies namely,
(1.4) 
where we have identified the entity as the temperature () of the black hole along with the identification of as the entropy (S) of the black hole^{3}^{3}3For an alternative derivation based on exact differentials, see [4]. [5][6].
However, in the early seventies soon after the discovery of the four laws of black hole mechanics, most of the researchers at that time viewed this analogy merely as a mathematical analogy. The reason for this was clear: Black holes in the framework of classical General Relativity do not radiate. It took time for people to realize that there indeed lies a deep physical meaning behind the above identifications (1.4) until Hawking came out with his dramatic discovery [7] which unleashed the fact that due to quantum particle creation effects a black hole is capable of radiating to infinity all species of particles with a perfect black body spectrum at temperature,
(1.5) 
Soon after the discovery of Hawking it became clear that the quantity actually plays the role of the physical temperature for a black hole. Thus the discovery of Hawking firmly established the fact that black holes may be identified as thermodynamic objects once quantum mechanical effects are incorporated in the framework of classical general relativity. This is usually known as semiclassical regime of black hole physics. Under this semiclassical framework one can therefore study various thermodynamic aspects of black holes, like thermal stability, phase transition, black hole evaporation etc.
The identification of black holes as thermodynamic objects with physical temperature and entropy opened a gate to a new realm. Once black holes are identified as thermodynamic objects, it is very natural to ask whether they also behave as familiar thermodynamic systems. Studying phase transition in black holes would be naturally one of the fascinating topics in this regard. Since the birth of black hole thermodynamics, the issue of phase transition in black holes has been found to be playing a significant role towards the understanding of several crucial properties of black holes including the statistical origin of its entropy.
Occurrence of phase transition in black holes had been first investigated by Davies and Hut in the late seventies [8][9]. Their analysis revealed the fact that Schwarzschild black holes in asymptotically flat space could be found in thermal equilibrium with the surrounding thermal radiation at some equilibrium temperature () which is precisely the Hawking temperature. However this equilibrium is unstable as can be seen through the computation of the heat capacity () which comes out to be negative. This means that if the black hole absorbs some mass from outside its temperature would go down and as a result the rate of absorption would be more than the rate of emission and the black hole would continue to grow. Therefore a canonical ensemble can not be defined for a Schwarzschild black hole in asymptotically flat space. Moreover, in course of time it also became clear that for rotating and charged black holes the heat capacity at constant volume () exhibits various types of discontinuities which resemble phase transitions [10]. It was found that for rotating (or charged) black holes the heat capacity is not necessarily always negative. For sufficiently highly charged or rapidly rotating black holes one could find a region of stable equilibrium with .
Although the very first attempt to understand the thermodynamic behavior of black holes was commenced for space times that are asymptotically flat, still this could be regarded merely as a theoretical model. The reason for this lies in the fact that the concept of asymptotic flatness is not always the correct theoretical idealization and is hard to satisfy in reality. Therefore in order to develop a systematic theoretical description for black holes, in the early nineties people tried to construct gravitational theories within a bounded and finite spatial region of space time [11][16]. It was indeed found that for a finite temperature at the spatial boundary the heat capacity of the black hole comes out to be positive and therefore there is no inconsistency in defining the black hole partition function [11]. From these analysis soon it became clear that instead of studying the thermodynamic behavior of black holes in asymptotically flat spaces, one should actually consider spacetimes that are asymptotically curved, in other words spacetimes which admit a cosmological constant ().
If , then the resulting spacetime is an asymptotically de Sitter (dS) space. Like in the case of an asymptotically flat space, black holes in an asymptotically de Sitter space also emit particles at a temperature determined by its surface gravity. On top of it, in case of an asymptotically de Sitter space we have a cosmological horizon which also emits particles determined by its own surface gravity. Therefore a thermal equilibrium is possible only when these two surface gravities are equal, i.e; when the temperatures of both the event horizon as well the cosmological horizon are same. Later on Carlip and Vaidya had performed a detailed analysis of the thermal stability of a charged black hole in an asymptotically de Sitter space in [17]. For a supercooled black hole in an asymptotically de Sitter space they observed an interesting phase structure that has a line of first order transition which terminates on a second order point.
If , then the resulting space time is known as the anti de Sitter (AdS) space. In an AdS space the gravitational potential relative to any origin increases as one moves away from the origin. This means that an asymptotically AdS space acts as a confining box. As a result the thermal radiation remains confined close to the black hole and cannot escape to infinity. Although zero rest mass particles can escape to infinity but the incoming and outgoing fluxes at infinity are equal. Therefore one can always consider a canonical ensemble description for black holes at any given temperature ().
Investigating thermodynamic behavior of black holes in an asymptotically AdS space had been initiated due to Hawking and Page [18]. According to their observations there exists a phase transition (so called Hawking Page (HP) transition) between thermal AdS and AdS black holes in four dimensions. Later on the phase transition of AdS black holes in five dimensions attained renewed attention in the context of AdS/CFT duality, which shows that HP transition in the bulk may be identified as a confinement deconfinement transition in the boundary field theory [19]. In the conventional HP transition one generally starts with the thermal radiation in an AdS space. If the temperature of the thermal radiation is below certain minimum () then the only possible equilibrium is the thermal radiation with out black holes. On the other hand if the temperature is higher than this minimum then there could be two possible black hole solutions which could be in equilibrium with the thermal radiation. According to the findings of Hawking and Page, there exists a critical mass value () for black holes in AdS space. If the mass of the black hole is below this critical value () then it appears with a negative heat capacity. Which means that the lower mass black hole is unstable. Therefore it may either decay completely to thermal radiation or to a larger mass () black hole with a positive heat capacity which corresponds a locally stable phase. In the second case the heat capacity changes from negative infinity to positive infinity at the minimum temperature . Also there is a change in the dominance from thermal AdS to black holes at some temperature which corresponds to a change in sign in the free energy of the system. Since the discovery of such a nice phenomenon in AdS black holes, till date a number of investigations have been made regarding various thermodynamic aspects of AdS black holes [20][52].
