QED effects on phase transition and Ruppeiner geometry of Euler-Heisenberg-AdS black holes

The action of the EH theory with cosmological constant in four-dimensional spacetime is salazar1987duality

Here $g$, $\Lambda$, and $R$ are the determinant of the metric tensor, cosmological constant, and Ricci curvature tensor, respectively. The term $\mathcal{L}(F,G)$ is the Lagrangian of the nonlinear electrodynamics which depends on the electromagnetic invariants $F=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $G=\frac{1}{4}F_{\mu\nu}{{}^{*}F^{\mu\nu}}$. The Maxwell field strength $F_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}$, and $\mathcal{A}_{\mu}$ is the corresponding vector potential. The parameter $a$ appearing in formula (2) can be used to measure the strength of the QED correction, which is related to the mass and charge of the electron, and thus we call it the QED parameter. When $a$=0, the influence of the QED term will vanish.

For a static spherically symmetric charged EH-AdS black hole solution, the line element is Magos:2020ykt

where the metric function is given by

where $M$ and $Q$ are the mass and electric charge of the black hole, respectively. When $a=0$, this solution reduces to the Reissner-Nordström (RN) AdS black hole solution.

The radius $r_{+}$ of the outer black hole event horizon is the largest root of $f(r_{+})=0$, which can be obtained by solving

In term of $r_{+}$, the mass $M$ can be expressed as

Using the “Euclidean trick”, the Hawking temperature is

In the extended phase space, the cosmological constants $\Lambda$ was interpreted as the pressure via $P=-\frac{\Lambda}{8\pi}$ kastor2009enthalpy . Accordingly, the black hole mass $M$ will act as the enthalpy $H$ of the thermodynamic system rather than the internal energy. Then the first law has the following form

where $\mathcal{A}=\frac{\partial H}{\partial a}$ is the conjugate quantity to the QED parameter $a$. The entropy $S$, thermodynamic volume $V$, electric potential $\Phi$, and $\mathcal{A}$ can be calculated as

It is straightforward to confirm the following Smarr formula holds

From Eq. (7), the equation of state can be written as

In term of thermodynamic volume $V=\frac{4}{3}\pi r_{+}^{3}$, the equation of state reads

We plot the pressure $P$ as a function of the specific volume $v=2\,r_{+}$ for fixed temperature in Fig. 1 with $a=-1.5$ ($a<0$) and $a=1.0$ ($0\leq a\leq\frac{32}{7}Q^{2}$) being two representative examples, respectively. In Fig. 1, we can see that there are two extremal points (marked with red dots) for $T<T_{c}$. And these points divide the isothermal curve into three branches. Two of them are the small black hole and large black hole locating at the left and right sides of the isothermal curve respectively, while the middle branch is for the intermediate black hole. According to the heat capacity, these black hole branches with negative slope are thermodynamic stable, while those with positive slope are unstable. Thus the small and large black holes are stable and the intermediate black hole is unstable. Making the use of the Maxwell equal area law, one can construct two equal areas along each isothermal curve to obtain the phase transition point. On the other hand, with the increase of the temperature, these two extremal points get closer and coincide with each other at $T=T_{c}$. When $T>T_{c}$, there is no extremal point anymore. Quite differently, for a positive QED parameter, i.e., $a=1.0\in(0\leq a\leq\frac{32}{7}Q^{2})$, there are three extremal points in Fig. 1 for $T<T_{c}$. When the temperature approaches its critical value, two of them coincide, while the other one continues to exist even when the temperature is above its critical value. Below the critical temperature, the nonmonotonic behavior of the isothermal curve also allows one to construct two equal areas indicating the existence of the black hole phase transition.

The critical point can be determined by the following conditions

These conditions gives a third degree equation for $x=\left(\frac{6}{\pi}\right)^{2/3}V^{2/3}$

When $0\leq a\leq\frac{32}{7}Q^{2}$, the equation (17) has three real roots Magos:2020ykt

One notes that $x_{2}$ gives a negative volume, and it should be excluded. Therefore, only $k$=0 and 1 is allowed, and the critical temperature and pressure are given by

The critical volume is defined as $V_{c}=\frac{1}{6}\pi x_{k}^{3/2}$ depending on the charge $Q$ and QED parameter $a$. It is worth noting that when the QED parameter $a>\frac{32Q^{2}}{7}$, the equation (17) has no real root, and thus no critical point exists. For $a<0$, there is a real root $x_{0}$ ($k$=0) giving one critical point.

