Thermodynamic topology of 4D Dyonic AdS black holes in different ensembles

We study the thermodynamic topology of four dimensional dyonic Anti-de-Sitter(AdS) black hole in three different ensembles: canonical, mixed and grand canonical ensemble. While canonical ensemble refers to the ensemble with fixed electric and magnetic charges, mixed ensemble is an ensemble where we fix magnetic charge and electric potential. In the grand canonical ensemble, potentials corresponding to both electric and magnetic charges are kept fixed. In each of these ensembles, we first compute the topological charges associated with critical points. We find that while in both canonical and mixed ensembles, there exists one conventional critical point with topological charge $-1$, in the grand canonical ensemble, we find no critical point. Then, we consider the dyonic AdS black hole as topological defects in thermodynamic space and study its local and global topology by computing the winding numbers at the defects. We observe that while the topologies of the black hole in canonical and mixed ensembles are identical with total topological charge equaling $1$, in the grand canonical ensemble, depending on the values of potentials, the total topological charge is either equal to $0$ or $1$. In canonical and mixed ensembles, either one generation and one annihilation points or no generation/annihilation points are found. In the grand canonical ensemble, depending on the values of potentials, we find either one generation point or no generation/annihilation point. Thus, we infer that the topological class of $4$D dyonic AdS black hole is ensemble dependent.

where, G is Newton's gravitational constant.Accordingly, a thermodynamic volume V is defined conjugate to thermodynamic pressure P and the first law of black hole thermodynamics takes the revised form: where, T is the temperature, S is the entropy and Y i dx i is the i-th chemical potential term.
A recent addition to the study of criticality in black holes is the the idea of thermodynamic topology.Initiated in [33], in this novel approach, Duan's topological current φ-mapping theory [35] is invoked in the thermodynamic space of a black hole to study its criticality.Consequently, the critical points in the thermodynamic space are characterized with distinct topological charges and based on those charges, are classified into conventional and novel critical points.The key steps involved are summarized below : The temperature, T , of a black hole is expressed as a function of pressure P , entropy S and other thermodynamic parameters.T = T (S, P, x i ), where x i denotes other thermodynamic parameters.Then, pressure is, then, eliminated by using (∂ S T ) P,x i = 0, and a new potential Φ, known as Duan's potential is constructed.
The term '1/ sin θ' is introduced for the sake of convenience.A two dimensional vector φ = (φ θ , φ θ ) is defined following the framework of Duan's φ-mapping theory [34,35] as The presence of θ in the vector field φ ensures that the zero point of the vector field φ is always at θ = π/2.The critical points can be calculated using this criteria.Also the topological current, j µ , is conserved i.e., φ a (x i ) = 0.This construction ensures the existence of topological charge which for a given parameter region Σ is equal to Here, w i , j 0 and β i are the winding number of i-th zero points of φ, the density of the topological current and the Hopf index respectively.Critical points with topological charges −1 and +1 are referred as conventional critical point and novel critical point respectively.The total topological charge of a black hole is computed as the sum of individual charges associated with each critical point.Following the work in [33], analysis of thermodynamic topology has been extended to a number of black holes [36][37][38][39][40][41][42].
An alternative way to apply topology in black hole thermodynamics has also been proposed in [43].In this method, black hole solutions are regarded as defects in thermodynamic parameter spaces.These defects are then studied in terms of their winding numbers.The sign of the winding number of a defect has been linked to the thermodynamic stability of the corresponding black hole solution.The sum of the winding numbers is, now, termed as topological number based on which different black hole solutions are categorized.The analysis begins with the introduction of a generalized free energy F defined as follows: where E and S are the energy, entropy.τ is a quantity which has the dimension of time.A vector field φ is defined from F in the following way : The zero point of the vector φ is at Θ = π/2.The unit vector is defined as Corresponding to a given value of τ , the zero points of n 1 are computed.The winding numbers of each of these zero points are calculated.The topological number of a black hole is obtained by summing over the individual winding numbers of all the black hole branches.Following [43], study of black holes as topological defects has been extended to a number of black holes in [44][45][46][47][48][49][50][51][52][53][54].
Motivated by all the above mentioned works, in this paper, we extend the study of thermodynamic topology to 4d dyonic AdS black hole in different ensembles.Our primary focus is to understand the ensemble dependent nature of thermodynamic topology.For this, we carry out our analysis in three different ensembles: 1. canonical ensemble where both the electric and the magnetic charges are kept fixed, 2. mixed ensemble where electric potential and magnetic charge are kept fixed and 3. grand canonical ensemble where both electric and magnetic potentials are kept fixed.To begin with, in each of these ensembles, we figure out the critical points and compute their topological charges.Based on the sign of those charges, we classify them as conventional and novel critical points.Then we consider the black hole as a topological defect in each of these ensembles and find out its topological number, generation and annihilation points in the thermodynamic space.This paper is organized as follows.In section II, we begin with 4d dyonic AdS black hole in canonical ensemble and study its thermodynamic topology.This is followed by similar studies in mixed ensemble in section III and grand canonical ensemble in section IV.We conclude with our findings in section V.

