# Topological equivalence and phase transition rate in holographic
thermodynamics of regularized Maxwell theory

Tran N. Hung<EMAIL_ADDRESS>Phenikaa Institute for
Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen
Nghia, Ha Dong, Hanoi 12116, Vietnam  Cao H. Nam  nam.caohoang@phenikaa-
uni.edu.vn Phenikaa Institute for Advanced Study and Faculty of Fundamental
Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam

###### Abstract

Utilizing the holographic dictionary from the proposal that treats Newton’s
constant as a thermodynamic variable, we establish a thermodynamic topological
equivalence between the AdS black holes in the bulk and the thermal states in
the dual CFT. The findings further reveal that the thermodynamic topological
characteristics of the RegMax AdS black holes are strongly influenced by the
characteristic parameter of the regularized Maxwell theory. Additionally, we
investigate the phase transition between low and high entropy thermal states
within a canonical ensemble in the dual CFT. Our observations indicate that
the phase transition behavior of the thermal states mirrors that of the black
holes. By modeling the phase transition process as a stochastic process, we
are able to calculate the rates of phase transition between the thermal
states. This result enhances our understanding of the dominant processes
involved in the phase transition of the thermal states in the dual CFT.

##  I Introduction

In recent decades, theoretical studies on black holes have shifted from solely
gravitational considerations to exploring their microscopic nature, with the
goal of uncovering insights into the theory of quantum gravity. Black hole
thermodynamics, initiated by the groundbreaking works of Jacob Bekenstein and
Stephen Hawking in the 1970s, is one of the most active fields in this area.
Employing a semi-classical approach, Hawking derived the notable result that
black holes emit thermal radiation, referred to as Hawking radiation, due to
vacuum fluctuations near the event horizon Hawk1975 . He also proposed the
area theorem, which states that the area of a black hole’s event horizon can
never decrease Hawk1972 . Inspired by this theorem, Bekenstein conjectured
that black hole entropy is proportional to the area of the event horizon
Beken1972 . Furthermore, Bardeen, Carter, and Hawking established the four
laws of black hole thermodynamics Bacaha1973 . Accordingly, the first law of
black hole thermodynamics is formulated as follows

$\displaystyle\delta M=T\delta S+\Omega\delta J+\Phi\delta Q$ (1)

for a black hole of mass $M$, charge $Q$, and angular momentum $J$. However,
this equation does not contain a pressure-volume term $V\delta P$ which
typically presents in standard thermodynamics. In recent years, this problem
prompted the idea of the negative cosmological constant acting as the positive
pressure of the black hole via the following relationship Dolan2011

$\displaystyle P=-\frac{\Lambda}{8\pi G_{N}}=\frac{(D-1)(D-2)}{16\pi
G_{N}l^{2}},$ (2)

where $G_{N}$ stands for the Newton’s constant and $l$ is the $D$ dimension
AdS curvature radius. The thermodynamic volume is identified as the conjugate
variable of pressure. By incorporating these new quantities, the extended form
of the first law of thermodynamics and the corresponding Smarr relation for a
charged AdS black hole can be expressed as follows

$\displaystyle\delta M$ $\displaystyle=$ $\displaystyle T\delta S+V\delta
P+\Omega\delta J+\Phi\delta Q,$ (3) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{D-2}{D-3}(TS+\Omega J)+\phi Q-\frac{2}{D-3}PV.$ (4)

In this new framework, the mass $M$ has no longer meaning of internal energy
but reinterpreted as a chemical enthalpy Karatra2009 . In the extended phase
space, the thermodynamics of these AdS black holes exhibit a wide range of
phenomena, including Van der Waals-like phase transitions Kuma2012 ; Caca2013
; Nam2018 , reentrant phase transitions Alkuma2013 ; Kurihe2021 ; Cuxuzhu2023
, black hole heat engine John2014 ; Bhaye2017 ; Zhali2019 ; Nam2021 , and
Joule-Thompson expansions Okay2017 ; Molila2018 ; Nam2020 .

Despite these interesting results, the extended phase space thermodynamics
lacks a clear holographic interpretation. In the anti-de Sitter/conformal
field theory (AdS/CFT) correspondence, the thermodynamics of the gravitational
theory in asymptotically spacetime is equivalent to that of the dual CFT
Witt1998 ; Karatra2009 . Therefore, we expect that the thermodynamic variables
of AdS black holes should be dual to standard thermodynamic variables in the
CFT. However, the variable $\Lambda$ is not clearly interpretable in the
context of holographic duality. A variation of $\Lambda$ corresponds to
changing both the central charge $C$ (or the number of colors $N$) and the CFT
volume $\mathcal{V}$. Consequently, the first law of AdS black holes cannot be
straightforwardly related to the CFT first law Karob2015 ; Cokuma2021 .

To reestablish the extended phase space thermodynamics for holographic
consistency, Visser proposed considering Newton’s constant $G_{N}$ as a
thermodynamic variable Visser2022 . Incorporating variations of $\Lambda$ and
$G_{N}$, the extended first law of charged AdS black holes, along with the
generalized Smarr relation, are expressed as follows

$\displaystyle dM$ $\displaystyle=$ $\displaystyle\frac{\kappa}{8\pi
G_{N}}dA+\Omega dJ+\Phi dQ+\frac{\Theta}{8\pi G_{N}}d\Lambda-(M-\Omega J-\Phi
Q)\frac{dG_{N}}{G_{N}},$ (5) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{D-2}{D-3}\left(\frac{\kappa A}{8\pi G_{N}}+\Omega
J\right)+\Phi Q-\frac{1}{D-3}\frac{\Theta\Lambda}{4\pi G_{N}},$ (6)

where $\Theta$ is the quantity conjugate to the cosmological constant, which
can be defined as $\int_{\Sigma_{BH}}|\xi|dV-\int_{\Sigma_{AdS}}|\xi|dV$. In
this definition the domain $\Sigma_{BH}$ extends from the horizon to infinity,
and $\Sigma_{AdS}$ in the pure AdS integral extends across the entire
spacetime. Additionally, $\xi$ is the timelike Killing field.

