Higher dimensional dyonic black holes

Higher dimensional dyonic black holes

S. H. Hendi111 email address: hendi@shirazu.ac.ir, N. Riazi222 email address: nriazi@sbu.ac.ir, S. Panahiyan333 email address: shahram.panahiyan@uni-jena.de and B. Eslam Panah444 email address: behzad.eslampanah@gmail.com Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
Physics Department, Shahid Beheshti University, Tehran 19839, Iran
Helmholtz-Institut Jena, Fröbelstieg 3, Jena D-07743, Germany
ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy
Abstract

The paper at hand presents a novel class of dyonic black holes in higher dimensions through a new proposal for the electromagnetic field tensor. The black hole solutions are extracted analytically and their geometrical/physical properties are studied. In addition, the details regarding thermodynamical structure and phase transition behavior of the solutions for different cases are investigated: i) general case, ii) constant electric field, iii) constant magnetic field, and iv) constant electric and magnetic fields. It will be shown that depending on the picture under consideration, the thermodynamical properties are modified. To have better picture regarding the phase transitions, the concept of the extended phase space is employed. It will be shown that in the absence of the electric field, magnetic black holes present van der Waals like phase transition. Furthermore, it will be highlighted that for super magnetized black holes, no phase transition exists.

I Introduction

Gravity in higher dimensions has been of interest for several decades. This is due to fact that specific range of the advanced physics theories requires existence of the higher dimensions in order to address different issues in the nature. To name a few, one can point out: I) String, superstring theories and in general M-theory, which have been the most celebrated theories towards unification of the all fundamental forces in nature Antoniadis (). II) Braneworlds theories which have been employed to address fundamental issues of gravity such as hierarchy problem Randall (). III) Kaluza-Klein compactification which was the pioneering proposal regarding unification of the gravity and electromagnetism Kaluza (); Klein (). Through all of the mentioned thoeries, the necessity of existence of the higher dimensionality were highlighted and employed.

In the context of black holes, it was shown that although the laws of black hole mechanic are universal, the properties of black hole are dimension dependent Thermodynamics (). Reissner-Nordstrom (RN) black holes in the context of string theory plays an important role in understanding the black hole entropy near extremal limits Strominger (), hence and in the context of this theory, Hawking temperature, radiation rate and entropy for these black holes have been studied and it was proposed that quantum evolution of black hole does not lead to information loss Callan (). Hereupon, the study of black holes in higher dimensions has attracted many authors. For example; the generalizations of Schwarzschild and Kerr black holes to arbitrary extra dimensions have been investigated in refs. Tangherlini () and Myers (), respectively. The existence of black rings and Saturns in higher dimensions have been studied Emparan (); Elvang (); Iguchi (). The thermodynamics and stability of higher dimensional Kerr-anti de Sitter black hole has been addressed in ref. Carter ().

In ref. Kim (), the ultraviolet divergent structures of the matter field in a higher dimensional RN black hole has been studied and the contributions to Bekenstein-Hawking entropy by using the Pauli-Villars regularization method was addressed. Gravitation with superposed Gauss–Bonnet terms and black object solutions in higher dimensions have been obtained in ref. GB (). Uniqueness and non-uniqueness of static (un)charged black holes and black p-branes in higher dimensions have been surveyed Gibbons (). Topology of black holes’ event horizons in higher dimensions has been investigated in ref. Helfgott (). Hawking emission of gravitons and generalization of Hawking’s black hole topology theorem to higher dimensions have been obtained, respectively in refs. Cardoso () and Galloway (). Quasinormal modes of Schwarzschild CardosoII () and Kerr Kao () black holes in higher dimensions have been studied. In addition, the production of higher-dimensional black holes in future colliders becomes a conceivable possibility in scenarios involving large extra dimensions and TeV-scale gravity Cavaglia (). In addition, as mathematical objects, black hole spacetimes are among the most important Lorentzian Ricci-flat manifolds in any dimension EmparanII ().

