J. E. Méndez-Delgado & J. García-Rojas The abundance discrepancy in ionized
nebulae: which are the correct abundances?

Planetary Nebulae: a Universal Toolbox in the Era of Precision Astrophysics
2023 10.1017/xxxxx 384

Proceedings IAU Symposium

# The abundance discrepancy in ionized nebulae: which are the correct
abundances?

J. E. Méndez-Delgado1 & J. García-Rojas2,3 1Astronomisches Rechen-Institut,
Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12-14,
D-69120 Heidelberg, Germany
2Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
3Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Tenerife, Spain

###### Abstract

Ionized nebulae are key to understanding the chemical composition and
evolution of the Universe. Among these nebulae, H ii regions and planetary
nebulae are particularly important as they provide insights into the present
and past chemical composition of the interstellar medium, along with the
nucleosynthetic processes involved in the chemical evolution of the gas.
However, the heavy-element abundances derived from collisional excited lines
(CELs) and recombination lines (RLs) do not align. This longstanding
abundance-discrepancy problem calls into question our absolute abundance
determinations. Which of the lines (if any) provides the correct heavy-element
abundances? Recently, it has been shown that there are temperature
inhomogeneities concentrated within the highly ionized gas of the H ii
regions, causing the reported discrepancy. However, planetary nebulae do not
exhibit the same trends as the H ii regions, suggesting a different origin for
the abundance discrepancy. In this proceedings, we briefly discuss the state-
of-the-art of the abundance discrepancy problem in both H ii regions and
planetary nebulae.

###### keywords:

ISM: abundances, (ISM:) HII regions, (ISM:) planetary nebulae: general

††editors: O. De Marco, A. Zijlstra, R. Szczerba, eds.

##  1 Introduction

At the beggining of the universe, baryonic matter was composed exclusively of
hydrogen, helium and small amounts of lithium (Peebles, 1966a, b; Wagoner,
1973). The rest of the heavier elements were formed later through
nucleosynthetic processes related to the life and death of stars (Bethe, 1939;
Caughlan, 1965; Truran & Cameron, 1971). Due to this interconnection between
star formation, nucleosynthesis, and chemical composition, the study of the
chemical abundances of ionized nebulae in the interstellar medium (ISM) has
been crucial in understanding galaxies formation and evolution (Peimbert et
al., 2017; Maiolino & Mannucci, 2019). For this purpose, star-forming regions,
known as H ii regions, and planetary nebulae (PNe) are particularly important.
The former are clouds of gas ionized by the ultraviolet radiation emitted by
newly formed massive O type or early B type stars (Osterbrock & Ferland,
2006). The latter are clouds of gas formed by the ejections of matter from low
mass stars between $0.8M_{\odot}$ to $8M_{\odot}$ in their final stage of
evolution, being ionized by the hot stellar remnant (Osterbrock & Ferland,
2006).

H ii regions reflect the current chemical composition of the interstellar
medium (ISM) (Shaver et al., 1983; Arellano-Córdova et al., 2020; Méndez-
Delgado et al., 2022; Groves et al., 2023), while PNe show the composition of
the ISM at the time of the formation of the original star, along with
nucleosynthetic changes that occurred during its lifetime (Péquignot et al.,
2000; Leisy & Dennefeld, 2006; Karakas & Lattanzio, 2014; Delgado-Inglada et
al., 2015). These characteristics make them very useful for complementary
analysis of the distribution of their chemical composition in galaxies.
(Magrini et al., 2016; Esteban & García-Rojas, 2018; Stanghellini & Haywood,
2018; Espinosa-Ponce et al., 2022).

However, there is a problem, and it is a big one: we are not certain about the
correct abundance of heavy elements like O, C, N, Ne, collectively known as
“metals”. Since the pioneering works of Bowen & Wyse (1939) and Wyse (1942),
it has been known that the abundances of these elements determined from the
bright collisionally excited lines (CELs) (e.g., [O iii] 5007, 4959) are
systematically lower than those inferred from the faint recombination lines
(RLs) (e.g., O ii 4649, 4650). The ratio between both estimates is known as
the ”Abundance Discrepancy Factor” (ADF), and it has been found to be around a
factor of 2-4 in H ii regions (García-Rojas & Esteban, 2007) but can reach
values of more than 500 in some PNe (Wesson et al., 2003). The adoption of one
metallicity or another can imply significant changes in the interpretation we
would make about the chemical evolution of galaxies. Therefore, it is
important to consider: which (if any) are the correct abundances?

##  2 The favourite scenarios

Since the emergence of this long-standing problem, several ideas have been
proposed to explain the observed ADF. For instance, fluorescent excitations
contaminating heavy element RLs (Seaton, 1968); errors in the atomic
parameters that model the emissivity of spectral lines (Rodríguez & García-
Rojas, 2010); the presence of a $\kappa$ velocity distribution in free
electrons instead of a Maxwellian one (Nicholls et al., 2012); temperature
inhomogeneities (Peimbert, 1967) and chemical inhomogeneities in the gas
(Torres-Peimbert et al., 1990; Liu et al., 2000).

The last two scenarios have gained particular relevance in the overall study
of chemical abundances in H ii regions (Peimbert, 1967, 2003; O’Dell et al.,
2003; Esteban et al., 2004; García-Rojas & Esteban, 2007; Stasińska et al.,
2013; Méndez-Delgado et al., 2023b; Méndez-Delgado et al., 2023a) and PNe
(Torres-Peimbert et al., 1990; Liu et al., 2000, 2006; Fang & Liu, 2011;
García-Rojas et al., 2022). Both scenarios are supported by solid theoretical
foundations, but the observational evidence to accept or discard either of
these scenarios (or both) remains a matter of debate.

###  2.1 Temperature inhomogeneities

To explain differences between the values obtained from several temperature
diagnostics in ionized nebulae, Peimbert (1967) proposed the existence of
temperature inhomogeneities within the ionized gas. Since the CEL-emissivities
have an exponential dependency on temperature (Osterbrock & Ferland, 2006), in
the case of internal variations, their emissions will be substantially
enhanced in the hotter volumes. On the other hand, RLs have emissivities
depending on $\sim T_{\rm e}^{-1}$, so their emission is favored in cooler
regions, although with substantially smaller bias. Mathematically, this
implies that the temperature inferred from the CEL intensity ratios (such as
[O iii] 4363/5007) will be systematically higher than the average gas
temperature within the volume where the CEL-emitter ion coexists (in the case
of [O iii] 4363/5007, the ion would be O2+). In such case, the temperature
structure of each ionization volume will be characterized by two parameters:
$T_{\rm 0}$, which is the average gas temperature, and $t^{2}$, which
quantifies how inhomogeneous the temperature is. The mathematical formalism is
presented in equations (1) and (2).