We observe that the $\Lambda$ and $G_{N}$ variations in (5) cannot be combined
into a single term $d(\Lambda/G_{N})$. This indicates that the standard
interpretation of the extended first law in terms of the pressure
$P=-\Lambda/8\pi G_{N}$ is inconsistent when Newton’s constant varies. This
inconsistency can be resolved by considering the central charge $C$ as a
thermodynamic variable in the dual CFT. According to the AdS/CFT
correspondence, the central charge of the dual CFT is related to both the AdS
radius and Newton’s constant, $C\sim l^{D-2}/G_{N}$. Therefore, variations of
$C$ in the CFT could lead to variations of both $\Lambda$ and $G_{N}$ in the
bulk.

Using Smarr relation and inserting $d\Lambda/\Lambda=-2dl/l$, we can rewrite
the extended first law as

$\displaystyle dM$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2\pi}d\left(\frac{A}{4G_{N}}\right)+\Omega
dJ+\frac{\Phi}{l}d(Ql)-\frac{M}{D-2}\frac{dl^{D-2}}{l^{D-2}}$ (7)
$\displaystyle+$ $\displaystyle\left(M-\frac{\kappa A}{8\pi G_{N}}-\Omega
J-\frac{\Phi}{l}Ql\right)\frac{d(l^{D-2}/G_{N})}{l^{D_{2}}/G_{N}}.$

Inserting the standard holographic dictionary for CFTs living on the boundary,
the extended CFT first law is expressed as follows Visser2022

$\displaystyle dE=\mathcal{T}d\mathcal{S}-pd\mathcal{V}+\phi d\mathcal{Q}+\mu
dC+\Omega d\mathcal{J},$ (8)

where $E,p$, and $\mathcal{V}$ are the CFT energy, pressure, and volume,
respectively. The advantage of this new rewriting is that all the terms in the
first law have a thermodynamic interpretation in the dual CFT.

Models of non-linear electrodynamics (NLEs) have long been investigated as
classical attempts to regularize the field of a point charge in Maxwell’s
theory Boin1933 ; Soro2022 . The important idea was to modify the
electromagnetic Lagrangian so that the point charge could exhibit a finite
field strength and finite self-energy. Among these, perhaps the most renowned
NLE theory is the Born-Infeld theory Boin1933 ; Boin1934 . This theory
features finite self-energy for point charges, possesses electromagnetic
duality, and approximates Maxwell’s electrodynamics in the weak field limit.
It was later discovered that Born-Infeld-type Lagrangians also emerge in the
low-energy regime of string theory Fratse1985  and in D-brane physics
Leigh1989 . Recently, Einstein gravity coupled with NLEs has garnered
significant scholarly attention due to its potential in addressing the
singularity problem associated with black holes Bardeen1968 ; Beagar2000 ;
Fanwa2016 .

In this paper, we explore the intriguing characteristics of the holographic
thermodynamics of AdS black hole solutions within the regularized Maxwell
theory (RegMax theory), a NLE theory first introduced in Taham2021 . This
theory, which admits the Maxwell limit, offers the simplest regularization of
the point charge field and its self-energy. Aside from Maxwell’s theory, it
stands as a unique model of non-linear electrodynamics that provides radiative
solutions in the Robinson-Trautman class. Remarkably, slowly rotating black
holes or charged accelerated black holes are also naturally found in the
RegMax theory Taham2021 ; Haku2023 . The spherical black hole solutions and
their thermodynamic properties within RegMax theory were thoroughly
investigated in Haku2023 . Their findings demonstrate that the characteristic
parameter of RegMax theory significantly influences the thermodynamic behavior
of the black holes.

This paper is organized as follows. In the next section, we summarized
interesting features of RegMax AdS black holes. Sect. III provides an overview
of the established holographic thermodynamics and defines the thermodynamic
quantities of the dual thermal states of RegMax AdS black holes. In Sect.
III.A, we demonstrate a general result that AdS black holes exhibit
topological thermodynamic equivalence between the bulk and the dual CFT by
employing the generalized free energy landscape. Detailed calculations of the
winding numbers in the topological thermodynamic space of RegMax AdS black
holes are provided. Sect. III.B investigates the phase transition features in
canonical ensembles of the thermal states in the dual CFT. We observe that the
thermodynamic behavior of the thermal states is strongly dependent on the
characteristic parameter in RegMax theory. We identify Van der Waals-like
behavior in the phase transition and determine the phase transition rates
using the thermal potential framework Zmxu2021  and Kramer’s escape rate from
stochastic processes Zwan2001 . Finally, we make conclusions in Sect. IV.