In summary, there are two characteristic types of phase transition. The first type is for $a<0$, which admits one critical point, and the phase transition is the similar to the gas-liquid phase transition of VdW fluid. For the other type with QED parameter $0<a<\frac{32Q^{2}}{7}$, two critical points can be found, and we shall see there exists the reentrant phase transition.

In the extended phase space, we can introduce the reduced quantities $\tilde{P}$, $\tilde{T}$, $\tilde{V}$, $\tilde{v}$

Then the equation of state (15) takes the following form

where $\rho_{c}=\frac{P_{c}v_{c}}{T_{c}}$. In the canonical ensemble, the Gibbs free energy $G=H-TS$, which reads

In general, the first-order phase transition can be determined by the swallow tail behavior of the Gibbs free energy. In the following sections, we shall study the phase transition via the behavior of the Gibbs free energy.

For this case, we take $Q=1.0$ and $a=-1.5$ for example. The Gibbs free energy $G$ is plotted as the temperature $T$ in Fig. 2 for different values of the pressure. When $P<P_{c}$, we observe the characteristic swallow tail behavior indicating the coexistence of the black hole phase transition. As the pressure increases, the shape of the swallowtail becomes smaller, and it shrinks to a point when $P=P_{c}$. Further increasing the temperature, the behavior completely disappears. Then the Gibbs free energy turns to a smooth function of the temperature.

In order to clearly show the phase transition, we describe the Gibbs free energy in Fig. 2 with $P<P_{c}$. After a simple calculation, we find that the small and large black hole branches in blue and red solid curves have positive heat capacity, and thus they are thermodynamic stable. Whereas these branches in dashed curve are unstable or metastable. Considering that a system always prefers a state of low Gibbs free energy, the system will undergo a first-order phase transition from a small black hole (SBH) phase to a large black hole (LBH) phase with the increase of the temperature. The phase transition point exactly locates at the intersection of the swallow tail behavior. Thus it is easy to see that this phase transition is similar to the liquid-gas phase transition of a VdW fluid.

The phase structures of the charged EH-AdS black hole are given in Fig. 3, from which we find that it shares a similar phase diagram with the VdW fluid. In the $P$-$T$ diagram, the first-order coexistence curve of the small and large black holes starts at $\tilde{T}=0$ and ends at the critical point, which divides the plane into two regions corresponding to the small and large black hole phases, respectively. Considering that the equation of state is not applicable in the shadow region, we use the spinodal curve marked with the blue dashed curve to distinguish the metastable phase from the coexistence phase of the black hole. The spinodal curve is determined by

Obviously, the spinodal curve meets the coexistence curve at the critical point.

For this case, two critical points can be found. The corresponding phase transition is the reentrant phase transition including a zeroth-order and a first-order phase transitions at a certain region of the temperature or pressure. Here we shall study the reentrant phase transition for the charged EH-AdS black hole.

As expected, the first-order phase transition can be determined by constructing two equal areas according to the Maxwell’s equal area law. As we shown in Fig. 1, the isotherm curves exhibit a nonmonotonic behavior in $P$-$v$ plane. So we can construct two equal areas between two stable black hole branches. However, as pointed out in Ref. wei2015clapeyron , the equal areas should be constructed in $P$-$V$ plane rather than $P$-$v$ plane. Alternatively, the Maxwell’s equal area law also holds in the $\tilde{T}$-$\tilde{S}$ plane for giving pressure both in ordinary or reduced parameter space

where $\tilde{T}_{0}$ denotes the reduced temperature of phase transition and $\tilde{S}_{1}$ and $\tilde{S}_{2}$ are the reduced entropy of the corresponding coexistence small and large black holes. Taking $\tilde{P}$=0.4 as an example, the Maxwell’s equal area law is fulfilled in the $\tilde{T}$-$\tilde{S}$ plane in Fig. 4. The black curves denote the small black hole and large black hole branches, respectively. The red curves are for the two metastable branches, the superheated small black hole branch and the supercooled large black hole branch. The blue curve with a negative slope is an unstable branch, and it is substituted by a horizontal line according to the equal area law. The area under the isobaric curve from $\tilde{S}_{1}$ to $\tilde{S}_{2}$ is required to equal the area enclosed by the rectangle under the isothermal horizontal line, so the areas of the two shadow regions are equal. These two black dots are the spinodal points, which separate the metastable branches from the unstable branch. Then the pressure and temperature corresponding to the isothermal horizontal line are just that of the phase transition point.

