# Thermodynamic Topology of $D=4,5$ Horava Lifshitz Black Hole in Two
Ensembles

Bidyut Hazarika1<EMAIL_ADDRESS>Prabwal Phukon1,2
<EMAIL_ADDRESS>$1.$Department of Physics,Dibrugarh University,
Dibrugarh,Assam,786004.
$2.$Theoretical Physics Division, Centre for Atmospheric Studies, Dibrugarh
University, Dibrugarh,Assam,786004.

###### Contents

  1. I Abstract
  2. II Introduction:
  3. III  Horava Lifshitz black hole
  4. IV Thermodynamic Topology of $D=4,5$ HL black hole in Fixed $\epsilon$ ensemble
    1. IV.1 For Spherical Horizon $\left(k=1\right)$
      1. IV.1.1 Case I : For D=5
      2. IV.1.2 Case II : For D=4
    2. IV.2 For flat horizon (k=0)
      1. IV.2.1 Case I : For D=5
      2. IV.2.2 Case II: For D=4
    3. IV.3 For hyperbolic horizon ( k=-1)
      1. IV.3.1 Case I : For D=5
      2. IV.3.2 Case II : For D=4
  5. V Thermodynamic topology in fixed $\zeta$ ensemble
    1. V.1 For spherical horizon (k=1)
      1. V.1.1 Case I : For D=5
      2. V.1.2 Case II : For D=4
    2. V.2 For hyperbolic horizon(k=-1)
      1. V.2.1 Case I : For D=5
      2. V.2.2 Case II : For D=4
  6. VI Conclusion

##  I Abstract

We study the thermodynamic topology of four and five dimensional Horava
Lifshitz (HL) black holes in Horava gravity. These exotic black hole solutions
belong to a special class of of black holes whose thermodynamics exhibit a
line of (continuous) second order phase transitions known as $\lambda$ phase
transitions akin to those observed in the superfluidity of liquid ${}^{4}He$.
To analyze their thermodynamic topology, we treat the Horava Lifshitz (HL)
black holes as topological defects in their thermodynamic spaces and compute
the winding numbers at those defects. We work in two different ensembles:
fixed $\epsilon$ ensemble and fixed $\zeta$ ensemble, where $\epsilon$ is a
parameter of the HL black holes and $\zeta$ is its conjugate parameter. In the
fixed $\epsilon$ ensemble, three different horizon types are considered : the
spherical horizon for $k=+1$, the flat horizon for $k=0$, and the hyperbolic
horizon for $k=-1$. In the fixed $\zeta$ ensemble, two different horizon types
are considered : the spherical horizon for $k=+1$, and the hyperbolic horizon
for $k=-1$. Fixed $\zeta$ ensemble could not be defined in case of flat
horizon with $k=0$. In $D=5$, in the fixed $\epsilon$ ensemble, the total
topological charge for a black hole with spherical horizon is $+1$ for
$\epsilon<1$ and $0$ for $\epsilon\geq 1$. At the critical value
$\epsilon_{c}=0.9428$, a $\lambda$-line phase transition occurs. Similar
observations are made in the $D=4$, fixed $\epsilon$ ensemble, spherical
horizon case where the critical value is $\epsilon=0.9785$. The total
topological charge for $5D$ and $4D$ HL black holes with flat horizon remains
$1$, irrespective of the values of $\epsilon$ and pressure $P$. In case of 5D
HL black hole with hyperbolic horizon in the fixed $\epsilon$ ensemble, the
total topological charge is $0$ for $0\leq\epsilon<1$ and is $1$ for
$\epsilon\geq 1$. One facet of the study in fixed $\epsilon$ ensemble is that
the change in pressure has no significant impact on determining the overall
thermodynamic topology. It primarily depends on the parameter $\epsilon$ only.
In the fixed $\zeta$ ensemble, pressure $P$ becomes a crucial parameter in
determining the overall thermodynamic topology of HL black holes. In fixed
$\zeta$ ensemble ,$5D$ black hole with spherical horizon posseses a total
topological charge of $+1$ or $0$, depending on the values of $P$ and $\zeta$.
Contrastingly, for $D=4$ black holes with spherical horizon, the total
topological charge is always $1$ for all values of $\zeta$ and pressure. For
$5D$ and $4D$ black hole with hyperbolic horizon, the total topological charge
is either $+1$ or $0$, depending on the values of $\zeta$ and pressure. Hence,
we infer that the thermodynamic topology of Horava Lifshitz black holes in
$D=4,5$ are highly influenced by the type of horizon, choice of ensemble,
values of pressure and parameters $\epsilon$ or $\zeta$.

##  II Introduction:

Since its inception in 1970s, black hole thermodynamics [1, 2, 3, 4, 5, 6, 7,
8, 9] has continued to evolove and has remained an active area of interest in
physics. One of the directions in which a lot of focus has been attributed to
over the last decade or so is the framework of extended black hole
thermodynamics [10, 11, 12, 13] . In this approach, the cosmological constant
is treated as a counterpart to pressure in ordinary thermodynamics. For anti-
de-Sitter (AdS) black hole with cosmological constant $\Lambda$, the
thermodynamic pressure is given by,

$P=-\frac{\Lambda}{8\pi}$

The extended black hole thermodynamics of a number of black holes has been
extensively studied throwing new insights about the rich phase structure of a
wide range of black holes[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].
A recent addition to the studies on black hole thermodynamics is the idea of
thermodynamic topology. Introduced in [29] and further developed in [30, 65],
thermodynamic topology refers to the topology of the thermodynamic parameter
space of a black hole. It is inspired mainly by the works of Duan [27, 28] in
the context of a relativistic particle system.

In this approach, black hole solutions are treated as topological defects in
their thermodynamic space and the associated local and global topologies are
studied by computing the winding numbers at these defects.The black holes are
then classified according to their total winding number or topological charge.
It has been widely reported in literature that based on the topological
number, all black hole solutions can be classified into three topological
classes. Moreover, the thermal stability of a black hole has also been linked
with the sign of the winding number.The studies on the thermodynamic topology
have been extended to a number of interesting black hole systems [31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69]. The key
concepts associated with thermodynamic topology that will be used in this work
are stated below.

One of the fundamental ideas in topology is the concept of topological
defects. If a vector field $\chi(\vec{a})$ is such that at some point,
$\vec{a}=\vec{b}$ ,

$\left(\chi(\vec{a})\right)_{\vec{a}=\vec{b}}=0$ (1)

then, such a zero point of $\chi(\vec{a})$ can be connected to a topological
defect. In [30], it was proposed that all physical black hole solutions can be
interpreteted as zero points of the tensor field $\gamma_{\mu\nu}$, where,

$\gamma_{\mu\nu}=G_{\mu\nu}-\frac{8\pi G}{c^{4}}T_{\mu\nu}$

Accordingly, a topological charge can be defined for each black hole solution.
In black hole thermodynamics, this idea is further extended by constructing a
vector field using the generalized off-shell free energy.The off shell free
energy for an black hole with arbitrary mass is given by,

$\displaystyle\mathcal{F}=E-\frac{S}{\tau}$ (2)

Here, $E$ and $S$ denotes the energy and entropy of the black hole,
respectively.The time scale parameter $\tau$ can be thought of as the inverse
of the temperature of the cavity that encloses the black hole. Here rather
than the mass, the time parameter $\tau$ can change freely.The free energy
takes its on-shell form when $\tau$ satisfies

$\tau=\frac{1}{T}$ (3)

where, $T$ is the equilibrium temperature at the surface of the cavity. This
is the exact condition where a black hole solution identifies with a zero
point of the tensor field $\gamma_{\mu\nu}$. Using this generalized free
energy, a vector field is defined in the following way :

$\displaystyle\phi=\left(\phi^{r},\phi^{\Theta}\right)=\left(\frac{\partial\mathcal{F}}{\partial
r_{+}},-\cot\Theta~{}\csc\Theta\right)$ (4)

Through a straightforward computation, it can be demonstrated that the zero
point of $\phi$ is $\theta=\pi/2$ and $\tau=1/T$. This validates the fact that
a black hole solution corresponds to a zero point of the vector $\phi$.
Consequently, we can assign a topological charge to each black hole
solution.The topological quantity that is linked to the zero point of a field
is its winding number or topological charge.Topological charge can be
calculated by constructing a topological current $j^{\mu}$ which is conserved
i.e $\partial_{\mu}j^{\mu}=0$ [29].The expression for topological current is
given by

$j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\partial_{\nu}n^{a}\partial_{\rho}n^{b}\hskip
42.67912pt\mu,\nu,\rho=0,1,2$ (5)

where, $\partial_{\nu}=\frac{\partial}{\partial^{\nu}}$ and
$x^{\nu}=(\tau,r_{+},\Theta)$.The unit vectors $\left(n^{1},n^{2}\right)$ are
worked out as follows:

$n^{1}=\frac{\phi^{r}}{\sqrt{(\phi^{r})^{2}+(\phi^{\Theta})^{2}}}\hskip
14.22636pt\text{and}\hskip
14.22636ptn^{2}=\frac{\phi^{\theta}}{\sqrt{(\phi^{r})^{2}+(\phi^{\Theta})^{2}}}$

The expression for current $j^{\mu}$ can be rewritten using the Jacobi tensor,
$\epsilon^{ab}J^{\mu}\left(\frac{\phi}{x}\right)=\epsilon^{\mu\nu\rho}\partial_{\nu}\phi^{a}\partial_{\rho}\phi^{b}$
and the two dimensional Laplacian Green function,
$\Delta_{\phi^{a}}\ln{||\phi||}=2\pi\delta^{2}\left(\phi\right)$ as,

$j^{\mu}=\delta^{2}(\phi)J^{\mu}\left(\frac{\phi}{x}\right)$ (6)

it is evident from equation (6) that $j^{\mu}$ is nonzero only at
$\phi^{a}(x^{i})=0$ where $x^{i}$ is the $i^{th}$ solution of $\phi^{a}$. We
denote these solutions as $z_{i}$. The $0^{th}$ component of topological
current is the current density given by,

$j^{0}=\sum_{i=1}^{N}\beta_{i}n_{i}\delta^{2}(\vec{x}-\vec{z_{i}})$

where $\beta_{i}$ is the Hopf index. From the current density,the
corresponding total topological number or charge for a parameter region
$\Sigma$ is obtained by,

$\displaystyle
W=\int_{\Sigma}j^{0}d^{2}x=\sum_{i=1}^{N}\beta_{i}n_{i}=\sum_{i=1}^{N}w_{i}$
(7)

