[a]Kirill Boguslavski

# Nonperturbative excitations in overoccupied gluon plasmas

###### Abstract

Motivated by the early-time dynamics of the quark-gluon plasma in high-energy
heavy-ion collisions, we extract gluonic spectral functions of overoccupied
gauge theories far from equilibrium using classical-statistical lattice
simulations and linear response theory. In 3+1 dimensions we find that the
spectral function exhibits quasiparticle excitations at all momenta that are
mostly consistent with perturbative hard-thermal loop predictions, while
partially showing nonperturbative deviations [1]. In contrast, the structure
of excitations in 2+1 dimensions is nontrivial and nonperturbative [2, 3].
These nonperturbative interactions lead to broad excitation peaks in the
spectral function, demonstrating the absence of soft quasiparticles in these
theories. This also suggests that there may be significant nonperturbative
corrections present in systems with large momentum anisotropy, which are
relevant to phenomenological applications in heavy-ion collisions.

##  1 Introduction

Motivated by heavy-ion collision experiments, our goal is to study microscopic
properties of QCD out of equilibrium from first principles, and to assess
their impact on the description of the quark-gluon plasma. This can be done
efficiently by studying the spectral functions $\rho(\omega,p)$ of gluons or
quarks since they encode the full single-particle excitation spectrum.

Having the initial stage of heavy-ion collisions in mind [4, 5], we use a
weak-coupling ($g^{2}\ll 1$) far from equilibrium approach where we study a
highly occupied gluon plasma with a large gluon distribution function
$f(p)\sim 1/g^{2}$ for soft momenta below a characteristic scale $p\lesssim
Q$. In this case both nonperturbative lattice and perturbative diagrammatic
methods are available. Focusing on the former, we run classical-statistical
lattice simulations in real time and extract the spectral functions. Our
results are then compared to perturbative hard-thermal loop (HTL) calculations
extended to out-of-equilibrium situations to assess which properties of the
spectral functions are beyond HTL results. Our main questions involve:

  * $\star$

What are the spectral functions in heavy-ion collisions at early times?

  * $\star$

Do they have general features that are common in other non-equilibrium states
or in thermal equilibrium at high temperatures?

  * $\star$

Are nonperturbative extensions of kinetic theory or anisotropic HTL necessary
for a faithful description of the dynamics?

To address these questions, we extract spectral functions in a linear-response
framework in classical-statistical simulations. Here we apply our framework in
Minkowski space-time to isotropic 3+1 dimensional plasmas and (effectively)
2+1 dimensional plasmas, corresponding to the limit of extreme momentum
anisotropy. Details of our approach and results can be found in Refs. [2, 3,
1]. Planned extensions of these studies are outlined in the Conclusion.

##  2 Method & setup

We consider $\mathrm{SU}(N_{c})$ Yang-Mills theory with classical action

$S_{\mathrm{Y}M}[A]=-\frac{1}{4}\,\int
d^{d+1}x\;F_{a}^{\mu\nu}[A]F^{a}_{\mu\nu}[A]$ (1)

in $d$ spatial dimensions. Simulations are performed for $N_{c}=2$ in a gauge-
covariant formulation with link fields
$U_{j}(x)\approx\exp\left(ig\,a_{s}A_{j}(x)\right)$ discretized on a cubic
lattice of size $N_{s}^{3}$ with lattice spacing $a_{s}$.

Motivated by different initial stages in heavy-ion collisions, we study three
models for highly occupied gluonic systems (here in a Minkowski space-time for
simplicity): isotropic 3+1 dimensional system (i.e., $d=3$ in (1)), a purely
2+1 dimensional plasma ($d=2$), and a Glasma-like 2+1 dimensional system. The
latter extends the 2+1D model by including an adjoint scalar field $\phi_{a}$,
and corresponds to a $3+1D$ system with independence of the $x^{3}$
coordinate.

In each case we initialize the system with large occupation numbers (with
$E=\partial_{t}A$)

$\displaystyle f(t{=}0,p\lesssim Q)\sim\frac{1}{g^{2}}\gg 1\;,\qquad
f(t,p)=\frac{\langle|E_{T}(t,p)|^{2}\rangle}{p}\;.$ (2)

###  2.1 Classical-statistical real-time lattice simulations

For classical-statistical lattice simulations as employed in our studies, we
first set the initial conditions by choosing fields randomly that satisfy
$\langle E_{T}^{*}(0,\mathbf{p})E_{T}(0,\mathbf{q})\rangle\propto
p\,f(0,p)\,(2\pi)^{3}\delta(\mathbf{p}-\mathbf{q})$ with transverse fields
$E_{T}^{j}p_{j}=0$ and vanishing $\langle
E_{L}^{*}(0,\mathbf{p})E_{L}(0,\mathbf{q})\rangle=0$. We set the gauge fields
$A$ similarly, and restore the Gauss law initially if needed. Each of the
initial configurations $\\{U(0,\mathbf{x}),E(0,\mathbf{x})\\}$ is evolved by
solving the classical field equations in a leapfrog scheme in temporal gauge
$A_{0}=0$, i.e., $U_{0}=\mathds{1}$. An observable $O[U,E]$ that depends on
the fields is computed at a time $t>0$ by averaging over the initial
configurations