$T_{0}(X^{i+})=\frac{\int T_{\rm e}n_{\rm e}n(X^{i+})dV}{\int n_{\rm
e}n(X^{i+})dV},$ (1) $t^{2}(X^{i+})=\frac{\int[T_{\rm
e}-T_{0}(X^{i+})]^{2}n_{\rm e}n(X^{i+})dV}{T_{0}(X^{i+})^{2}\int n_{\rm
e}n(X^{i+})dV}.$ (2)

In these equations, in a specific volume of the nebula, $dV$, $T_{\rm e}$,
$n_{\rm e}$ are the electron temperature and density and $n(X^{i+})$ is the
ion’s particle density.

Under this paradigm, when the estimated temperature from CEL intensity ratios
(e.g., $T_{\rm e}$([O iii] 4363/5007)) is adopted to calculate chemical
abundances relative to H using CEL/RL intensity ratios (e.g., [O iii]
5007/H$\beta$), the overestimation of temperature will result in an
underestimation of ionic abundances. Conversely, if chemical abundances
relative to H are calculated using RL/RL intensity ratios (e.g., O ii V1
4649+/H$\beta$), both lines will cancel their linear dependence on
temperature, and the derived ionic abundances will be correct.

Despite the simplicity and elegance of this proposal, its adoption as a
generalized explanation for the observed ADF has not been universal, and
various arguments against the existence of temperature variations have been
put forward. Although classical photoionization models predict the existence
of temperature inhomogeneities, these are too small ($t^{2}\sim 0.004$)
(Kingdon & Ferland, 1995; Stasińska & Szczerba, 2001; Ercolano et al., 2007;
Ferland et al., 2017) to account for the observed ADF in H ii regions
($t^{2}\sim 0.04$) or PNe ($t^{2}\sim 0.06$) (Peimbert et al., 2017). An
exception might be nebula models with very high metallicity, where temperature
gradients are more pronounced (Stasińska, 2005). Temperature inhomogeneities
in the gas tend to balance out over relatively short timescales (Ferland et
al., 2016), so their existence requires an additional physical mechanism to
create and maintain them. Additionally, some PNe (Liu et al., 2000; Corradi et
al., 2015; García-Rojas et al., 2022) exhibit ADF values that are too large
and appear irreconcilable with the $t^{2}$ formalism.

However, observational evidence seems to favour temperature variations as the
cause of the ADF in H ii regions with precise measurements of O ii RLs. For
instance, García-Rojas & Esteban (2007) found that the ADF of various ions in
several Galactic H ii regions depends on the excitation energy of the atomic
levels giving rise to CELs, which is one of the predictions of the formalism
by Peimbert (1967). Recently, Méndez-Delgado et al. (2023b) analyzed 32
spectra from extragalactic H ii regions, 20 from Galactic H ii regions, and 8
from Galactic ring nebulae with simultaneous detections of O ii RLs, $T_{\rm
e}$([O iii] 4363/5007) and $T_{\rm e}$([N ii] 5755/6584). As shown in Fig1,
these authors demonstrated that $t^{2}(\text{O}^{2+})$, measured by comparing
O ii RLs and [O iii] CELs (Peimbert & Peimbert, 2013), is linearly correlated
with the temperature difference $T_{\rm e}$([O iii] 4363/5007) $-$ $T_{\rm
e}$([N ii] 5755/6584). This shows that $t^{2}(\text{O}^{2+})$ is indeed
related to the gas temperature structure. Moreover, it shows that temperature
variations are concentrated in regions of higher ionization degree.

Figure 1: Relationship between $t^{2}(\text{O}^{2+})$, measured from O ii RLs
and [O iii] CELs, with $T_{\rm e}$([O iii] 4363/5007) and $T_{\rm e}$([N ii]
5755/6584) in a sample of 32 extragalactic H ii regions, all the existing ones
in the literature with reliable measurements of all the mentioned indicators.
Reproduced from Fig. 1 and extended data Fig. 2 of Méndez-Delgado et al.
(2023b).

The fact that temperature variations are concentrated in the zones of higher
ionization, closer to the ionizing stars of the gas, points to stellar
feedback as a possible cause of temperature variations. Phenomena such as
shocks, winds, or even variations in the ionizing spectrum of stars can cause
significant inhomogeneities in the temperature of the surrounding gas
(Peimbert et al., 1991; Perez, 1997; Ercolano et al., 2007). These are not
entirely understood physical phenomena, and their incorporation into
photoionization models is not widespread. Méndez-Delgado et al. (2023b) then
analyzed 8 spectra of ring nebulae, bubbles of ionized gas created by the
ejection of material from young massive stars. Five of these analyzed spectra
are from NGC 6888, a nebula associated with a Wolf–Rayet WN6 star, and three
others from NGC 7635 associated with an O6.5(n)fp star (Esteban et al., 2016).
The authors found that the observed trend between $t^{2}(\text{O}^{2+})$ and
$T_{\rm e}$([O iii] 4363/5007) $-$ $T_{\rm e}$([N ii] 5755/6584) in the ring
nebulae is very similar to what is observed in H ii regions. Interestingly,
the spectra of NGC 6888 exhibit the most extreme temperature variations
($t^{2}(\text{O}^{2+})$ $>$ 0.08), while the values for NGC 7635 are lower
($t^{2}(\text{O}^{2+})$ $<$ 0.06).

Figure 2: Figure analogous to Figure 1 but considering Galactic PNe.
Reproduced from Fig. 10 (ArXiv version) of Méndez-Delgado et al. (2023b).