Studying phase transition in black holes from a completely new perspective has been initiated recently in [38] where the idea of ClausiusClapeyronEhrenfest’s equations [53] have been successfully implemented in the context of black hole thermodynamics. Subsequently this analysis has been further extended to analyze the phase transition occurring in charged (RN) AdS and Kerr AdS black holes [39][40]. The motivation behind such an analysis is the following: Although the discontinuity in the specific heat (during phase transition in black holes [26],[30]) is sufficient to indicate the onset of a continuous higher order transition, but certainly it is not a systematic or complete way to correctly identify the nature of this phase transition, i.e; we cannot conclude uniquely whether the phase transition is a genuine second order or any higher order transition. Therefore we need certain prescription in order to identify the nature of this phase transition correctly. In ordinary thermodynamics similar situations are dealt within the framework of Ehrenfest’s equations. For a genuine second order phase transition both the Ehrenfest’s relations are found to be satisfied simultaneously [53].
Although the analysis carried out in [38][40] was proved to be quite successful in order to classify the phase transition correctly, nevertheless it had some shortcomings. The reason for this was the lack in analytic techniques while computing the Ehrenfest’s equations near the critical point. Therefore one of the major motivations of the present thesis is to fill up this gap by developing suitable analytic scheme in order to compute the Ehrenfest’s equations. As a matter of fact, based on exact analytic computations, in the present thesis we show that it is indeed possible to verify both the Ehrenfest’s equations exactly at the critical point and thereby to fix the actual nature of the phase transition unambiguously.
An alternative approach to study thermodynamics is using statespace geometry. In 1979 George Ruppeiner [54] proposed a geometrical way to study thermodynamics and statistical mechanics. Different aspects of thermodynamics and statistical mechanics of the system are encoded in the metric of the state space. Consequently various thermodynamic properties of the system can be obtained from the properties of this metric and curvature. It is successful in describing the behavior of classical systems [54] and recently it has been found to play an important role in explaining black hole thermodynamics [55][56].
In conventional thermodynamic systems, phase transition phenomena plays an important role in order to explore thermodynamic properties of various systems near the critical point of phase transition. The main objective in the theory of phase transition is to study the behavior of a given system in the neighborhood of the critical point, which is characterized by the discontinuity in some thermodynamic variable. In usual thermodynamics various physical quantities (like heat capacity, compressibility etc.) pertaining to a system suffer from a singularity near the critical point. The effect of diverging correlation length near the critical point manifests as divergences of these thermodynamic quantities. This is known as static critical phenomena. It is customary in usual (equilibrium) thermodynamics to express all these singularities by a set of static critical exponents [57][58], which determine the qualitative nature of the critical behavior of a given system near the critical point. The effect of diverging correlation length in case of non equilibrium thermodynamics results in a critical slowing down near the critical point which is parametrized by the dynamic critical exponents, and these two sets of exponents could be related to each other.
In case of a second order phase transition the so called static critical exponents are found to be universal in a sense that apart from a few factors, like spatial dimensionality, symmetry of the system etc. they do not depend on the details of the interaction. As a matter of fact different physical systems may belong to the same universality class. Critical exponents are also found to satisfy certain scaling laws, which in turn implies that not all the critical exponents are independent. These scaling relations are related to the scaling hypothesis for thermodynamic functions. Since black holes are well defined thermodynamic objects, therefore it will be quite natural to study all these issues for black holes particularly defined in an asymptotically AdS space.
Studying the critical behavior in black holes had been commenced long ago considering the charged and rotating black holes in asymptotically flat space time [59][64]. In [61], the critical exponents for the KerrNewmann black holes were calculated for the first time in order to check the validity of the scaling laws for black holes near the critical point. Recently the thermodynamics of black holes in AdS space has attained much attention in the context of AdS/CFT duality, and the critical phenomena might play a crucial role in order to gain some insights regarding this duality. Like in the case of asymptotically flat space time, a number of attempts have also been made in order to explore the critical behavior of rotating and charged black holes both in the de Sitter (dS) and anti de Sitter (AdS) space times [17], [65].
Recently gravity theories in AdS space has attained renewed attention in the context of AdS/CFT (or, gauge/gravity ) duality [66][70], where it has been realized that the physics of black holes in the bulk AdS spacetime plays a crucial role in order to explore the underlying physics behind the various critical phenomena (like phase transition, scaling behavior etc.) occurring in a CFT living on the boundary of the AdS bulk. This fact has been established by Witten long back when he showed that the phase transition taking place between the thermal AdS space at low temperatures and the Schwarzschild AdS black hole at high temperatures could be realized as the confinement/deconfinement transition in the language of boundary CFT [19]. The central idea behind such an analysis is based on a remarkable conjecture, which states that:
With AdS boundary conditions, string theory is completely equivalent to a gauge theory living on the AdS boundary at infinity.
In the early seventies, long before the discovery of AdS/CFT duality, based on the structure of perturbation theory, t’ Hooft first argued that expansion of certain gauge theory could be viewed as a theory of strings [71]. Since then it took more than twenty years to develop a solid theoretical platform for this idea to be more precise. Gradually it became clear that AdS/CFT duality is the manifestation of the fact that the theory of Quantum Gravity with AdS boundary conditions is holographic: It could be completely described by degrees of freedom living in a lower dimensional space [72].
Besides explaining various strongly coupled phenomena in gauge theories, people have also tried to explore the other non gravitational areas of physics using AdS/CFT duality, for example it is found that the so called Hall Effect and the Nernst Effect could be explained using AdS/CFT duality [73][74]. The basic idea that lies behind such an analysis is that a non gravitational system at thermal equilibrium could be well described by a black hole in the AdS bulk with the Hawking temperature . Moreover it has been confirmed that one can actually compute the electrical conductivity and the other transport properties for a non gravitational system through linear perturbations in black holes. Inspired from these early success, people have further tried to extend the domain of applicability of AdS/CFT duality by asking the following question:
Is it possible to find out a gravitational theory that describes superconductivity?