$O(t)=\frac{1}{\\#k}\sum_{k}O[U(t),E(t)]\,.$ (3)

###  2.2 Self-similar turbulent attractors

Figure 1:  Gluon distribution function $f_{E}\equiv f$ as a function of
momentum of a Glasma-like plasma. Left: Different initial conditions lead to
the same attractor (shown at $Qt=4000$). Right: Rescaling $f$ and $p$ with
universal power-laws of time signals self-similarity. Figures taken from Ref.
[3].

Each of the highly occupied systems approaches a self-similar attractor with
defining property

$\displaystyle f(t,p)=(Qt)^{\alpha}f_{s}\left((Qt)^{\beta}p\right).$ (4)

The scaling exponents $\alpha$, $\beta$ and scaling function $f_{s}(p)$ are
universal, insensitive to details of initial conditions, and given by

$\displaystyle\beta=-1/5\;,$ $\displaystyle\alpha=3\beta\qquad\text{for 2+1D
and Glasma-like \cite[cite]{[\@@bibref{Number}{Boguslavski:2019fsb}{}{}]}}$
(5) $\displaystyle\beta=-1/7\;,$ $\displaystyle\alpha=4\beta\qquad\text{for
isotropic 3+1D
\cite[cite]{[\@@bibref{Number}{Berges:2008mr,Kurkela:2012hp}{}{}]}.}$ (6)

This is shown in Fig. 1 for the example of 2+1D Glasma-like simulations. In
the left panel, different initial conditions ($Qt=0$) lead to the same
universal distribution at a later time ($Qt=4000$). In the right panel,
distributions at different times fall on top of each other after rescaling
with power-laws of time with the universal exponents in (5).

###  2.3 Spectral and statistical correlation functions

Due to Eq. (4), the evolution of self-similar attractors is known. We extract
the spectral and statistical correlation functions at these attractor states
because their scaling properties allow us to study the dependence of the
correlators on time and momentum scales. For instance, if a quantity in the
spectral function scales like a mass $\sim Q(Qt)^{\beta}$, it is likely that
the soft scale plays an important role. We will return to this below.

Hiding indices, the spectral function is defined as the commutator of fields

$\rho(x^{\prime},x)=\frac{i}{N_{c}^{2}-1}\left\langle\left[\hat{A}(x^{\prime}),\hat{A}(x)\right]\right\rangle$
(7)

and we also denote $\dot{\rho}=\partial_{t}\rho$ with $t\equiv x^{0}$. The
statistical correlation function $\langle EE\rangle$ ($\equiv\ddot{F}$) is in
general, out of equilibrium, independent of $\dot{\rho}$ and defined as the
anti-commutator of fields

$\langle
EE\rangle(x^{\prime},x)=\frac{1}{2(N_{c}^{2}-1)}\left\langle\left\\{\hat{E}(x^{\prime}),\hat{E}(x)\right\\}\right\rangle\,.$
(8)

To extract the correlation functions from our simulations, we first fix the
residual gauge freedom to Coulomb-type gauge
$\left.\partial^{j}A_{j}\right|_{t}=0$ at time $t$ and Fourier transform the
correlations in $t^{\prime}-t$ and $\mathbf{x}^{\prime}-\mathbf{x}$ to
frequency $\omega$ and momentum $\mathbf{p}$. Instead of doing this at fixed
$\bar{t}=\frac{1}{2}(t+t^{\prime})$, we keep $t$ fixed, which requires the
approximation $t\approx\bar{t}\gg\Delta t\equiv t^{\prime}-t$ that is
satisfied in our simulations. The statistical correlator can be measured as

$\langle EE\rangle(t^{\prime},t,p)=\frac{1}{N_{c}^{2}-1}\left\langle
E(t^{\prime},\mathbf{p})E^{*}(t,\mathbf{p})\right\rangle\,.$ (9)

###  2.4 Nonperturbative computation of spectral function $\rho$

Figure 2: Visualization of the method to compute $\rho$ (see [1]).

In order to measure the spectral function, we combine classical-statistical
simulations with linear response theory, as introduced in Ref. [1] and
visualized in Fig. 2 on the left.

At time $t$ we perturb the fields with an instantaneous source
$j(t^{\prime},\mathbf{p})=j_{0}(\mathbf{p})\,\delta\left(t^{\prime}-t\right)$.
Splitting $A(t,\mathbf{x})\mapsto A(t,\mathbf{x})+\delta A(t,\mathbf{x})$ due
to the perturbation, the background field $A$ keeps following the classical
Yang-Mills equations while for the linear response field $\delta A$ we solve
linearized equations of motion formulated in a gauge-covariant form in Ref.
[8] that are consistent with Gauss law conservation by construction. The
solution is linked to the retarded propagator $\langle\delta
A(t^{\prime},\mathbf{p})\rangle=G_{R}(t^{\prime},t,\mathbf{p})\,j_{0}(\mathbf{p})$
and to the spectral function via
$\theta(t^{\prime}-t)\rho(t^{\prime},t,p)=G_{R}(t^{\prime},t,p)$.