If the parameter region does not enclose any zero point, the total topological
number or charge is $0$. Corresponding to a given value of $\tau$, first the
zero points are located and the corresponding winding numbers are computed. By
summing over the winding numbers of all the zero points for a given $\tau$,
the total topological number can be obtained.This method of calculating
topological number or charge is called Duan’s $\phi$ mapping technique.
Another method for calculating topological number is suggested in [65],where
using residue theorem, the winding number$(w_{i})$ for each solution can be
calculated as,

$\displaystyle
w_{i}=\frac{Res\mathcal{R}(z_{i}}{|Res\mathcal{R}(z_{i}|}=Sign[Res\mathcal{R}(z_{i}]$
(8)

where the rational complex function can be written as

$\mathcal{R}(z)=\frac{1}{\tau-\mathcal{G}(z)}$ (9)

where,$\mathcal{G}(z)$ is the solution for$\tau$ satisfying the equation
$(\frac{\partial\mathcal{F}}{\partial r_{+}}=0$ with $r_{+}$ replaced by $z$ .
The total winding number or the topological charge, $W$ is given by
$W=\sum_{i}w_{i}$. As we have mentioned earlier, based on the topological
number $W$, all black hole solutions can be classified into three topological
classes.
In this work, we extend the study of thermodynamic topology to $D=4,5$ Horava-
Lifshitz blac holes in Horava gravity. These special black holes are also
known as superfluid black holes whose thermodynamics exhibit a line of
(continuous) second order phase transitions known as $\lambda$ phase
transitions similar to those observed in the superfluidity of liquid
${}^{4}He$. Horava gravity is considered to be a probable candidate for
quantum gravity at extremely high energies [70]. There has been a lot of
interest in the Horava-Lifshitz (HL) black hole solutions, thermodynamics, and
phase transitions [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81] due to their
rich phase structures. In this paper, we address the issues related to the
dependence of thermodynamic topology on dimension, type of horizon
thermodynamic parameters and the choice of ensemble.

This paper is organized into the following sections: in Section III, we
briefly introduce the Horava Lifshitz black hole and its thermodynamic
quantities. In Section IV, we analyze the thermodynamic topology of $D=4,5$ HL
black holes in fixed $\epsilon$ ensemble, where $\epsilon$ is a parameter of
the black hole solution. We do so for three types of horizon : spherical, flat
and hyperbolic. In Section V, we extend our analysis to examine fixed $\zeta$
ensemble, where $\zeta$ is a parameter conjugate to $\epsilon$. Finally, the
conclusions are presented in SectionVI.

##  III  Horava Lifshitz black hole

The space-time metric for $D=d+1$ dimensional Horava Lifshitz AdS black hole
is given by[72, 80],

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{d-1,k}$

$f(r)=k+\frac{32\pi
Pr^{2}}{(d-1)d\left(1-\epsilon^{2}\right)}-4r^{2-\frac{d}{2}}\sqrt{\frac{64\pi^{2}P^{2}\epsilon^{2}r^{d}}{(d-1)^{2}d^{2}\left(1-\epsilon^{2}\right)^{2}}+\frac{\pi(d-2)MP}{d\left(1-\epsilon^{2}\right)}}$

where, $k$ can take three values $+1,0,-1$ corresponding to spherical, flat
and hyperbolic horizons respectively. $d$ is the spatial dimension. $M$ and
$P$ are the mass and thermodynamic pressure of the black hole respectively.
Here, $\epsilon$ is a constant parameter that appears in the action for HL
gravity [80]. While $\epsilon=0$ corresponds to the detailed balance
condition, $\epsilon=1$ takes the action back to the general relativity
regime. Mass, $M$ is given by the condition $f\left(r_{+}\right)=0$, where
$r=r_{+}$ is the location of the horizon.

$M=\frac{r_{+}^{d-4}\left(d^{2}k(\epsilon+1)-dk(\epsilon+1)+32\pi
Pr_{+}^{2}\right)\left(d^{2}(k-k\epsilon)+dk(\epsilon-1)+32\pi
Pr_{+}^{2}\right)}{16\pi(d-2)(d-1)^{2}dP}$ (10)

where $r_{+}$ is the event horizon radius The entropy is computed to be

$S=\begin{cases}4\pi
r_{+}^{2}\left(1+\frac{dk\left(1-\epsilon^{2}\right)\ln(r_{+})}{8\pi
Pr_{+}^{2}}\right)+S_{0}&\text{d=3}\\\ \frac{16\pi
r_{+}^{d-1}\left(\frac{d(d-1)^{2}k\left(1-\epsilon^{2}\right)}{32\pi(d-3)Pr_{+}^{2}}+1\right)}{(d-2)(d-1)^{2}}+S_{0}&\text{$d\geq
3$}\par\end{cases}$ (11)

Where, $S_{0}$ is an integration constant. In this work, we fix $S_{0}=0$ for
our analysis.

##  IV Thermodynamic Topology of $D=4,5$ HL black hole in Fixed $\epsilon$
ensemble

In this section, we study the thermodynamic topology of $D=4,5$ Horava
Lifshitz black holes in an ensemble where the parameter $\epsilon$ is kept
fixed. The section is organized into three subsections corresponding to
spherical, hyperbolic and flat horizon cases.

###  IV.1 For Spherical Horizon $\left(k=1\right)$

####  IV.1.1 Case I : For D=5

For five dimensional HL black hole with spherical horizon, the free energy,
$\mathcal{F}=E-S/\tau$, is given by

$\mathcal{F}=\frac{64\pi^{2}Pr_{+}^{3}(Pr_{+}\tau-1)+24\pi
r_{+}\left(2Pr_{+}\tau+3\epsilon^{2}-3\right)-9\tau\left(\epsilon^{2}-1\right)}{72\pi
P\tau}$ (12)

The components of the vector $\phi$ are found to be

$\phi^{r}=\frac{\partial\mathcal{F}}{\partial
r_{+}}=\frac{12kPr_{+}\tau+9k\epsilon^{2}-9k+32\pi P^{2}r_{+}^{3}\tau-24\pi
Pr_{+}^{2}}{9P\tau}$ (13) $\phi^{\Theta}=-\cot\Theta~{}\csc\Theta$ (14)

The unit vectors $\left(n^{1},n^{2}\right)$ are computed using the following
prescription :

$n^{1}=\frac{\phi^{r}}{\sqrt{(\phi^{r})^{2}+(\phi^{\Theta})^{2}}}\hskip
14.22636pt\text{and}\hskip
14.22636ptn^{2}=\frac{\phi^{\theta}}{\sqrt{(\phi^{r})^{2}+(\phi^{\Theta})^{2}}}$

The expression for $\tau$ corresponding to zero points is obtained by setting
$\phi^{r}=0$.

$\tau=\frac{3\left(8\pi Pr_{+}^{2}-3\epsilon^{2}+3\right)}{4Pr_{+}\left(8\pi
Pr_{+}^{2}+3\right)}$ (15)

Critical points are computed by applying the following conditions :

$\frac{d\tau}{dr_{+}}=0\hskip 11.38092pt,\hskip
11.38092pt\frac{d^{2}\tau}{dr_{+}^{2}}=0$ (16)

These conditions result in the two equations stated below:

$64\pi P^{2}r_{+}^{2}\text{{\hbox{\tau}}}+4P\left(8\pi
Pr_{+}^{2}+3\right)\text{{\hbox{\tau}}}-48\pi Pr_{+}=0$ (17) $192\pi
P^{2}r_{+}\text{{\hbox{\tau}}}-48\pi P=0$ (18)

By solving the above equations, we end up with the expressions for critical
pressure, $P_{c}$ and critical time scale parameter,$\tau_{c}$ given by,

$P_{c}=\frac{1}{8\pi r_{+}^{2}}\hskip 11.38092pt,\hskip
11.38092pt\tau_{c}=2\pi r_{+}$ (19)

Putting these expressions in eq.$\left(\ref{eos}\right)$, the critical value
of the parameter $\epsilon$ is obtained

$\epsilon_{c}=\frac{2\sqrt{2}}{3}\approx 0.9428$

The horizon radius $r_{+}$ is plotted against $\tau$ at this critical value,
$\epsilon_{c}=0.9428$ in Figure.1. The pressure, $P$ is set equal to $0.01$.
We can clearly see a $\lambda$ line transition between a small black hole and
a large black hole.

Figure 1: $\tau$ vs $r_{+}$ plot for $5D$ HL black hole with spherical horizon
at $\epsilon=\epsilon_{c}=0.9428$ with $P=0.01$. The $LBH$ and $SBH$ regions
represent the large and small black hole branches respectively.

(a)

(b)

(c)

Figure 2:  Plots for $5D$ HL black hole with spherical horizon at
$\epsilon=0.9$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$
plot, figure $\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=5$ . The red arrows represent the vecor field
$n$. The zero point is located at $r_{+}=14.3328.$ In figure $\left(c\right)$,
computation of the winding number for the contour around the zero point
$\tau=5$ and $r_{+}=14.3328$ is shown.

At the same value of pressure, $P=0.01$, but at $\epsilon=0.90$ which is below
the value of $\epsilon_{c}$, the horizon radius $r_{+}$ is plotted against
$\tau$ in Figure.2(a). Here, we observe a single black hole branch whose
points are also the zero points of $\phi$. For $\tau=5$, the zero point is
located at $r_{+}=14.3328$. This is also confirmed from the vector plot of $n$
in the $r_{+}-\theta$ plane as shown in Figure.2(b). For finding out the
winding number/topological number associated with this zero point, we perform
a contour integration around $r_{+}=14.3328$ which is shown in Figure.2(c). It
reveals that the topological charge in this case is equal to $+1$. We have
explicitly verified that the topological charge of any zero point on the black
hole branch remains the same and is equal to $+1$.

(a)

(b)

(c)

Figure 3:  Plots for $5D$ HL black hole with spherical horizon at
$\epsilon=0.975$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs
$r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field $n$ on a
portion of $r_{+}-\theta$ plane for $\tau=10$ . The zero points are located at
$r_{+}=0.5408,1.495,5.463$. In figure $\left(c\right)$, computation of the the
winding numbers for the contours around the zero points
$r_{+}=0.5408,1.495,5.463$ are shown in black colored solid line, blue colored
dashed line and red colored solid line respectively.