##  II RegMax AdS black holes

In the NLE theories, the non-linear electrodynamics is minimally coupled to
Einstein’s gravity described by the following action Haku2023

$\displaystyle I=\frac{1}{16\pi G_{N}}\int
d^{4}x\sqrt{-g}(R+4\mathcal{L}-2\Lambda),$ (9)

where $\Lambda$ is the cosmological constant. $\mathcal{L}$ is the non-linear
electromagnetic Lagrangian which is defined as a function of the two
electromagnetic invariants

$\displaystyle\mathcal{S}=\frac{1}{2}F_{\mu\nu}F^{\mu\nu},\quad\mathcal{P}=\frac{1}{2}F_{\mu\nu}(*F^{\mu\nu}),$
(10)

where the electromagnetic field strength
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. RegMax theory
belongs to a restricted class of NLEs, whose Lagrangian is only
$\mathcal{L}(\mathcal{S})$ as follows

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-2\alpha^{4}\left(1-3\log(1-s)+\frac{s^{3}+3s^{2}-4s-2}{2(1-s)}\right),$
(11) $\displaystyle s$ $\displaystyle\equiv$
$\displaystyle\left(-\frac{\mathcal{S}}{\alpha^{4}}\right)^{\frac{1}{4}}\in(0,1).$
(12)

The theory is characterized by a parameter $\alpha>0$ that has dimension
(length)-1/2. In the weak field approximation, $\alpha\rightarrow\infty$, the
theory reduces to the Maxwell theory due to
$\mathcal{L}\rightarrow\mathcal{L}_{M}=-\frac{1}{2}\mathcal{S}$.

The RegMax AdS black hole is characterized by the static spherically symmetric
metric reads

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (13)

where the metric function as

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-2\alpha^{2}|Q|+\frac{4\alpha|Q|\sqrt{|Q|}-6m}{3r}+4r\alpha^{3}\sqrt{|Q|}$
$\displaystyle-4\alpha^{4}r^{2}\log\left(1+\frac{\sqrt{|Q|}}{r\alpha}\right)+\frac{r^{2}}{l^{2}}.$

The solution exhibits a singularity at $r=0$, characterized by divergent Ricci
and Kretschmann scalars as follows

$\displaystyle R$ $\displaystyle=$
$\displaystyle-\frac{4|Q|\alpha^{2}}{r^{2}}+O(1/r),$ (15)
$\displaystyle\mathcal{K}$ $\displaystyle=$ $\displaystyle
R_{\alpha\beta\gamma\sigma}R^{\alpha\beta\gamma\sigma}$ (16) $\displaystyle=$
$\displaystyle\frac{16(2\alpha|Q|^{3/2}-3m)^{2}}{3r^{6}}+O(1/r^{5}).$

Nonetheless, the severity of the Kretschmann scalar’s divergence can be
mitigated by selecting specific values for the mass parameter $m$ and the
charge $Q$.

The values of parameters determine whether the solution is a black hole with
one or two horizons, or a naked singularity. We can expand the metric function
around the origin as follows

$\displaystyle
f=\frac{2(M_{m}-m)}{r}+1-2\alpha^{2}|Q|+4\alpha^{3}r\sqrt{|Q|}+O(r^{2}),$ (17)

where

$\displaystyle M_{m}=\frac{2\alpha|Q|\sqrt{|Q|}}{3}.$ (18)

For $m>M_{m}$, the solution exhibits a single horizon, characteristic of a
Schwarzschild-like (S-type) black hole. Conversely, for $m<M_{m}$, the
solution manifests as a Reissner–Nordström-like (RN-type) black hole. In the
marginal case where $m=M_{m}$, the metric function attains a finite value at
the origin, $f_{0}(r=0)=1-2|Q|\alpha^{2}$. If this function is positive, the
RN-type solution is a naked singularity.

The thermodynamic mass $M$ can be defined as the mass parameter $m$

$\displaystyle M=m$ $\displaystyle=$
$\displaystyle\frac{2}{3}|Q|\sqrt{|Q|}\alpha+\frac{1-2|Q|\alpha^{2}}{2}r_{h}+2\sqrt{|Q|}\alpha^{3}r_{h}^{2}$
$\displaystyle+\frac{r_{h}^{3}}{2l^{2}}-2\alpha^{4}r_{h}^{3}\log\left(1+\frac{\sqrt{|Q|}}{\alpha
r_{h}}\right).$

The temperature and entropy are calculated by the standard formulae and read

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{f^{\prime}(r_{h})}{4\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\sqrt{|Q|}\alpha^{3}+\frac{1-2|Q|\alpha^{2}}{4\pi
r_{h}}+\frac{3r_{h}}{4\pi l^{2}}$
$\displaystyle+\frac{\sqrt{|Q|}\alpha^{4}r_{h}}{\pi(\sqrt{|Q|}+\alpha
r_{h})}+\frac{3}{\pi}\alpha^{4}r_{h}\log\left(1+\frac{\sqrt{|Q|}}{\alpha
r_{h}}\right),$ $\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{\pi
r_{h}^{2}}{G_{N}}.$ (21)

Because the solution is asymptotically an AdS black hole, we can define the
pressure and conjugate volume as Kumate2017

$\displaystyle P$ $\displaystyle=$ $\displaystyle-\frac{\Lambda}{8\pi G_{N}},$
(22) $\displaystyle V$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial P}\right)=\frac{4}{3}\pi r_{h}^{3}.$ (23)

From these quantities, the first law in the thermodynamic extended phase space
is written as

$\displaystyle dM=TdS+\Phi dQ+VdP+\mu_{\alpha}d\alpha.$ (24)

This is accompanied by the Smarr relation

$\displaystyle M=2TS+\Phi Q-2VP-\frac{1}{2}\mu_{\alpha}\alpha,$ (25)

where the parameter $\alpha$ is considered as an $\alpha-$polarization
potential.