While the spectral function in thermal equilibrium is usually computed by
using the statistical correlation function and relating it to the spectral
function via the fluctuation dissipation relation (FDR) [9, 10], a separate
and independent computation of the spectral function like ours is crucial out
of equilibrium since the existence of a FDR is not guaranteed. Similar methods
to the one for gauge theories described here have also been developed for
scalar systems far from equilibrium [11, 12] and recently introduced for
fermionic spectral functions in Ref. [13]. There the Dirac field and the Dirac
equation play the role of the linear response field and linearized equation of
motion, respectively.

##  3 Numerically extracted spectral functions

###  3.1 Gluon spectral function in isotropic 3+1D plasmas

Figure 3: Correlations in isotropic 3+1D systems. Left: Spectral function of
transversely polarized gluons $\rho_{T}(\omega,p)$ for different momenta in
relative time $\Delta t$ and frequency domains. Right: Normalized correlations
$\ddot{F}_{T}/\ddot{F}_{T}(t,t)\equiv\langle E_{T}E_{T}\rangle/\langle
E_{T}(t)E_{T}(t)\rangle$ and $\dot{\rho}_{T}$ for different momenta. Figures
taken from Ref. [1].

Figure 4: Damping rates $\gamma_{\alpha}(t,p)/Q$ of an isotropic 3+1D plasma
in the self-similar regime as a function of momentum. Left: For transverse and
longitudinal polarizations ($\alpha=T,L$) and different discretizations at
fixed time $Qt=1500$. Right: Transverse damping rate $\gamma_{T}(t,p)$
rescaled by $(t/t_{\text{ref}})^{3/7}$ as a function of rescaled momentum at
different times, with $Qt_{\text{ref}}=1500$. Figures taken from Ref. [1].

We start with Fig. 3 where we discuss the gluonic spectral function of an
isotropic 3+1D plasma at a self-similar attractor [1]. In the left panel we
show the spectral function of transversely polarized gluons $\rho_{T}$ at
different momenta $p$ as functions of relative time $\Delta t\equiv
t^{\prime}-t$ (left) and frequency $\omega$ (right). The spectral function is
shown to exhibit two distinguishable features: a narrow Lorentzian
quasiparticle peak at the position $\omega_{T}(p)$ (dispersion relation) with
the width $\gamma_{T}(p)$ (damping rate), and the Landau cut region at
frequencies $|\omega|<p$. Perturbative results at leading order (LO) HTL
(black dashed lines) describe the main features well. The existence of a
generalized fluctuation dissipation relation (FDR) between statistical and
spectral correlations

$\frac{\langle EE\rangle_{\alpha}(t,\omega,p)}{\langle
EE\rangle_{\alpha}(t,\Delta
t{=}0,p)}\approx\frac{\dot{\rho}_{\alpha}(t,\omega,p)}{\dot{\rho}_{\alpha}(t,\Delta
t{=}0,p)}\,,$ (10)

with $\ddot{F}\equiv\langle EE\rangle$,
$\dot{\rho}_{T}(\omega,p)\approx\omega\rho_{T}(\omega,p)$,
$\dot{\rho}_{T}(t,\Delta t{=}0,p)=1$ and $\alpha=T,L$ corresponding to
transverse and longitudinal polarizations, is shown in the right panel of Fig.
3 using transversely polarised gluons as an example.

The damping rates are extracted from $\dot{\rho}_{\alpha}(\omega,p)$ by
performing fits with a Lorentzian shape and are shown in Fig. 4. One observes
in the left panel that $\gamma_{\alpha}$ agree for both polarizations for
momenta below the mass $p\lesssim m\approx 0.15\,Q$, despite having different
dispersion relations (see Fig. 12 in [1]). Comparing the values, one finds a
separation of scales

$\gamma_{\alpha}(t,p)\ll\omega_{\alpha}(t,p)\ll Q\,,$ (11)

where $\omega(t,p{=}0)\equiv\omega_{\mathrm{pl}}\sim m$. For comparison, we
have included curves from different simulations to show the insensitivity to
discretization parameters $N_{s}$ and $a_{s}$ of our results.

The finite peak width is a subleading order effect in HTL and is of order
$\gamma_{\alpha}(t,p)\sim g^{2}T_{*}(t)\sim Q(Qt)^{-3/7}$ in the self-similar
regime. The point $\gamma_{\rm HTL}(p{=}0)$ corresponds to the HTL result at
$p=0$ [14]. In the right panel of Fig. 4, $\gamma_{T}$ is rescaled with
$g^{2}T_{*}(t)$ and becomes time-independent when plotted as a function of
rescaled momentum. This confirms $\gamma_{\alpha}(t,p)\sim g^{2}T_{*}(t)$ and
implies that $\frac{\gamma_{\alpha}(t,p)}{\omega(t,p{=}0)}\sim(Qt)^{-2/7}$
decreases with time, making the quasiparticle peaks even more narrow.