Another motivation for considering higher dimensions is related to the AdS/CFT correspondence which relates the properties of a black hole in d-dimensions with those of a quantum field theory in (d-1)-dimensions Aharony (). In other words, we can extract the properties of a complex system in quantum field theory by using the properties of black holes in one higher dimension. Among the different set ups for AdS/CFT studies, the ones including a magnetic field has been of special interests. Historically speaking, Hartnoll and Kovtun in their pioneering work incorporated a background magnetic field and studied low frequency charge transport and Hall conductivity Hartnoll (). The magnetic field was included by considering a type of black holes known as dyonic black holes. Later, it was shown that large dyonic black holes in anti-de Sitter spacetime are dual to stationary solutions of the equations of relativistic magnetohydrodynamics on the conformal boundary of AdS Caldarelli (). In addition, the dyonic black holes was employed to induce the effects of external magnetic field on superconductors. It was shown that the size of condensate for the superconductor is magnetic field dependent in a manner which is a reminiscent of the Meissner effect Albash (). Furthermore, the holographical properties of the dyonic dilatonic black branes including transport coefficients, Hall conductance, DC longitudinal conductivity and response were investigated in ref. Goldstein (). So far, a large number of publications was dedicated to study systems including dyonic black holes in different contexts (for a very incomplete list, we refer the reader to refs. Dyonic1 (); Dyonic2 (); Dyonic3 (); Dyonic4 (); Dyonic5 (); Dyonic6 (); Dyonic7 (); Dyonic8 (); Dyonic9 (); Dyonic10 (); Dyonic11 (); Dyonic12 (); Dyonic13 (); Dyonic14 (); Dyonic15 (); Dyonic16 (); Dyonic17 (); Dyonic18 (); Dyonic19 (); Dyonic20 (); Dyonic21 (); Dyonic22 (); Dyonic23 (); Dyonic24 ()).

In this paper, we introduce a novel approach for constructing electrically-magnetically charged black holes, or simply put dyonic black hole holes, in higher dimension. Our main motivation is to propose a simplified method for constructing higher dimensional dyonic black holes in a manner that magnetic and electric parts of it are stand alone properties. We expand our study to thermodynamical properties of the black holes in order to understand the physical and geometrical properties of the solutions in details. We consider four distinctive cases which correspond to four different scenarios in the context of AdS/CFT correspondence; I) General case: in which no specification is given about electric and magnetic field. II) Constant electric field: which corresponds to immersing the magnetically charged black holes in a finite electric field. III) Constant magnetic field: which corresponds to considering the electrically charged black holes in the presence of finite external magnetic field. IV) Constant electric and magnetic fields: which corresponds to immersing the black holes in a field consisting finite electric and magnetic fields. We will show how the magnetization, electrification and dimensionality of the solutions affect physical properties of the black holes including the behaviors of temperature, enthalpy and heat capacity. In addition, we will explore the possibility of the van der Waals like phase transition for the four mentioned cases and investigate the effects of magnetization, electrification and dimensionality on van der Waals like critical points and phase transition. Among different benefits of our proposal, we can point out two important ones: I) The set up is easy to understand and could be employed without going into trouble of introducing complex system of the equations. II) The set up provides the possibility of including higher dimensional gravities such as Lovelock gravity with simplicity. We intend to provide the possibility of studying the effects of magnetism on holographical systems in higher dimensions (with or without higher dimensional gravity theories). The set up also could be employed to study the effects of non-finite magnetic/electric field on holographical systems.

The structure of the paper is as follows: first, the action and field equations are introduced. The metric function is obtained and, geometrical properties and conditions regarding the existence of black holes are investigated. In section III, thermodynamical quantities of interest in this paper are introduced and obtained. Sections IV-VII are, respectively, dedicated to investigation of four different cases including: General case, constant electric field, constant magnetic field and, constant electric and magnetic fields. The paper is concluded with some closing remarks.

Ii Basic Equations

The main goal of this paper is construction of the novel dyonic black holes in higher dimensions. Our motivation comes from interesting properties of the dyonic black holes specially in holographical aspects, DC conductivity string theory, etc. Here, we introduce higher dimensional dyonic black holes which could be employed to understand the DC effects in higher dimensions. Furthermore, we are providing the possibility of studying higher dimensional theories of gravity such as Lovelock gravity in the presence of dyonic configuration as well.