Before we try to answer this question, let us first have a quick review on conventional superconductors like Cu, Al, Nb etc. The conventional BCS theory [75] for superconductors tells us that below certain critical temperature () there happens to be a second order phase transition from the ordinary conducting phase to a superconducting phase and the electrons with opposite spin are combined to form a charged boson called the Cooper pair. Besides having its remarkable achievements, the BCS theory has also its limitations. For example it cannot provide any satisfactory theoretical explanation that will help us to understand the properties of a new class of high superconductors which were first discovered in 1986 [76]. Recently a new class of high superconductors were discovered in 2008 which are found to be layered and the superconductivity is found to be associated with two dimensional planes [77]. Although many people believe that electron pairs do form in these high materials while actually we still do not have any proper theoretical explanation regarding the pairing mechanism that is responsible for the high superconductivity.
Surprisingly over the past few years, based on the fundamental principles of gauge/gravity duality, many people have been able to establish a remarkable connection between General Relativity and the phenomena like superconductivity. The minimal ingredient required to build such a theory is the following: In order to build a superconductor we need the notion of a critical temperature () and there must be two distinct phases which meet at . Speaking more specifically in order to have a conductor/superconductor type phase transition occurring at we must have a charged condensate (that results in a superconductivity) for . On the gravity side the role of temperature is played by the black hole where the (Hawking) temperature of the black hole is precisely the temperature of the field theory living in the boundary. On the other hand, in order to have a charged condensate in the boundary field theory one must include charged scalar field in the bulk gravitational theory. Therefore all these arguments suggest that in order to describe a conductor/superconductor like transition in the boundary field theory one must have a (charged) black hole in the bulk that possesses scalar hair for and no hair for . This indeed suggests that in the bulk gravitational theory we must have a corresponding phase transition occurring at where two distinct phases meet. For we have a charged hairy black hole configuration which corresponds to a superconducting phase in the boundary field theory where as on the other hand for we have planar Reissner Nordstrom AdS solution which corresponds to an ordinary conducting phase in the boundary field theory. Under the frame work of AdS/CFT duality it is easy to show that the phase transition occurring at is indeed a second order phase transition. Theory of superconductors that are built under the frame work of AdS/CFT duality are known as Holographic Superconductors. Interestingly enough such a holographic model of superconductivity has been found to shed some light on various mysterious issues like the pairing mechanism that is responsible for high superconductivity. At this stage it must be noted that, one must use AdS boundary conditions in order to build a gravitational theory that includes (charged) hairy black hole configuration at low temperatures. The reason for this is simple: In an AdS space charged particles cannot escape out to infinity because the negative cosmological constant acts like a confining box and as result there could be a charged condensate outside the event horizon.
It was Gubser [78][79] who first argued that the gravity dual of a superconductor could be found through the mechanism of spontaneous symmetry breaking near the black hole event horizon which results in a condensation of scalar hair at a temperature (T) that is less than certain critical value (). This critical value () below which the scalar hair forms may be identified as the critical temperature corresponding to a second order phase transition from a normal phase to a superconducting phase in the dual field theory. It is in fact the local symmetry breaking in the bulk which corresponds to a global (or sometimes a weakly gauged) symmetry breaking in the dual field theory residing at the boundary of the AdS space and thereby inducing a superconductivity^{4}^{4}4 For excellent reviews on this subject one may consult [80][82].. Later on, this idea was further developed and systematically extended by Horowitz et al. [83][88], who have found that such a simple gravitational dual can indeed reproduce all the standard features of the conventional superconductors [89]. A number of important observations have been made in this regard which may be put as follows [90][100] :
It has been found that the condensate^{5}^{5}5 is the vacuum expectation value of some operator in the field theory living on the boundary of the AdS and is dual to the scalar field added in the bulk. Moreover plays the role of an order parameter in the boundary field theory. () is non zero only at sufficiently low temperatures and is found to be acquiring a first non trivial value other than zero at . The behavior of the condensate at low temperatures is surprisingly found to be similar to that obtained from the BCS theory. Near the critical temperature () it is found that,
(1.6) 
which confirms the standard mean field behavior as predicted by the LandauGinzburg theory^{6}^{6}6In chapter 4, we will derive this relation using perturbation technique. .
From the computation of the free energy it has been further confirmed that the free energy for the hairy configuration is always lower than that of the black hole configuration with out scalar hair which in the present case corresponds to the planar Reissner Nordstrom AdS spacetime.
The optical conductivity is found to be constant for . On the other hand for a pronounced gap appears at low frequency. Most importantly it has been observed that exactly at there appears to be a delta function which indeed suggests the presence of an infinite dc conductivity that is the unique feature of all superconductors.
At low temperatures the ratio of the width of the gap in the optical conductivity () to the critical temperature () is found to be,
(1.7) 
Interestingly for certain high cuprates this ratio has been found to be exactly eight [101].
Apart from these issues recently there have been some attempts in the search for the magnetic response response in holographic superconductors. It was shown that holographic superconductors are type II in nature and both the vortex and droplet states have been constructed in this regard [102][109].
The holographic model of a superconductor that was initially constructed was based on  wave symmetry. Latter on it was found that the theory may be further modified which could still result in a dual description of a superconductor with a  wave symmetry. This was obtained by replacing the original Maxwell action and the charged scalar field by a gauge field in the bulk theory. These are known as holographic  wave superconductors where the order parameter is a vector and the conductivity is strongly anisotropic in a manner that is consistent with the  wave nodes on the fermi surface [110].
1.2 Outline of the thesis
The present thesis is based on the works [111][118] and aims to explore the issue of phase transition in black holes under various circumstances. The whole content of this thesis may be divided into two parts. The first part of it (Chapter 2  Chapter 3 ) is focused towards the application of basic thermodynamic principles to AdS black holes. Based on the thermodynamic analogy, during these works [111][112], we have developed a unique method to study the thermodynamic stability of black holes in AdS space. It is shown that using the notion of thermal stability in ordinary thermodynamics one can indeed distinguish between different phases of a black hole that is undergoing a phase transition. On top of it, incorporating the idea of Ehrenfest’s scheme [38][40] from usual thermodynamics to black hole thermodynamics, we have shown a novel way to determine the nature of this phase transition.