Figure 5:  Normalized transverse (left) and longitudinal (right) correlations
$\dot{\rho}_{\alpha}(\omega,p)/\dot{\rho}_{\alpha}(\Delta t{=}0,p)$ and
$\langle EE\rangle_{\alpha}(\omega,p)/\langle EE\rangle_{\alpha}(\Delta
t{=}0,p)$ in 2+1D systems for different momenta. Green dashed lines correspond
to HTL results. Figures taken from Ref. [2].  Figure 6:  Left: Dispersion
relation $\omega_{T}(t,p)/\omega_{\mathrm{pl}}(t)$ and right: damping rate
$\gamma_{T}(t,p)/\omega_{\mathrm{pl}}(t)$ of the Glasma-like system at times
$Qt=200$, $500$, $2000$ as a function of $p/\omega_{\mathrm{pl}}(t)$. Figures
taken from Ref. [2].  Figure 7:  Gluon spectral function in 2+1D classical
thermal equilibrium with $f=T/\omega(p)$ (from [2]).

###  3.2 Limit of extreme anisotropy: spectral functions in 2+1D plasmas

Figure 8: Blue: 3+1D distribution, red: 2+1D limiting distribution.

So far, we have discussed the situation of an isotropic 3+1 dimensional system
with distribution $f(t,p)$. However, in heavy-ion collisions, the distribution
function can be very anisotropic at initial stages, with the limiting case
being an effectively 2+1 dimensional system with distribution
$f(t,p_{T},p_{z}{=}0)$, as illustrated in Fig. 8 on the left. We will again
employ Minkowski metric for simplicity, as in the 3+1D case, and extract
unequal-time correlation functions from our simulations at the corresponding
self-similar attractor (see Ref. [2] for a more detailed discussion of our
results).

The normalized spectral and statistical correlation functions are shown in
Fig. 5 for both transverse and longitudinal polarizations. Both types of
correlators lie on top of each other to good accuracy, confirming the validity
of the generalized FDR (10) also for 2+1 dimensional systems.

However, there are several differences to the 3+1D case: the resonances are
broad peaks and the correlators remain finite even for $\omega\to 0$. They
have a non-Lorentzian shape that is well approximated by a Gaussian
distribution, as shown in Fig. 5. The green dashed lines correspond to the HTL
results that would be expected in 2+1D, but fail to reproduce our
nonperturbative simulation results. The latter is not too surprising since, as
argued already in Ref. [3], 2+1 dimensional gauge systems lack the scale
separation that is required for a hard loop approximation underlying HTL
calculations.

More concretely, we show in Fig. 6 the transverse dispersion relation and
damping rate of the 2+1D Glasma-like system at different times. Rescaling all
dimensionful quantities by the plasmon mass $\omega_{\mathrm{pl}}(t)$, one
finds that both become time-independent. This leads to

$\displaystyle\gamma(t,p)$ $\displaystyle\sim$
$\displaystyle\omega_{\mathrm{pl}}(t)\,,$ (12) $\displaystyle\gamma(t,p)$
$\displaystyle\sim$ $\displaystyle\omega(t,p)\qquad\text{for}\quad
p\lesssim\omega_{\mathrm{pl}}\,.$ (13)

Due to their short lifetime, this implies that no quasiparticles exist for
$p\lesssim\omega_{\mathrm{pl}}$. Note that these statements remain true also
at later times, in sharp contrast to the scale separation we observed for
isotropic 3+1D gluonic plasmas in Eq. (11).

We find the same properties also in classical thermal equilibrium, as shown in
Fig. 7. Running simulations with a classical thermal distribution
$f(p)\approx\frac{T}{\omega(p)}$, we observe that $\rho(\omega,p)$ behaves
qualitatively similarly to our far-from-equilibrium case above: one finds
broad gluonic excitations of non-Lorentzian shape with
$\gamma(p)\sim\omega_{\mathrm{pl}}$, HTL provides a poor description, and for
$\omega\to 0$ the spectral function $\dot{\rho}_{T}=\omega\rho_{T}$ remains
finite at low $p$.

This indicates that the extracted properties in 2+1D theories are not special
to the non-equilibrium self-similar regime but also emerge in the time-
translation invariant classical thermal equilibrium, and may be even found in
more general 2+1 dimensional states.

##  4 Conclusion

In this work we have presented a non-perturbative tool developed in [1] to
extract gluonic spectral functions in highly occupied non-Abelian plasmas far
from equilibrium in classical-statistical lattice simulations, and that we
have applied to 3+1 and (effectively) 2+1 dimensional systems at classical
self-similar attractors (see Refs. [2, 3, 1] for details).