II. 4D DYONIC ADS BLACK HOLE IN CANONICAL ENSEMBLE
The four dimensional, asymptotically anti-de Sitter, dyonic black holes solution has its origin in maximal gauged supergravity .Such a black hole carries both electric and magnetic charges.The dyonic black hole solution can be obtained by the reduction of five dimensional Kaluza-Klein theory and it has some very interesting properties [55][56][57][58][59][60].A simpler solution of dyonic AdS black hole can be obtained by varying the Reissner-Nordström action with a cosmological constant [61].The four dimensional, asymptotically anti de-Sitter, dyonic black hole metric is given by : where, Here, q e , q m , M and l are electric charge, magnetic charge, mass of the black hole and the AdS radius respectively.Thermodynamic pressure, P , is related to the AdS radius as The mass, M and the entropy, S of the black hole are given by (in the following expressions, r + denotes the horizon radius)

II.1. Topology of 4d dyonic AdS black hole thermodynamics in canonical ensemble
To study the topology of dyonic AdS black hole thermodynamics, we write the temperature as a function of pressure, horizon radius, electric and magnetic charges.
Use of the condition ∂r + T ∂r + S qe,qm,P = 0 leads us to an expression for pressure, P .
Plugging P in (15), we get rid of the pressure term and the temperature, T takes the following form : A thermodynamic function Φ is defined as, The vector components of the vector field φ = (φ r + , φ θ ) are and The normalized vector components are and The normalized vector n = φ r + ||φ|| , φ θ ||φ|| has been plotted in Figure 1.This figure shows the vector plot of n in a r + vs θ plane for dyonic AdS black hole.For this plot, we have fixed q e = q m = 1.The black dot represents the critical point (CP 1 ).To calculate the critical point we set θ = π/2 in (21) and equate this to zero.The critical point is located at (r + , θ) = ( 6(q 2 e + q 2 m ), π/2) or at (r For the calculation of the topological charge of the critical point, a contour C parametrized by ϑ ∈ (0, 2π) is defined [33] as follows: We construct two contours C 1 and C 2 where the first contour encloses the critical point CP 1 and the second contour is outside the critical point.For these contours we choose (a, b, r 0 ) = (0.6, 0.2, 2 √ 3) and (0.6, 0.4, 5).The deflection of the vector field n along the contour C is, The topological charge is, then, equal to, Q = 1 2π Ω(2π).For the critical point CP 1 enclosed by the contour C 1 , the topological charge has been found to be, Q CP 1 = −1.This is a conventional critical point.Since the contour C 2 does not enclose any critical point, it corresponds to zero topological charge.Thus, the total topological charge is, Q = −1.The behaviour of Ω is shown in Figure 2 4d dyonic AdS black hole in canonical ensemble has the following equation of state : The corresponding critical points are given by, It can be clearly seen that the critical radius in (26) exactly matches the critical point obtained from thermodynamic topology, (r + , θ) = ( 6(q 2 e + q 2 m ), π/2).As mentioned above this is a conventional critical point with topological charge −1.To see the nature of the critical point we plot the phase structure (isobaric curves) around it in Figure 3.The location of the critical point in the isobaric curve is shown as a black dot.The red curve is the isobaric curve for P = P c .The green curves above and below the red curve are respectively for P > P c and P < P c .The blue dashed curvet describes the extremal points and is plotted using (17).From Figure 3, it is observed that for P < P c , the small and large black hole phases are separated by the unstable region (the negative slope region of the isobaric curves or the region enclosed by the two extremal points corresponding to each isobaric curve).Different phases of the dyonic AdS black hole in canonical ensemble disappear at the critical point.Hence, the critical point CP 1 can be thought of as a phase annihilation point.