The investigation of the phase transition phenomena has been further pushed forward during the works [113][115] where a general method has been prescribed in order to calculate the critical exponents for the charged AdS black holes near the critical point of the phase transition. The novelty of the present approach is that it could be used to calculate the critical exponents for any other black hole that possesses a discontinuity in the heat capacity near the critical point. As a non trivial exercise, in [115] this general method has been extended in order to compute the critical exponents for the charged topological black holes in the HořavaLifshitz theory of gravity. The computation of these critical exponents are crucial in a sense that it helps us to explore the scaling behavior of black holes near the critical point and thereby to justify the validity of the scaling laws in the context of black hole thermodynamics.
The rest part of the thesis (Chapter  Chapter ) is devoted towards the study of the phase transition phenomena in planar AdS black holes under the framework of AdS/CFT duality. These works [116][118] consider a gravitational theory coupled to the Born Infeld (BI) Lagrangian in an asymptotically AdS space where the planar BIAdS black hole becomes unstable to develop a scalar hair at low temperatures. This implies that in the bulk we have a phase transition
BI AdS Charged black hole with scalar hair
The mechanism that is responsible for such a transition is the spontaneous breaking of symmetry at low temperatures. As a consequence of this in the boundary field theory there happens to be a superconducting phase transition which is thereby termed as holographic superconductor. During these works [116][118], based on analytic calculations and using the AdS/CFT dictionary several crucial properties of these holographic superconductors have been investigated in the probe limit, for example, various physical entities like, the critical temperature (), order parameter, the critical magnetic field etc. have been computed near the critical point of the phase transition.
The whole thesis consists of 6  chapters, including this introductory part. Chapter wise summary is given below.
Chapter 2: Thermodynamics of Phase Transition In AdS Black Holes: In this chapter, based on the fundamental concepts of thermodynamics and considering a grand canonical framework we present a detailed analysis of the phase transition phenomena in AdS black holes. The method is in line with that used a long ago to understand the liquidvapor phase transition where the first order derivatives of Gibbs potential are discontinuous and ClausiusClapeyron equation is satisfied. We have extended this idea to black holes in order to explore the vexing issue of phase transition phenomena that occurs between a lower mass black hole with negative specific heat to a higher mass black hole with positive specific heat. The critical point has been marked by the discontinuity in the corresponding heat capacity (i.e; second order derivative in the Gibbs potential). Moreover, using Ehrenfest’s scheme of standard thermodynamics, we have analytically checked that the phase transition occurring in these black holes is indeed a second order phase transition. Chapter 3: Critical Phenomena In Charged AdS Black Holes: In this chapter, considering a canonical ensemble we have further extended our analysis in order to investigate the scaling behavior in charged AdS black holes near the critical point(s). We have explicitly computed all the static critical exponents associated with various thermodynamic entities (like, heat capacity, isothermal compressibility etc.) and found that all these exponents indeed satisfy so called thermodynamic scaling laws near the critical point(s). We have also checked the Generalized Homogeneous Function (GHF) hypothesis [57] for charged AdS black holes and found its compatibility with thermodynamic scaling laws. Going beyond the usual framework of Einstein gravity, we have also investigated the phase transition of the static, topological charged black hole solution with arbitrary scalar curvature in the HořavaLifshitz theory of gravity at the Lifshitz point . The analysis has been performed using the canonical ensemble frame work; i.e. the charge was kept fixed. We have found (a) for both and , there is no phase transition, (b) while case exhibits the second order phase transition within the physical region of the black hole. The critical point of second order phase transition was obtained by the divergence of the heat capacity at constant charge. Near the critical point, we have computed various critical exponents. It has been also observed that they satisfy the usual thermodynamic scaling laws. Chapter 4: AdS/CFT Duality And Phase Transition In Black Holes: Holographic Superconductors: In this chapter, based on the SturmLiouville (SL) eigenvalue problem, we have analytically investigated various properties of holographic swave superconductors in the background of both SchwarzschildAdS and GaussBonnet AdS spacetimes in the framework of BornInfeld (BI) electrodynamics. Using perturbation technique, we have explicitly computed the relation between the critical temperature and the charge density and also found that both the BornInfeld (BI) coupling parameter as well as the Gauss Bonnet (GB) coupling parameter indeed affect the formation of scalar hair at low temperatures. It has been observed that the condensation is harder to form for higher values of BI/GB coupling parameters. The critical exponent associated with the order parameter has been found to be 1/2, which is in good agreement to that with the universal mean field value. The analytic results thus obtained are found to be in good agreement to that with the existing numerical results. Chapter 5: Magnetic Response In Holographic Superconductors: In this chapter, based on a different analytic scheme known as the Matching technique [91], several properties of holographic wave superconductors have been investigated in the presence of an external magnetic field and considering various higher derivative (non linear) corrections to the usual Maxwell action. Explicit expressions for the critical temperature as well as the  wave order parameter have been obtained in the probe limit. It has been found that below certain critical magnetic field strength () there exists a superconducting phase. Most importantly it is observed that the value of this critical field strength () indeed gets affected due to the presence of higher derivative corrections to the usual Maxwell action. Chapter 6: Summary and Outlook: Finally, in chapter6 we present the summary of our findings along with some future prospects.
Chapter 2 Thermodynamics of Phase Transition In AdS Black Holes
2.1 The gibbsian approach
It is well known that black holes behave as thermodynamic systems. The laws of black hole mechanics become similar to the usual laws of thermodynamics after appropriate identifications between the black hole parameters and the thermodynamical variables [3]. Nonetheless black holes show some exotic behaviors as a thermodynamic system, such as (i) entropy of black holes is proportional to area and not the volume, (ii) their temperature diverges to infinity when mass tends to vanish due to Hawking evaporation. In fact these exotic behaviors enhance the scope of alternative studies on black holes. The thermodynamic properties of black holes were further elaborated in early eighties when Hawking and Page discovered a phase transition between the thermal AdS and the Schwarzschild AdS black hole [18]. Thereafter this subject has been intensely studied in various viewpoints [20][52].