In general, we find that even far from equilibrium there exists a generalized
fluctuation-dissipation relation that relates spectral and statistical
correlation functions. However, there are crucial differences between the
studied systems. On the one hand, the gluonic spectral function of an
isotropic 3+1 dimensional plasma is well described by HTL perturbation theory.
Our method allowed us to determine the damping rates of transverse and
longitudinal quasiparticle excitations as functions of momentum from first
principles. These were much smaller than the dispersion relation, showing that
quasiparticles are long-lived and a perturbative description is, indeed,
justified.

In contrast, the gluonic spectral functions of 2+1 dimensional plasmas agree
poorly with HTL predictions. They exhibit broad non-Lorentzian excitation
peaks with a width of the order of the plasmon mass. This is observed for
different times and even in classical thermal equilibrium, indicating that
these properties are not special to the classical self-similar attractor, but
may be even found in more general 2+1 dimensional states. As a consequence of
the broad peaks, momentum modes up to the mass scale have only short-lived
excitations and have to be treated non-perturbatively.

These results are relevant for the description of gluonic excitations at low
momenta in heavy-ion collisions and for anisotropic HTL perturbation theory.
They show that 2+1D plasmas have genuinely nonperturbative properties. Since
the effectively 2+1D system stems from an extreme momentum anisotropy of the
plasma, our results suggest that there may be significant nonperturbative
corrections present also in systems with finite but large anisotropy, which
are relevant to phenomenological applications in heavy-ion collisions.

To approach these questions in more detail, the simulations will be extended
to expanding and anisotropic plasmas to study what may be lacking in the
current kinetic [15, 16, 17] and HTL descriptions. Moreover, we study how
nonperturbative properties in correlation functions affect transport
coefficients, which has been started in Ref. [18].

## Acknowledgments

The author is grateful to A. Kurkela, T. Lappi and J. Peuron for collaboration
on different projects presented here. He would also like to thank the
organizers for the interesting conference. This research was funded in part by
the Austrian Science Fund (FWF) project P 34455-N. The author wishes to
acknowledge the CSC - IT Center for Science, Finland, and the Vienna
Scientific Cluster (VSC) for computational resources.

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# The Phase Transition of Reissner-Nordstr$\ddot{o}$m Black Holes

Bobin Li<EMAIL_ADDRESS>Department of physics, School of Science,
Lanzhou University of Technology, Lanzhou 730050, P.R.China

###### Abstract

Under the framework of thermodynamics, the phase transition of the black hole
is a general issue in general relativity. In this work, the phase transition
of charged black holes is discussed carefully. The metric tensor of
thermodynamics is redefined in the charged black hole, based on the Ruppeiner
geometry. With the well-defined metric tensor of thermodynamics, the scalar
curvature of the charged black hole is obtained. It is indicated that the
scalar curvature is diverged and infinite when the mass M or charge Q are set
by some values, and it is shown that the charged black hole suffers from a
phase transition. At the same time, there is a phase transition from small
mass to large mass or from small to high charged state. It is shown that the
phase transition of a charged black hole is a common and general process and
this work is meaningful for the construction of microscopic states of black
holes.

##  I introduction

Under the framework of thermodynamics, the phase transition of the black hole
is a general issue in general relativity. The thermodynamics and stability of
charged black holes in anti-de Sitter space have been extensively studied in
the literature. Fernando 1  focused on the thermodynamics of static
electrically charged Born-Infeld black holes in AdS space, noting that small
black holes are unstable while large black holes are stable. Myung et al. 2
compared the thermodynamic quantities of Born-Infeld-anti-de Sitter black
holes with those of Reissner-Norstr$\ddot{o}$m-AdS and Schwarzschild-AdS black
holes. Niu et al. 3  studied the phase transition of Reissner-
Nordstr$\ddot{o}$m black holes in anti-de Sitter spacetime, finding identical
exponents for an RN-AdS black hole and a Van der Waals liquid gas system. Wei
et al. 4 ; 7  explored the phase structure and equilibrium state space
geometry of charged topological Gauss-Bonnet black holes in anti-de Sitter
spacetime. Cai et al. 5  investigated the P-V criticality and phase transition
in the extended phase space of charged Gauss-Bonnet black holes in AdS space,
highlighting the appearance of criticality and phase transition in certain
dimensions and conditions. Ma et al. 6  discussed the existence condition and
second-order phase transition of RN-dS black holes. Hendi et al. 9  studied
the phase transition and thermodynamic geometry of Einstein-Maxwell-dilaton
black holes, identifying two types of phase transitions based on parameter
values. Xu et al. 8  examined the P-V criticality and phase transition in the
extended phase space of black holes in massive gravity, showing a black hole-
to-vacuum phase transition in certain cases. Overall, these studies contribute
to a deeper understanding of the phase transition behavior of charged black
holes in anti-de Sitter spacetime 10 .