##  III Topological and holographic thermodynamics in RegMax theory

In the context of the AdS/CFT correspondence, the dual field theory resides on
the conformal boundary of the asymptotically AdS bulk spacetime. According to
the prescription provided in Gukpo1998 , the CFT metric is defined by the AdS
metric evaluated on the boundary, adjusted by a Weyl factor. Consequently, the
line element of the CFT can be expressed as Cokuma2022

$\displaystyle ds^{2}=\frac{R^{2}}{l^{2}}(-dt^{2}+R^{2}d\Omega_{2}^{2}),$ (26)

where $d\Omega_{2}^{2}$ is the differential element of a 2-sphere and $R$ is
the curvature radius of the boundary. The spatial volume of the boundary
sphere is defined by the boundary curvature radius as follows

$\displaystyle\mathcal{V}=\Omega_{2}R^{2},$ (27)

where $\Omega_{2}$ is the volume of the unit 2-sphere. For this choice of CFT
metric, the holographic dictionary is written as follows

$\displaystyle E$ $\displaystyle=$ $\displaystyle
M\frac{l}{R},\quad\mathcal{T}=\frac{\kappa}{2\pi}\frac{l}{R},\quad\mathcal{S}=S=\frac{A}{4G_{N}},\quad
C=\frac{l^{2}}{4G_{N}}$ (28) $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{\Phi}{l}\frac{l}{R},\quad\mathcal{Q}=Ql,\quad\tilde{\mu}_{\alpha}=\frac{\mu_{\alpha}}{l}\frac{l}{R},\quad\tilde{\alpha}=\alpha
l.$

Using this holographic dictionary, we can infer the extended first law and
Euler relation for the boundary CFT as follows

$\displaystyle dE$ $\displaystyle=$ $\displaystyle\mathcal{T}d\mathcal{S}+\phi
d\mathcal{Q}+\tilde{\mu}_{\alpha}d\tilde{\alpha}-pd\mathcal{V}+\mu dC,$ (29)
$\displaystyle E$ $\displaystyle=$
$\displaystyle\mathcal{T}\mathcal{S}+\phi\mathcal{Q}-\frac{1}{2}\tilde{\mu}_{\alpha}\tilde{\alpha}+\mu
C.$ (30)

The pressure and the chemical potential are defined as

$\displaystyle p$ $\displaystyle=$ $\displaystyle\frac{E}{2\mathcal{V}},$ (31)
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{1}{C}\left(E-\mathcal{T}\mathcal{S}-\phi\mathcal{Q}+\frac{1}{2}\tilde{\mu}_{\alpha}\tilde{\alpha}\right).$
(32)

Finally, we explicitly determine the thermodynamic variables of the dual CFT.
For convenience, we introduce the dimensionless parameters

$\displaystyle x\equiv\frac{r_{h}}{l},\quad y\equiv\alpha\sqrt{l}.$ (33)

Using the holographic dictionary, we can define the energy, the temperature
and the entropy of the CFT as follows

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{R}\left[2Cx+2Cx^{3}+\frac{|\mathcal{Q}|}{3}\sqrt{\frac{|\mathcal{Q}|}{C}}y-|\mathcal{Q}|xy^{2}+4\sqrt{|\mathcal{Q}|C}x^{2}y^{3}\right.$
$\displaystyle\left.-8Cx^{3}y^{4}\log\left(1+\sqrt{\frac{|\mathcal{Q}|}{C}}\frac{1}{2xy}\right)\right],$
$\displaystyle\mathcal{T}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
R}\left[3x+\frac{1}{x}-\frac{|\mathcal{Q}|}{2C}\frac{y^{2}}{x}+4\sqrt{\frac{|\mathcal{Q}|}{C}}y^{3}+2\sqrt{\frac{|\mathcal{Q}|}{C}}y^{3}\left(1+\sqrt{\frac{|\mathcal{Q}|}{C}}\frac{1}{2xy}\right)^{-1}\right.$
$\displaystyle\left.-12xy^{4}\log\left(1+\sqrt{\frac{|\mathcal{Q}|}{C}}\frac{1}{2xy}\right)\right],$
$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle 4\pi Cx^{2}.$ (36)

###  III.1 Thermodynamic topology equivalence between bulk and boundary

In topological thermodynamic spaces, the black hole and the corresponding
boundary CFT states are considered as defects. Utilizing Duan’s topological
current theory, we can determine the local and global topological winding
numbers of these defects Weilima2022 . In this section, we establish that the
topological numbers are identical in the bulk and the boundary. In other
words, the thermodynamic states of the black holes are topologically
equivalent to the thermodynamic states of the CFTs.

Prior to specific calculations, we wish to make a few remarks on the
topological current method in the thermodynamic topology of black holes. We
define a vector field $\phi$, using the generalized free energy
$F=M-\frac{S}{\tau}$, as follows

$\displaystyle\phi=(\phi^{1},\phi^{2})\equiv\left(\frac{\partial F}{\partial
r_{h}},-\cot\Theta\csc\Theta\right),$ (37)

where the parameter $0\leq\Theta\leq\pi$ for convenience. The defects,
referred to as zero points, correspond to the conditions
$\Theta=\frac{\pi}{2}$ and $\frac{\partial F}{\partial r_{h}}=0$, which is
equivalent to $\tau=T^{-1}$, where $T$ represents the black hole temperature.
The topological current as