However, Méndez-Delgado et al. (2023b) tested this same hypothesis in Galactic
PNe, and these do not exhibit a similar correlation as in H ii regions, as
shown in Fig. 2. This indicates that the cause of the ADF in H ii regions and
PNe is different. Although obvious, it is important to note that even if
temperature variations alone are not able to explain the observed ADF in PNe,
this does not rule out their presence and impact on the determination of
chemical abundances in these objects. The ADF in PNe may be caused by various
linked physical phenomena. For example, the observational evidence presented
by Richer et al. (2022) for NGC 6153 shows that while most of the ADF in this
region may be due to chemical inhomogeneities, temperature variations are not
negligible, finding values of $t^{2}(\text{O}^{2+})\approx 0.03$, which is
typically found in H ii regions.

###  2.2 Chemical inhomogeneities

An alternative explanation to the ADF is the existence of chemical
inhomogeneities. If in a gas with a typical nebular temperature of $\sim
10000$K there are gas components of different metallicity, they will induce
temperature variations (Zhang et al., 2007). However, if the metallicity of
these inclusions is extremely high compared to their hydrogen abundance, their
temperature can drop to a few thousand kelvin or even lower. Under such
circumstances, the flux of O ii RLs may surpass the flux of the optical [O
iii] CELs (Liu, 2003), and a different mathematical formalism than that
proposed by Peimbert (1967) must be applied (note the $dV$ dependence on Eq.
(2)), as the emission from RLs and CELs will essentially come from different
volumes (Stasińska, 2002).

The existence of these high-metallicity clumps has been particularly important
to explain cases of extreme ADF (Liu et al., 2000; Yuan et al., 2011; Corradi
et al., 2015; Gómez-Llanos & Morisset, 2020; García-Rojas et al., 2022) that
have been associated with the existence of central binary stellar systems
(Corradi et al., 2015; Jones et al., 2016; Wesson et al., 2018). Narrow-band
imaging and Integral Field Unit (IFU) observations of Galactic PNe have been
particularly crucial (García-Rojas et al., 2016; Ali & Dopita, 2019; García-
Rojas et al., 2022) as they have revealed morphological differences in the
distribution of heavy-element RLs and their optical CEL counterparts. In
addition, there are indications of kinematical differences between [O iii]
optical CELs and O ii RLs (Richer et al., 2013, 2022; Peña et al., 2017),
which strengthens this scenario.

However, although this scenario provides a plausible explanation for the
extreme ADF in some PNe, it does not entirely answer the initial question of
which of the abundances, based on CELs or RLs, is correct. One of the most
interesting proposals to assess how important the error in chemical abundance
determinations is under this scenario is the ”Abundance Contrast Factor (ACF)”
formalism (see Gómez-Llanos & Morisset, 2020, and the proceeding by Morisset
et al. in this series). The ACF measures the proportion of flux emitted by the
cold component relative to the total emission. If the cold component is
extremely hydrogen-poor, heavy-element RLs will have the most significant
errors. However, H i RLs can also have a significant bias by increasing their
emission in the cold component, affecting the inferred abundances from heavy-
element CEL to H i RL intensity ratios.

Even in cases where the high-metallicity clumps are so hydrogen-poor that the
impact on H i RLs is negligible, abundances observationally determined with
CELs may not be reliable. Besides potential existence of real temperature
inhomogeneities in the “hot” gas component emitting CELs, the existence of
extremely cold clumps emitting a large amount of heavy-element RLs could bias
the $T_{\rm e}$ values inferred from auroral to nebular CEL ratios by
contaminating the emission of auroral lines (e.g., [O iii] $\lambda$4363, [N
ii] $\lambda$5755) (Rubin, 1986; Gómez-Llanos et al., 2020). In order to
“decontaminate” the temperatures determined with CEL ratios, it is necessary
to know precisely the temperature and ionic abundances of the high-metallicity
clump that is emitting the “RL-contamination”. This requires determining the
physical conditions using heavy-element RLs such as O ii
$\lambda$4089/$\lambda$4649 to determine $T_{\rm e}$ or O ii
$\lambda$4661/$\lambda$4649 to determine $n_{\rm e}$ (Wesson et al., 2005;
Peimbert & Peimbert, 2013; Storey et al., 2017). This involves measuring
heavy-element RLs that are comparatively fainter than those traditionally used
to determine chemical abundances. A correction like the one proposed by Liu et
al. (2000) in their equations (1) and (2) using physical conditions and ionic
abundances determined in the “hot” gas component is conceptually incorrect,
and it has not been demonstrated to be “less incorrect” than not making any
correction.

In addition to the difficulty of explaining the creation and survival of high-
metallicity clumps (Henney & Stasińska, 2010), extrapolating the existence of
chemical inhomogeneities to explain the ADF in all PNe, not just extreme
cases, could imply the existence of a wide variety of ACF values. If the high-
metallicity component is not so hydrogen-poor, the emission of both CELs and
RLs can come from both volumes. Such a scenario would imply that both
estimates of CELs and RLs using the standard “direct method” are essentially
incorrect. On the other hand, there are still some interesting questions about
the observational evidence of the high-metallicity components. For example,
García-Rojas et al. (2022) has shown that the emission of O ii RLs has a
different morphology than its collisionally excited counterparts, as shown in
Fig. 3. This has been argued as evidence of the presence of a cold gas
component with a temperature of $\sim 800$ K. However, excellent MUSE maps
show that highly ionized CELs such as [Cl iv] or [Ar IV] have a similar
distribution to O ii RLs. This does not seem to be a contrast problem since
weaker lines than O ii $\lambda$4649, such as [Cl iii] $\lambda$5518, have a
distribution similar to that of [O iii] CELs. Since both Cl and Ar are third-
row elements in the periodic table, this also does not seem to be a
recombination contribution to these CELs (Barlow et al., 2003). How could a
clump with a temperature of $\sim 800$ K could emit these highly ionized CELs?

Figure 3:  MUSE distribution of the emission of O ii 4649 and [O iii]
$\lambda$4959 in the high-ADF Galactic planetary nebula NGC 6778 taken from
García-Rojas et al. (2022). The morphological differences are evident and have
been interpreted as evidence of the existence of components with different
metallicity in the nebula.

Figure 4:  MUSE distribution of the emission of [Cl iv] $\lambda$7531 and [Ar
iv] $\lambda$4740 in the high-ADF Galactic planetary nebula NGC 6778 taken
from García-Rojas et al. (2022). The morphology of these highly ionized CELs
are more similar to those of heavy element RLs (see Fig.3).