The phase transition of black holes has been a topic of interest in various
gravitational theories. Myung 11  studied the phase transition between black
holes with scalar hair and topological black holes, showing that the
transition is second-order based on differences in free energies. Wei et al.
12  explored the thermodynamic geometry and phase transition of the Kehagias-
Sfetsos black hole in deformed Hořava-Lifshitz gravity, suggesting an
effective repulsive interaction similar to an ideal gas of fermions. Ma 13
investigated the thermodynamic phase transition of black holes in
asymptotically safe gravity by introducing spatially finite boundary
conditions and found similarities to the thermodynamic phase structure of RN-
AdS black holes. Galante et al. 14  discussed a phase transition driven by
chemical potential in Einstein-Maxwell theory conformally coupled to a scalar
field, leading to an asymptotically anti-de Sitter black hole. Hennigar et al.
15  presented a phase transition in asymptotically anti-de Sitter hairy black
holes in Lovelock gravity, showcasing a "λ-line" phase transition in black
hole thermodynamics. Feng et al. 16  studied the thermodynamics and phase
transition of a rainbow Schwarzschild black hole in rainbow gravity, analyzing
critical behavior and thermodynamic stability. Rodrigue et al. 17
investigated the thermodynamics and Hawking radiation of Schwarzschild black
holes with quintessence-like matter and deficit solid angle, deriving
expressions for temperature and specific heat. Bhattacharya et al. 18  focused
on the extremal phase transition of black holes, showing that the phase
transition occurs only in the microcanonical ensemble and calculating critical
exponents. Xu et al. 19  extended the study of the relationship between the
photon sphere and thermodynamic phase transition to charged Born-Infeld-AdS
black holes, identifying different cases of phase transitions based on the
Born-Infeld parameter. Lastly, Cao et al. 20  discussed the thermodynamic
phase structure of the complex Sachdev-Ye-Kitaev model and charged black holes
in deformed JT gravity, drawing parallels to the van der Waals-Maxwell phase
transition theory. Overall, these studies contribute to our understanding of
the phase transitions of black holes in various gravitational theories.

Chaloshtary et al. 21  explored the critical behavior and reentrant phase
transition for logarithmic nonlinear charged black holes in massive gravity.
Wu et al. 22  identified a crucial parameter that characterizes the
thermodynamic properties of black holes in massive gravity, affecting the
existence of different black hole phases. Zhou et al. 23  investigated the
impact of the generalized uncertainty principle on black hole thermodynamics
and phase transitions in a cavity, observing the Gross-Perry-Yaffe phase
transition for both large and small black hole configurations. Furthermore,
Zhou et al. 24 also studied the phase transition and microstructures of five-
dimensional charged Gauss-Bonnet-AdS black holes in the grand canonical
ensemble, highlighting the constancy of interaction at the phase transition.
Li et al. 25  proposed that the kinetics of Hawking-Page phase transitions
could be governed by a reaction-diffusion equation, with Hawking evaporation
playing a crucial role in facilitating the phase transition. Xu et al. 26
demonstrated how the Landau functional can reflect the physical process of
black hole phase transitions, with the splitting and transformation of the
global minimum corresponding to different types of phase transitions. Lastly,
Cao et al. investigated the thermodynamic phase structure of the complex
Sachdev-Ye-Kitaev model and charged black holes in deformed JT gravity,
drawing parallels between these models and the van der Waals-Maxwell liquid-
gas model. The presence of charge was found to influence the Hawking-Page
phase transition in neutral black holes. Overall, these studies contribute to
our understanding of the phase transitions of black holes in various
theoretical frameworks and gravitational backgrounds.

This work meticulously examines the phase transition of charged black holes.
The metric tensor of thermodynamics is reinterpreted in the charged black hole
using the Ruppeiner geometry. The scalar curvature of a charged black hole can
be calculated using the well-defined thermodynamic metric tensor. When the
mass M or charge Q is set to certain levels, the scalar curvature diverges and
becomes infinite, and the charged black hole undergoes a phase transition. At
the same moment, a phase transition occurs from small mass to large mass, or
from low to high charged state. The section II and III reveal that the metric
tensor and entropy of Reissner-Nordstr$\ddot{o}$m black holes, and the section
IV introduce the ruppeiner geometry and metric tensor of thermodynamics.
Finally, in the section V, the scalar curvature of thermodynamics is
calculated, based on metric tensor of thermodynamics. It is demonstrated that
a charged black hole’s phase transition is a common and general phenomenon.
and this work is meaningful for the construction of microscopic states of
black holes.