$\displaystyle
j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\partial_{\nu}n^{a}\partial_{\rho}n^{b},\quad\mu,\nu,\rho=0,1,2,$
(38)

where $\partial_{\nu}=\frac{\partial}{\partial x^{\nu}}$ with
$x^{\nu}=(\tau,r_{h},\Theta)$ and the unit vector
$n^{a}=\frac{\phi^{a}}{||\phi^{a}||}$ $(a=1,2)$. Using the Jacobi tensor
$\epsilon^{ab}J^{\mu}\left(\frac{\phi}{x}\right)=\epsilon^{\mu\nu\rho}\partial_{\nu}\phi^{a}\partial_{\rho}\phi^{b}$,
we can reexpress the current as

$\displaystyle j^{\mu}=\delta^{2}(\phi)J^{\mu}\left(\frac{\phi}{x}\right).$
(39)

The winding numbers of the $i$-th zero point $z_{i}$ can be defined as
$\omega_{i}=\beta_{i}\eta_{i}$, where the positive Hopf index $\beta_{i}$
counts the number of loops that $\phi^{a}$ makes in the vector space when
$x^{\mu}$ goes around the zero point $z_{i}$, and the Brouwer degree
$\eta_{i}=$ sign$(J^{0}|_{z_{i}})=\pm 1$. From the definition of the Jacobi
tensor, we find that

$\displaystyle
J^{0}|_{z_{i}}=\partial_{1}\phi^{1}\partial_{2}\phi^{2}|_{z_{i}}.$ (40)

On the other hand, $\Theta$ has the value of $\frac{\pi}{2}$ at the zero
points, thus $\partial_{2}\phi^{2}|_{z_{i}}=1$. Therefore, the winding number
$\omega_{i}$ can be simply calculated as follows

$\displaystyle\omega_{i}=\text{sign}(J^{0}|_{z_{i}})=\text{sign}(\partial_{1}\phi^{1}|_{z_{i}})=\text{sign}\left(\frac{\partial^{2}F}{\partial
r_{h}^{2}}|_{z_{i}}\right).$ (41)

Based on the preceding analysis, we can infer that the zero points align with
the extrema of the generalized free energy. Specifically, local maxima,
associated with unstable black holes, have a winding number of -1, while local
minima, corresponding to metastable or stable black holes, exhibit a winding
number of 1. The global topological winding number $W$ is derived by summing
all the winding numbers $\omega_{i}$

$\displaystyle W=\sum_{i=1}^{N}\omega_{i}.$ (42)

Hence, under the assumption of smoothness for the generalized free energy, the
global winding number $W$ only takes three values, -1, 0, and 1, in accordance
with the conjecture proposed in Weilima2022 .

In holographic thermodynamics of the canonical ensemble, we define two
generalized free energies for the bulk and the dual CFT, respectively, as
follows

$\displaystyle F$ $\displaystyle=$ $\displaystyle M-\frac{S}{\tau},$ (43)
$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle
E-\frac{\mathcal{S}}{\tilde{\tau}},$ (44)

where the parameters $\tau$ and $\tilde{\tau}$ can be considered as the
inverse temperatures of the ensembles. The holographic dictionary (28) gives
us the equality $\mathcal{F}=F\frac{l}{R}$. For the grand canonical ensemble,
the generalized free energies $W=M-\frac{S}{\tau}-\Phi Q$ and
$\mathcal{W}=E-\frac{\mathcal{S}}{\tilde{\tau}}-\phi\mathcal{Q}$ also have the
same relation, $\mathcal{W}=W\frac{l}{R}$. Because the two generalized free
energies differ by only one factor, we can conclude, based on previous
remarks, that there is thermodynamic topological equivalence between the bulk
and the CFT.

The extended phase space thermodynamics of the RegMax AdS black holes,
encompassing both the canonical and grand canonical ensembles, was
investigated in Haku2023 . Notably, the phase transition properties exhibit a
strong dependence on the dimensionful parameter $\alpha$, particularly when it
crosses the critical value. In this section, we aim to elucidate how the
thermodynamic topology of these solutions varies with changes in this
parameter. To achieve this, we will determine the topological winding numbers
of the RegMax AdS black hole in detail. The generalized free energy of the
black hole in the canonical ensemble is given by

$\displaystyle F$ $\displaystyle\equiv$ $\displaystyle M-\frac{S}{\tau}$
$\displaystyle=$
$\displaystyle\frac{2}{3}|Q|\sqrt{|Q|}\alpha+\frac{1-2|Q|\alpha^{2}}{2}r_{h}+2\sqrt{|Q|}\alpha^{3}r_{h}^{2}$
$\displaystyle+\frac{r_{h}^{3}}{2l^{2}}-2\alpha^{4}r_{h}^{3}\log\left(1+\frac{\sqrt{|Q|}}{\alpha
r_{h}}\right)-\frac{\pi r_{h}^{2}}{G_{N}\tau}.$

For simplicity we fix other parameters as $l=1$, $G_{N}=1$, and $Q=1$. From
the condition $\frac{\partial F}{\partial r_{h}}=0$, we can obtain the
parameter $\tau$ of the zero points $z_{i}$ as follows

$\displaystyle\tau=-\frac{4\pi
r_{h}(1+r_{h}\alpha)}{1-2\alpha^{2}+(\alpha+6\alpha^{3})r_{h}+3(1+4\alpha^{4})r_{h}^{2}+3\alpha
r_{h}^{3}-12\alpha^{4}(1+\alpha r_{h})r_{h}^{2}\log\left[1+\frac{1}{\alpha
r_{h}}\right]}.$ (46)

We find some interesting properties of $\tau$ following this formula. The
denominator has one positive solution $r_{h}>0$ if
$\alpha>\frac{1}{\sqrt{2}}$, and it is always positive if
$\alpha<\frac{1}{\sqrt{2}}$. From equation (46) we obtain $\tau\rightarrow 0$
when $r_{h}\rightarrow 0$ in both cases. Additionally, if
$\alpha=\alpha_{c}=\frac{1}{\sqrt{2}}$, $\tau\rightarrow\pi\sqrt{2}$ as
$r_{h}\rightarrow 0$. Through numerical calculations, we also determine that
$\tau$ has two local extrema if $\frac{1}{\sqrt{2}}<\alpha<\alpha_{l}\simeq
0.713$ and has no extrema if $\alpha>\alpha_{l}$. The zero-point curves are
depicted in the $\tau-r_{h}$ plane, as shown in Fig. 1.