Dyonic black holes enjoy existence of magnetic charge as well as electric charge in their structures. It is worthwhile to mention that the set up which is going to be introduced here, provides the possibility of having magnetic field for black holes without introduction of the electric field. For simplicity, we consider Einstein Lagrangian in the presence of the cosmological constant as the gravitational sector of the action. As for the matter field, we simply consider the Maxwell Lagrangian with modified vector potential. Therefore, the -dimensional action will be given by

(1)

in which, is the Ricci scalar, refers to the cosmological constant and is the electromagnetic field tensor. It is worthwhile to mention the possibility of generalization of this action to include higher dimensional theories of gravity, scalar-tensor field theories and nonlinear electromagnetic field. The magnetic charge, hence the dyonic property lies within the structure of electromagnetic tensor. The -dimensional metric with topological boundary of and , is given by

(2)

in which is the line element of a -dimensional hypersurface with the constant curvature and volume with the following explicit form

(3)

In order to have consistent field equations with magnetic charge included, we modify the electromagnetic tensor with the following non-zero components

(4)

in which and are respectively, electric and magnetic charges, and

(5)

The is representing the electric part of the electromagnetic field while is related to the magnetic part. There are several issues that must be pointed out; first of all, the electromagnetic field tensor has been generated by both electric and magnetic charges separately. Therefore, it is possible to cancel out the electric part by setting and have magnetic black hole solutions. Second, we have restricted the magnetic field to one direction which is a common practice for the magnetic charges. Finally, except for -dimensional case which has constant magnetic field, for large values of the , magnetic field will vanish similar to the electric field which is physically expected. One of the important properties of the proposed electromagnetic tensor is that magnetic field is a stand alone property. In other words, even in the absence of electric charge, one can construct magnetic black holes without resorting to complex field equations.

Using the variational principle, it is a matter of calculation to reach the following field equation

(6)

which by considering metric (2) and non-zero components of the electromagnetic field tensor (4), one finds

Solving these two equations with respect to metric function, one obtains as

(7)

where is geometrical mass related to total mass of the black hole. By setting , metric function yields

(8)

which was previously obtained in ref. dyonicmassless (). This shows that our proposal includes other types of dyonic black holes as well.

Existence of the black hole solutions depends on satisfaction of specific conditions simultaneously: i) existence of the singularity, and ii) presence of at least one horizon which covers the singularity and is known as event horizon.

The existence of singularity could be determined by divergencies of the curvature scalars. One of the well known curvature scalars is the Kretschmann scalar. It is a matter of calculation to find the Kretschmann for these solutions in the following form

(9)

where

The Kretschmann has the following limit

(10)

Eq. 10 confirms the existence of a curvature singularity at . By series expansion of this curvature scalar for small values of , one can find the following relation

in which are dimension dependent coefficients. It is interesting to note that the singularity is affected by both electric and magnetic fields with same order of magnitude. In the absence of electric field, the dominant term on singular behavior is the magnetic charge. This highlights the contribution of the magnetic part. It is worthwhile to mention that power of the singularity is stronger in higher dimensions and the divergency is reached faster compared to lower dimensions.

Existence of the cosmological constant in the solutions, provides specific complexity which prevents us to extract the root(s) of the metric function, hence event horizon, analytically. In the absence of the cosmological constant (), the roots of metric function are obtained as

(11)

which shows that under certain conditions, two distinct roots for the metric function may exists. The first condition comes from positivity of the expression under square root function. In other words, in order to have real roots for the metric function, the square root function must be positive valued which results into the following condition

(12)

This condition has several points which must be highlighted; considering that , and are positive values, the mentioned condition is valid only for spherical case, . In other words, for the horizon flat () and hyperbolic horizon (), the square root function is always positive valued and mentioned condition is satisfied irrespective of choices for , and . If this condition is violated, no real valued root exists for the metric function which indicates that such solutions is a naked singularity. Therefore, it is safe to state that naked singularity only exists for the spherical cases while for horizon flat and hyperbolic horizon, obtained solutions definitely enjoy at least one root, hence event horizon in their structure. It is interesting to note that for Ricci flat solutions (), the inner horizon goes to , and therefore, the singularity is null.