##  II The general black hole

$\displaystyle ds^{2}$
$\displaystyle=-(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}})dt^{2}+(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\varphi^{2}$ (1)

The equation (1) is metrics tensor of Reissner-Nordstr$\ddot{o}$m black holes
where $(M,Q)$ is mass and charge of black hole

##  III The entropy of charged black hole

The entropy of Reissner-Nordstr$\ddot{o}$m black holes as follows

$S(M,Q)=\pi[(M+\sqrt{M^{2}-Q^{2}})^{2}]$ (2)

##  IV Ruppeiner geometry

According to the thermodynamic fluctuation theory

$S=S_{o}+\frac{1}{2}(\partial_{\mu}\partial_{\nu}S)|_{0}\Delta x^{\mu}\Delta
x^{\nu}+\cdots$ (3)

According to Ruppeiner geometry, the metric tensor of thermodynamics is as
follows

$g_{\mu\nu}=\frac{\partial^{2}S}{\partial x_{\mu}\partial
x_{\nu}}\;\;\;where\;x_{\mu},x_{\nu}=M,Q\;(\mu,\nu=1,2)$ (4)

And

$[g_{\mu\nu}]=\left[\begin{array}[]{cc}\frac{\partial^{2}S}{\partial M\partial
M}&\frac{\partial^{2}S}{\partial M\partial Q}\\\ \frac{\partial^{2}S}{\partial
Q\partial M}&\frac{\partial^{2}S}{\partial Q\partial Q}\end{array}\right]$ (5)

##  V Scalar curvature R

###  V.1 Riemannian Geometry

According to basic general relativity, the scalar curvature is as follows

$\displaystyle R$ $\displaystyle=-\frac{1}{\sqrt{g}}[\frac{\partial}{\partial
M}(\frac{g_{QM}}{g_{MM}}\frac{1}{\sqrt{g}}\frac{\partial g_{MM}}{\partial
Q}-\frac{1}{\sqrt{g}}\frac{\partial g_{QQ}}{\partial M})$
$\displaystyle+\frac{\partial}{\partial Q}(\frac{2}{\sqrt{g}}\frac{\partial
g_{QM}}{\partial Q}-\frac{1}{\sqrt{g}}\frac{\partial g_{MM}}{\partial
Q}-\frac{g_{MQ}}{g_{MM}}\frac{1}{\sqrt{g}}\frac{\partial g_{MM}}{\partial
M})]$ (6)

###  V.2 Scalar curvature of thermodynamics

$\displaystyle R(M,Q)$
$\displaystyle=\frac{3M^{2}{\left(6M^{8}+6M^{7}Q-27M^{6}Q^{2}-30M^{5}Q^{3}-M^{4}Q^{4}\right)}}{Q{\left(M-Q\right)}^{2}{\left(M+Q\right)}\sigma_{1}}$
$\displaystyle+\frac{3M^{2}{\left(34M^{3}Q^{5}+47M^{2}Q^{6}+6MQ^{7}+15Q^{8}\right)}}{Q{\left(M-Q\right)}^{2}{\left(M+Q\right)}\sigma_{1}}$
$\displaystyle-\frac{6M{\left(M+Q\right)}^{2}{\left(3M^{5}-6M^{2}Q^{3}+MQ^{4}-2Q^{5}\right)}}{Q\sqrt{M^{2}-Q^{2}}\sigma_{1}}$
(7)

where$\sigma_{1}={\left(-3M^{4}+6M^{2}Q^{2}+Q^{4}\right)}^{2}$

##  VI Result and discussion

###  VI.1 Phase transition from small mass to large mass

Figure 1: The diagram of scalar curvature $R$ with mass $M$ ($M>Q$). The
critical
mass$M_{C}(Q=0.1)=0.15,$$M_{C}(Q=0.5)=0.75,$$M_{C}(Q=0.9)=1.35,$$M_{C}(Q=1.3)=1.95$

When the charge $Q\rightarrow 0$, that is the case of Schwarzschild black
hole, there will be no phase transition and the critical mass
$M_{C}\rightarrow 0$. For given value of charge $Q$ of black hole, there are
one critical mass $M_{C}$ corresponding to the divergence of scalar curvature
$R$. And it is indicated that there is a phase transition from small mass to
big mass for a charged black hole.

Due to$M>Q$ of charged black hole, scalar curvature $R$ actually has one
critical point to be divergent. For different charged states of black hole,
there are different critical mass $M_{C}$ of phase transition. Shown in Fig.1,
$M_{C}(Q=0.1)=0.15,$ $M_{C}(Q=0.5)=0.75,$ $M_{C}(Q=0.9)=1.35,$
$M_{C}(Q=1.3)=1.95$.

###  VI.2 Phase transition from small charged to high charged state

Figure 2: The diagram of scalar curvature $R$ with charged $Q(Q<M)$. The
critical
charge$Q_{C}(M=0.1)=0.07,$$Q_{C}(M=0.5)=0.33,$$Q_{C}(M=1.0)=0.67,$$Q_{C}(M=1.3)=0.87$

For given value of mass $M$ of black hole, there are one critical charge
$Q_{C}$ corresponding to the divergence of scalar curvature $R$. And it is
indicated that there is an phase transition from small charged to high charged
state for a charged black hole.