Figure 1: The zero-point curves are shown in the $\tau-r_{h}$ plane. The black
dots represent the reversal positions of the curves. They divide the curves
into different branches: small, intermediate, and large black holes.

For $\alpha\leq\alpha_{c}$, the collection of zero points in the $\tau-r_{h}$
plane forms curves as illustrated in the left panel of Fig. 1. There exists a
maximum value $\tau_{m}$ (black dots) corresponding to the minimum temperature
of the black hole, $T_{m}$. For $\tau<\tau_{m}$ (or $T>T_{m}$), the
generalized free energy $F$ behaves as shown in the left panel of Fig. 2. Two
black dots represent two extrema, corresponding to two zero points at a
constant $\tau$ in the $\tau-r_{h}$ plane. The stable black hole,
corresponding to the minimum, has a positive winding number 1, while the
unstable black hole, corresponding to the maximum, has a negative winding
number -1. Thus, the global winding number of the topological thermodynamic
space is zero.

Figure 2: The generalized free energy $F$ versus the horizon radius $r_{h}$.
The left, middle, and right panel for $\alpha\leq\alpha_{c}$,
$\alpha_{c}<\alpha<\alpha_{l}$, and $\alpha\geq\alpha_{l}$, respectively.

For values of $\alpha$ greater than the critical value $\alpha_{c}$, the zero-
point curves are depicted in the right panel of Fig. 1. Notably, no phase
transition occurs when $\alpha$ is greater than or equal to $\alpha_{l}$.
However, in the range $\alpha_{c}<\alpha<\alpha_{l}$, the curves exhibit two
inflection points corresponding to distinct values of $\tau$,
$\tau_{1}<\tau_{2}$. These inflection points mark changes in the direction of
variation of $\tau$. In the case $\tau_{1}<\tau<\tau_{2}$, the generalized
free energy $F$ is illustrated in the middle panel of Fig. 2. Specifically,
the behavior of $F$ reveals two local minima and one local maximum. The first
local minimum, observed at small horizon radii, characterizes a metastable
black hole. Conversely, the second local minimum, associated with large
horizon radii, corresponds to a stable black hole. The local maximum, situated
at intermediate horizon radii, describes an unstable black hole that rapidly
transitions to more stable states. This behavior bears similarities to the Van
der Waals fluid. Finally, we can compute the local winding numbers for the
metastable, unstable, and stable black holes, yielding values of 1, -1, and 1,
respectively. Thus, the global winding number is equal to 1.

###  III.2 Holographic thermodynamics in the dual CFT

Employing the holographic dictionary, we study the extended thermodynamics of
the thermal states in the dual CFT. We examine the phase transition in a
canonical ensemble with fixed values of $(\mathcal{Q},\mathcal{V},C)$,
considering its dependence on the parameter $y$, which is related to the
characteristic parameter $\alpha$ of the regular Maxwell theory via Equation
(33). Utilizing Kramer’s escape rate from stochastic processes, we calculate
the phase transition rates of the thermal states within the thermal potential
framework proposed in Zmxu2021 .

####  III.2.1 Phase transition in canonical ensemble

In the canonical ensemble with fixed $(\mathcal{Q},\mathcal{V},C)$, we have
the free energy as

$\displaystyle G$ $\displaystyle\equiv$ $\displaystyle
E-\mathcal{T}\mathcal{S}$ $\displaystyle=$ $\displaystyle\frac{1}{R}\left[Cx-
Cx^{3}+\frac{\mathcal{Q}}{3}\sqrt{\frac{\mathcal{Q}}{C}}y-\frac{\mathcal{Q}}{2}xy^{2}-2\sqrt{\mathcal{Q}C}x^{2}y^{3}\left(1+\sqrt{\frac{\mathcal{Q}}{C}}\frac{1}{2xy}\right)^{-1}\right.$
$\displaystyle\left.+4Cx^{3}y^{4}\log\left(1+\sqrt{\frac{\mathcal{Q}}{C}}\frac{1}{2xy}\right)\right]$

In this section, we study the dependence of the free energy $G$ on the
parameter $y$, which is directly related to $\alpha$ via Equation (33), while
keeping $R=1$, $C=1$, and $\mathcal{Q}=1$. For this scenario, the temperature
$\mathcal{T}$ and the free energy $G$ are re-expressed as follows

$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left[3x+\frac{2-y^{2}}{2x}+4y^{3}+2y^{3}\left(1+\frac{1}{2xy}\right)^{-1}-12xy^{4}\log\left(1+\frac{1}{2xy}\right)\right],$
(48) $\displaystyle G$ $\displaystyle=$ $\displaystyle
x-x^{3}+\frac{1}{3}y-\frac{1}{2}xy^{2}-2x^{2}y^{3}\left(1+\frac{1}{2xy}\right)^{-1}+4x^{3}y^{4}\log\left(1+\frac{1}{2xy}\right).$
(49)