We repeat the analysis keeping pressure constant at $P=0.01$ with
$\epsilon=0.975$ which is above the value of $\epsilon_{c}$. The horizon
radius $r_{+}$ is plotted against $\tau$ in Figure.3(a). Here, we observe
three black hole branches: a small, an intermediate and a large black hole
branch. For $\tau=10$, three zero points are located a
t$r_{+}=0.5408,1.495,5.463$. This is again confirmed from the vector plot of
$n$ in the $r_{+}-\theta$ plane as shown in Figure.3(b). As shown in
Figure.3(c), the winding numbers corresponding to $r_{+}=0.5408,1.495,5.463$
(represented by the black colored solid line, the blue colored dashed line and
the red colored solid line respectively) are found to be $+1$, $-1$ and $+1$
respectively. In fact, all the points on the small black hole and large black
hole branches are found to have winding number equal to $+1$. The winding
number of the intermediate branch is found to be $-1$. The sum of the winding
numbers of the three branches gives us the total topological charge of the
black hole, which in this case equals $1-1+1=1$. From Figure.3(a), we can also
see a generation point at $\tau=11.453,r_{+}=3.0695$ and an annihilation point
at $\tau=9.143,r_{+}=0.864$ which are shown as black dots.

(a)

(b)

(c)

Figure 4:  Plots for $5D$ HL black hole with spherical horizon at $\epsilon=1$
with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$ plot, figure
$\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=8$ . The zero points are located at
$r_{+}=1.519,7.855$. In figure $\left(c\right)$, computation of the the
winding numbers for the contours around the zero points $r_{+}=1.519,7.855$
are shown in black colored and red colored lines respectively.

We now consider the GR limit by setting $\epsilon=1$ . The pressure is again
kept at $P=0.01$. The horizon radius $r_{+}$ is plotted against $\tau$ in
Figure.4(a). Here, we observe two black hole branches: a small and a large
black hole branch. For $\tau=8$, the zero points are located at
$r_{+}=1.519,7.855$. The vector plot of $n$ in the $r_{+}-\theta$ plane in
Figure.4(b) demonstrates the same. As shown in Figure.4(c), the winding
numbers corresponding to $r_{+}=1.519,7.855$ (black colored and red colored
lines respectively) are found to be $-1$ and $+1$ respectively. Hence, the
total topological charge equals $1-1=0$. We locate one generation point at
$\tau=10.854$ and $r_{+}=3.454$ as seen in Figure.4(a).

For, all the cases discussed above, we also studied the dependence of
topological charge on pressure. We found that, for a given value of
$\epsilon$, the topological charge of $5D$ HL black hole with spherical
horizon remains constant at different values of pressure.

Therefore, for $5D$ HL black hole with spherical horizon in the fixed
$\epsilon$ ensemble, below a critical value of $\epsilon=\epsilon_{c}=0.9428$,
we find one black hole branch with topological charge $+1$. Above
$\epsilon_{c}$, but below the GR limit, $\epsilon=1$, we observe three black
hole branches but the topological charge still remains $+1$. In the GR limit
with $\epsilon\geq 1$, two black hole branches are found with total
topological charge equaling $0$. Variation of pressure has no impact on the
topological charge. We summarize our results in the table below:

Table 1: Summary of results for $5D$ HL black hole with spherical horizon in fixed $\epsilon$ ensemble. $\epsilon$ | $P$ | No of black hole branches | Topological charge | No of generation points | No of annihilation points
---|---|---|---|---|---
0.90 | 0.005 | 1 | 1 | 0 | 0
0.90 | 0.01 | 1 | 1 | 0 | 0
0.90 | 0.1 | 1 | 1 | 0 | 0
0.975 | 0.005 | 3 | 1-1+1=1 | 1 | 1
0.975 | 0.01 | 3 | 1-1+1=1 | 1 | 1
0.975 | 0.1 | 3 | 1-1+1=1 | 1 | 1
1 | 0.005 | 2 | -1+1=0 | 1 | 0
1 | 0.01 | 2 | -1+1=0 | 1 | 0
1 | 0.1 | 2 | \- 1+1=0 | 1 | 0

####  IV.1.2 Case II : For D=4

For four dimensional HL black hole with spherical horizon, the free energy is
given by

$\mathcal{F}=\frac{4\pi
r_{+}^{2}(4Pr_{+}\tau-3)}{3\tau}+\frac{3\left(\epsilon^{2}-1\right)\log(r_{+})}{2P\tau}-\frac{3\left(\epsilon^{2}-1\right)}{16\pi
Pr_{+}}+2r_{+}$ (20)

The components of the vector $\phi$ are found to be

$\phi^{r}=\frac{3k^{2}\tau\left(\epsilon^{2}-1\right)+8\pi
kr_{+}\left(4Pr_{+}\tau+3\epsilon^{2}-3\right)+128\pi^{2}Pr_{+}^{3}(2Pr_{+}\tau-1)}{16\pi
Pr_{+}^{2}\tau}$ (21) $\phi^{\Theta}=-\cot\Theta~{}\csc\Theta$ (22)

The expression for $\tau$ corresponding to zero points is obtained as:

$\tau=\frac{8\pi\left(-3kr_{+}\epsilon^{2}+3kr_{+}+16\pi
Pr_{+}^{3}\right)}{3k^{2}\epsilon^{2}-3k^{2}+32\pi
kPr_{+}^{2}+256\pi^{2}P^{2}r_{+}^{4}}$ (23)

Critical points can be computed by following the same procedure as explained
in the case of 5D HL black holes,resulting in the equations:

$P_{c}=P_{c}=-\frac{k\left(-3\epsilon^{2}\pm\sqrt{\epsilon^{2}\left(9\epsilon^{2}-8\right)}+2\right)}{32\pi
r_{+}^{2}},\hskip 11.38092pt\tau_{c}=\frac{\pm 3\pi
r_{+}\left(3\epsilon\sqrt{\left(9\epsilon^{2}-8\right)}+\left(9\epsilon^{2}-4\right)\right)}{2k}$
(24)

Putting these expressions in eq.$\left(\ref{eos4}\right)$, the critical value
of the parameter $\epsilon$ is obtained

$\epsilon_{c}=\sqrt{\frac{4}{9}+\frac{8}{9\sqrt{3}}}\approx 0.978593$

The horizon radius $r_{+}$ is plotted against $\tau$ at this critical value,
$\epsilon_{c}=0.9785$ in Figure.5. The pressure, $P$ is set equal to $0.01$.
We see a $\lambda$ line transition between a small black hole and a large
black hole for the case of $D=4$ case also.

Figure 5: $\tau$ vs $r_{+}$ plot for $4D$ HL black hole with spherical horizon
at $\epsilon=\epsilon_{c}=0.9785$ with $P=0.01$. The $LBH$ and $SBH$ regions
represent the large and small black hole branches respectively.

(a)

(b)

(c)

Figure 6:  Plots for $4D$ HL black hole with spherical horizon at
$\epsilon=0.9$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$
plot, figure $\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=8$ . The red arrows represent the vecor field
$n$. The zero point is located at $r_{+}=5.785.$ In figure $\left(c\right)$,
computation of the the winding number for the contour around the zero point
$\tau=8$ and $r_{+}=5.785$ is shown.

At the same value of pressure, $P=0.01$, but at $\epsilon=0.90$ which is below
the value of $\epsilon_{c}$, the horizon radius $r_{+}$ is plotted against
$\tau$ in figure.6(a). Here, we observe a single black hole branch whose
points are also the zero points of $\phi$. For $\tau=8$, the zero point is
located at $r_{+}=5.785$. This is also confirmed from the vector plot of $n$
in the $r_{+}-\theta$ plane as shown in figure 6(b). By performing contour
integration around $r_{+}=5.785$ which is shown in figure 6(c) we get the
topological charge equal to $+1$.

(a)

(b)

(c)

Figure 7:  Plots for $4D$ HL black hole with spherical horizon at
$\epsilon=0.99$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$
plot, figure $\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=11$ . The zero points are located at
$r_{+}=0.5358,0.7419,3.44$. In figure $\left(c\right)$, computation of the the
winding numbers for the contours around the zero points
$r_{+}=0.5358,0.7419,3.44$ are shown in black colored solid line, blue colored
dashed line and red colored solid line respectively.

We repeat the analysis keeping pressure constant at $P=0.01$ with
$\epsilon=0.99$ which is above the value of $\epsilon_{c}$. The horizon radius
$r_{+}$ is plotted against $\tau$ in figure 7(a). Here, we observe three black
hole branches: a small, an intermediate and a large black hole branch. For
$\tau=11$, three zero points are located a t$r_{+}=0.5358,0.7419,3.44$. This
is again confirmed from the vector plot of $n$ in the $r_{+}-\theta$ plane as
shown in Figure.7(b). As shown in figure 7(c), the winding numbers
corresponding to $r_{+}=0.5358,0.7419,3.44$ (represented by the black colored
solid line, the blue colored dashed line and the red colored solid line
respectively) are found to be $+1$, $-1$ and $+1$ respectively. Here also all
the points on the small black hole and large black hole branches are found to
have winding number equal to $+1$. and the winding number of the intermediate
branch is found to be $-1$. The total topological charge of the 4D black hole
is equals to $1-1+1=1$. From figure 7(a), we can also see a generation point
at $\tau=13.0563,r_{+}=1.8047$ and an annihilation point at
$\tau=10.8284,r_{+}=0.6225$ which are shown as black dots.

(a)

(b)

(c)

Figure 8:  Plots for $4D$ HL black hole with spherical horizon at $\epsilon=1$
with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$ plot, figure
$\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=10$ . The zero points are located at
$r_{+}=0.9929,4.007$. In figure $\left(c\right)$, computation of the winding
numbers for the contours around the zero points $r_{+}=0.9929,4.007$ are shown
in black colored and red colored lines respectively.