We depict the Gibbs free energy for these two types of phase transitions in Fig. 5. In Fig. 5, the small/large black hole phase transition is analyzed for the reduced pressure $\tilde{P}=0.4$. As the temperature increases, the system jumps directly from the SBH phase (red solid line) to the LBH phase (blue solid line), and the volume of the black hole directly undergoes a sudden change when the temperature is equal to the phase transition temperature. For the black hole branches described by the black dashed lines, their heat capacity is negative indicating thermodynamically instability. Note that although these black hole branches described by the dotdashed lines in the Fig. 5 have a positive heat capacity, they do not have the lowest free energy for a fixed temperature, and thus they are metastable black hole branches. In addition to the small/large black hole phase transition, another new zeroth-order phase transition emerges in Fig. 5. When the zeroth-order phase transition occurs, the black hole system jumps from the LBH phase to the SBH phase. Moreover, not only the volume of a black hole will undergo sudden changes, but the Gibbs free energy will also change drastically. In short, the black hole system firstly undergoes a zeroth-order phase transition from LBH phase to SBH phase and then returns to LBH phase through the small/large black hole phase transition with the increase of the temperature. This phase transition of such pattern is called the reentrant (large/small/large BH) phase transition.

The phase diagrams of reentrant phase transition are shown in Fig. 6. Similar to the case of the VdW phase transition type, the coexistence curve (red solid line) divides the parameter space into two regions in Fig. 6. Above and below the coexistence curve are the SBH phase and the LBH phase, respectively. The zeroth-order phase transition line marked with green solid line connects the first-order coexistence curve and the minimum temperature curve. The phase diagram is also exhibited in $\tilde{P}$-$\tilde{v}$ plane in Fig. 6. The critical point divides the coexistence curve into left and right parts corresponding to the coexistence small black hole and the large black hole, respectively. The small and large black hole regions locate at the left and right.

In Fig. 7, we plot the difference of the volume $\Delta V=V_{l}-V_{s}$ among the first-order phase transition as a function of the temperature and pressure, respectively. It is obvious that below the critical point, $\Delta V$ has a finite value indicating a sudden change during the black hole phase transition. However, when the critical point is reached, $\Delta V$ vanishes, which means that one could not distinguish the small and large black holes anymore. Considering the behavior of $\Delta V$, it can be regarded as an order parameter to characterize the phase transition.

Near the critical point, the critical exponent of $\Delta V$ can be calculated, see Ref. Magos:2020ykt . However, in the calculation, the specific volume $v$ is used in constructing the equal area law, which as we pointed out is inappropriate. Here we would like to make use of the thermodynamic volume $V$ instead, and to see whether the result keeps unchanged. Using the relationship between thermodynamic volume and specific volume $V=\frac{\pi v^{3}}{6}$, we get $\tilde{V}=\tilde{v}^{3}$. Then the formula (21) can be expressed as

where $\rho_{c}=\frac{P_{c}v_{c}}{T_{c}}$. Let the first term in the previous equation be the function $h(\tilde{V})$, so (25) can be written as

The Taylor expansion in the vicinity of the critical point at $\tilde{V}=1$ and $\tilde{T}=1$ is written as

From the definition of the critical point, the equation (II.3) must meet the following conditions

Therefore, Eq. (II.3) can be simplified to

Taking $\omega=\tilde{V}-1$ and $t=\tilde{T}-1$, the previous expression become

where we have defined $C=\frac{14}{81\,\rho_{c}}-\frac{1}{6}h^{(3)}(1)$. From Maxwell’s equal area law we obtained

where $\omega_{l}$ and $\omega_{s}$ are the volumes of the coexistence small and large black hole respectively. For an isothermal process, the pressure $P_{s}$ is equal to $P_{l}$, i.e. $\tilde{P}_{s}(t,\omega)=\tilde{P}_{l}(t,\omega)$, which reads

Combining (31) and (32), we have

The critical exponent $\beta$ is related to the order parameter $\eta=|t|^{\beta}$, namely with the change of volume at the phase transition for a given isotherm process,

Therefore the critical exponent $\beta$ is

As a result, we can see that the choose of the specific volume or thermodynamic volume does not change the value of the critical exponent $\beta$. However when applying the Maxwell equal area law, we should choose the thermodynamic volume rather than the specific volume.

