Topology of the charged AdS black hole in restricted phase space

For the four-dimensional RN-AdS black hole, its metric has the following form

where $l$ is related to the cosmological constant via $l^{2}=-3/\Lambda$. As $M>\bar{q}/\sqrt{G}$, the function $h(r)$ has two distinct real zeros at $r=r_{\pm}$, $r_{+}$ corresponds to the black hole event horizon radius. The black hole mass parameter can be described as

In the restricted phase space for the RN-AdS black hole, the macro states are characterized by the following three pairs of dual thermodynamical parameters $(S,T),~{}(Q,\Phi),$ and $(\mu,N)$. The corresponding first law of thermodynamics and smarr formula read

with

If the parameters $S,~{}Q,~{}N$ are scaled as $S\rightarrow\lambda S,~{}Q\rightarrow\lambda Q,~{}N\rightarrow\lambda N$, from eq. (7) we can see that the mass parameter will also be scaled as $M\rightarrow\lambda M$, and $T,~{}\Phi,~{}\mu$ will not be scaled. Hence the first order homogeneity of the mass parameter and the zeroth order homogeneity of $T,~{}\Phi,~{}\mu$ are clear as crystal. Note that the zeroth order homogeneous functions are intensive the critical point

Introducing the relative parameters $s=S/S_{c},~{}t=T/T_{c},~{}q=Q/Q_{c}$ and a Legendre transform of eq. (7), the relative parameter of free energy is given as Gao2022a

In order to uncover the thermodynamical topology, the vector field mapping $\phi:~{}X=\{(s,\theta)|0<s<\infty,$ $0<\theta<\pi\}\rightarrow\mathbb{R}^{2}$ is defined as Wei2020 ; Wei2022 ; Wei2022a ; Wu2023 ; Wu2023a

When the parameter $s$ is used to characterize AdS black hole, it becomes the first parameter of the domain of the mapping $\phi$. Another parameter $\theta$ serves as an auxiliary function and it is utilized to construct the second component of the mapping. The component $\phi^{\theta}$ is divergent at $\theta=0,~{}\pi$, thus the direction of the vector points outward there. It is obvious that the zero point of $\phi$ corresponds to the black hole with the temperature of $T=\tau^{-1}$ as $\theta=\pi/2$. Therefore, the zero point of the mapping can be used to characterize the black hole solution with a given parameter $\tau$. Based on the Duan’s $\phi-$mapping topological current theory Duan1979 ; Duan1984 , the zero points of the mapping $\phi$ is linked to the topological number. The topological number can be obtained by the weighted sum of the zero points of the mapping $\phi$. The weight of each zero point is determined by its nature. A saddle point has a weight of negative one, and conversely, the weight of an extremum point is one. A topological current can be described as the following form

where $n$ is the unit vector $(n^{s},n^{\theta})$ with $n^{s}=\phi^{s}/\parallel\phi\parallel$ and $n^{\theta}=\phi^{\theta}/\parallel\phi\parallel$. And it satisfies the conserve law: $\partial_{\mu}j^{\mu}=0$. As shown in Refs. Duan1998 ; Fu2000 , the topological current is a $\delta$-function of the field configuration

where the three-dimensional Jacobian reads $\epsilon^{ab}J^{\mu}\left(\frac{\phi}{x}\right)=\epsilon^{\mu\nu\rho}\partial_{\nu}\phi^{a}\partial_{\rho}\phi^{b}$. As $\phi^{a}(x_{i})=0$, the topological current equals to zero. The topological number in a parameter region $\Sigma$ can be calculated by the following expression

where $j^{0}=\sum\limits^{\bar{N}}_{i=1}\beta_{i}\eta_{i}\delta^{2}\left(\overrightarrow{x}-\overrightarrow{s}_{i}\right)$ is the density of the topological current, $\beta_{i}$ is the Hopf index which is always positive. $s_{i}$ is the $i$-th zero point of the mapping $\phi$, $\eta_{i}=signJ^{0}(\phi/x)_{s_{i}}=\pm 1$ is the Brouwer degree. For the $i$-th zero point of the vector, the winding number $w_{i}$ is determined by the stability of black hole state.