We now consider the GR limit by setting $\epsilon=1$ . The pressure is again
kept fixed at $P=0.01$. The horizon radius $r_{+}$ is plotted against $\tau$
in figure 8(a). Here, we observe two black hole branches: a small and a large
black hole branch. For $\tau=10$, the zero points are located at
$r_{+}=0.9929,4.007$. The vector plot of $n$ in the $r_{+}-\theta$ plane in
figure 8(b) demonstrates the same. As shown in figure 8(c), the winding
numbers corresponding to $r_{+}=0.9929,4.007$ (black colored and red colored
lines respectively) are found to be $-1$ and $+1$ respectively. Hence, the
total topological charge equals $1-1=0$. We locate one generation point at
$\tau=12.5331$ and $r_{+}=1.9947$ as seen in Figure.8(a).
Similar to the outcomes in the case of a 5D HL black hole, pressure does not
significantly influence the determination of the topological number for
various values of $\epsilon$.
Therefore it can be inferred that for $4D$ HL black hole with spherical
horizon in the fixed $\epsilon$ ensemble, below a critical value of
$\epsilon=\epsilon_{c}=0.9785$, we find one black hole branch with topological
charge $+1$. Above $\epsilon_{c}$, but below the GR limit, $\epsilon=1$, we
observe three black hole branches but the topological charge still remains
$+1$. In the GR limit with $\epsilon\geq 1$, two black hole branches are found
with total topological charge equaling $0$. We summarize our results in the
table below:

Table 2: Summary of results for $4D$ HL black hole with spherical horizon in fixed $\epsilon$ ensemble. $\epsilon$ | $P$ | No of black hole branches | Topological charge | No of generation points | No of annihilation points
---|---|---|---|---|---
0.90 | 0.008 | 1 | 1 | 0 | 0
0.90 | 0.05 | 1 | 1 | 0 | 0
0.90 | 0.2 | 1 | 1 | 0 | 0
0.99 | 0.008 | 3 | 1-1+1=1 | 1 | 1
0.99 | 0.05 | 3 | 1-1+1=1 | 1 | 1
0.99 | 0.2 | 3 | 1-1+1=1 | 1 | 1
1 | 0.008 | 2 | -1+1=0 | 1 | 0
1 | 0.05 | 2 | -1+1=0 | 1 | 0
1 | 0.2 | 2 | \- 1+1=0 | 1 | 0

###  IV.2 For flat horizon (k=0)

####  IV.2.1 Case I : For D=5

For flat horizon , we put $k=0$ in the expression for mass and entropy
.Finally obtained the free energy for $5D$ HL black hole with flat horizon as
:

$\mathcal{F}=\frac{8\pi r_{+}^{3}(Pr\tau-1)}{9\tau}$

followed by-

$\phi_{r}=\frac{8\pi r_{+}^{2}(4Pr_{+}\tau-3)}{9\tau}\hskip
28.45274pt\text{and}\hskip 28.45274pt\tau=\frac{3}{4Pr_{+}}$

For $5D$ HL black hole with flat horizon ,from the expression for $\tau$ it is
clear that $\tau$ is independent of $\epsilon$ ,hence no criticality will be
observed for this case.

(a)

(b)

(c)

Figure 9:  Plots for $5D$ HL black hole with flat horizon at $\epsilon=0.96$
with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$ plot, figure
$\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=6$ . The red arrows represent the vecor field
$n$. The zero point is located at $r_{+}=12.5.$ in figure $\left(c\right)$,
computation of the winding number for the contour around the zero point
$\tau=6$ and $r_{+}=12.5$ is shown.

Here, we observe a single black hole branch whose points are also the zero
points of $\phi$. For $\tau=6$, the zero point is located at $r_{+}=12.5$.
This is also confirmed from the vector plot of $n$ in the $r_{+}-\theta$ plane
as shown in figure 9(b). By performing contour integration around $r_{+}=12.5$
which is shown in figure 9(c) we get the topological charge equal to
$+1$.Infact for any value of pressure and $\epsilon$ the topological charge
remains constant.

####  IV.2.2 Case II: For D=4

For $D=4$, we get the expression for free energy of 4D HL black hole with flat
horizon as:

$\mathcal{F}=\frac{4\pi r_{+}^{2}(4Pr\tau-3)}{3\tau}$

followed by-

$\phi^{r}=\frac{8\pi r_{+}^{2}(4Pr_{+}\tau-3)}{9\tau}\hskip
28.45274pt\text{and}\hskip 28.45274pt\tau=\frac{3}{4Pr_{+}}$

For $4D$ HL black hole with flat horizon ,the $\tau$ is independent of
$\epsilon$ similar to the case of 5D HL black hole ,hence no criticality will
be observed for this case also.

(a)

(b)

(c)

Figure 10:  Plots for $4D$ HL black hole with flat horizon at $\epsilon=0.96$
with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$ plot, figure
$\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=6$ . The red arrows represent the vecor field
$n$. The zero point is located at $r_{+}=8.333.$ In figure $\left(c\right)$,
computation of the winding number for the contour around the zero point
$\tau=6$ and $r_{+}=8.333$ is shown.

Here also, we observe a single black hole branch whose points are also the
zero points of $\phi$. For $\tau=6$, the zero point is located at
$r_{+}=8.333$. This is also confirmed from the vector plot of $n$ in the
$r_{+}-\theta$ plane as shown in Figure.10(b).By performing contour
integration around $r_{+}=12.5$ which is shown in Figure.10(c)we get the
topological charge equal to $+1$.Here also for any value of pressure and
$\epsilon$ the topological charge remains constant.

###  IV.3 For hyperbolic horizon ( k=-1)

####  IV.3.1 Case I : For D=5

The free energie expression for 5D HL black hole with hyperbolic horizon is
found to be:

$\mathcal{F}=\frac{64\pi^{2}Pr_{+}^{3}(Pr_{+}\tau-1)-24\pi
r_{+}\left(2Pr_{+}\tau+3\epsilon^{2}-3\right)-9\tau\left(\epsilon^{2}-1\right)}{72\pi
P\tau}$

and the expression for $\phi_{r}$ can be found out as-

$\phi_{r}=\frac{32\pi P^{2}r_{+}^{3}\tau-12Pr_{+}(2\pi
r_{+}+\tau)-9\epsilon^{2}+9}{9P\tau}$

The zero points of the vector is obtained by setting $\phi_{r_{+}}=0$ and
followed by getting a relation between $\tau$ and $r_{+}$ as-

$\tau=\frac{3\left(8\pi Pr_{+}^{2}+3\epsilon^{2}-3\right)}{4Pr_{+}\left(8\pi
Pr_{+}^{2}-3\right)}$

For 5D HL black hole with hyperbolic horizon,we do not get any critical
conditions hence there is no superfluid $\lambda$ phase transition observed.It
is important to note that while calculating the winding number some
restrictions regarding the values of $r_{+}$ and $\epsilon$ ,are needed to be
taken care of [78]. From Figure 11 it can be observed that we can have values
of r and $\epsilon$ only from the shaded portions due to positive temperature
condition.

Figure 11: The relations between $\epsilon^{2}$ and $r$ for the positive
temperature for $k=-1$ in $D=5.$Temperature is positive only on the shaded
portions.We have taken $P=0.01$.

Thus while calculating winding number from the $\tau$ vs $r_{+}$ curve for a
particular value of $\epsilon^{2}$ and pressure $P$ we will have to take
values of $r_{+}$ from the allowed range(shaded portions).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12: Plots for $5D$ HL black hole with hyperbolic horizon at
$\epsilon^{2}=0.4$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs
$r_{+}$ plot for $0\leq r_{+}\leq 2.67619$ range and $\left(d\right)$ shows
the same for the range $3.4549\leq r\leq 10$.figure $\left(b\right)$ and
figure $\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=15$ in the range $0\leq r_{+}\leq 2.67619$ and
$3.4549\leq r_{+}\leq 10$ respectively.The zero points are located at
$r_{+}=5.994$ and $r_{+}=1.996$ . In figure $\left(c\right)$ and
$\left(f\right)$, computation of the contours around the zero points for
$r_{+}=5.994$ and $r_{+}=1.996$ are shown in red and black colored solid lines
respectively.

The horizon radius $r_{+}$ is plotted against $\tau$ in Figure.12(a) and
Figure.12(b). Here, we observe two separate or discontinous black hole
branches. One for the range $0\leq r\leq 2.67619$(Figure 12(a)) and other for
the range $3.4549\leq r\leq 10$(Figure.12(b)).We will do the analysis for
calculating the winding numbers by keeping the pressure $P$ ,$\epsilon^{2}$
and $\tau$ fixed at $P=0.01$,$\epsilon^{2}=0.4$ and $\tau=15$.For this
combination of values we will observe two zero point of vector field $n$ at
$r_{+}=1.996$ and $r_{+}=5.994$ which lied in two distinct shaded portion as
shown in Figure 11. Figure.12(c) and Figure 12(f) suggests that the winding
numbers corresponding to $r_{+}=1.9968,5.9947$ (represented by the black
colored solid line and the red colored solid line respectively) are found to
be $+1$, and $-1$ respectively. The sum of the winding numbers of the two
discontinous branches gives us the total topological charge of the black hole,
which in this case equals $1-1=0$.in conclusion of the above study we can say
there will be always two allowed range of $r_{+}$ values for any
$\epsilon^{2}$ value,for which we will get two branches.For the lower branch
we will have winding number $-1$ and for the upper branch the winding number
will be $1$.The total topological charge is always $0$.Pressure do not play
any significant role here also similar to the case of 5D black hole with
spherical horizon
We now consider the GR limit by setting $\epsilon=1$ . The pressure is again
kept at $P=0.01$. The horizon radius $r_{+}$ is plotted against $\tau$
considering the allowed range for $r_{+}$. in Figure.13(a). Here, we observe
only one black hole branches.For $\tau=15$, the zero point is located at
$r_{+}=6.76458$. The vector plot of $n$ in the $r_{+}-\theta$ plane in
figure13(b) demonstrates the same. As shown in Figure.15(c), the winding
number corresponding to $r_{+}=6.76458$ (purple colored line) are found to be
$+1$. Hence, the total topological charge equals $1$.It is seen that similar
to the spherical horizon case ,in GR limit the topological charge changes.For
5D HL black hole with hyperbolic horizon the topological charge changes from
$0$ to $1.$ which is opposite to the transition $1\to 0$ in case of 5D HL
black hole with spherical horizon.The results are summarized in the table3.