Here, we firstly give a brief introduction for the Ruppeiner geometry and then calculate the curvature scalar for the charged EH-AdS black hole.

Let us consider a thermodynamically isolated system in equilibrium with the total entropy $S$. The system is divided into two subsystems, the small system we consider and its large environment. Their entropies are denoted by $S_{S}$ and $S_{E}$ with $S_{S}\ll S_{E}\sim S$. We suppose that the total entropy of the system is described by two independent thermodynamic variables $x_{0}$ and $x_{1}$. The total entropy of the system can be written as

For an equilibrium system, the entropy reaches its maximum locally. We perform Taylor expansion in the neighborhood of this local maximum $(x^{\mu}=x^{\mu}_{0})$

where a zeroth-order term $S_{0}$ is the local maximum of entropy at $x^{\mu}_{0}$. The entropy of an isolated system in equilibrium is conserved under virtual change. This shows that the first derivative of the entropy vanishes, and thus we get

It is worth noting that $S_{E}$ is a thermodynamic extensive quantity, and it is the same order of magnitude as the entropy of the entire system. Therefore, its derivative with respect to the intensive quantity $x^{\mu}$ is much smaller than the derivative of $S_{S}$, which can be ignored. Therefore, the probability of finding the system in the internals ($x_{0}$, $x_{0}+dx_{0}$) and ($x_{1}$, $x_{1}+dx_{1}$) will be

where $k_{B}$ is Boltzmann’s constant. With (37), the line element of Ruppeiner geometry measuring the distance between two neighboring fluctuation states can be written as

Because $\Delta l^{2}$ can measure the distance between two neighboring fluctuation states, the thermodynamic metric $g_{\mu\nu}^{R}$ potentially contains some information about the microstructure of the system.

When we choose the thermodynamic coordinates $x^{\mu}$ to be the temperature $T$ and the volume $V$, the Helmholtz free energy is the thermodynamic potential. The corresponding line element can be written as follows Wei:2019uqg

where $C_{V}=T(\partial_{T}S)_{V}$ is the heat capacity at constant volume. Using the convention in the literature ruppeiner1995riemannian , we can directly calculate the corresponding scalar curvature of the line element

It should be noted that the sign of scalar curvature characterizes the type of interaction between two microscopic molecules in a given system ruppeiner2010thermodynamic , i.e. $R>0$ and $R<0$ represent dominated repulsive interaction and attractive interaction, respectively. For VdW fluid, the study shows that it has a negative scalar curvature indicating the dominated attractive interaction among these underlying molecules. In addition, when $R=0$, the interaction of repulsion and attraction reaches equilibrium dolan2015intrinsic .

For the charged EH-AdS black hole, the heat capacity at constant volume vanishes. We adopt the treatment of Ref. Wei:2019yvs , the new normalized scalar curvature $R_{N}$ is

Using the equation of state (15), the normalized scalar curvature of the charged EH-AdS black hole reads

where

We plotted the scalar curvature $R_{N}$ as a function of $\tilde{V}$and $\tilde{T}$ in Fig. 8 for the small/large black hole phase transition and reentrant phase transition. It can be observed that the surface of the normalized scalar curvature is concave where the scalar curvature diverges. Besides, compared with the VdW type phase transition, a new divergence occurs in the reentrant phase transition when the reduced volume $\tilde{V}$ is close to zero.

As we shown above, there is the small/large black hole phase transition of VdW-like for the case.