In the canonical ensemble, substituting eq. (9) into eq. (10), we have

The zero points are determined by the expression of $\phi^{s}=0$. So we have

It is obvious that there are two extremes: $\tau_{min,max}=\frac{2\left(1\mp\sqrt{1-q^{2}}\right)^{3/2}}{3(1\mp\sqrt{1-q^{2}})-q^{2}}$. As $q=1$, $\tau_{min}=\tau_{max}=\tau_{c}=1$. Note that the generation point satisfies the constraint conditions: $\frac{\partial\tau}{\partial s}=0,~{}\frac{\partial^{2}\tau}{\partial s^{2}}>0$, and the annihilation point obeys the conditions: $\frac{\partial\tau}{\partial s}=0,~{}\frac{\partial^{2}\tau}{\partial s^{2}}<0$. The zero points of $\phi^{s}$ in the diagram of $\tau-s$ is displayed in Fig. 1. $\tau_{min}$ is related with the generation point, and $\tau_{max}$ is of the annihilation point. As $\tau_{min}<\tau<\tau_{max}$, there are three intersection points for the RN-AdS black hole in the canonical ensemble. The black, red dashed, blue lines are for the low-potential black hole (LPSB), intermediate-potential black hole (IPBH), and high-potential black hole (HPBH), respectively. They are just the stable, unstable, and stable black hole states. The corresponding winding numbers are 1, -1, 1, so the topological number is $W=1-1+1=1$ as $\tau_{min}<\tau<\tau_{max}$. The intersection points exactly are satisfying the equation $\tau=1/T$. For $\tau=\tau_{min,max}$, the three intersection points for the RN black hole coincide. And they will disappear as $\tau<\tau_{min}$ or $\tau>\tau_{max}$, the three branches black holes reduce to the one stable black hole and the topological number is still one. These results are consistent with that in Ref. Fan2023 .

Now we proceed to investigate the topology of the RN-AdS black hole in the grand canonical ensemble. For the given values of $\Phi$ and $N$, from eq. (7) the minimum temperature reads Gao2022a

the corresponding minimum entropy, mass parameter, and free energy are

By introducing the relative parameters $t=T/T_{min},~{}s=S/S_{min},~{}m=M/M_{min},~{}f=F/F_{min}$, we can obtain the express

so the component of the mapping $\phi^{s}$ reads

Hence, we have

There exist one extreme: $\tau_{extr}=\sqrt{\frac{\left(1-l^{4}\Phi^{4}\right)}{\left(1+l^{4}\Phi^{4}\right)}}$. The zero points of $\phi^{s}$ in the diagram of $\tau-s$ is displayed in Fig. 2. $\tau_{extr}$ is related with the annihilation point. As $\tau<\tau_{extr}$, there are two intersection points for the RN-AdS black hole in the grand canonical ensemble. The red and blue lines are for the low-potential black hole (LPSB) and high-potential black hole (HPBH), respectively. They are just the stable and unstable black hole states. The corresponding winding numbers are 1, -1, so the topological number is $W=1-1=0$ as $\tau<\tau_{extr}$. For $\tau=\tau_{extr}$, the two intersection points for the RN black hole coincide. Furthermore, there is no black hole as $\tau>\tau_{extr}$.

Setting $(d,k)=(5,1)$, the eq. (21) becomes

This is the black hole solution in the Einstein-Hilbert gravity theory, and the entropy is $S=A/4G$ with the Newton constant $G=3\pi G_{(5,1)}/4$. Through the transformations

the temperature and mass parameter of the black hole can be rewritten as

It is transparent that when $\widetilde{S}\rightarrow\lambda\widetilde{S},~{}\widetilde{Q}\rightarrow\lambda\widetilde{Q},~{}N\rightarrow\lambda N$, the mass parameters behaves as $M\rightarrow\lambda M$, while the temperature is kept unchanged under these re-scalings. From these thermodynamical parameters, we can obtain the critical point

With the relative parameters $t=T/T_{c},~{}s=S/S_{c},~{}q=Q/Q_{c},~{}\widetilde{\phi}=\widetilde{\Phi}/\widetilde{\Phi}_{c},~{}m=M/M_{c},~{}f=F/F_{c}$, the relative free energy reads

From the above equation, the components of the mapping $\phi$ can be written as

The zero points of the mapping $\phi$ can be calculated by $\phi^{s}=0$. We solve it and get

In the Fig. 3, we can see that there are three branches as $\tau_{min}<\tau<\tau_{max}$, the large black hole branch for $\tau<\tau_{min}$, and the small black hole branch for $\tau>\tau_{max}$. That means the existence of a phase transition in the region of $\tau_{min}<\tau<\tau_{max}$. The winding numbers of the large black hole branch for $\tau<\tau_{min}$ and small black hole branch for $\tau>\tau_{max}$, which are stable, both read $w=1$. And the corresponding topological number equals to one. While for the system in the region of $\tau_{min}<\tau<\tau_{max}$, there exist the stable large and small black hole, and the unstable intermediate black hole, thus the corresponding winding number are $w=1,1,-1$. The topological number is $W=1$. These results are consistent with that for the RN-AdS black hole in the canonical ensemble.