(a)

(b)

(c)

Figure 13:  Plots for $5D$ HL black hole with hyperbolic horizon at $\epsilon=1$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field $n$ on a portion of $r_{+}-\theta$ plane for $\tau=15$ . The zero point is located at $r_{+}=6.76458$. In figure $\left(c\right)$, computation of the contours around the zero point $r_{+}=6.76458$ is shown in purple colored solid lines. Table 3: Summary of results for $5D$ HL black hole with hyperbolic horizon in fixed $\epsilon$ ensemble. $\epsilon^{2}$ | $P$ | No of black hole branches | Topological charge
---|---|---|---
0.40 | 0.008 | 2 | -1+1=0
0.40 | 0.05 | 2 | -1+1=0
0.40 | 0.1 | 2 | -1+1=0
0.6 | 0.008 | 2 | -1+1=0
0.6 | 0.05 | 2 | -1+1=0
0.6 | 0.1 | 2 | -1+1=0
1 | 0.008 | 1 | 1
1 | 0.05 | 1 | 1
1 | 0.1 | 1 | 1

####  IV.3.2 Case II : For D=4

The free energie expression for 4D HL black hole with hyperbolic horizon is
found to be:

$\mathcal{F}=\frac{16}{3}\pi
Pr_{+}^{3}-\frac{3\left(\epsilon^{2}-1\right)\log(r_{+})}{2P\tau}-\frac{3\left(\epsilon^{2}-1\right)}{16\pi
Pr_{+}}-\frac{2r_{+}(2\pi r_{+}+\tau)}{\tau}$

The expression for $\phi_{r}$ can be found out as-

$\phi_{r}=\frac{128\pi^{2}Pr_{+}^{3}(2Pr\tau-1)-8\pi
r\left(4Pr_{+}\tau+3\epsilon^{2}-3\right)+3\tau\left(\epsilon^{2}-1\right)}{16\pi
Pr_{+}^{2}\tau}$

The zero points of the vector is obtained by setting $\phi_{r}=0$ and followed
by getting a relation between $\tau$ and $r$ as-

$\tau=\frac{8\pi\left(16\pi
Pr_{+}^{3}+3r\epsilon^{2}-3r_{+}\right)}{256\pi^{2}P^{2}r_{+}^{4}-32\pi
Pr_{+}^{2}+3\epsilon^{2}-3}$

For 4D HL black hole with hyperbolic horizon also,we do not get any critical
conditions hence there is no phase transition observed.While calculating the
winding number some restrictions regarding the values of $r_{+}$ and
$\epsilon$ are observed here as well From figure 14 it can be observed that we
can have values of r and $\epsilon$ only from the shaded portions due to
positive temperature condition.

Figure 14: The relations between $\epsilon^{2}$ and $r$ for the positive
temperature for $k=-1$ in $D=4.$Temperature is positive only on the shaded
portions.We have taken $P=0.01$.

Thus while calculating winding number from the $\tau$ vs $r_{+}$ curve for a
particular value of $\epsilon$ and pressure $P$ we will have to take values of
$r$ from the allowed range(shaded portions).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 15: Plots for $4D$ HL black hole with hyperbolic horizon at
$\epsilon^{2}=0.2$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs
$r_{+}$ plot for $0\leq r_{+}\leq 2.18509$ range and $\left(d\right)$ shows
the same for the range $2.37861\leq r\leq 12$.figure $\left(b\right)$ and
figure $\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=5$ in the range $0\leq r_{+}\leq 2.18509$ and
$2.37861\leq r_{+}\leq 12$ respectively.The zero points are located at
$r_{+}=0.2041$ and $r_{+}=2.10093$ for the range $0\leq r_{+}\leq 2.18509$ and
$r_{+}=9.9259$ for the range $2.37861\leq r_{+}\leq 12$. In figure
$\left(c\right)$ computation of the contours around the zero points for
$r_{+}=0.2041$ and $r_{+}=2.10093$ are shown in blue and black colored solid
lines respectively.In $\left(f\right)$ computation of the contour around the
zero points for $r_{+}=9.9259$ is shown in red colored solid line.

For $P=0.01$,$\epsilon^{2}=0.4$ the horizon radius $r_{+}$ is plotted against
$\tau$ in Figure.15(a) and Figure.15(b). Here, we observe two separate or
discontinous black hole branches. One for the range $0\leq r_{+}\leq
2.18509$(Figure 15(a)) and other for the range $2.37861\leq r_{+}\leq
12$(Figure.15(b)).For the case of 4D HL black hole depending upon the value of
$\tau$ there will be one or three solution of the vector field $n$.Firstly we
will do the analysis for calculating the winding numbers by keeping the
pressure $P$ ,$\epsilon^{2}$ and $\tau$ fixed at $P=0.01$,$\epsilon^{2}=0.2$
and $\tau=5$ where three solution of the vector field is observed.For this
combination of values we will observe three zero point of vector field $n$ at
$r_{+}=0.2041$, $r_{+}=2.10093$ and $r_{+}=9.9259$ which lied in two distinct
shaded portion as shown in Figure 14. Figure.15(c) and Figure 15(f) suggests
that the winding numbers corresponding to $r_{+}=0.2041$, $r_{+}=2.10093$ and
$r_{+}=9.9259$ (represented by the black,blueband the red colored solid line
respectively) are found to be $+1$, and $-1$ and $+1$ respectively. The sum of
the winding numbers of the two discontinous branches gives us the total
topological charge of the black hole, which in this case equals
$1-1+1=1$.Again for the same set values of $\epsilon^{2}$ and $P$ but for
different $\tau$ value where there exist only one solution of vector field $n$
,the winding number/topological charge is still 1 as all the single zero
points will be on the red curve shown in Figure15(d) and for that curve ,the
topological charge is $1$.In conclusion of the above study we can say there
will be always two allowed range of $r_{+}$ values for any $\epsilon^{2}$
value,for which we will get two branches for some values of $\tau$ and one
branch for some values of $\tau$.For the lower branch we will have winding
number $0$ and for the upper branch the winding number will be $1$.The total
topological charge is always $1$.

(a)

(b)

(c)

Figure 16:  Plots for $4D$ HL black hole with hyperbolic horizon at
$\epsilon=1$ with $P=0.01$. Figure $\left(a\right)$ shows $\tau$ vs $r_{+}$
plot, figure $\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=15$ . The zero point is located at
$r_{+}=4.266$. In figure $\left(c\right)$, computation of the contours around
the zero point $r_{+}=4.266$ is shown in magenta colored solid lines.

We now consider the GR limit by setting $\epsilon=1$ . The pressure is again
kept at $P=0.01$. The horizon radius $r_{+}$ is plotted against $\tau$
considering the allowed range for $r_{+}$. in Figure.16(a). Here, we observe
only one black hole branches.For $\tau=15$, the zero point is located at
$r_{+}=4.266$. The vector plot of $n$ in the $r_{+}-\theta$ plane in
Figure.16(b) demonstrates the same. As shown in Figure.16(c), the winding
number corresponding to $r_{+}=4.266$ (magenta colored line) are found to be
$+1$. Hence, the total topological charge equals $1$.Strikingly it is seen
that unlike 5D black holes with any horizon types,in GR limit the topological
charge does not change for 4D HL black hole with hyperbolic horizon.Pressure
do not play any significant role here also similar to the case of 4D black
hole with spherical horizon.The results are summarized in the table 4

Table 4: Summary of results for $4D$ HL black hole with hyperbolic horizon in fixed $\epsilon$ ensemble. $\epsilon^{2}$ | $P$ | No of black hole branches | Topological charge
---|---|---|---
0.2 | 0.008 | 3 | -1+1+1=1
0.2 | 0.05 | 3 | -1+1+1=1
0.2 | 0.1 | 3 | -1+1+1=1
0.4 | 0.008 | 3 | -1+1+1=1
0.4 | 0.05 | 3 | -1+1+1=1
0.4 | 0.1 | 3 | -1+1+1=1
1 | 0.008 | 1 | 1
1 | 0.05 | 1 | 1
1 | 0.1 | 1 | 1

##  V Thermodynamic topology in fixed $\zeta$ ensemble

###  V.1 For spherical horizon (k=1)

####  V.1.1 Case I : For D=5

In fixed $\zeta$ ensemble, we define a parameter $\zeta$ conjugate to
$\epsilon$ and keep it fixed. The relation between these two can be found out
by solving $\zeta=\frac{dM}{d\epsilon}$. The new mass of the system is given
by the following transformation-

$\tilde{M}=M-\zeta\epsilon$

Using equation (5) and substituting $\epsilon=-\frac{4\pi P\zeta}{k^{2}}$ we
get

$\tilde{M}=\frac{k^{2}}{8\pi P}+\frac{2kr_{+}^{2}}{3}+\frac{8}{9}\pi
Pr_{+}^{4}+\frac{2\pi P\zeta^{2}}{k^{2}}$ (25)

The new entropy for D=5 is given by-

$\tilde{S}=\frac{kr_{+}}{P}+\frac{8\pi
r_{+}^{3}}{9}-\frac{16\pi^{2}Pr_{+}\zeta^{2}}{k^{3}}$ (26)

The modified free energy can be calculated using-

$\tilde{\mathcal{F}}=\tilde{M}-\frac{\tilde{S}}{\tau}$ (27)

or

$\tilde{\mathcal{F}}=\frac{16\pi^{2}Pr_{+}\zeta^{2}}{k^{3}\tau}+\frac{2\pi
P\zeta^{2}}{k^{2}}+\frac{k^{2}}{8\pi
P}+k\left(\frac{2r_{+}^{2}}{3}-\frac{r_{+}}{P\tau}\right)+\frac{8\pi
r_{+}^{3}(Pr_{+}\tau-1)}{9\tau}$ (28)

The components of the vector field are obtained as:

$\phi^{r}=\frac{\partial\tilde{\mathcal{F}}}{\partial r_{+}}=\frac{32\pi
k^{3}P^{2}r_{+}^{3}\tau-24\pi
k^{3}Pr_{+}^{2}+12k^{4}Pr_{+}\tau-9k^{4}+144\pi^{2}P^{2}\zeta^{2}}{9k^{3}P\tau}$
(29)

and

$\phi^{\theta}=-\cot\Theta\csc\Theta$ (30)

Finally we obtain the expression for $\tau$ as-

$\tau=\frac{3\left(8\pi
k^{3}Pr_{+}^{2}+3k^{4}-48\pi^{2}P^{2}\zeta^{2}\right)}{4k^{3}Pr_{+}\left(3k+8\pi
Pr_{+}^{2}\right)}$ (31)

for $k=1$ i.e for spherical horizon, equation 31 takes the form

$\tau=\frac{3\left(8\pi
k^{3}Pr_{+}^{2}+3k^{4}-48\pi^{2}P^{2}\zeta^{2}\right)}{4k^{3}Pr_{+}\left(3k+8\pi
Pr_{+}^{2}\right)}$

It is important to note that in this paper we have only considered the range
$0\leq\epsilon^{2}\leq 1$,If values of $\epsilon^{2}$ is equal to 1 then HL
gravity returns back to general relativity and $\epsilon$=0 corresponds to the
detailed-balance condition.Based on this restriction a range is defiend for
$\zeta$ as $0\leq\zeta^{2}\leq 63.389$(pressure is kept fixed as P=0.01).
In fixed $\zeta$ ensemble the $\lambda$ phase transition property disappears
and other interesting phenomenon observed is : while transforming to grand
canonical ensemble , we lost one topological horizon i.e the flat horizon as k
can not take zero as a value here.