II.2. Dyonic AdS black hole solution as topological thermodynamic defects in canonical ensemble
Now, we proceed to study the dyonic AdS black hole solution in canonical ensemble as topological thermodynamic defects.Using the mass and entropy of the black hole from ( 13) and ( 14) in (7), the generalized free energy is found to be, The vector field components of the vector given by ( 8) are and The corresponding unit vectors are and These unit vectors are plotted and used to locate the zero points by setting Θ = π/2 in n 1 (see (30)) and equating it to zero.For example, setting q e /r 0 = 1, q m /r 0 = 1, P r 2 0 = 0.0002 and τ /r 0 = 30, we can find one zero point (ZP 1 ) located at (r + /r 0 , Θ) = (80.8742,π/2).Here, r 0 is an arbitrary length scale which is determined by the size of a cavity that surrounds the black hole.The value of pressure is taken below the critical pressure P c .The representation of the unit vectors along with the zero point is shown in Figure 4a.The winding number or the topological charge corresponding to this zero point is computed following the prescription stated in the previous section and found to be w = +1.Similarly, keeping the same charge and pressure configuration, corresponding to τ /r 0 = 50, we find three zero points ZP 2 , ZP 3 and ZP 4 with winding numbers +1, −1 and +1 respectively.These are shown in Figure 4b.For τ /r 0 = 100, a solitary zero point ZP 5 with winding number +1 is observed (Figure 4c).
An analytic expression for τ corresponding to zero points can be otained by setting φ r + = 0.
A plot of r + vs τ obtained above is shown in Figure 5.The points on this curve are the zero points of φ r + .Here, we have fixed q e /r 0 = 1, q m /r 0 = 1 and P r 2 0 = 0.0002 (below the critical pressure P c ).
In Figure 5, three different black hole branches are clearly visible.The branch τ < τ b corresponds to the large black hole region.The winding number for any zero point on this branch is found to be w = +1.Similarly, winding number w = +1 is also observed for any zero point on the branch τ > τ a which corresponds to the small black hole region.The branch τ a < τ < τ b represents the intermediate black hole region and winding number for any zero point on this branch is equal to w = −1.The topological number is, hence, W = +1 − 1 + 1 = +1.We explicitly computed the specific heats at the three branches and found that the branches with winding number +1 have positive specific heat (thermodynamically stable) and the branch with winding number −1 has negative specific heat(thermodynamically unstable).
For a value of pressure, P r 2 0 = 0.01, which is above the critical pressure P c , and q e /r 0 = q m /r 0 = 1, the plot of r + vs τ is shown in Figure 6b.In this case, the plot exhibits only one branch corresponding to stable black hole region with positive specific heat.The winding numbers of zero points on this branch is computed to be w = +1.The topological number is, hence, W = +1.Notably, we don't find any generation/annihilation point in this case.The zero point for τ /r 0 = 30, q m /r 0 = q e /r 0 = 1, P r 2 0 = 0.01 is shown in Figure 6a.What we have seen is that the topological number of 4d dyonic AdS black hole in canonical ensemble is not altered by a variation in pressure.
We repeated our analysis by changing the values of q e and q m .We found that the topological number was always equal to (W = +1) for all the charge configurations.As an example, the r + vs τ curve) for q e /r 0 = 0.1, q m /r 0 = 0.1 and P r 2 0 = 0.04 is shown in Figure 7.The topological number of 4d dyonic AdS black hole in canonical ensemble is not influenced by a variation in charge configuration. ) and zero point of φ r + for pressure P r 2 0 = 0.01 (above the critical pressure P c ).