The gibbsian approach to the phase transitions is based on the ClausiusClapeyronEhrenfest’s equations [53]. These equations allow for a classification of phase transitions as first order or continuous (higher order) transitions. For a first order transition the first order derivatives (entropy and volume) of the Gibbs free energy is discontinuous and ClausiusClapeyron equation is satisfied. Liquid to vapor transition is an ideal example of such a phase transition. Similarly for a second order transition Ehrenfest’s relations are satisfied. However, this specific tool concerning the nature of the phase transition has never been systematically explored for black holes. For the past couple of years there have been a series of papers [38][40],[111][112] which have systematically applied these concepts to study phase transitions in black holes. Despite of some conceptual issues mentioned in the earlier paragraph, it is found that black holes not only allow us to implement above basic ideas but also support them. In normal physical systems a verification of the ClausiusClapeyron or Ehrenfest relation is possible subjected to the design of a table top experiment. However black holes are found to be like a theorist’s laboratory, where these ideas on phase transition could be proved without performing any table top experiment.
From all of these works [38][40],[111][112], it is indeed evident that in case of black hole phase transition we do not encounter any discontinuity in the first order derivative of Gibbs energy and hence ClausiusClapeyron equation was not needed. However, because of the discontinuity in second order derivatives, Ehrenfest’s relations for black holes were first derived in [38]. Because of infinite divergences of various physical entities at the critical point, numerical methods were adopted in order to check the validity of the Ehrenfest’s relations. From these analyses it was found that away from the critical point, only the first Ehrenfest’s relation was satisfied [38][39], whereas on the other hand, close to the critical point both relations were satisfied [40]. Nevertheless it should be emphasized that such a numeric check has a drawback. Since in this method one assigns a numerical value for the angular velocity or electric potential beforehand and then check the Ehrenfest relations, every time this value is changed it is necessary to repeat the numerical analysis. This is a limitation for generalizing the result for other cases. On top of it, because of the infinite divergences in various physical entities near the critical point, it is indeed very difficult to estimate the physical quantities exactly at the critical point and therefore the whole numerical exercise loses its significance. Such limitations in the numerical analysis should provide us enough motivation towards developing certain analytic steps that would essentially capture the physics of black hole phase transition near the critical point.
Based on the papers [111][112], in this chapter we present an elegant analytic technique to treat the singularities near the critical points of the phase transition curves. This is important for further studies on black hole phase transition using our techniques. The example of the dimensional charged (RN) AdS and Schwarzschild AdS black holes are worked out in details. Moreover, we have also checked the validity of the Ehrenfest’s relations for the rotating (Kerr) black holes in dimensions. We show analytically that at the critical point both Ehrenfest’s relations are satisfied exactly. This confirms the onset of a second order phase transition for these black holes. Explicit expressions for the critical temperature, critical mass etc. are given. This phase transition is characterized by the divergence of the specific heat at the critical temperature. The sign of specific heat is different in two phases and they essentially separate two branches of AdS black holes with different mass/ horizon radius. The branch with lower mass (horizon radius) has a negative specific heat and thus falls in an unstable phase. The other branch with larger mass (horizon radius) is locally stable since it is associated with a positive specific heat and also positive Gibbs free energy.
Before we proceed further, let us briefly mention about the organization of this chapter. To start with, it will be nice to have a quick review on the conventional liquidvapor transition in the framework of standard ClausiusClapeyron equations which is done in section 2.2. In the next section we make a detail discussion on the basic thermodynamic structure of the phase transition occurring in dimensional charged (RN) AdS black holes and as an appropriate limit we obtain the corresponding results for the Schwarzschild AdS case. In section 2.4, based on the Ehrenfest’s scheme of standard thermodynamics we perform a detail analysis in order to identify the nature of the phase transition in these black holes. In section 2.5 we carryout an identical analysis for the dimensional rotating (Kerr) AdS black holes and discuss the results in brief. Finally we conclude in section 2.6.
2.2 Liquid to vapour phase transition
Liquid and vapour are just two phases of matter of a single constituent. When heated upto the boiling point liquids vapourise and at this point the slope of the coexistence curve (pressure versus temperature plot) obeys the ClausiusClapeyron equation, given by
(2.1) 
where and are the difference between the entropy and volume of the constituent in two different phases.
Note that the derivation of the above relation comes from the definition of the Gibbs potential ( is the internal energy), where it is assumed that the first order derivatives of with respect to the intrinsic variables ( and ) are discontinuous and it comes out that their specific values in two phases give the slope of the coexistence curve at the phase transition point. This is an example of the first order phase transition which clearly differs from the case that we will be studying next.
2.3 Phase transition in higher dimensional AdS black holes
We start our analysis with ReissnerNordström black holes in dimensions. ReissnerNordström black holes are charecterised by their mass (M) and charge (Q). The solution for dimensional RNAdS space time with a negative cosmological constant is defined by the line element [34],
(2.2) 
where,
(2.3) 
Here is related to the ADM mass ( ) of the of the black hole as,
(2.4) 
where is the volume of unit sphere. The parameter is related to the electric charge as,
(2.5) 
The entropy of the system is defined as,
(2.6) 
where is the radius of the outer event horizon defined by the condition . The electrostatic potential difference between horizon and infinity is given by
(2.7) 
Using (2.5) and (2.6) we can further express (4.58) as,
(2.8) 
From the condition and using (2.4), (2.5), (2.6) and (2.8) we can express the black hole mass as,
(2.9) 
Using the first law of black hole mechanics , the Hawking temperature may be obtained as,
(2.10)  
At this stage, it is customary to mention that in the following analysis the various black hole parameters have been interpreted as respectively, where the symbols have their standard meanings. Also, with this convention, the parameter does not appear in any of the following equations.
Using the above relation (2.10) we plot the variation of entropy () against temperature () in Figure (2.1). In order to obtain an explicit expression for the temperature () corresponding to the turning point we first set . From this condition and using (2.10) we find
(2.11) 
Furthermore we find that,
(2.12) 
Therefore for . Hence the temperature corresponding to the entropy represents the minimum temperature for the system. Substituting (2.11) into (2.10) we find the corresponding minimum temperature for RNAdS space time to be
(2.13) 
In the chargeless limit (), we obtain the corresponding (minimum) temperature
(2.14) 
for Schwarzschild AdS space time in dimension. For equation (2.14) yields
(2.15) 
which was obtained earlier by Hawking and Page in their original analysis^{1}^{1}1Note that does not appear since as explained before, it has been appropriately scaled out. [18].