(a)

(b)

(c)

Figure 17:  Plots for $5D$ HL black hole with spherical horizon in fixed
$\zeta$ ensemble at $\zeta=7.2$ with $P=0.01$. Figure $\left(a\right)$ shows
$\tau$ vs $r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field
$n$ on a portion of $r_{+}-\theta$ plane for $\tau=15$ . The red arrows
represent the vecor field $n$. The zero point is located at $r_{+}=1.719.$ In
figure $\left(c\right)$, computation of the contour around the zero point
$\tau=15$ and $r_{+}=1.719$ is shown.

Firstly the horizon radius $r_{+}$ is plotted against $\tau$ for different
values $\zeta$ keeping the pressure $P$ fixed.It is seen that depending on the
values of $\zeta$ 5D black hole exhibit different number of branches.For
$\zeta$ value smaller than a critical value it shows only one branch and no
phase transition.In Figure.17(a)one such scenario is being demonstrated. Here,
we observe a single black hole branch whose points are also the zero points of
$\phi$. For $\tau=15$, the zero point is located at $r_{+}=1.719$. This is
also confirmed from the vector plot of $n$ in the $r_{+}-\theta$ plane as
shown in Figure.17(b). For finding out the winding number/topological number
associated with this zero point, we perform a contour integration around
$r_{+}=1.719$ which is shown in Figure.17(c). It reveals that the topological
charge in this case is equal to $+1$. We have explicitly verified that the
topological charge of any zero point on the black hole branch remains the same
and is equal to $+1$.

(a)

(b)

(c)

Figure 18:  Plots for $5D$ HL black hole with spherical horizon in fixed
$\zeta$ ensemble at $\zeta=7.7$ with $P=0.01$. Figure $\left(a\right)$ shows
$\tau$ vs $r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field
$n$ on a portion of $r_{+}-\theta$ plane for $\tau=11$ . The zero points are
located at $r_{+}=$. In figure $\left(c\right)$, computation of the contours
around the zero points $r_{+}=0.657,1.813,4.346$ are shown in black colored
solid line, blue colored dashed line and red colored solid line respectively.

We repeat the analysis keeping pressure constant at $P=0.01$ with $\zeta=7.7$
which is above the critical value.The horizon radius $r_{+}$ is plotted
against $\tau$ in Figure.17(a). Here, we observe three black hole branches: a
small, an intermediate and a large black hole branch. For $\tau=10$, three
zero points are located a t$r_{+}=0.5408,1.495,5.463$. This is again confirmed
from the vector plot of $n$ in the $r_{+}-\theta$ plane as shown in
Figure.17(b). As shown in Figure.17(c), the winding numbers corresponding to
$r_{+}=0.657,1.813,4.346$ (represented by the black colored solid line, the
blue colored dashed line and the red colored solid line respectively) are
found to be $+1$, $-1$ and $+1$ respectively.The sum of the winding numbers of
the three branches gives us the total topological charge of the black hole,
which in this case equals $1-1+1=1$. From Figure.3(a), we can also see a
generation point at $\tau=11.6571,r_{+}=2.9323$ and an annihilation point at
$\tau=10.2052,r_{+}=1.02765$ which are shown as black dots.

(a)

(b)

(c)

Figure 19:  Plots for $5D$ HL black hole with spherical horizon in fixed
$\zeta$ ensemble at $\zeta=8$ with $P=0.01$. Figure $\left(a\right)$ shows
$\tau$ vs $r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field
$n$ on a portion of $r_{+}-\theta$ plane for $\tau=8$ . The zero points are
located at $r_{+}=1.6366,7.8313$. In figure $\left(c\right)$, computation of
the contours around the zero points $r_{+}=1.6366,7.8313$ are shown in black
colored and red colored lines respectively.

We now consider the case $\zeta=8$ which is beyond the defined limit of
$0\leq\zeta^{2}\leq 63.389$.The pressure is again kept at $P=0.01$. The
horizon radius $r_{+}$ is plotted against $\tau$ in Figure.19(a). Here, we
observe two black hole branches: a small and a large black hole branch. For
$\tau=8$, the zero points are located at $r_{+}=1.6366,7.8313$. The vector
plot of $n$ in the $r_{+}-\theta$ plane in Figure.19(b) demonstrates the same.
As shown in Figure.19(c), the winding numbers corresponding to
$r_{+}=1.6366,7.8313$ (black colored and red colored lines respectively) are
found to be $-1$ and $+1$ respectively. Hence, the total topological charge
equals $1-1=0$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 20: Effect of variation of pressure on number black hole branches and
total topological charge.The $\zeta$ value is kept fixed at $\zeta=7.2$ and
for different values of pressure $r_{+}$ vs $\tau$ graphs are plotted from
which we calculate the totaltopological charge.First ,second and third row
corresponds to $P=0.0098,0.0109,0.02$ respectively.

Above analysis is followed by plotting the horizon radius $r_{+}$ against
$\tau$ for different values pressure $P$ keeping $\zeta$ value fixed. Unlike
fixed $\epsilon$ ensemble ,here pressure becomes an important quantity.The
number of black hole branch along with the total topological charge is changed
when pressure is increased and $\zeta$ is kept fixed.Figure 20 confirms the
same where variation of pressure is shown and $\zeta$ is kept fixed at
$\zeta=7.2$.It is seen that when the pressure is lowered from $P=0.01$ to
$P=0.0098$ then the number of branch(Figure 20(a)) and total topological
remains the same(Figure 20(c)) .When pressure is slightly increased to
$P=0.0109$ then first order phase transition takes place(Figure 20(d)).However
the total topological charge remains the same i.e $1$(Figure 20(f)).When
pressure is increased further to $P=0.02$ then the number of black hole branch
decreases to two(Figure 20(g)) and the total topological charge becomes
zero(Figure 20(i)).Variations of the topological charge and number of black
hole branches with $\zeta$ and $P$ are shown in Table 5

Table 5: Summary of results for $5D$ HL black hole with spherical horizon in fixed $\zeta$ ensemble. $\zeta$ | $P$ | No of black hole branches | Topological charge | No of generation points | No of annihilation points
---|---|---|---|---|---
7.1 | 0.008 | 1 | 1 | 0 | 0
7.1 | 0.0109 | 3 | 1-1+1=1 | 1 | 1
7.1 | 0.02 | 2 | -1+1=0 | 1 | 0
7.5 | 0.008 | 1 | 1 | 0 | 0
7.5 | 0.0105 | 3 | 1-1+1=1 | 1 | 1
7.5 | 0.0109 | 2 | 1-1=0 | 1 | 0
7.8 | 0.008 | 1 | 1 | 0 | 0
7.8 | 0.0101 | 3 | -1+1+1=1 | 1 | 1
7.8 | 0.0105 | 2 | \- 1+1=0 | 1 | 0

####  V.1.2 Case II : For D=4

In this case ,substituting $\epsilon=-\frac{8\pi Pr_{+}\zeta}{3k^{2}}$ we get
the new mass as-

$\tilde{M}=\frac{3k^{2}}{16\pi Pr_{+}}+2kr_{+}+\frac{16}{3}\pi
Pr_{+}^{3}+\frac{4\pi Pr_{+}\zeta^{2}}{3k^{2}}$ (32)

Also,the entropy for fixed $zeta$ ensemble in 4D is given by-

$\tilde{S}=\log(r_{+})\left(\frac{3k}{2P}-\frac{32\pi^{2}Pr_{+}^{2}\zeta^{2}}{3k^{3}}\right)+4\pi
r_{+}^{2}$ (33)

The modified free energy is-

$\tilde{\mathcal{F}}=\frac{1}{48}\left(\frac{8\log(r_{+})\left(64\pi^{2}P^{2}r_{+}^{2}\zeta^{2}-9k^{4}\right)}{k^{3}P\tau}+\frac{64\pi
Pr_{+}\zeta^{2}}{k^{2}}+\frac{9k^{2}}{\pi Pr_{+}}+96kr_{+}+\frac{64\pi
r_{+}^{2}(4Pr_{+}\tau-3)}{\tau}\right)$ (34)

The components of the vector field are obtained as:

$\phi^{r}=\frac{1}{48}\left(\frac{8\left(64\pi^{2}P^{2}r_{+}^{2}\zeta^{2}-9k^{4}\right)}{k^{3}Pr_{+}\tau}-\frac{9k^{2}}{\pi
Pr_{+}^{2}}+\frac{1024\pi^{2}Pr_{+}\zeta^{2}\log(r)}{k^{3}\tau}+\frac{64\pi
P\zeta^{2}}{k^{2}}+96k+256\pi Pr_{+}^{2}+\frac{128\pi
r_{+}(4Pr_{+}\tau-3)}{\tau}\right)$ (35)

and

$\phi^{\theta}=-\cot\Theta\csc\Theta$ (36)

And finally the expression for $\tau$ as-

$\tau=\frac{8\pi\left(-48\pi
k^{3}Pr_{+}^{3}-9k^{4}r_{+}+64\pi^{2}P^{2}r_{+}^{3}\zeta^{2}+128\pi^{2}P^{2}r_{+}^{3}\zeta^{2}\log(r_{+})\right)}{k\left(-768\pi^{2}k^{2}P^{2}r_{+}^{4}-96\pi
k^{3}Pr_{+}^{2}+9k^{4}-64\pi^{2}P^{2}r_{+}^{2}\zeta^{2}\right)}$ (37)

For spherical horizon

$\tau=\frac{8\pi
r_{+}\left(64\pi^{2}P^{2}r_{+}^{2}\zeta^{2}+128\pi^{2}P^{2}r_{+}^{2}\zeta^{2}\log(r)-48\pi
Pr_{+}^{2}-9\right)}{64\pi^{2}P^{2}r_{+}^{2}\left(12r_{+}^{2}+\zeta^{2}\right)+96\pi
Pr_{+}^{2}-9}$

Similar to 5D case, the study is confined within the range $0\leq\zeta^{2}\leq
63.389$(for pressure $P=0.01$) and here also the $\lambda$ phase transition
property disappears.In 4D also the flat horizon does not exit.

(a)

(b)

(c)

Figure 21:  Plots for $4D$ HL black hole with spherical horizon in fixed
$\zeta$ ensemble at $\zeta=7.2$ with $P=0.01$. Figure $\left(a\right)$ shows
$\tau$ vs $r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field
$n$ on a portion of $r_{+}-\theta$ plane for $\tau=40$ . The red arrows
represent the vecor field $n$. The zero point is located at $r_{+}=1.259.$ In
figure $\left(c\right)$, computation of the contour around the zero point
$\tau=40$ and $r_{+}=1.259$ is shown.