The coupling between topological factor and the electric and magnetic charges is another important issue. Here, we see that the presence of magnetic charge affected the condition regarding the existence of real valued roots. On the other hand, we see that the presence of magnetic charge has its own effects on the position of root as well. In the absence of the electric charge, the magnetic charge for spherical case upholds the mentioned condition. Returning to obtained root for metric function (11), we see that it is possible to have two real positive valued roots provided that the following inequality holds for negative branch

(13)

Now, considering that for , the square root function is positive valued, one can state that for these two cases, the mentioned condition (13) is violated. Therefore, it is safe to conclude that for , only one root exists. On the contrary, for spherical horizon, it is possible to have three cases: i) violation of the condition (12) which results in naked singularity. ii) if , then positive and negative branches of the obtained roots coincide which results in extreme black hole solutions (existence of one root). iii) satisfaction of mentioned conditions ((12) and (13)) which results in existence of two roots for metric function.

Since it was not possible to obtain the root of metric function in the presence of cosmological constant, we have employed numerical method to plot diagrams (see Fig. 1). Evidently, the existence of root for metric function and its number is a function of the magnetic charge. By suitable choices of this quantity, it is possible to have naked singularity, extreme black holes (one root) and two roots with larger root being event horizon. is linearly related to magnetic charge. Its value determines the power of magnetic field. Considering this fact, Fig. 1 confirms a very important fact: the super magnetized solutions suffers the absence of horizon. In other words, the super magnetized solutions are naked singularity.

Our final study in this section is regarding the asymptotic behavior of the solutions. To investigate this, it is sufficient to study the behavior of curvature scalar for large . It is a matter of calculation to show that the obtained Kretschmann scalar will have following behavior

(14)

in which and are dimension dependent coefficients. Evidently, the dominant term in this limit is term which indicates that the asymptotic have AdS/dS behavior depending on the sign of cosmological constant. In the absence of the cosmological constant, the dominant term for asymptotic behavior will be geometrical term in following form

(15)

which shows that the effects of matter field (electric and magnetic fields) on asymptotic behavior is of secondary importance in the absence of the cosmological constant ( and are dimension dependent coefficients).

In conclusion, we established that these solutions enjoy the presence of a singularity which is located at the origin and depending on choices of different parameters, this singularity could be covered by one or two horizons. The role of the magnetic charge on singular behavior and properties of the horizon(s) was pointed out. In addition, the asymptotic behavior was investigated and it was shown that in full form, it is AdS/dS depending on negativity/positivity of the cosmological constant. In the next section, we derive conserved quantities and study the effects of magnetic term and higher dimensions on the thermodynamical behavior of the system.

Figure 1: versus for , , , , ; (continuous line), (dashed line) and (dashed-dotted line).

Iii Thermodynamic properties

In the previous section, we established the fact that our solutions could be interpreted as black holes. Having black hole solutions, it is possible to calculate thermodynamic quantities and study their thermodynamical behavior. Here, our main focus is on obtaining thermodynamical properties and employing the first law of thermodynamics to study various properties of these black holes.

The entropy of Einsteinian black holes could be extracted by using the area law, which leads to

(16)

in which is the outer horizon (the largest positive real root of metric function). The total electric charge could be obtained through the use of the Gauss law which leads to

(17)

The same method could be employed to calculate total magnetic charge which is

(18)

It is interesting to note that although the electric and magnetic parts of are completely different, their conserved charges are of the same nature which are arisen from Gauss’s law. Besides, the total mass of these black holes could be calculated by using the ADM (Arnowitt-Deser-Misner) method which leads to

(19)

By evaluating the metric function on the horizon, one can also find the geometrical mass. Using obtained entropy (16), electric (17) and magnetic charges (18), one can obtain the following Smarr like formula for these black holes

(20)

Using the obtained mass together with the first law of black hole thermodynamics

(21)

one can extract the electric and magnetic potentials respectively in the following forms

(23)

Recently, there has been a renewed proposal for the cosmological constant; which is taken not as a fixed parameter, instead, it is regarded as a thermodynamic variable known as dynamical pressure Kubiznak (). The relation between these two quantities is given by

(24)

which by replacing it in Smarr like formula, one can obtain

(25)

This consideration modifies the role of the mass from internal energy to enthalpy and the first law of black hole thermodynamics (21) will be modified into

(26)

Using this relation, it is a matter of calculation to extract the corresponding volume

(27)

which is geometrically expected. The temperature of black hole is generally obtained through the concept of surface gravity which is given by

(28)

where is a Killing vector. Considering the fact that our solutions are static, the Killing vector will be , and therefore, temperature is calculated as

(29)

By replacing the cosmological constant with its correspondence pressure in temperature (29), it is possible to extract an equation of state. Using the equation of state and checking the existence of its inflection point

(30)

one is able to extract the critical points. Furthermore, it is possible to obtain the free energy by using the following relation

(31)

The last thermodynamic quantity of interest is the heat capacity which could be used to determine thermal stability and possible phase transition of the black holes. This quantity is given by

(32)

In next sections, we will investigate thermodynamical behavior of the solutions for the following ensembles; i) general case, ii) constant electric field, iii) constant magnetic field, and iv) constant electric and magnetic fields.