##  3 Conclusions

The systematic difference between heavy element abundances determined with RLs
and CELs, the so-called abundance discrepancy problem, is one of the most
interesting and important enigmas in the study of the chemical composition of
ionized nebulae. This long-standing problem has significant implications for
models of galaxy formation and evolution, and until its complete
clarification, these models may be called into question. However, over the
years, significant progress has been made thanks to deep observations of
ionized nebulae collected over several decades by numerous researchers
worldwide.

The evidence suggests that the ADF may have different origins in H ii regions
and PNe. Recent observational studies seem to favor the presence of
temperature inhomogeneities in highly ionized gas as the cause of the ADF in H
ii regions, in consistency with the formalism of Peimbert (1967). In this
case, heavy element RLs appear to provide the correct chemical abundances,
being 2$-$4 times higher than those obtained using CELs with the “direct
method” (which assumes homogeneous electron temperature, see Méndez-Delgado et
al., 2023b).

The case of PNe appears more intriguing. The evidence suggests that the
presence of temperature variations alone is not able to quantitatively explain
the observed ADFs in many PNe. In these regions, especially in those with the
highest ADF values, there are indications that components of gas with
different metallicity could exist. In fact, narrow-band imaging and IFU
observations reveal that heavy-element RLs have morphological distributions
different from their collisionally excited counterparts, which is
qualitatively consistent with this scenario. However, the quantitative impact
of the presence of high-metallicity clumps in the gas remains an open question
which is now being addressed by several research groups (Gómez-Llanos &
Morisset, 2020; García-Rojas et al., 2022). Finally, although the
differentiated morphological distribution between heavy-element RLs and CELs
is evident in PNe with high ADFs, it is important to understand exactly why
this is also observed in highly ionized CELs such as [Cl iv] or [Ar iv]. More
detailed IFU observations of PNe considering the morphological distribution of
RLs and CELs of more ions will shed light on this interesting problem.

JEMD thanks the organizers for their invitation to this excellent symposium as
an invited speaker. JEMD gratefully acknowledge funding from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) in the form of an
Emmy Noether Research Group (grant number - KR4598/2-1, PI Kreckel). JG-R
acknowledges financial support from the Canarian Agency for Research,
Innovation and Information Society (ACIISI), of the Canary Islands Government,
and the European Regional Development Fund (ERDF), under grant with reference
ProID2021010074, and from grant P/308614 financed by funds transferred from
the Spanish Ministry of Science, Innovation and Universities, charged to the
General State Budgets and with funds transferred from the General Budgets of
the Autonomous Community of the Canary Islands by the MCIU. JG-R also
acknowledges funds from the Spanish Ministry of Science and Innovation
(MICINN) through the Spanish State Research Agency, under Severo Ochoa Centres
of Excellence Programme 2020-2023 (CEX2019-000920-S).

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††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Topology and phase transition for EPYM AdS black hole in thermal potential

Yun-Zhi Du, Huai-Fan Li111the corresponding author, Yu-Bo Ma, Qiang Gu
Department of Physics, Shan xi Da tong University, Da tong 037009, China
Institute of Theoretical Physics, Shan xi Da tong University, Da tong 037009,
China

###### Abstract

As we all know the local topological properties of thermodynamical systems can
be expressed by the winding numbers as the defects. The topological number
that is the sum of all winding numbers can be used to classify the global
topological nature of thermodynamical systems. In this paper, we construct a
kind of thermal potential and then put the Einstein-power-Yang-Mills AdS black
hole in it. Through the analysis of the geometric characteristics of the
thermal potential based on the complex analysis we find the topological number
is an invariant that is same as shown in the way of the Duan’s $\phi$-mapping
topological current [Sci. Sin. 9, 1072 (1979)]. Furthermore, we adopt the
Kramer’s escape rate method to investigate the intensity of the first-order
phase transition.

##  I Introduction

Since the 70’s of last century, black holes have been discovered to be both
strong gravitational and thermodynamic systems, satisfying the four laws of
thermodynamics. Subsequently, Hawking and Page Hawking1983  presented that
there exists a phase transition between a pure AdS spacetime and a stable
large black hole state, i.e., Hawking-Page phase transition, which was
explained to a confinement/deconfinement phase transition in the gauge theory
Witten1998 . That makes black hole be paid more and more attention. Since the
lack of pressure in the traditional black hole thermodynamics, the authors
Kastor2009  expanded black hole thermodynamics into the expanded phase space
though regarding the negative cosmological constant as pressure. These leads
that black hole thermodynamics have attracted lots of attention. Especially
the thermodynamical properties of AdS black holes in the expanded phase space
had been widely considered Hendi2017a ; Hennigar2017a ; Frassin ; Kubiznak2012
; Cai2013 ; Ma2017 ; Banerjee2017 ; Mann1207 ; Wei2015 ; Bhattacharya2017 ;
Zeng2017 ; Zhang1502 ; Du2021 ; Zhang2020 . Recently, in order to make black
hole thermodynamics more like the ordinary thermal systems, people proposed
the holographic thermodynamics Visser2022 ; Ahmed2023  and the restricted
phase space thermodynamics Zhao2022 ; Gao2022 ; Gao2022a ; Du2023  of AdS
black holes to give a holographic interpretation of black hole thermodynamics.
Furthermore, through the $\phi$-map topological flow theory Duan1979  the
authors proposed black hole solutions are regarded as defects which are
described by winding numbers Wei2022 . The wingding number of the local stable
black hole state is one, and for the local unstable black hole state it is
negative one. The topological number is the sum of all wingding numbers to
reveal the global topological nature of black hole system. All the black hole
systems may be classified by topological numbers. In this work we will apply
these results to the topological property of the Einstein-power-Yang-Mills AdS
black hole.

On the other hand, the study of the phase transition of a thermodynamic system
is usually based on the heat capacity, Gibbs free energy, as well as the
Maxwell’s equal area law. Recently, the authors Xu2021 ; Xu2023  proposed a
new way to investigate the phase transition of thermodynamic systems, i.e.,
though constructing a kind of thermal potential by the temperature of the
ensemble, black hole systems are put in such unique thermal potential. We can
obtain the corresponding thermodynamic properties of black holes directly by
the geometric characteristics of the thermal potential. Black holes will be in
such a thermal potential by the driven of the thermal fluctuations. Hence the
complex structure of the phase transition for black holes can be obtained by
the method of the complex analysis. Furthermore, the detailed description of
the phase transition process of the charged AdS black holes were also
presented in Xu2023 . In this manuscript, we will construct a kind of thermal
potential for the charged AdS black holes with non-linear source, then use the
complex analysis method and the results in Ref. Wei2022  to uncover the nature
of topological number for the different black hole systems. And through the
Kramer’s escape rate method Risken1988  we will examine the first-order phase
transition process to exhibit the corresponding rate of phase transition.