The horizon radius $r_{+}$ is plotted against different values $\tau$ for
different set of values of pressure $P$ and $\zeta$ and it is seen that for
all values of $P,\zeta$, 4D black hole has only one branch.In Figure.21(a) one
such scenario is being demonstrated. Here, we observe a single black hole
branch whose points are also the zero points of $\phi$. For $\tau=40$, the
zero point is located at $r_{+}=1.259$. This is also confirmed from the vector
plot of $n$ in the $r_{+}-\theta$ plane as shown in Figure.21(b). For finding
out the winding number/topological number associated with this zero point, we
perform a contour integration around $r_{+}=1.259$ which is shown in
Figure.21(c). It reveals that the topological charge in this case is equal to
$+1$. We have explicitly verified that for any value of pressure and $\zeta$
the topological charge of any zero point on the black hole branch remains
constant which is equal to $+1$.The table 6 shows the same.

Table 6: Summary of results for $4D$ HL black hole with spherical horizon in fixed $\zeta$ ensemble. $\zeta$ | $P$ | No of black hole branches | Topological charge
---|---|---|---
7.1 | 0.008 | 1 | 1
7.1 | 0.0109 | 1 | 1
7.1 | 0.02 | 1 | 1
7.5 | 0.008 | 1 | 1
7.5 | 0.0105 | 1 | 1
7.5 | 0.0109 | 1 | 1
7.8 | 0.008 | 1 | 1
7.8 | 0.0101 | 1 | 1
7.8 | 0.0105 | 1 | 1

###  V.2 For hyperbolic horizon(k=-1)

####  V.2.1 Case I : For D=5

For hyperbolic horizon we put $k=-1$ in the expression 29 and 31 which gives

$\phi^{r}=\frac{\partial\tilde{\mathcal{F}}}{\partial
r_{+}}=\frac{-144\pi^{2}\zeta^{2}P^{2}+32\pi
P^{2}r_{+}^{3}\tau-12Pr_{+}\tau-24\pi Pr_{+}^{2}+9}{9P\tau}$ (38)

and

$\tau=\frac{-3\left(8\pi
Pr_{+}^{2}+3-48\pi^{2}P^{2}\zeta^{2}\right)}{-4Pr_{+}\left(-3+8\pi
Pr_{+}^{2}\right)}$ (39)

For 5D HL black hole with hyperbolic horizon in fixed $\zeta$ ensemble also,we
do not get any critical conditions hence there is no phase transition
observed.While calculating winding number from the $\tau$ vs $r_{+}$ curve for
a particular value of $\zeta$ and pressure $P$ we will have to take values of
$r$ from the allowed range where temperature is positive.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 22: Plots for $5D$ HL black hole with hyperbolic horizon in fixed
$\zeta$ ensemble.Here $\zeta=7.6$ with $P=0.01$. Figure $\left(a\right)$ and
$\left(d\right)$ shows $\tau$ vs $r_{+}$ plot for two allowed range of values
of $r_{+}$ where temperature is positive.Figure $\left(b\right)$ and figure
$\left(b\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane for $\tau=5$ for both the allowed range.The zero points
are located at $r_{+}=0.0.7118$ and $r_{+}=15.6966$ for lower and upper
allowed region respectively. In figure $\left(c\right)$ computation of the
contours around the zero points for $r_{+}=0.0.7118$ and $r_{+}=15.6966$ are
shown in black and red colored solid lines respectively.

It is being observed that at a fixed pressure $P=0.01$ in case of 5D HL black
hole with hyperbolic horizon in fixed $\zeta$ ensemble have two types of
topological charge $0$ and $1$.At a fixed pressure,for a range of $\zeta$
value we observe two black hole branch in two allowed range of $r_{+}$ where
temperature is positive. For this scenario the total topological charge is
equals to $-1+1=0$. One of such scenario is shown in figure 22 where the
horizon radius $r_{+}$ is plotted against $\tau$ for $\zeta=7.6$ and pressure
$P=0.01$.In Figure.22(a) and Figure.22(b) $\tau$ vs $r_{+}$ plot is
represented for two different allowed range of $r_{+}$ for same set of values
of $\zeta$ and $P$.For the case of $\tau=5$, there will two zero points of the
vector field $n$ at $r_{+}=0.2041$, $r_{+}=2.10093$ as shown in Figure 22(b)
and figure 22(e). Figure.22(c) and figure 22(f) suggests that the winding
numbers corresponding to $r_{+}=0.2041$, $r_{+}=2.10093$ (represented by the
black and the red colored solid line respectively) are found to be $-1$, and
$+1$ respectively. The sum of the winding numbers of the two discontinous
branches gives us the total topological charge of the black hole, which in
this case equals $1-1=0$.In conclusion of the above study we can say there
will be always two allowed range of $r_{+}$ values for a range of $\zeta$
value at a fixed pressure,for which we will get two branches for some values
of $\tau$.For the lower branch we will have winding number $-1$ and for the
upper branch the winding number will be $1$.The total topological charge is
always $0$.

(a)

(b)

(c)

Figure 23:  Plots for $5D$ HL black hole with hyperbolic horizon in fixed
$\zeta$ ensemble,at $\zeta=8$ with $P=0.01$. Figure $\left(a\right)$ shows
$\tau$ vs $r_{+}$ plot, figure $\left(b\right)$ is the plot of vector field
$n$ on a portion of $r_{+}-\theta$ plane for $\tau=5$ . The zero point is
located at $r_{+}=15.7648$. In figure $\left(c\right)$, computation of the
contours around the zero point $r_{+}=15.7648$ is shown in orange colored
solid lines.

We now consider a higher range of values of $zeta$ for the same value of
pressure. Taking $\zeta=8$,horizon radius $r_{+}$ is plotted against $\tau$
considering the allowed range for $r_{+}$ in Figure.23(a). Here, we observe
only one black hole branches.For $\tau=5$, the zero point is located at
$r_{+}=15.7648$. The vector plot of $n$ in the $r_{+}-\theta$ plane in
Figure.23(b) demonstrates the same. As shown in Figure.15(c), the winding
number corresponding to $r_{+}=15.7648$ (orange colored line) are found to be
$+1$. Hence, the total topological charge equals $1$.
Above analysis is followed by plotting the horizon radius $r_{+}$ against
$\tau$ for different values pressure $P$ keeping $\zeta$ value fixed at
$\zeta=7.6$.The number of black hole branch along with the total topological
charge is changed when pressure is slightly varied and $\zeta$ is kept
fixed.Figure 24 confirms the same where variation of pressure is shown and
$\zeta$ is kept fixed at $\zeta=7.6$.It is seen that when the pressure is
lowered from $P=0.01$ to $P=0.001$ then the number of branch(Figure 24(a)) and
total topological remains the same(Figure 24(c)) .When pressure is slightly
increased to $P=0.05$ then total topological charge becomes zero. place(Figure
24(d)).However the total topological charge remains the same i.e $1$(Figure
LABEL:40f).Variations of the topological charge and number of black hole
branches with $\zeta$ and $P$ are shown in Table 7

Table 7: Summary of results for $5D$ HL black hole with hyperbolic horizon in fixed $\zeta$ ensemble. $\zeta$ | $P$ | No of black hole branches | Topological charge
---|---|---|---
7.2 | 0.035 | 1 | 1
7.2 | 0.096 | 2 | -1+1=0
7.8 | 0.008 | 2 | -1+1=0
7.8 | 0.07 | 1 | 1
8 | 0.0058 | 2 | -1+1=0
8 | 0.05 | 1 | 1

(a)

(b)

(c)

(d)

(e)

(f)

Figure 24: Plots for $5D$ HL black hole with hyperbolic horizon in fixed
$\zeta$ ensemble.Here $\zeta=7.6$ is kept fixed and pressure is varied.In
figure $\left(a\right)$ and $\left(d\right)$ shows $\tau$ vs $r_{+}$ plot for
pressure $P=0.001$ and $P=0.05$ respectively.Figure $\left(b\right)$ and
figure $\left(e\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane .For figure $\left(b\right)$ $\tau$ value is taken to be
$\tau=50$ and the zero points are located at $r_{+}=10.7482$ and
$r_{+}=15.1483$ for lower and upper allowed region respectively. In figure
$\left(c\right)$ computation of the contours around the zero points for
$r_{+}=0.0.7118$ and $r_{+}=15.6966$ are shown in black and red colored solid
lines respectively.For figure $\left(e\right)$ $\tau$ value is taken to be
$\tau=1.5$ and the zero point is located at $r_{+}=13.1782$ .In figure
$\left(f\right)$ computation of the contours around the zero points for
$r_{+}=13.1782$ is shown.

####  V.2.2 Case II : For D=4

For 4D HL black hole, we put $k=-1$ in the expression 35 and 37 which gives

$\phi^{r}=\frac{4\pi P\left(12r_{+}^{2}\tau-8\pi
r_{+}\phi^{2}+\tau\zeta^{2}\right)}{3\tau}+\frac{3(8\pi r-\tau)}{16\pi
Pr_{+}^{2}\tau}-\frac{64\pi^{2}Pr_{+}\zeta^{2}\log(r_{+})}{3\tau}-\frac{8\pi
r_{+}}{\tau}-2$ (40)

and

$\tau=\frac{8\pi
r\left(64\pi^{2}P^{2}r_{+}^{2}\zeta^{2}+128\pi^{2}P^{2}r_{+}^{2}\zeta^{2}\log(r_{+})+48\pi
Pr_{+}^{2}-9\right)}{64\pi^{2}P^{2}r_{+}^{2}\left(12r_{+}^{2}+\zeta^{2}\right)-96\pi
Pr_{+}^{2}-9}$ (41)

For 4D HL black hole with hyperbolic horizon in fixed $\zeta$ ensemble,we do
not get any critical conditions hence there is no superfluid $\lambda$ phase
transition observed.While calculating winding number from the $\tau$ vs
$r_{+}$ curve for a particular value of $\zeta$ the positive temperature
condition needs to be keep in mind in this case also.It is being observed that
at a fixed pressure $P=0.01$ in case of 4D HL black hole with hyperbolic
horizon in fixed $\zeta$ ensemble have two types of topological charge $0$ and
$1$.For very high value of $\zeta$ we observe topological charge is equal to
zero.Two of such scenario where topological charge is one,is shown in figure
25 and figure 26 where the horizon radius $r_{+}$ is plotted against $\tau$
for $\zeta=0.2$ and $\zeta=11.1$ respectively keeping pressure constant at
$P=0.01$

(a)

(b)

(c)

Figure 25: Plots for $4D$ HL black hole with hyperbolic horizon in fixed
$\zeta$ ensemble.Here $\zeta=0.2$ with $P=0.01$. Figure $\left(a\right)$ and
$\left(d\right)$ shows $\tau$ vs $r_{+}$ plot for two allowed range of values
of $r_{+}$ where temperature is positive.Figure $\left(b\right)$ is the plot
of vector field $n$ on a portion of $r_{+}-\theta$ plane for $\tau=10$ for
both the allowed range.The zero points are located at $r_{+}=0.5934,2.3551$
and $r_{+}=4.581$ for lower and upper allowed region respectively. In figure
$\left(c\right)$ computation of the contours around the zero points for
$r_{+}=0.5934,2.3551$ and $r_{+}=4.581$ are shown in black,blue and red
colored solid lines respectively.