Unfortunately, it was not possible to extract all the properties for mentioned cases in arbitrary dimension, analytically. Therefore, we consider -dimensional black holes (as a prototype higher dimensional solutions) as a case study to understand the effects of higher dimensions on properties of solutions.

Iv General case

Here, we consider unspecified electric and magnetic fields and study the general behavior. Using obtained mass (25), it is a matter of calculation to show that mass/enthalpy for this case is given by

(33)

As it was pointed out, it is not possible to obtain roots of the mass/enthalpy for arbitrary , analytically. Therefore, we consider -dimensional solution as a case study. So, in 5-dimensions, the root is given by

(34)

in which .

In addition, the high energy limit and asymptotical behavior of the mass are given by

(36)

The high energy limit of the mass (enthalpy) for this case is governed by magnetic and electric charge terms. This indicates that in the absence of the electric field, the dominant term in the high energy limit of the mass is magnetic charge of the solutions. The effects of electric and magnetic charges are of the same order. On the other hand, the asymptotic behavior is governed by the pressure term which is essentially the cosmological constant term. In general, one can state that for small black holes, the effects of matter field, hence the magnetic and electric charges become dominant over other quantities contributing to the mass, whereas for large black holes, the significant effect comes from the pressure term. This difference in the limiting behavior could be employed to determine the size of black hole, although there are other matters that should be considered before making a final statement. But, the important subject is that here, the contribution of the magnetic charge becomes significant as we study the high energy limit. Considering that for both limits, the dominant terms are positive valued with power of the horizon radius presented for different terms, one expects a minimum for the mass. Plotting mass versus confirms this statement (see left panel of Fig. 8). The minimum takes place where the topological term, , becomes dominant which is the case for medium black holes. This indicates that the smallest mass (enthalpy) available for these black holes belongs to medium black holes.

Using Eq. (29) or , it is possible to extract the temperature as

(37)

The root of this quantity could not be calculated for -dimensional case, whereas for the -dimensional case it is obtained as

(38)

in which . The high energy limit and asymptotic behavior of the temperature are given by

(40)

Here too, similar to enthalpy, the dominant term in high energy limit includes electric and magnetic charges but contrary to enthalpy case, its sign is negative. On the other hand, the asymptotic behavior of the temperature is governed by the pressure term whereas for medium black holes, the topological factor determines the behavior of temperature. Since the high energy limit has negative sign and asymptotic behavior is positive, it is expected that there is at least one real valued positive root for temperature which could be observed in plotted diagrams (see middle panel of Fig. 2). Studying the diagrams for temperature confirms that depending on the choices of magnetic charge, temperature: i) could be an increasing function of horizon radius with one root. ii) could have one extremum and one root. iii) could have one root with one minimum and one maximum which are located after the root. Existence of the extremum for temperature confirms the existence of divergency for the heat capacity. In other words, extrema are where the heat capacity acquires divergencies, and therefore, they are phase transition points. Here, we see that modification in magnetic charge results in existence/absence of extremum for the temperature. In addition, since before root, the temperature is negative and solutions are not physical, one can conclude that modification in magnetic charge changes the valid range for existence of physical black holes. The root of temperature is an increasing function of the magnetic charge.

It is a matter of calculation to obtain the equation of state for this case as

(41)

Using the concept of inflection point (30), one can obtain the following equation for calculating the critical horizon radius

(42)

in which the critical horizon radius, temperature and pressure are, respectively, obtained as

(43)
(44)
(45)

Evidently, the critical horizon radius is an increasing function of the magnetic charge while critical temperature and pressure are decreasing function of it. This indicates that for magnetized black holes, phase transition takes place in larger horizon radius for smaller pressure and temperature. Therefore, one can conclude that for super magnetized black holes (large magnetic charge), van der Waals like phase transition takes place in very small values of temperature and pressure but for large volume. This enables one to understand the effects of magnetization on van der Waals like behavior of the black holes. In order to confirm the existence of van der Waals like behavior, we have plotted the following diagrams for the pressure (see Fig. 3). Evidently, by variation of the magnetic charge, the pressure can acquire i) one extremum which indicates existence of critical behavior taking place at one point. ii) two extrema which show the existence of phase transition over range (rather than a single point) iii) without any extremum which is interpreted as the absence of critical behavior.