Due to $M>Q$ of charged black hole, scalar curvature $R$ actually has one
critical point to be divergent. For different mass of black hole, there are
different critical charge $Q_{C}$ of phase transition. Shown in Fig.2,
$Q_{C}(M=0.1)=0.07,$ $Q_{C}(M=0.5)=0.33,$ $Q_{C}(M=1.0)=0.67,$
$Q_{C}(M=1.3)=0.87$.

###  VI.3 Phase transition Overview

Figure 3: The diagram of scalar curvature R with mass M and charge Q

From overview diagram $R\sim QM$, there are one critical lines on plane
$QM(M>Q)$. And the critical line means that the phase of charged black hole
would be transform from one state to another state. Figure 3(b) shows that
there are three phase zone I ($M>Q$,impossible zone), zone II and zone III.
For fixed charge $Q$ or fixed mass $M$ of charged black hole, the phase state
will transfer from phase zone II to zone III. And this situation is like the
phase transition of the ordinary matter, for example of transition between
liquid and gas phase.

The critical lines on plane $QM$ means where the scalar curvature $R$ will
diverge and infinite. But it is shown that it is discontinuous and a series of
islands for where scalar curvature $R$ is divergent, and it is interesting
features for phase transition of charged black hole. The critical lines is

$M_{c}=\frac{3}{2}Q\;\;\;and\;\;\;Q_{c}=\frac{2}{3}M$ (8)

In the usual situation, the value of scalar curvature $R$ is equal to zero,
and it indicates that there is no interaction between the microscopic elements
of the charged black hole, and the microscopic elements are free and are like
the ideal gas under the condition of thermodynamic fluctuation theory. In the
critical point of phase transition, the value of scalar curvature $R$ is less
than $0$ $(R<0)$, and it is indicated that there is an attractive force
between microscopic elements of the charged black hole. the more approach the
critical point, the stronger this attractive force. The correlation length
will increase from local to global between microscopic elements of the charged
black hole. and it is shown that the microscopic structure of the charged
black hole would rearrange and transform from one phase to another phase.

###  VI.4 The critical exponents of scalar curvature $R$

Under the condition of $M_{c}=\frac{3}{2}Q$, scalar curvature $R$ is expand as
follows

$\displaystyle R(M)|_{M=\frac{3}{2}Q}\approx$ $\displaystyle
const_{2}(Q)\cdot(M-\frac{3}{2}Q)^{1}-const_{3}(Q)\cdot(M-\frac{3}{2}Q)^{2}+O(2)$
(9)

Under the condition of $M_{c}=\frac{3}{2}Q$,

$R(M)|_{M=\frac{3}{2}Q}\approx const(Q)\cdot(M-\frac{3}{2}Q)^{\nu}+O(\nu)$
(10)

the critical exponents of scalar curvature $R$ is $\nu=1$

##  VII Conclusion

The black hole phase transition is a general relativity problem within the
domain of thermodynamics. The phase transition of Reissner-Nordstr$\ddot{o}$m
black holes is thoroughly examined in this article. Based on the Ruppeiner
geometry, the metric tensor of thermodynamics is redefined in the Reissner-
Nordstr$\ddot{o}$m black holes. The charged black hole’s scalar curvature can
be found using the well-defined metric tensor of thermodynamics. It is
demonstrated that the charged black hole experiences a phase transition where
the scalar curvature is divergent and infinite, under the conditions that the
mass $M$ or charge $Q$ are fixed by certain values, that there is a phase
transition from a small to a large mass or from a small to a highly charged
state of Reissner-Nordstr$\ddot{o}$m black holes .

When the charge $Q\rightarrow 0$, the critical mass $M_{C}\rightarrow 0$ and
there won’t be a phase transition. There is one critical masses $M_{C}$ for a
fixed black hole charge $Q$, which correspond to the divergence of scalar
curvature $R$. Additionally, the critical line indicates that a charged black
hole’s phase would change from one state to another state. In phase parameter
space, It demonstrates that there are three phases parts: I(impossible zone),
II, and III. The phase state of a charged black hole will go from phase zone
II to zone III for fixed charge $Q$ or fixed mass $M$. This situation is like
the phase transition of ordinary matter, such as from liquid to gas phase. In
the area of phase zone II or zone III, the value of scalar curvature $R$ is
equal to zero, which means that there is no interaction between the
microscopic elements of Reissner-Nordstr$\ddot{o}$m black holes and resemble
an ideal gas under the condition of thermodynamic fluctuation theory. When the
scalar curvature R is smaller than zero $(R<0)$ at the crucial moment of phase
transition, it suggests that the microscopic elements of Reissner-
Nordstr$\ddot{o}$m black holes are attracted to each other, and this
attracting force becomes stronger the closer one gets to the critical point.
The microscopic elements of Reissner-Nordstr$\ddot{o}$m black holes will see
an increase in correlation length from local to global. It is shown that the
microscopic structure of the charged black hole would rearrange and transform
from one phase to another phase.

As a conclusion, It is shown that the phase transition of a charged black hole
is a common and general process and this work is meaningful for the
construction of microscopic states of black holes.

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