III. DYONIC ADS BLACK HOLE IN MIXED ENSEMBLE
In mixed ensemble, the electric potential φ e and the magnetic charge q m are kept constant.The electric potential φ e is defined as The mass and temperature, are, then modified as and for q e /r 0 = 0.1, q m /r 0 = 0.1 and P r 2 0 = 0.04.

III.1. Topology of dyonic AdS black hole in mixed ensemble
In this section, we study the thermodynamic topology of 4d dyonic AdS black hole in mixed ensemble.We begin by eliminating pressure from (35) which is then simplified to and the thermodynamic function Φ becomes The vector components of φ = (φ r + , φ θ ) are given by, and The vector φ is normalized and the components are and Now, we plot the normalized vector n = φ r + ||φ|| , φ θ ||φ|| in r + vs θ plane by fixing q m = 1 and φ e = 1/2 (see Figure 8).Here, we find a single critical point CP 2 at (r + , θ) = ( √ 6q m / 1 − φ 2 e , π/2) represented by the black dot.
The critical values are In this ensemble also, we see that the critical radius given in ( 43) is exactly the same as the conventional critical point which is (r + , θ) = ( √ 6q m / 1 − φ 2 e , π/2).We plot the T vs r + isobaric curve around the critical point in Figure 10.This figure shows that the the critical point (black dot) is on the isobaric curve for P = P c (red curve).The blue curve gives the extremal points of the isobaric curves and is plotted using (36).Similar to the canonical ensemble case, the number of phases clearly decreases with the increase of pressure P and disappears at the critical point CP 3 .This implies that the critical point CP 3 is a phase annihilation point.

III.2. Dyonic AdS black hole solution as topological thermodynamic defects in mixed ensemble
We now study the dyonic AdS black hole in mixed ensemble as a topological defect.As usual, we start with the generalized free energy potential In this ensemble Hence, (44) gives From ( 8), the vector components can easily be worked out resulting and We plot the normalized vector and figure out the zero points.The corresponding winding numbers are also calculated.For different values of τ /r 0 the zero points are shown in Figure 11.Here, we have set φ e = 1/2, q m /r 0 = 1, P r 2 0 = 0.001 (pressure value below the critical pressure P c ).While for τ /r 0 = 35, we find one zero point ZP 6 with winding number +1, for τ /r 0 = 45, we locate three zero points ZP 7 , ZP 8 and ZP 9 with winding numbers +1, −1 and +1 respectively.For τ /r 0 = 55 , we again encounter one zero point ZP 1 0 with winding number +1.The zero points of the component φ r + = 0 are given by the equation The zero points of φ r + in τ /r 0 vs r + /r 0 plane for dyonic AdS black hole in mixed ensemble below the critical pressure P c .
For φ e = 1/2, q m /r 0 = 1, P r 2 0 = 0.001 the resulting r + vs τ graph is plotted in Figure 12.Similar to the canonical ensemble case, below critical pressure, we have three branches of the τ curve in the regions for τ < τ b , τ a < τ < τ b and τ > τ a .The first and third branch are the large black hole and small black hole region and the zero points on these two branches have w = +1 and positive specific heat.The other branch represents intermediate black hole region and the zero points in this region have w = −1 and negative specific heat.Thius, the topological number is W = +1 − 1 + 1 = +1.The generation and annihilation points for φ e = 1/2, q m /r 0 = 1, P r 2 0 = 0.001 is found at τ /r 0 = τ a /r 0 = 41.5688 and τ /r 0 = τ b /r 0 = 46.9484respectively.
Above critical pressure P c , the same plot is shown in Figure 13b.Here, we chose P r 2 0 = 0.02, q m = 1 and φ e = 1/2.In this case, the unstable region disappears and the τ (r + ) curve corresponds to stable black hole region.The winding number for each point on the curve is w = +1 and the topological number is hence W = +1.The unit vector and the zero point for τ /r 0 = 35, P r 2 0 = 0.02, q m /r 0 = 1 and φ e = 1/2 is shown in Figure 13a.
We repeated the exercise altering the values of φ e and q m /r 0 and observed that the topological number for all the combinations was identical and equal to W = +1.The zero points of φ r + is shown in Figure 14 for φ e = 0.1, q m /r 0 = 0.1 and P r 2 0 = 0.04.