It is now possible to interpret the two branches of the curve, separated by the point , corresponding to two phases. Since entropy is proportional to the mass of the black hole, therefore phase 1 corresponds to a lower mass black hole and phase 2 corresponds to a higher mass black hole. We also observe that both of these phases exist above the (minimum) temperature . To obtain an expression for the mass () of the black hole separating the two phases we substitute (2.11) into (2.9) which yields,
(2.16) 
Phase 1 corresponds to a black hole with mass while phase 2 corresponds to a black hole with mass .
In the chargeless limit () we obtain the corresponding critical mass for the Schwarzschild AdS black hole in dimension,
(2.17) 
Substituting into (2.17) yields
(2.18) 
which is the result obtained earlier in [18].
A deeper look at the plot further reveals that the slope of the curve gradually changes it’s sign around , thereby clearly indicating the discontinuity in the specific heat () that is associated with the occurrence of a continuous higher order transition at . All these features become more prominent from the following analysis. In order to do that we first compute the specific heat at constant potential (), which is the analog of (specific heat at constant pressure) in conventional systems. This is found to be
(2.19)  
The critical temperature (at which the heat capacity () diverges) may be found by setting the denominator of (2.19) equal to zero, which exactly yields the value of entropy () as found in (2.11). Substituting this value into (2.10) it is now trivial to show that (2.13) corresponds to the critical temperature at which diverges. This diverging nature of at may also be observed from Figure (2.2) where we plot heat capacity () against the temperature (). From this plot we find that the heat capacity () changes from negative infinity to positive infinity at [26]. One can further note that the smaller mass black hole (phase 1) corresponds to an unstable since it posses a negative specific heat (), whereas the larger mass black hole falls into a stable phase (phase 2) as it corresponds to a positive heat capacity ().
It is now customary to define grand canonical potential which is known as Gibbs free energy. For RNAdS black hole this is defined as where the last term is the analog of term in conventional systems. Using (2.8), (2.9) and (2.10) one can easily write as a function of (),
(2.20) 
Since in the above relation we cannot directly substitute entropy () in terms of temperature () therefore in order to obtain the plot we first note the variation of Gibbs free energy () against entropy () using the above relation (2.20).
As a next step, (from Figure (2.3)) we note the values of for different values of and obtain the corresponding values of from Figure (2.1). Finally we put all those data together in order to obtain the following numerical plot (fig. (2.4)).
From Figure (2.4) we note that the value of for phase 1 is greater than that of phase 2. Therefore the larger mass black hole falls into a more stable phase than the black hole with smaller mass. We also observe that is always positive for . This also corroborates the findings of [18] for the Schwarzschild AdS black hole.
In order to calculate we first set (in 2.20) which yields
(2.21) 
Substituting this into (3.11) we finally obtain
(2.22) 
In the chargeless limit () we obtain the corresponding temperature for the Schwarzschild AdS black hole in dimension
(2.23) 
Substituting into (2.23) yields
(2.24) 
which reproduces the result found earlier in [18].
2.4 Ehrenfest’s equations
Classification of the nature of phase transition using Ehrenfest’s scheme is a very elegant technique in standard thermodynamics. In spite of its several applications to other systems [119] [123], it is still not a widely explored scheme in the context of black hole thermodynamics although some attempts have been made recently [38][40]. In the remaining part of this paper we shall analyze and classify the phase transition phenomena in RN AdS black holes by exploiting Ehrenfest’s scheme.
Looking at the () graph (fig (2.1)) we find that entropy is indeed a continuous function of temperature . Therefore the possibility of a first order phase transition is ruled out. However, the infinite divergence of specific heat near the critical point (see fig (2.3)) strongly indicates the onset of a higher order (continuous) phase transition. The nature of this divergence is further illuminated by looking at the graph (fig (2.5)).
We now exploit Ehrenfest’s scheme in order to understand the nature of the phase transition. Ehrenfest’s scheme basically consists of a pair of equations known as Ehrenfest’s equations of first and second kind [53, 57]. For a standard thermodynamic system these equations may be written as [53]
(2.26)  
(2.27) 
For a genuine second order phase transition both of these equations have to be satisfied simultaneously.
Let us now start by considering the RNAdS black holes. Bringing the analogy () between the thermodynamic state variables and various black hole parameters, we are now in a position to write down the Ehrenfest’s equations for this system as [39],
(2.28)  
(2.29) 
where, is the analog of volume expansion coefficient and is the analog of isothermal compressibility. Their explicit forms are given by,
(2.30)  
(2.31) 
Observe that the denominators of (2.19), (2.30), and (2.31) are all identical. Hence , and diverge at the critical point since, as already discussed, this point just corresponds to the vanishing of the denominator in . However as shown below, the R.H.S. of (2.28) and (2.29) which involves the ratio of these quantities, remain finite at the critical point.
Using (2.10), the L.H.S. of the first Ehrenfest’s equation (2.28) may be found as (at the critical point )
(2.32) 
In order to calculate the right hand side, we first note that
(2.33) 
Therefore the R.H.S. of (2.28) becomes,
(2.34) 
Using (2.8) we calculate the R.H.S. of the above equation and obtain,
(2.35) 
Equations (2.32) and (2.35) clearly reveal the validity of the first Ehrenfest’s equation. Remarkably we find that the divergence in is canceled with that of in the first equation.
In order to evaluate the L.H.S. of (2.29) we first note that, since , therefore,
(2.36) 
Since specific heat diverges at the critical point () therefore it is evident from (2.19) that . Also from (2.8) we find that, has a finite value when evaluated at . Therefore the first term on the R.H.S. of (2.36) vanishes. This is a very special thermodynamic feature of AdS black holes, which may not be true for other systems. Therefore under this circumstances we obtain,
(2.37) 
Using the thermodynamic identity,
(2.38) 
we find,
(2.39) 
Therefore R.H.S. of (2.29) may be found as
(2.40) 
This indeed shows the validity of the second Ehrenfest’s equation. Also we find that divergences in and get canceled in the second equation like in the previous case. Using (2.28), (2.37) and (2.40) the Prigogine Defay (PD) ratio [123][124] may be found to be,
(2.41) 
Hence the phase transition occurring at is a second order equilibrium transition [119, 120],[122, 123]. This is true in spite of the fact that the phase transition curves are smeared and divergent near the critical point.