As well we know, for linear charged AdS black holes at the critical point
there exists a scaling symmetry Johnson2018 ; Johnson2018a , the entropy is
proportional to the square of charge, the pressure is proportional to the
minus square of charge, and the temperature is proportional to the negative
power of charge. So for the non-linear charged AdS black holes, whether the
similar scaling symmetry still exists? Additional, with the existence of the
infinite self-energy of point like charges in Maxwell’s theory Born1934 ;
Anninos2009 ; Cai2008 , the linear theory will bring in non-linearities
Birula1970  as the field is strong. An interesting non-linear generalization
of charged AdS black holes involves a Yang-Mill field exponentially coupled to
Einstein gravity, i.e., the Einstein-power-Yang-Mills (EPYM) theory. Several
thermodynamical features of the EPYM AdS black hole in extended phase space
have recently been studied Du2021 ; Zhang2015 ; Moumni2018 .

Inspired by these, we mainly investigate the relation between topological
number and phase transitions of the EPYM AdS black hole. This work is
organized as follows. In Sec. II, we would like to briefly review the
construction of the thermal potential. In Sec. III, we investigate the
topology of the EPYM AdS black holes with different values of model parameters
through the complex analysis, and check out the corresponding results as the
system undergoing a first-order phase transition by the comments in Ref.
Wei2022 . Furthermore, based on the Kramer’s escape rate method Risken1988
the rates of first-order phase transition processes for this system are also
exhibited in Sec. IV. A brief summary is given in Sec. V.

##  II Construction of thermal potential

For a canonical ensemble with temperature $T$, it can be consist of all kinds
of states, among them one state can be regarded a real black hole in
equilibrium when the ensemble temperature equals the black hole hawking
temperature. The real black hole state is one of the solutions for the
Einstein field equation while others are not. For a black hole thermodynamic
system, we can construct the thermal potential Xu2021 ; Xu2023

$\displaystyle U=\int\left(T_{+}-T\right)dS,$ (1)

where $T_{+}$ and $S$ are the hawking temperature and the entropy of black
hole, respectively. The ensemble temperature $T$ is treated as an independent
constant, which can take any positive value in any way. For a black hole, its
hawking temperature $T_{+}$ can be written as
$T_{+}(S,~{}q,~{}\Lambda,~{}J,~{}...)$. In this work, we just consider the
case of $T=T_{+}$. In addition, according to the first law of thermodynamics
$dM=T_{+}dS-PdV+\phi dq+...$, the thermal potential becomes

$\displaystyle U=M+PV-TS.$ (2)

Formally, we can see that the thermal potential is equivalent to the free
energy landscape of AdS black hole systems.

The integrand in the above definition (1) of thermal potential can be
understood as the deviation of all possible states in the canonical ensemble
from the real black hole state in equilibrium. In other words, in the
equilibrium state, the thermal potential exhibit the following extreme
behavior,

$\displaystyle\frac{dU}{dS}=0,~{}~{}~{}\text{or}~{}~{}~{}T=T_{+}.$ (3)

Through the construction of the thermal potential (1), we can put a black hole
in the thermal potential field $U$. Due to the thermodynamic fluctuations, the
black hole will exhibit the stochastic behavior in this thermal potential
field, which can reflect some thermodynamic characteristics of black holes.

Moreover, we can investigate the stability of black hole by this thermal
potential, in which the extreme minimum stands for the stable state and the
local extreme maximum is for the unstable state, i.e.,

$\displaystyle\delta\left(\frac{dU}{dS}\right)_{T=T_{+}}=\left(\frac{\partial
T_{+}(S,~{}q,~{}J,~{}\Lambda,...)}{\partial
S}\right)_{q,~{}J,~{}\Lambda,~{}...}\delta S.$ (4)

When the right hand of the above equation is positive, the black hole will be
in stable state, while the negative right hand corresponds to the unstable
state, which are shown in Fig. 1. It is obviously that the lowest point (the
green point) is the most stable state in the entire canonical ensemble. For
different parameters of the black hole, the extreme point of the thermal
potential constantly changes, which corresponds to the changes between the
black hole state and other unknown states. In the framework, the authors had
investigated the microscopic phase transition mechanism of the charged AdS
black holes and found that the phase transition of the large and small black
holes exhibits severely asymmetric features, which fills the gap in the
analysis of stochastic processes in the first-order phase transition rate
problem of AdS black holes Xu2023 .

Figure 1: The schematic diagram of thermal potential, where the green-point
represents the global minimum (the large black hole), the black-point
represents the local maximum (the middle black hole) and the red-point
represents the local minimum (the small black hole). Except for these three
extreme points, the other positions stand for unknown states.

##  III Topology of EPYM AdS black hole system

In the four-dimensional AdS spacetime with the Einstein-power-Yang-Mills
gravity, the action had been given in Refs. Zhang2015 ; Corda2011 ;
Mazharimousavi2009 ; Lorenci2002

$\displaystyle I=\frac{1}{2}\int
d^{4}x\sqrt{g}\left(R-2\Lambda-\mathcal{F}^{\gamma}\right)$ (5)

with the Yang-Mills(YM) invariant $\mathcal{F}$ and the YM field
$F^{(a)}_{\mu\nu}$

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle
Tr\left[F^{(a)}_{\mu\nu}F^{(a)\mu\nu}\right],$ (6) $\displaystyle
F^{(a)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A^{(a)}_{\nu}-\partial_{\nu}A^{(a)}_{\mu}+\frac{1}{2\xi}C^{(a)}_{(b)(c)}A^{(b)}_{\mu}A^{(c)}_{\nu}.$
(7)