In Figure.25(a) $\tau$ vs $r_{+}$ plot is represented for two different
allowed range of $r_{+}$ for same set of values of $\zeta$ and $P$.For the
case of $\tau=10$, there will three zero points of the vector field
$n$.Figure.25(c) suggests that the sum of the winding numbers of the two
discontinous branches gives us the total topological charge of the black hole,
which in this case equals $1-1+1=1$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 26:  Plots for $4D$ HL black hole with hyperbolic horizon in fixed
$\zeta$ ensemble,at $\zeta=11.1$ with $P=0.01$.Figure $\left(a\right)$ is the
$\tau$ vs $r_{+}$ plot for the two allowed region where temperature is
positive. Figure $\left(b\right)$ shows $\tau$ vs $r_{+}$ plot for the lower
allowed branch($0\leq r_{+}\leq 1.2$) and figure (c) is the corresponding
vector field $n$ on a portion of $r_{+}-\theta$ plane for $\tau=10$ . Figure
$\left(d\right)$ shows $\tau$ vs $r_{+}$ plot for the upper allowed
branch($1.2\leq r_{+}\leq 300$) and figure (d) is the corresponding vector
field $n$ on a portion of $r_{+}-\theta$ plane for $\tau=180$. In figure
$\left(e\right)$, computation of the contours around the zero points are shown
in different colored solid lines.Red and black colored solid line corresponds
to figure (c) and rest are related to figure (e)

An interesting case is represented in figure.26(a) where $\tau$ vs $r_{+}$
plot is represented for two different allowed range of $r_{+}$ for
$\zeta=11.1$(which is beyond our considered range) and $P=0.01$.As figure
26(a) suggests in one of the two region where temperature is positive$(1.2\leq
r_{+}\leq 300)$ we observe two black hole branches.For $\tau=10$, there will
be two zero points on the vector field $n$ for this case(figure
26(c)).Figure.26(f) suggests that the sum of the winding numbers for this
region gives us the total topological charge equals to $1-1=0$.On the other
allowed region we observe three black hole branches i.e small,intermediate and
large black hole branches. For $\tau=180$, there will three zero points on the
vector field $n$ for this case(figure 26(e)).Figure.26(f) suggests that the
sum of the winding numbers for this region gives us the total topological
charge equals to $1-1+1=1$. We now consider a higher range of values of $zeta$
for the same value of pressure. Taking $\zeta=20$,horizon radius $r_{+}$ is
plotted against $\tau$ considering the allowed range for $r_{+}$ in
Figure.27(a). Here, we observe two black hole branches.For $\tau=100$, the
contour integration is operated around the points $r_{+}=0.5$ and
$r_{+}=1.75$. As shown in Figure.27(c), the winding number corresponding to
$r_{+}=0.5$ (black colored line) are found to be $+1$ and the winding number
corresponding to $r_{+}=1.75$ (red colored line) are found to be $-1$ . Hence,
the total topological charge equals $1-1=0$.

(a)

(b)

(c)

Figure 27: Plots for $5D$ HL black hole with hyperbolic horizon in fixed
$\zeta$ ensemble.Here $\zeta=7.6$ is kept fixed and pressure is varied.In
figure $\left(a\right)$ and $\left(d\right)$ shows $\tau$ vs $r_{+}$ plot for
pressure $P=0.001$ and $P=0.05$ respectively.Figure $\left(b\right)$ and
figure $\left(e\right)$ is the plot of vector field $n$ on a portion of
$r_{+}-\theta$ plane .For figure $\left(b\right)$ $\tau$ value is taken to be
$\tau=50$ and the zero points are located at $r_{+}=10.7482$ and
$r_{+}=15.1483$ for lower and upper allowed region respectively. In figure
$\left(c\right)$ computation of the contours around the zero points for
$r_{+}=0.0.7118$ and $r_{+}=15.6966$ are shown in black and red colored solid
lines respectively.For figure $\left(e\right)$ $\tau$ value is taken to be
$\tau=1.5$ and the zero point is located at $r_{+}=13.1782$ .In figure
$\left(f\right)$ computation of the contours around the zero points for
$r_{+}=13.1782$ is shown.

Above analysis is followed by investigating the effect of variation of
pressure pressure on the number of black hole branch along with the total
topological charge.In this case we see similar pattern as that of 5D HL black
hole with hyperbolic horizon which is : when the pressure is lowered ,the
number of branch and total topological remains the same and when pressure is
increased sufficiently then total topological charge becomes zero.Variations
of the topological charge and number of black hole branches with $\zeta$ and
$P$ are shown in Table 8.

Table 8: Summary of results for $4D$ HL black hole with hyperbolic horizon in fixed $\zeta$ ensemble. $\zeta$ | $P$ | No of black hole branches | Topological charge
---|---|---|---
0.2 | 0.01 | 3 | 1-1+1=1
0.2 | 0.05 | 2 | 1-1+1=1
0.2 | 1 | 2 | -1+1=0
11.1 | 0.01 | 5 | 1-1+1-1+1=1
11.1 | 0.005 | 2 | -1+1=0
14 | 0.01 | 4 | -1+1-1+1=0

##  VI Conclusion

In this paper, we studied the thermodynamic topology of four and five
dimensional Horava Lifshitz (HL) black holes, also referred to as superfluid
black holes. In our analysis, the Horava Lifshitz (HL) black holes are treated
as topological defects in their thermodynamic spaces. The winding numbers at
those defects were calculated in order to understand the topology of these
spaces. We primarily focused our study in two different ensembles: fixed
$\epsilon$ ensemble and fixed $\zeta$ ensemble, where $\epsilon$ is a
parameter of the HL black holes and $\zeta$ is its conjugate parameter. In the
fixed $\epsilon$ ensemble, three different horizon types were worked out: the
spherical horizon for $k=+1$, the flat horizon for $k=0$, and the hyperbolic
horizon for $k=-1$. In the fixed $\zeta$ ensemble, two different horizon types
were worked out : the spherical horizon for $k=+1$, and the hyperbolic horizon
for $k=-1$. Fixed $\zeta$ ensemble could not be defined in case of flat
horizon with $k=0$.

We find that in the fixed $\epsilon$ ensemble for 5D HL black holes with
spherical horizon, the total topological charge is 1 for $\epsilon<1$, and it
becomes 0 when $\epsilon\geq 1$. Notably, a superfluid $\lambda$ transition is
observed at $\epsilon=0.9428$. In the $0\leq\epsilon<1$ range, variations in
the number of branches in the $\tau-r$ curve are found. For $\epsilon<0.9428$,
only one branch is visible, while for $\epsilon>0.9428$, three branches are
observed. However, the total topological charge remains unchanged and remains
+1 for both $\epsilon<0.9428$ and $\epsilon>0.9428$ cases. In the General
Relativity (GR) limit ($\epsilon=1$) and for values $\epsilon\geq 1$, the
total topological charge is 0.

Similar observations are made for 4D HL black holes with a spherical horizon
in the fixed $\epsilon$ ensemble. For $\epsilon<0.9785$, one branch is
observed, whereas for $\epsilon$ greater than $0.9785$ but less than $1$,
three branches appear. The total topological charge is +1 for both
$\epsilon<0.9785$ and $\epsilon>0.9785$ cases. In the GR limit ($\epsilon=1$)
and for values $\epsilon\geq 1$, the total topological charge is 0.

For both 5D and 4D HL black holes with a flat horizon ($k=0$), the total
topological charge is always 1 for all values of $\epsilon$ and $P$.

In the case of a 5D HL black hole with a hyperbolic horizon, the total
topological charge is zero for $\epsilon<1$ at all values of $P$. In the GR
limit, the total topological charge changes to 1.

However, for a 4D HL black hole with a hyperbolic horizon, the total
topological charge is always 1 (even in the GR limit) at all values of $P$.

One notable finding in the fixed $\epsilon$ ensemble is that the change in
pressure has no significant impact on determining the overall thermodynamic
topology. It primarily depends on the parameter $\epsilon$.

Moving to the fixed $\zeta$ ensemble, for a 5D black hole with a spherical
horizon, the total topological charge is either +1 or 0. Interestingly,
pressure $P$ becomes a crucial parameter in determining the overall
thermodynamic topology of HL black holes in the fixed $\zeta$ ensemble. For a
fixed pressure, we observed a range of $\zeta$ values resulting in a
topological charge of $1$ and another range resulting in a charge of 0.
Exploring further, we observed that, for the same $\zeta$ value, increasing
pressure led to changes in the total topological charge. The number of
branches in the $\tau-r_{+}$ curve also changes based on the values of $P$ and
$\zeta$, but the total topological number remains either 1 or 0.

In contrast to its $5D$ counterpart, $4D$ HL black hole with a spherical
horizon consistently maintains a total topological charge of $+1$ at all
values of $\zeta$ and pressure. We did not observe the superfluid $\lambda$
phase transition property in this ensemble.

In both 5D and 4D HL black holes with a hyperbolic horizon in the fixed
$\zeta$ ensemble, the total topological charge takes values $1$ or $0$,
depending on the combination of $\zeta$ and $P$ values.

Hence, from our findings, we conclude that, the thermodynamic topology of
$D=4,5$ HL black holes is highly influenced by the type of horizon, choice of
ensemble, pressure, and the parameters, $\epsilon$ or $\zeta$. The roles of
thermodynamic parameters and pressure in determining the thermodynamic
topology of HL black holes change based on the ensemble we work in and the
space time dimension of the black hole.

Extending this work to other black hole systems and modified gravity theories
could yield more generalized conclusions regarding the relationship between
topological number and horizon topology in black holes. We plan to address
these issues in our future work.

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