The behavior of the temperature of the thermal state is depicted by the
$\mathcal{T}-x$ curves for different values of $y$ in Fig. (3). We observe
that the curves exhibit the same behavior for large $x$. However, they become
distinct for small $x$. At the critical value $y=y_{c_{1}}=\sqrt{2}$, the
temperature curve approaches a definite value of $\frac{2\sqrt{2}}{\pi}$ as
$x\rightarrow 0$. For $y<y_{c_{1}}$, the temperature becomes positively
indefinite as $x\rightarrow 0$. For $y>y_{c_{1}}$, the temperature curves
approach zero at specific values of $x$. In this case, we identify another
critical value $y_{c_{2}}\approx 2.795$ at which the following system of
equations has a solution

$\displaystyle\frac{\partial\mathcal{T}}{\partial
x}=0,\quad\frac{\partial^{2}\mathcal{T}}{\partial x^{2}}=0.$ (50)

For $y_{c_{1}}<y<y_{c_{2}}$, the temperature curves exhibit two local extrema,
which may be associated with a Van der Waals-like phase transition. A clearer
understanding of this phase transition can be achieved by examining the
behavior of the free energy.

Figure 3: The $\mathcal{T}-x$ curves of the canonical ensemble for different
values of $y$.

The free energy versus temperature curves are illustrated in Fig. (4). For
$y_{c_{1}}<y<y_{c_{2}}$, the free energy exhibits a swallowtail shape. Along
the curve, the value of $x$ increases from the leftmost point. According to
Equation (36), small black holes correspond to low entropy thermal states in
the dual CFT, while large black holes correspond to high entropy thermal
states. Therefore, the self-intersection point of the curve signifies a first-
order phase transition between low and high entropy thermal states. The phase
transition rate for this case will be calculated in the next section. When
$y>y_{c_{2}}$, the curve of the free energy becomes monotonic, indicating a
single-phase state. For $y\leq y_{c_{1}}$, the curve exhibits a cusp at a
minimum temperature. At this cusp, a second-order phase transition occurs
between the low entropy states (in the upper branch) and high entropy states
(in the lower branch). We observe that the behavior of the free energy of the
thermal states is similar to that of the RegMax AdS black holes studied in
Haku2023 .

Figure 4: The free energy $G$ versus the horizon radius $T$.

####  III.2.2 Phase transition rates

We consider a canonical ensemble comprising a large number of states,
including the thermal states in the dual CFT. The phase transition can be
understood as the rearrangement of thermal states in the ensemble caused by
thermal fluctuations. Using Kramer escape rate from stochastic processes
Zwan2001 ; Xuwu2023  and the thermal potential framework described in Zmxu2021
, we investigate the phase transition rates between the low and high entropy
thermal states in the case of swallowtail free energy. The thermal potential
of the canonical ensemble is defined as follows Zmxu2021

$\displaystyle U=\int\left(\mathcal{T}-t\right)d\mathcal{\mathcal{S}},$ (51)

where $\mathcal{T}$ and $\mathcal{S}$ are the temperature and the entropy of a
thermal state in the CFT, respectively, as defined in (III) and (36), and $t$
is the ensemble temperature. The integrand under consideration can be
interpreted as the deviation between all possible states within the canonical
ensemble and the dual thermal states. The extreme points of the thermal
potential exhibit the equilibrium states corresponding to the thermal states

$\displaystyle\frac{dU}{d\mathcal{S}}=0\rightarrow t=\mathcal{T}.$ (52)

Figure 5: The schematic diagram of the thermal potential in a Van der Waals-
like phase transition is illustrated. The left panel displays the
$G-\mathcal{T}$ curve, where points $A,B$, and $C$ represent three states with
the same temperature $\mathcal{T}_{0}$, satisfying
$\mathcal{T}_{1}<\mathcal{T}_{0}<\mathcal{T}_{2}$. The right panel presents
the $U-\mathcal{S}$ curve at temperature $\mathcal{T}_{0}$, with these states
corresponding to the extreme points.

The behavior of the thermal potential $U$ in a Van der Waals-like phase
transition can be depicted in Fig. 5. Points $A,B$ and $C$ represent three
thermal states at the same temperature $\mathcal{T}_{0}$. Points $A$ and $C$
correspond to local minima on the $U-\mathcal{S}$ curve, indicating stable
thermal states. Point $B$ represents a local maximum on this curve, signifying
an unstable thermal state. A thermal state at point $A$ can overcome the
potential barrier at point $B$ and transition to point $C$. Another state can
also revert from $A$ to $C$. The thermal states behave as Brownian particles
in the potential $U$. We observe that the transition rates depend on the
barrier heights $E_{AB}=U(x_{B})-U(x_{A})$ and $E_{CB}=U(x_{B})-U(x_{C})$.
Assuming that the barrier heights are significantly larger than the
temperature, the transition rates from $A$ to $C$ and from $C$ to $A$ can be
expressed using the Kramer’s escape rate formula, respectively,

$\displaystyle
r_{ac}=\frac{\sqrt{|U^{\prime\prime}(x_{A})U^{\prime\prime}(x_{B})|}}{2\pi}\exp{-\frac{U(x_{B})-U(x_{A})}{D}},$
$\displaystyle
r_{ca}=\frac{\sqrt{|U^{\prime\prime}(x_{C})U^{\prime\prime}(x_{B})|}}{2\pi}\exp{-\frac{U(x_{B})-U(x_{C})}{D}},$
(53)

where $D$ is the constant diffusion coefficient, which is proportional to the
ensemble temperature Zwan2001 . We select the value of $D$ to ensure that the
barrier heights are significantly greater than the temperature. However, if
$D$ is too small, the exponential factors will become very large, causing the
rates to approach zero. Thus, an appropriate value for $D$ is chosen to be
0.01.