2.5 Kerr AdS black hole
In this section we discuss the phase transition for dimensional rotating (Kerr) AdS black holes under the framework of Ehrenfest’s equations. For the KerrAdS black hole one can easily perform a similar analysis and describe phase transition for that case. The graphical analysis shown above for RNAdS black hole physically remains unchanged for the KerrAdS case. In particular one can calculate the inverse temperature,
(2.42) 
and the condition gives,
(2.43) 
By substituting the positive solution of this polynomial in (5.63) one finds the minimum temperature . Furthermore for the KerrAdS black hole the heat capacity diverges exactly where (2.43) holds [40] which corresponds to the minimum temperature (). In the irrotational limit reproduces the known result of [18]. Results for and can also be calculated by a similar procedure. These expressions are rather lengthy and hence omitted. Moreover there are no new insights that have not already been discussed in the RNAdS example.
Ehrenfest’s set of equations for KerrAdS black hole are given by [40],
(2.44)  
(2.45) 
The expressions for specific heat (), analog of the volume expansion coefficient () and compressibility () are all provided in [40]. Once again, they all have the same denominator (like the corresponding case for RNAdS). Considering the explicit expressions given in [40] and using the same techniques we find that both sides of (2.44) and (2.45) lead to an identical result, given by,
(2.46) 
This shows that the phase transition for KerrAdS black hole is also second order.
2.6 Discussions
In this chapter, based on standard thermodynamics, we have systematically developed an approach to analyze the phase transition phenomena in Reissner Nordstrom AdS black holes in arbitrary dimensions. In suitable limits the results for the Schwarzschild AdS black hole are obtained. This phase transition is characterized by divergences in specific heat, volume expansivity and compressibility near the critical point^{2}^{2}2It is important to point out the difference between the phase transition we discuss in this chapter to that with the conventional HawkingPage (HP) phase transition. Our analysis is strictly confined to the critical temperature () where the specific heat () acquires an infinite divergence. On the other hand HawkingPage transition occurs at a temperature () where Gibbs free energy () changes its sign [28][30]. While the HP transition is first order, the one is considered here is second order.. Furthermore, within the gibbsian approach, and using a grand canonical ensemble, we have provided a unique way (which is valid in any dimension greater than two) to analyze both the quantitative and qualitative features of this phase transition. From our analysis it is clear that such a transition takes place from a black hole with smaller mass () to a black hole with larger mass (), where denotes the critical mass. Also, we have found that the black hole with smaller mass falls in an unstable phase since it has negative specific heat, whereas that with the larger mass possesses a positive specific heat and thereby corresponds to a stable phase. We have provided completely generalized expressions for the critical temperature () and critical mass () that characterize this phase transition. Also, we have explicitly calculated the temperature () which is associated with the change in sign in Gibb’s free energy (). Following our approach, we have established a universal relation (2.25) between and that is valid in any arbitrary dimension ().
Finally and most importantly, employing the Ehrenfest’s scheme we have resolved the vexing issue regarding the nature of phase transition in Reissner Nordstrom AdS black holes. In order to do that we have analytically checked the validity of both the Ehrenfest’s equations near the critical point. This clearly suggests that the phase transition that is associated with the divergence in the heat capacity at the critical point () is indeed a second order equilibrium transition. A similar analysis for the dimensional rotating (Kerr) AdS black holes has been performed as well and results identical to the charged (RN) AdS case are obtained.
From the entire analysis of this chapter it is indeed quite evident that during the black hole phase transition one should actually encounter some sort of singularity in various thermodynamic entities (like, heat capacity, isothermal compressibility etc.). According to the basic principles of statistical mechanics such a divergence could be regarded as the manifestation of certain diverging correlation length near the critical point of the phase transition curve. Therefore in order to further advance the study of the phase transition, one should check the scaling behavior of black holes near the critical point and find out the corresponding universality class. We shall discuss this issue in the next chapter.
Chapter 3 Critical Phenomena In Charged AdS Black Holes
3.1 Critical phenomena and black holes
In ordinary thermodynamics, a phase transition occurs whenever there is a singularity in the free energy or one of its derivatives. The corresponding point of discontinuity is known as the critical point of phase transition. In Ehrenfest’s classification of phase transition, the order of the phase transition is characterized by the order of the derivative of the free energy that suffers discontinuity at the critical points. For example, if the first derivative of the free energy is discontinuous then the corresponding phase transition is first order in nature. On the other hand, if the first derivatives are continuous and second derivatives are discontinuous then the transition may be referred as higher order or continuous. These type of transitions correspond to a divergent heat capacity, an infinite correlation length, and a power law decay of correlations near the critical point.
The primary aim of the theory of phase transition is to study the singular behavior of various thermodynamic entities (for example heat capacity) near the critical point. It order to do that one often expresses these singularities in terms of power laws characterized by a set of static critical exponents [57][58]. Generally these exponents depend on a few parameters of the system, like, (1) the spatial dimensionality () of the space in which the system is embedded, (2) the range of interactions in the system etc. To be more specific, it is observed that for systems possessing short range interactions, these exponents depend on the spatial dimensionality (). On the other hand for systems with long range interactions these exponents become independent of , which is the basic characteristic of a mean field theory and is observed for the case in usual systems. It is interesting to note that the (static) critical exponents are also found to satisfy so called thermodynamic scaling laws [57][58] which apply to a wide variety of thermodynamical systems, from elementary particles to turbulent fluid flow. Such studies have also been performed, albeit partially, in the context of black holes [59][65]. In this chapter we aim to provide a detailed analysis of these issues and show how the study of critical phenomena in black holes is integrated with the corresponding studies in other areas of physics adopting a mean field approximation.