Using obtained thermodynamical quantities, it is possible to calculate the free energy as

(46)

Considering that term is horizon radius independent, for high energy limit, this term becomes dominant, whereas, for asymptotic behavior, the dominant term is pressure term. Finally, the heat capacity of this case is obtained as

(47)

Divergencies of the heat capacity could not be extracted analytically for arbitrary dimension, while for the -dimensional case, divergency occurs at

(48)

where .

In addition, based on series expansion, we find that high energy limit and asymptotic behavior of the heat capacity are given by

(50)

The effect of the magnetic charge on the obtained divergency for the heat capacity is evident. This shows that phase transition points, root and stability conditions of the black holes are modified due to the magnetic charge. If certain conditions are satisfied, the heat capacity could acquire two divergencies (see right panel of Fig. 2). Between the divergencies, solutions suffer from thermal instability, since the heat capacity is negative valued. But comparing this case with plotted diagrams for the pressure, one can see that extrema of pressure coincide with divergencies of the heat capacity. Therefore, the region between divergencies is where physical black holes are absent. As a result, one can draw the following conclusions regarding thermal stability of the solutions: i) in the absence of divergencies, stable black holes exist after root of the heat capacity which is the same as root of the temperature. ii) in the case of one divergency, there is a phase transition between medium stable black holes and larger stable ones taking place at the divergency. Before root of the heat capacity (temperature), solutions are unstable and non-physical (due to negative temperature). iii) in case of two divergencies for the heat capacity, the phase transition takes place over a region between small stable black holes and large stable ones. Once more, we emphasize that between two divergencies, due to thermodynamical concepts, no physical black holes exist.

Interestingly, the high energy limit of heat capacity and asymptotic behavior of it depends only on horizon radius with some factors. In other words, the high energy limit and asymptotic behavior of the heat capacity are depending neither on electric and magnetic charges nor on pressure. It seems that for the heat capacity, these two limits are only affected by the gravitational part of the action, since there is no trace of matter field and cosmological constant generalization present in them. It is worthwhile to mention that the second dominant term in high energy limit of the heat capacity includes electric and magnetic charges which highlights the contribution of magnetic charge.

Figure 2: (left panel), (middle panel) and (right panel) versus for , , and ; (continuous line), (dashed line) and (dashed-dotted line).

Figure 3: versus for , , and ; (continuous line), (dashed line) and (dashed-dotted line).

V Constant electric field

In this section, we work in an ensemble where the temporal component of the electromagnetic tensor, , is constant everywhere. Therefore, we can replace the electric charge, by its value at the horizon with following relation

(51)

So, the mass/enthalpy of this case is given by

(52)

in which, its roots could not be obtained analytically, except for the -dimensional case which are

(53)

where and . Also the high energy limit and asymptotic behavior are given by

(55)

The high energy limit of mass/enthalpy of this case is governed by the magnetic charge. This means that for small black holes the magnetization has a significant role on the behavior of enthalpy. That being said, one can see that due to the presence of magnetic charge (none constant magnetic field), the enthalpy for vanishing the horizon radius diverges. Comparing this case with previous one, it can be stated that the presence of magnetic charge with constant electric field leads to the high energy limit of enthalpy becomes modified at a significant level. The enthalpy is an increasing function of the magnetic charge. Taking a closer look at the high energy limit and asymptotic behavior, one can see that for medium range of horizon radius dominant terms of enthalpy are both electric potential and topological term. Interestingly, by choosing

(56)

for , it is possible to cancel out all the effects of electric field on enthalpy. It is clear that such a situation could only occur for black holes with hyperbolic horizon. This specific behavior is rooted in the assumption of electric field being constant. In order to have a better picture regarding thermodynamic behavior of the enthalpy for this case, we have plotted qa diagram (see left panel of Fig. 4). Evidently, for constant electric field and in the absence of magnetic charge, enthalpy is an increasing function of horizon radius. The situation is modified in the presence of magnetic charge. In this case, the enthalpy acquires a minimum. Depending on choices of different parameters (specially topological factor), the enthalpy can have no root, one root, or two roots. It is notable that existence of two roots and presence of a minimum originates from contributions of the magnetic charge.