IV. DYONIC ADS BLACK HOLE IN GRAND CANONICAL ENSEMBLE
In the grand canonical ensemble, both the electric potential φ e and the magnetic potential φ m are kept fixed.
The relevant thermodynamic parameters of dyonic AdS black hole in grand canonical ensemble are given by, The zero points of φ r + in τ /r 0 vs r + /r 0 plane for dyonic AdS black hole in mixed ensemble for φ e = 0.1, q m /r 0 = 0.1 and P r 2 0 = 0.04. and IV.1.Topology of dyonic AdS black hole in grand canonical ensemble Now, we proceed to study the topology of dyonic AdS black hole thermodynamics in grand canonical ensemble.First, we eliminate pressure from (53) using ∂r + T ∂r + S φe,φm,P = 0.
The thermodynamics function Φ = T / sin θ is, therefore, The components of vector field φ = (φ r + , φ θ ) are and The normalized vector components are and On following the procedure discussed earlier, we find that this system does not have any critical points (see Figure 15a).Also, since the contour does not enclose any critical point therefore, the Ω(ϑ) function reaches 0 at ϑ = 2π.(see Figure 15b) There is no critical point.IV.2.Dyonic AdS black hole solution as topological thermodynamic defects in grand canonical ensemble Now, we identify dyonic AdS black hole as a topological defect in the thermodynamic we can use the following general free energy potential: Following ( 8), we calculate the vector components Accordingly, we find the unit vectors and For τ /r 0 = 110, φ e = 0.1, φ m = 0.1 and P r 2 0 = 0.0001, we find two zero points ZP 11 and ZP 12 with winding numbers −1 and +1 respectively.These are shown in Figure 16.Hence, the topological number of dyonic AdS black hole in grand canonical ensemble is 0. The expression for τ representing zero points is given by For φ e = 0.1, φ m = 0.1 and P r 2 0 = 0.0001, r + /r 0 vs τ /r 0 plot is shown in Figure 17.In this case, unlike what we saw in canonical and mixed ensembles, the plot exhibits two black hole branches in the regions τ < τ a and τ > τ a .The former branch represents unstable black hole region whereas the later branch represents stable black hole region.Any zero point in the unstable and stable region has winding numbers w = −1 and w = +1 respectively.The topological number is hence W = 0   18).The topological number, therefore, is +1.In this case, we do not see any generation or annihilation point.Thus, in the grand canonical ensemble, for4d dyonic AdS black hole, we have two topological numbers.When φ 2 e + φ 2 m < 1, the topological number is 0 in contrast to what we had in canonical and mixed ensembles.When φ 2 e + φ 2 m > 1, the topological number is 1, same as the ones in canonical and mixed ensembles.