Here,
$Tr\left[F^{(a)}_{\mu\nu}F^{(a)\mu\nu}\right]=\sum^{3}_{\mu\nu}F^{(a)}_{\mu\nu}F^{(a)\mu\nu}$,
$R$ and $\gamma$ are the scalar curvature and a positive real parameter,
respectively, $C^{(a)}_{(b)(c)}$ represent the structure constants of three-
parameter Lie group $G$, $\xi$ is the coupling constant, and $A^{(a)}_{\mu}$
represents the SO(3) gauge group Yang-Mills (YM) potentials defining by the
Wu-Yang (WY) ansatz Balakin2016 ; Balakin2007 ; Balaki2007 . Variation of the
action with respect to the spacetime metric $g_{\mu\nu}$ yields the field
equations

$\displaystyle
G^{\mu}_{~{}\nu}+\Lambda\delta^{\mu}_{~{}\nu}=T^{\mu}_{~{}\nu},$ (8)
$\displaystyle
T^{\mu}_{~{}\nu}=-\frac{1}{2}\left(\delta^{\mu}_{~{}\nu}\mathcal{F}^{\gamma}-4\gamma
Tr\left[F^{(a)}_{\nu\rho}F^{(a)\mu\rho}\right]\mathcal{F}^{\gamma-1}\right).$
(9)

It is obviously that the EPYM theory with $\gamma=1$ can reduce to the
standard Einstein-Yang-Mills theory Habib2007 ; Mazharimousavi2008 . In the
following we mainly pay attention on the role of the non-linear YM charge
parameter $\gamma$. With the WY ansatz for the YM field, the invariant for
this field becomes

$\displaystyle
Tr\left[F^{(a)}_{\mu\rho}F^{(a)\mu\rho}\right]=\frac{q^{2}}{r^{4}},$ (10)

which will lead to the disappearance of the structure constants described the
non-Abelian property of the YM gauge field.

The four-dimensional static spherical symmetry EPYM black hole solution in the
AdS spacetime reads Yerra2019

$\displaystyle
f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}+\frac{(2q^{2})^{\gamma}}{2(4\gamma-3)r^{4\gamma-2}}.$
(11)

Here, $q$ is the YM charge, the non-linear YM charge parameter satisfies
$\gamma\neq 0.75$ and $\gamma>0$. In the extended phase space the cosmological
constant $\Lambda$ is interpreted as the pressure $P=-\frac{\Lambda}{8\pi}$,
whose dual quantity is volume $V=\frac{4\pi r_{+}^{3}}{3}$. Since the YM
charge term of this theory with $\gamma=1$ have the same form as the Maxwell
charge term in the Einstein-Maxwell-Yang-Mills (EMYM) theory, the role of the
YM charge will be similar as that of the Maxwell charge in EMYM theory. The
only difference between them is that they have different gauge groups, i.e.,
the YM field is of SO(3), the Maxwell field is of U(1). These results are
consistent with that in Ref. Moumni2018 . With the above metric function, the
black hole hawking temperature, mass parameter, and entropy are, respectively,

$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r_{+}}\left(1+8\pi
Pr_{+}^{2}-\frac{(2q^{2})^{\gamma}}{2r_{+}^{4\gamma-2}}\right),~{}S=\pi
r_{+}^{2}$ (12) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{r_{+}}{2}\left(1+\frac{8\pi
Pr_{+}^{2}}{3}+\frac{(2q^{2})^{\gamma}}{2(4\gamma-3)r_{+}^{4\gamma-2}}\right).$
(13)

Therefore, the thermal potential of the EPYM AdS black hole is expressed as

$\displaystyle U$ $\displaystyle=$ $\displaystyle\int\left(T_{+}-T\right)dS$
(14) $\displaystyle=$ $\displaystyle\frac{1}{2}r_{+}-T\pi
r_{+}^{2}+\frac{4}{3}P\pi
r_{+}^{3}-\frac{\left(2q^{2}\right)^{\gamma}}{4(3-4\gamma)}r_{+}^{3-4\gamma}$

Since the various black hole states are located at the extreme points of the
thermal potential function, so we have

$\displaystyle h(r_{+})=\frac{dU(r_{+})}{dS(r_{+})}=-T+\frac{1}{4\pi
r_{+}}+2Pr_{+}-\frac{(2q^{2})^{\gamma}}{8\pi}r_{+}^{1-4\gamma}$ (15)

We can transform the black hole thermodynamic properties into solving the
zero-points of the above function $h(r_{+})$. To investigate the whole zero-
points properties of the thermal potential function, the method of complex
analysis should be applied to make the real thermal potential to the one with
complex continuation, i.e., $h(r_{+})\rightarrow h(z)$. The different black
hole states correspond to the positive real zero-points of $h(z)$. Its
analytical function with complex continuation has the following form

$\displaystyle
h(z)=\frac{1}{z^{4\gamma-1}}\left(-Tz^{4\gamma-1}+\frac{z^{4\gamma-2}}{4\pi}+2Pz^{4\gamma}-\frac{(2q^{2})^{\gamma}}{8\pi}\right).$
(16)

In the complex analysis, the Argument Principle is often applied to calculate
the number of zero-points for analytic functions. For a meromorphic function
$h(z)$ in a closed cuntour $C$, which is analytically nonzero on $C$, we have

$\displaystyle N(h,C)-P(h,C)=\frac{1}{2\pi
i}\oint_{C}\frac{h^{\prime}(z)}{h(z)}dz=\frac{\Delta_{C}arg~{}h(z)}{2\pi},$
(17)

where $N(h,C)$ and $P(h,C)$ are the number of zero-points and poles for the
analytic function $h(z)$, $arg~{}h(z)$ stands for the argument of $h(z)$. With
a transformation $\omega=h(z)$, the above equation can be rewritten as the
number of rotations of $\omega$ around the origin of $C^{\prime}$ along with
the complex variable $z$ moving around $C$. Here $C^{\prime}$ is the image
curve of $C$ after this transformation. Thus the topological number can be
denoted as

$\displaystyle W:=\frac{1}{2\pi
i}\oint_{C^{\prime}}\frac{d\omega}{\omega}=\frac{1}{2\pi
i}\oint_{C^{\prime}}\frac{h^{\prime}(z)}{h(z)}dz.$ (18)

Specially, for the analytic function $h(z)$ without poles inside $C$, i.e.,
$P(h,C)=0$, the topological number equals to the number of zero-points
$W=N(h,C)$. Here we should point out that $N(h,C)$ is determined by the order
of the complex various $z$. For the AdS EPYM black hole system, for
convenience we require that the powers of all order of $z$ must be positive
integers, i.e., $\gamma=\frac{n+2}{4}$ with the positive integer $n$ ($n>1$).
The equation (19) becomes