The temperature for this case is obtained as

(57)

The root of the temperature could not be obtained analytically for general -dimensional case, but in - dimensions, it is given by

(58)

where , in which and are

(60)

For high energy limit and asymptotic behavior we have

(62)

The dominant term in the high energy limit is the magnetic term with negative sign. For medium black holes, topological and electric potential terms are governing the behavior of temperature. Finally, the asymptotic behavior is governed by pressure, hence cosmological term. It is worthwhile to mention that for , the effects of topological and electric potential cancel each other. This could only take place for spherical black holes (). Since the high energy limit of the temperature is negative valued and asymptotic behavior is positive, one can confirm that there exists at least one root for the temperature. To show this, we have plotted it in middle panel of Fig. 4. In the absence of magnetic charge, temperature has a minimum. By taking a non-zero value for the magnetic charge, one can find the following behavior for temperature: There exists a critical magnetic charge, at which the temperature acquires one extremum. For magnetic charges less than this critical magnetic charge, , the temperature has two exterma: one minimum and one maximum. On the other hand, for , no extremum is available for the temperature. Remembering that extrema are where the heat capacity diverges, one can state that the presence of magnetic charge enables the possibility of existence of van der Waals like behavior for these black holes, provided by mentioned cases. In the absence of magnetic charge, temperature indeed has a minimum, but the type of phase transition for these two cases are different: while one is van der Waals like phase transition the other one (absence of magnetic charge) does not enjoy this type of phase transition. This highlights the importance of the contribution of the magnetic charge. It is worthwhile to mention that there exists a root for temperature which is an increasing function of the magnetic charge.

The equation of state for the pressure is obtained in the following form

(63)

Using the properties of the inflection point, one can derive the following relation governing the critical horizon radius

(64)

which leads to

(65)

It is a matter of calculation to obtain the critical temperature and pressure as

(66)
(67)

in which

Evidently, here, the critical horizon radius is a decreasing function of the electric potential and magnetic charge while it is an increasing function of the topological parameter (if taken as a continuous variable). In previous case, the critical horizon radius was a decreasing function of the electric charge but the presence of the electric charge was in the denominator of the critical horizon radius. But here, the electric potential is in the numerator of the obtained critical horizon radius with negative sign. The positive real valued critical horizon radius only exits for the spherical case. On the other hand, critical temperature and pressure are decreasing functions of the magnetic charge and electric potential. Here, the obtained critical temperature and pressure show a significant modification compared to the previous case. This indicates that consideration of the electric field being constant resulted in different classes of black holes which have different thermodynamical properties, therefore critical structure. We should point it out that in the absence of electric field, the black holes will have critical behavior which indicates that magnetic black holes have also van der Waals like behavior in their structure considering the set up proposed in this paper. In order to have a better picture regarding the effects of magnetic charge on this case, we have plotted Fig. 5. Evidently, in the absence of magnetic charge, no van der Waals like behavior is present for the pressure vs horizon radius. In this case, pressure has a root and maximum which appears after root. Therefore, for this case, there is a region of negative pressure before root. After such root, there exists a maximum which marks the existence of a phase transition which is not van der Waals like. Later, in studying the heat capacity diagrams, we will see what type of phase transition these diagrams represent. In the presence of magnetic charge, the behavior of pressure diagrams is modified. There exists a critical magnetic charge, in which for , pressure has one minimum and one maximum. The van der Waals like phase transition takes place over this region. For , the pressure acquires an extremum which is a critical point. In other words, this is the case in which phase transition at a single point. For , pressure will be a decreasing function of the horizon radius without any extremum, hence critical point.

It is possible to obtain the free energy of this case in following form

(68)

The final subject of interest in this section is the heat capacity. It is a matter of calculation to obtain the heat capacity as

(69)

in which, for the -dimensional case, one can obtain the corresponding divergencies as

(70)

where , in which and are

(72)

It is worthwhile to mention that the high energy limit and the asymptotic behavior of the heat capacity for this case are, respectively, given by