The goal of the present chapter is to carry out a detailed analytic computation in order to explore the scaling behavior of black holes near the critical point(s)^{1}^{1}1The critical points of the phase transition are characterized by the discontinuities in the heat capacity at constant charge . of the phase transition curve in various theories of gravity. To do that, in our analysis we consider examples both from the usual Einstein gravity as well as from the HořavaLifshitz theory of gravity. This eventually helps us to make a systematic comparison between the scaling behavior of black holes in various theories of gravity as well as to identify the corresponding universality class.
The entire content of this chapter is based on the papers [113][115] and could be divided mainly into two parts. In the first part, based on a canonical framework, we aim to study the critical behavior of charged black holes taking the particular example of BornInfeld AdS (BI AdS) black holes in dimensions. Results obtained in the above case smoothly translate to that of the dimensional RN AdS case in the appropriate limit. We compute the static critical exponents associated with this phase transition and verify the scaling laws. Interestingly enough these laws are found to be compatible with the static scaling hypothesis of usual statistical systems [57][58].
In the remaining part of this chapter we make a detail analysis on the phase transition and scaling behavior of charged topological black holes in the HořavaLifshitz theory of gravity at the Lifshitz point . Based on a canonical framework (i.e; keeping the charge () of the black hole fixed [34]), we investigate the critical behavior of topological charged black holes in HořavaLifshitz theory of gravity at the Lifshitz point . The black hole solution was given in [125]. Although all the thermodynamic quantities were evaluated earlier [125], a detailed study of the nature of phase transition is still lacking. Particularly the issue regarding the scaling behavior of (charged) black holes has never been investigated so far in the framework of HořavaLifshitz theory of gravity. This essentially gives us enough motivation to carry out a systematic analysis of the phase transition as well as the scaling behavior of topological black holes in the HořavaLifshitz theory of gravity. The present thesis therefore aims to explore the phase transition phenomena considering all the three cases taking . We observe the following interesting features: There is no Hawking Page transition [18] for black hole with . For , there is a upper bound in the value of the event horizon, above which the temperature becomes negative. This indicates that above this critical value of horizon, the black hole solution does not exist. We will call this valid range as the physical region. Within the physical region, interesting phase structure could be observed for the hyperbolic charged black holes (). In this particular case we observe the second order transition. This is different from the usual one. In Einstein gravity, the Hawking Page transition mostly occurs for the case, while usually there is no such transition for the case [28, 29].
Finally, we explicitly calculate all the static critical exponents associated with the second order transition, and check the validity of thermodynamic scaling laws near the critical point. Interestingly enough it is found that the hyperbolic charged black holes () in the HořavaLifshitz theory of gravity fall under the same universality class as the black holes having spherically symmetric topology () in the usual Einstein gravity. The values of these critical exponents indeed suggest a universal mean field behavior in black holes which is valid in both Einstein as well as HořavaLifshitz theory of gravity.
The plan of the chapter is as follows: In section 3.2, we make a qualitative as well as quantitative analysis of the various thermodynamic entities for dimensional BI AdS black holes in order to have a meaningful discussion on their critical behavior in the subsequent sections. In section 3.3, we explicitly compute the static critical exponents in order to check the validity of the scaling laws both for the BI AdS and RN AdS black holes in dimensions. We begin in section 3.4 by giving a brief introduction of the charged topological black hole solutions in HořavaLifshitz gravity and use it in section 3.5 to study the different thermodynamic quantities as well as the phase transition. In section 3.6, the critical exponents near the critical point(s) are being evaluated and a brief discussion on the validity of the ordinary scaling laws is presented. Finally, we conclude in section 3.7.
3.2 Charged black holes in higher dimensional AdS space
For the past couple of decades gravity theories with BornInfeld (BI) action have garnered considerable attention due to its several remarkable features. For example, BornInfeld type effective actions, which arise naturally in open super strings and D branes are free from physical singularities [126]. Also, using the Born Infeld action one can in fact study various thermodynamic features of Reissner Nordstrom AdS black holes in a suitable limit. During the last ten years many attempts have been made in order to understand the thermodynamics of BornInfeld black holes in AdS space [127]. In spite of these efforts, several significant issues remain unanswered. Studying the critical phenomena is one of them. In order to have a deeper insight regarding the underlying phase structure for these black holes one needs to compute the critical exponents associated with the phase transition and check the validity of the scaling laws near the critical point. A similar remark also holds for the higher dimensional Reissner Nordstrom AdS (RN AdS) black holes.
In the present thesis therefore we aim to fill up this gap step by step. In order to do that we first compute the essential thermodynamic entities both for the BI AdS and RN AdS black holes which will be required in the next step to calculate the critical exponents for these black holes.
The action for the Einstein BornInfeld gravity in dimensions is given by [127],
(3.1) 
where,
(3.2) 
with . Here is the BornInfeld parameter with the dimension of mass and is the cosmological constant. It is also to be noted that for the rest of our analysis we set Newton’s constant .
By solving the equations of motion, the BornInfeld anti de sitter (BI AdS) solution may be found as,
(3.3) 
where,
(3.4) 
and is a hypergeometric function [128].
It is interesting to note that for the metric (3.4) has a curvature singularity at . It is in fact possible to show that this singularity is hidden behind the event horizon(s) whose location may be obtained through the condition . Therefore the above solution (3.3) could be interpreted as a space time with black holes [127].
In the limit and one obtains the corresponding solution for Reissner Nordstrom (RN) AdS black holes. Clearly this is a nonlinear generalization of the RN AdS black holes. Here is related to the mass () of the black hole as [127],
where is the volume of the unit sphere. Identical expression could also be found for the RN AdS case [34].
Using (LABEL:mq) one can rewrite (3.4) as,
(3.8) 
In order to obtain an explicit expression for the mass () of the black hole we set , which yields,
(3.9) 
where is the radius of the outer event horizon. Here the parameter is chosen in such a way so that or, in other words our analysis is carried out in the large limit. Also, such a limit is necessary for abstracting the results for RN AdS black holes obtained by taking . Therefore all the higher order terms from onwards have been dropped out from the series expansion of . Using (3.7) one can express the electrostatic potential difference () between the horizon and infinity as,
(3.10)  
where is the electric charge. Henceforth all our results are valid upto only.