$\displaystyle
h(z)=\frac{1}{z^{n+1}}\left(-Tz^{n+1}+\frac{z^{n}}{4\pi}+2Pz^{n+2}-\frac{(2q^{2})^{n+2}}{8\pi}\right),$
(19)

which have $n+2$ zero-points and a $n+1$-order pole in the entire complex
plane. Thus the topological number of this system is $W=1$ for different
values of $n$, i.e., it is independent with the non-linear YM charge
parameter. Note that for some certain values of the pressure and YM charge,
the EPYM AdS black hole will reduce to different black hole systems and the
corresponding topological number will change for different black hole systems
(see as Tab. 1). These results are consistent with that in Refs. Wei2022 ;
Fang2023

Table 1: Topological numbers of different black hole system with different values of $P,~{}q$ $P,~{}q$ |  $P=q=0$ Sch-BH |  $P=0,~{}q\neq 0$ RN-BH |  $P\neq 0,~{}q=0$ Sch-AdS-BH |  $P\neq 0,~{}q\neq 0$ EPYM-AdS-BH
---|---|---|---|---
$N(h,C)$ | 0 | $n+1$ | 2 | $n+2$
$P(h,C)$ | 1 | $n+1$ | 1 | $n+1$
$W$ | -1 | 0 | 1 | 1

In the following, we will verify the above viewpoint. For AdS EPYM black
holes, for any values of the non-linear YM charge parameter the critical
points read as Du2021

$\displaystyle(r_{+}^{c})^{4\gamma-2}$ $\displaystyle=$
$\displaystyle(2q^{2})^{\gamma}f(1,\gamma),~{}~{}~{}f(1,\gamma)=\gamma(4\gamma-1),$
(20) $\displaystyle T_{+}^{c}$ $\displaystyle=$
$\displaystyle\frac{2\gamma-1}{4\gamma-1}\frac{1}{\pi(2q^{2})^{\gamma/4\gamma-2}f^{1/4\gamma-2}(1,\gamma)},$
(21) $\displaystyle P^{c}$ $\displaystyle=$
$\displaystyle\frac{2\gamma-1}{16\pi\gamma(2q^{2})^{\gamma/2\gamma-1}f^{1/2\gamma-1}(1,\gamma)},$
(22) $\displaystyle G^{c}$ $\displaystyle=$ $\displaystyle
M(r_{+}^{c},~{}P^{c})-T_{+}^{c}S^{c}.$ (23)

For the sake of discussion, we introduce the dimensionless thermodynamic
quantities as follows

$\displaystyle
p\equiv\frac{P}{P^{c}},~{}~{}~{}~{}t\equiv\frac{T}{T_{+}^{c}},~{}~{}~{}~{}x\equiv\frac{r_{+}}{r_{+}^{c}},~{}~{}~{}~{}s\equiv\frac{S}{S^{c}},~{}~{}~{}~{}u\equiv\frac{U}{U^{c}},~{}~{}~{}~{}g\equiv\frac{G}{G^{c}}.$
(24)

Here
$U^{c}=\frac{1}{CQ^{\gamma}}\left(-\frac{12(2\gamma-1)}{4\gamma-1}+6Q^{2\gamma(1-\gamma)}+\frac{2\gamma-1}{\gamma}Q^{6\gamma(1-\gamma)}+\frac{3}{4\gamma-3}C^{4(\gamma-1)})\right)$
with $C=\left[\gamma(4\gamma-1)\right]^{1/2(1-\gamma)}$ and
$Q=(2q^{2})^{\frac{1}{2(1-2\gamma)}}$. Thus the thermal potential reads

$\displaystyle
u=\frac{1}{CQ^{\gamma}U^{c}}\left(-\frac{12(2\gamma-1)}{4\gamma-1}tx^{2}+6Q^{2\gamma(1-\gamma)}x+\frac{2\gamma-1}{\gamma}Q^{6\gamma(1-\gamma)}px^{3}+\frac{3}{4\gamma-3}C^{4(\gamma-1)})x^{3-4\gamma}\right).$
(25)

From Eq. (25) we can seen that there exist two key parameters ($p$ and $t$)
affecting the behaviors of the thermal potential. Generally, there are two
methods to obtain the information of the phase transition: one is the Gibbs
free energy, another is the Maxwell’s equal area law. Here we adopt the first
method to investigate the property of phase transition for this system. On the
other hand the authors Wei2022 ; Fang2023  proposed that the local topological
properties of thermodynamical systems can be reflected by the winding numbers
as the defects, i.e., for the local stable black hole state the winding number
equals to one, for the local unstable black hole state it becomes minus one.
The global topological nature can be classified by the topological number
which is the sum of all local winding numbers. In the following we will apply
indirectly these results to this system.

(a) $p=0.125$

(b) $p=0.125,~{}t=t_{1}$

(c) $p=0.125,~{}t_{1}<t<t_{2}$

(d) $p=0.125,~{}t=t_{2}$

(e) $p=0.125,~{}t_{2}<t<t_{3}$

(f) $p=0.125,~{}t=t_{3}$

Figure 2: The plots of the Gibbs free energy and the thermal potential versus
temperature with different values of the dimensionless pressure. In $g-t$
plane, the red line is the small black hole branch, the green line is the
middle black hole branch, and the blue line is the large black hole branch.
The parameters are set to $\gamma=1.25,~{}q=1/\sqrt{2}$.