V. CONCLUSION
In this work, we studied the thermodynamic topology of 4d dyonic AdS black hole in canonical, mixed and grand canonical ensembles.Canonical, mixed and grand canonical ensembles were formed by fixing electric and magnetic charges, magnetic charge and electric potential and potentials corresponding to both electric and magnetic charges respectively.In all the three ensembles, we evaluated the topological charges of the critical points in their thermodynamic spaces.We observed the presence of a solitary conventional critical point with topological charge −1 in both canonical and mixed ensembles.Contrastingly, in the grand canonical ensemble, no critical point was found.Next, we recognized the dyonic AdS black hole as topological defects in thermodynamic space and analyzed its local and global topology by calculating the winding numbers at the defects.We found that, in both canonical and mixed ensembles, the total topological charge was equal to 1, which was not altered by changes in thermodynamic parameters.In both these ensembles , either one generation and one annihilation points (below critical pressure) or no generation/annihilation points (above critical pressure) were seen.In the grand canonical ensemble, depending on the values of potentials, the total topological charge was found to be either equal to 0 (when φ 2 e + φ 2 m < 1) or 1 (when φ 2 e + φ 2 m > 1) .In this ensemble, we found either one generation point (when φ 2 e + φ 2 m < 1) or no generation/annihilation point ( when φ 2 e + φ 2 m > 1).
From our analysis, we conclude that 4d dyonic AdS black hole in canonical and mixed ensembles can be placed in the same thermodynamic topological class.However, the thermodynamic topology of 4d dyonic AdS black hole in grand canonical ensemble is different from those in the other two ensembles.Or in other words, the topological class of 4d dyonic AdS black hole is ensemble dependent.It will be interesting to extend the study of ensemble dependent thermodynamic topology to other black holes with rich phase structures.We plan to do so in our future works.

FIG. 1 :
FIG.1: Plot of the normalized vector field n in r + vs θ plane for dyonic AdS black hole in canonical ensemble.The black dot represents the critical point.

FIG. 3 :
FIG.3: Isobaric curves (red and green) of dyonic AdS black hole in canonical ensemble.Black dot represents the critical point.

FIG. 5 :FIG. 6 :
FIG.5:The zero points of φ r + in τ /r 0 vs r + /r 0 plane for dyonic AdS black hole in canonical ensemble for pressure less than the critical pressure P c .

FIG. 8 : 2 √ 2 ) 1 .FIG. 9 :
FIG.8: Plot of the normalized vector field n in r + vs θ plane for dyonic AdS black hole in mixed ensemble.The black dot represents the critical point.

FIG. 10 :
FIG.10: Isobaric curves (red and green) of dyonic AdS black hole in mixed ensemble.Red curve is the isobaric curve for P = P c .The isobaric curves above and below the red curve is for P > P c and P < P c respectively.Black dot represents the critical point.

55 FIG. 11 :
FIG.11: Unit vector n = (n 1 , n 2 ) shown in Θ vs r + /r 0 plane for P r 2 0 = 0.001 (pressure point below the critical pressure P c ).The black dots represent the zero points.
FIG.12:The zero points of φ r + in τ /r 0 vs r + /r 0 plane for dyonic AdS black hole in mixed ensemble below the critical pressure P c .
FIG.13: Plot of unit vector n = (n 1 , n 2 ) and zero point of φ r + for pressure P r 2 0 = 0.02 (above the critical pressure P c ).
Plot of the normalized vector field n in r + vs θ plane for Dyonic AdS black hole in grand canonical ensemble.
FIG.15: Plot of vector n in r + vs θ plane and the plot of deflection angle Ω(ϑ) along C 5 .
FIG.17:The zero points of φ r + in τ /r 0 vs r + /r 0 plane for dyonicAdSblack hole in grand canonical ensemble which is different from what we had in canonical and mixed ensembles.The creation point is located at τ /r 0 = τ a /r 0 = 126.604.Interestingly, for φ 2 e + φ 2 m > 1, we get only one black hole branch with winding numbert +1 (see Figure