In the grand canonical ensemble, using the eqs. (25), (26), and (29), we can obtain

then substituting them into eqs. (30) and (31), the temperature and mass parameter become

From the above equation, the minimum temperature is determined by the expression $\frac{\partial T}{\partial S}=0$, yielding

the corresponding minimum entropy, mass and free energy read

With the definitions $f=F/F_{min}=(M-TS)/F_{min},~{}1/\tau=t=T/T_{min},~{}s=S/S_{min}$, we can obtain the components of the mapping $\phi$

The zero points of the mapping $\phi$ can be calculated by $\phi^{s}=0$. We solve it and get

As $s=(1+2L^{3}\widetilde{\Phi}^{2})^{3/2}/(1-2L^{3}\widetilde{\Phi}^{2})^{3/2}$, $\tau=\tau_{max}=\frac{\sqrt{2}}{\left(\left[\frac{1+2L^{3}\widetilde{\Phi}^{2}}{1-2L^{3}\widetilde{\Phi}^{2}}\right]^{1/3}+\left[\frac{1+2L^{3}\widetilde{\Phi}^{2}}{1-2L^{3}\widetilde{\Phi}^{2}}\right]^{2/3}\right)}$. The zero points of $\phi^{s}$ in the diagram of $\tau-s$ is displayed in Fig. 4. $\tau_{max}$ is related with the annihilation point. As $\tau<\tau_{max}$, there are two intersection points for the AdS black hole under the $(5,1)$-model frame in the grand canonical ensemble. The red and blue lines are for the low-potential black hole (LPSB) and high-potential black hole (HPBH), respectively. They are just the stable and unstable black hole states. The corresponding winding numbers are 1, -1, so the topological number is $W=1-1=0$ as $\tau<\tau_{max}$. For $\tau=\tau_{max}$, the two intersection points for this type AdS black hole coincide. Furthermore, there is no black hole as $\tau>\tau_{max}$.

When adopting $(d,k)=(6,2)$, the Wald entropy for the charged spherically symmetric AdS black hole solution was given in Ref. Kong2022 as

With the definitions $\bar{S}=L^{2}S/N+\pi l^{2},~{}\bar{Q}=Q/N$, the corresponding radiation temperature and mass parameter are

From these thermodynamical parameters, the critical point reads

By introducing the relative parameters $f=F/F_{c}=(M-TS)/F_{c},~{}1/\tau=t=T/T_{c},~{}s=\bar{S}/\bar{S}_{c},~{}q=\bar{Q}/\bar{Q}_{c}$, the component of the mapping $\phi^{s}$ reads

The zero points of the mapping $\phi$ can be calculated by $\phi^{s}=0$. In the Fig. 5, we can see that there are three branches as $\tau_{min}<\tau<\tau_{max}$, the large black hole branch for $\tau<\tau_{min}$, and the small black hole branch for $\tau>\tau_{max}$. That means the existence of a phase transition in the region of $\tau_{min}<\tau<\tau_{max}$. The winding numbers of the large and small black hole branches, which are stable, both read $w=1$. And the corresponding topological number equals to one. While for the system in the region of $\tau_{min}<\tau<\tau_{max}$, there exist the stable large and small black hole, and the unstable intermediate black hole, thus the corresponding winding number are $w=1,1,-1$. The topological number is $W=1$. These results are consistent with that for the RN-AdS black hole and the AdS black hole under the $(5,1)$-model in the canonical ensemble.

In this part we will consider the topology of the charged AdS black hole for the case of $(6,2)$-model in the grand canonical ensemble. From eq. (24), we have

With the above equation, eqs. (25), (26), and the definition $\widetilde{S}=\bar{S}/(\pi l^{2})=\frac{L^{2}S}{\pi l^{2}N}+1,~{}\bar{Q}=Q/N$, the eqs. (44) and (45) can be rewritten as

Through the redefinitions $f(\widetilde{S})=F\L^{2}/(lN),~{}m(\widetilde{S})=M\L^{2}/(lN),~{}1/\tau=t(\widetilde{S})=TlN$, the components of the mapping $\phi$ read

From the zero-point equation of the mapping $\phi^{\widetilde{S}}=0$, we can obtain