The swallowtail behavior in the $g-t$ phase diagram and the corresponding
black hole states with different temperatures in $u-x$ planes are exhibited in
Fig. 2. As $t=t_{1}$ shown in Fig. 2(a) the free energies of the large and
middle black hole states are the same, for the small black hole state it is
lowest that indicates the system is in the stable small black hole state,
while as $t=t_{3}$ the free energies of the small and middle black hole states
are the same, the large black hole state is lowest. The corresponding results
are expressed in Figs. 2(b) and 2(f): the thermal potential only has a global
minimum that stands for the stable small black hole state when $t=t_{1}$ or
that stands for the stable large black hole state as $t=t_{3}$. For these two
cases there is one winding number of the stable black hole state, so the
topological number equals to one. From Fig. 2(d) it is obviously that when
$t=t_{2}$ (i.e., the first-order phase transition point) the thermal potential
have two global minima and a local maximum, and the thermal potentials of two
minima equal to each other. That is to say the system is in the coexistence of
the stable large and small black hole states, while the middle black hole
state is local unstable. Hence there are three winging numbers: 1, 1, -1, and
he topological number of this system undergoing a first-order phase transition
is one ($1+1-1=1$). For other two cases ($t_{1}<t<t_{2}$ and $t_{2}<t<t_{3}$)
shown in Figs. 2(c) and 2(e), the thermal potential have two local stable
minima and one local unstable maximum, the corresponding winding numbers of
these states are 1, 1, -1 and the topological number is one. Though above
analysis, for a given pressure no matter whether the system undergos a phase
transition, the topological number is an invariant ($W=1$). The results are
the same as that obtained by the complex analysis.

##  IV Rate of the first-order phase transition

Based on the black hole molecular hypothesis Wei2015 , the black hole phase
transition is the rearrangement of black hole molecules due to the thermal
fluctuation in the thermal potential. Fig. 2(e) might describe a molecular
rearrangement. We assume that the ensemble temperature is much lower than the
barrier height. Molecules will spend a lot of time near the potential minimum
(B), and only rarely will Brownian motion take them to the top of the barrier
(A). Once the molecule reaches the top of the barrier, it is likely to fall
equally to either side of the barrier. If it moves to the right-hand side, it
will rapidly fall to the other minimum (C), stay there for a while, and then
perhaps cross back to the original minimum (B). In order to depict the
crossing rate (B$\rightarrow$A or C$\rightarrow$A), we introduce the Kramer’s
rate Risken1988

$\displaystyle\nu=\frac{\sqrt{|u^{\prime\prime}(x_{min})u^{\prime\prime}(x_{max})|}}{2\pi}e^{-\frac{x_{max}-x_{min}}{D}},$
(26)

where $D$ is the constant diffusion coefficient, $x_{max}$ and $x_{min}$ are
the corresponding locations of the thermal potential extreme points and they
can be obtained by solving the equation of $u(x)^{\prime}=0$. ′′ stands for
the second derivative of the thermal potential $u(x)$. We define the crossing
rate from B to A (or from C to A) as $\nu_{1}$ (or $\nu_{2}$)

$\displaystyle\nu_{1}=\frac{\sqrt{|u^{\prime\prime}(x_{1})u^{\prime\prime}(x_{2})|}}{2\pi}e^{-\frac{x_{2}-x_{1}}{D}},~{}~{}\nu_{2}=\frac{\sqrt{|u^{\prime\prime}(x_{3})u^{\prime\prime}(x_{2})|}}{2\pi}e^{-\frac{x_{3}-x_{2}}{D}}.$
(27)

Namely, $\nu_{1}$ denotes the rate of phase transition from the small black
hole to the large black hole, $\nu_{2}$ is the one from the large black hole
to the small black hole. The phase transition rates and their difference
($\Delta\nu=\nu_{1}-\nu_{2}$) for different black hole states are depicted in
Fig. 3. When the ensemble temperature equals $t_{1}$ or $t_{3}$, the phase
transition rate is zero, which means that no phase transition emerges. With
the increasing of the ensemble temperature from $t_{1}$ to $t_{3}$, both two
rates show a trend of increasing first and then decreasing. At $t=t_{2}$ the
depths of thermal potential between two minima and local maximum, while the
corresponding phase transition rates are not equal to each other
($\nu_{1}>\nu_{2}$). This indicates the transition process from the small
black hole to the large black hole is far from that of the transition from the
large black hole to the small black hole. Two rates will be the same as
$t=t_{*}$, which indicates the phase transition reaches a dynamic equilibrium
at this time. The corresponding difference of two rates is positive for
$t_{1}<t<t_{*}$, it is zero as $t_{t}*$ and is negative in the range of
$t_{*}<t<t_{3}$. In short, the phase transition between different black holes
presents very asymmetric features, and the overall process of this system
initial being in the stable large black hole state is dominated by the phase
transition from a small black hole to a large black hole. On the contrary, if
the system is initially in the stable small black hole state the phase
transition becomes the one from the large black hole state to the small black
hole state.

(a) $p=0.125$

(b) $p=0.125$

Figure 3: The transition rates $\nu_{1,2}$ and their difference $\Delta\nu$
with respect to the ensemble temperature $t$ at the pressure $p=0.125$. The
parameters are set to $\gamma=1.25$, $q=1/\sqrt{2}$, and $D=20$.

##  V Discussions and Conclusions

Different black hole solutions are the different defects from the viewpoint of
thermodynamics, which can be characterized by different topological numbers
and belong to different topological classes. In this manuscript, we considered
the EPYM AdS black hole in a thermal bar with the thermal potential and the
whole system was in thermodynamics equilibrium. Based on the construction of
the thermal potential, we probed the topology of the whole system with
different black hole states by the method of Argument Principle. When the
system with $q=0,~{}\Lambda=0$ reduces to the Schwarzschild black hole, the
topological number is negative one. For the RN black hole system (i.e,
$q=0,~{}\Lambda<0$) the topological number is zero. For the charged AdS black
hole systems (i.e, $q\neq 0,~{}\Lambda<0$), for example RN-AdS black hole and
EPYM AdS black hole, the topological number is one.

Next we checked out these topological numbers by the method of the Duan’s
$\phi$-mapping topological current when the system was in different states. In
other words, for a certain black hole system its topological number is an
invariant and is independent of whether the system undergoes a phase
transition. Finally, in order to depict the direction of phase transition we
introduced the Kramer’s rate to investigate the crossing rate between local
stable large/small black hole states and local unstable middle black hole
state. The results indicated that if the system is initial in the stable large
black hole state, the process of phase transition is the from the small black
hole state to the large black hole state, and vice versa.

## Acknowledgements

We would like to thank Profs. Meng-Sen Ma and Zen-Ming Xu for their
indispensable discussions and comments. This work was supported by the
National Natural Science Foundation of China (Grant No. 12075143, 12375050),
the Natural Science Foundation of Shanxi Province, China (Grant No.
202203021221209), and the Teaching Reform Project of Shanxi Datong Universtiy
(Grant No. XJG2022234).

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