From equations (III), (36), and (51), by fixing $R=1,C=1$, and
$\mathcal{Q}=1$, we can integrate over the variable $x$ to derive the thermal
potential formula as

$\displaystyle U=-4\pi
tx^{2}+x\left(-y^{2}+2(1+x^{2}+2xy^{3})-8x^{2}y^{4}\log\left[1+\frac{1}{2xy}\right]\right).$
(54)

In the previous section, we observed that the free energy $G$ exhibits a
swallowtail shape when the dimensionless parameter $y$ satisfies the condition
$y_{c_{1}}<y<y_{c_{2}}$. Additionally, for each consistent value
$\mathcal{T}_{0}$, there exist two values $y_{min}$ and $y_{max}$ within the
range $[y_{c_{1}},y_{c_{2}}]$ such that a Van der Waals-like phase transition
occurs for $y_{min}<y<y_{max}$. For numerical calculations, given
$\mathcal{T}_{0}=0.27$, we receive $y_{min}=y_{c_{1}}$ and $y_{max}\approx
2.199$. The behavior of the thermal potential at $t=\mathcal{T}_{0}$ is
depicted in Fig. 6. This result demonstrates that the relative height between
the two local minima changes as $y$ varies. The left and right local minima
represent the low and high entropy states, respectively. The local maximum
corresponds to the unstable state, which rapidly transitions to the remaining
states. When $y=y_{0}\approx 1.761$, the two minima in the thermal potential
are equal. However, for $y<y_{0}$, the left minimum is lower than the right
minimum, indicating that the low entropy state is more stable than the high
entropy state. Conversely, for $y>y_{0}$, the right minimum is lower than the
left minimum, indicating that the high entropy state is more stable than the
low entropy state.

Figure 6: The thermal potential at $t=\mathcal{T}_{0}=0.27$. The left, middle,
and right panels show the behavior of the thermal potential for $y<y_{0}$,
$y=y_{0}$, and $y>y_{0}$, respectively.

We numerically calculate the two rates $r_{ac}$ and $r_{ca}$ in Equations (53)
as $y$ varies within the range $y_{min}<y<y_{max}$. The results are presented
in Fig. 7. For small values of $y$, the rate $r_{ac}$ is very low because the
barrier height $E_{AB}$ is large, as shown in the left panel of Fig. 6.
Conversely, the rate $r_{ca}$ is higher than $r_{ac}$ because the barrier
height $E_{CB}$ is smaller than $E_{AB}$. This result indicates that the
transition process from the high entropy state to the low entropy state
dominates the reverse process for small $y$. At $y=y*\approx 1.71$, the two
rates are equal, despite $E_{AB}$ still being larger than $E_{CB}$. The
barrier heights only become equal at $y=y_{0}>y*$, as depicted in the middle
panel of Fig. 6. This can be explained by the second-order derivatives in
Equations (53). In other words, the transition rates also depend on the
curvature of the potential at points $A$, $B$, and $C$. For $y>y*$, the rate
$r_{ac}$ surpasses $r_{ca}$, rapidly increases to a maximum, and then
decreases. Meanwhile, $r_{ca}$ gradually decreases to zero as $y$ increases.
We can conclude that the transition process from the low entropy state to the
high entropy state dominates the reverse process for large values of $y$.

Figure 7: The phase transition rates at $t=0.27$. The solid red curve
($r_{ac}$) represents the rate of transition from low entropy states to high
entropy states, while the solid blue curve ($r_{ca}$) depicts the rate of
transition from high entropy states to low entropy states.

##  IV  Conclusions

Using the holographic dictionary from the proposal that treats Newton’s
constant as a thermodynamic variable, we find a thermodynamic topological
equivalence between the AdS black holes in the bulk and the thermal states in
the dual CFT. Additionally, we employ Duan’s topological current theory to
derive the topological winding numbers from the behavior of the generalized
free energy. Assuming smoothness for the generalized free energy, we confirm
the conjecture proposed in Weilima2022  that the global winding number only
takes three values: -1, 0, and 1.

Our findings indicate that the characteristic parameter $\alpha$ of the
regularized Maxwell theory significantly influences the topological features
of RegMax AdS black holes. For $\alpha\leq\alpha_{c}=\frac{1}{\sqrt{2}}$, the
local winding numbers are 1 and -1, corresponding to stable and unstable
thermal states, respectively, resulting in a global winding number of zero.
For $\alpha\geq\alpha_{l}$, there is only a stable state with a winding number
of 1. In the remaining case, $\alpha_{c}<\alpha<\alpha_{l}$, the local winding
numbers for the metastable, unstable, and stable states are 1, -1, and 1,
respectively, yielding a global winding number of 1.

Employing the holographic dictionary, we investigate the phase transition in a
canonical ensemble of the dual thermal states with fixed values of
$(\mathcal{Q},\mathcal{V},C)$, considering its dependence on the
characteristic parameter of the regularized Maxwell theory. The results
indicate the existence of a Van der Waals-like phase transition when $y$ lies
between two critical constant values. Treating the phase transition as a
stochastic process, we quantitatively calculate the Kramer escape rates
between the stable thermal states. Our observations reveal that these rates
are closely related to the behavior of the thermal potential. This finding
provides a clearer understanding of the dominant processes in the phase
transition between the low and high entropy thermal states in the dual CFT.

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