In order to show the details, we take $Q=1.0$ and $a=-1.5$, and illustrate the normalized scalar curvature $R_{N}$ as a function of $\tilde{V}$ for fixed reduce temperature $\tilde{T}$= 0.4, 0.8, 1.0 and 1.2 in Fig. 9. For $T<T_{c}$, there are two divergent points of the normalized scalar curvature $R_{N}$. It can be found that these two points will come closer as the temperature increases. When the critical temperature is approached, these two points merge. When $T>T_{c}$, the normalized scalar curvature $R_{N}$ does not diverge while keeps finite values. In most of the parameter space, $R_{N}$ is negative, while in a small region of $\tilde{V}$ for $\tilde{T}$= 0.4 shown in the inset of Fig. 9, $R_{N}$ is positive. This phenomenon indicates that the dominated repulsive interaction may exist in some parameter regions.

From the analysis of the normalized scalar curvature $R_{N}$, it is easy to know that the divergent point of $R_{N}$ satisfies the spinodal point condition $(\partial_{V}P)_{T}=0$, which gives

After a simple calculation, the specific expression of the sign changing curve corresponding to $R_{N}=0$ reads

which indicates that the temperature of the sign changing curve is half of the temperature of the spinodal curve Wei:2019uqg ; Wei:2019yvs .

In Fig. 10, the sign changing curve (black dotdashed), spinodal curve (blue dashed), and the coexistence curve (red solid) are displayed. The shadow regions I and II have positive scalar curvature, while the other regions are for negative scalar curvature. Considering that the equation of state for the charged EH-AdS black hole in region II is no longer valid, this region needs to be excluded. However, in region I, the small black hole with high temperature still admits the dominated repulsive interaction. This is similar to the situation of the charged RN-AdS black holes, while different from the neutral black hole in Gauss-Bonnet gravity Wei:2019ctz ; Zhou:2020vzf .

We show the behavior of $R_{N}$ along the coexistence small and large black holes in Fig. 10. It is easy to see that both them go to negative infinity at the critical temperature indicating the existence of the critical exponent. Moreover, we can also find that $R_{N}$ of the coexistence small black hole is always above that of the large black hole. At low temperature, the coexistence small black hole has a positive value, which is consistent with that of Fig. 10.

Since the critical exponent can provide us with some universal properties, we now turn to calculate it for the scalar curvature at the critical point. Here it is natural to assume the normalized scalar curvature $R_{N}$ near the critical point has the following form johnston2014advances

or equivalently,

Via calculating $R_{N}$ near the critical point, we obtain the following fitting results

The numerical results (small red dots) and fitting results (blue solid lines) are shown in Fig. 11. It is obvious that the numerical and fitting results are highly consistent with each other. The slopes obtained by fitting the coexistence curves of small and large black holes are respectively $\alpha_{SBH}$=2.0365 and $\alpha_{LBH}$=1.9636. Taking into account the error of the numerical calculation, we can say that the critical exponent is $\alpha=2$, which is the same as that of the VdW fluid.

Moreover, using the intercept $\beta$ obtained by fitting results, we obtain a dimensionless constant

It is the same value with the charged AdS black holes and VdW fluids, and is also consistent with the result obtained in Ref. Magos:2020ykt via expanding the equation of state near the critical point at the first leading term.

We explore the underlying microstructure of the black hole that results in reentrant phase transition with $Q$=1 and $a$=1 by using the similar method. The behavior of $R_{N}$ against the reduced thermodynamic volume $\tilde{V}$ for a fixed temperature is shown in Fig. 12. Comparing with the VdW phase transition, we observe that the scalar curvature of the reentrant phase transition has an additional divergent point near the origin. The reason is when $T<T_{c}$, there are three extremal points on each isothermal curve, see Fig. 1. As the temperature increases, the evolution behavior of other two divergent points is very similar to the case of the VdW-like phase transition. Furthermore, when the temperature is higher than its critical value, the additional divergent point still exists. On the other hand, when the reduced volume is close to the origin, one observes a positive scalar curvature as expected.

The first-order coexistence curve, zeroth-order phase transition curve, spinodal curve, and sign changing curve are depicted in Fig. 13. In the shadow regions I, II, and III, the scalar curvature takes positive values. Compared with the VdW-like phase transition, the number of the shadow regions is three instead of two. It can be seen that the leftmost of the coexistence curve of small black hole ends at the spinodal curve and it indicates that there is a divergence. On the other hand, one should note that the black hole does not exist in region I.

We also plot the normalized scalar curvature $R_{N}$ along the coexistence small and large black hole curves as a function of temperature in Fig. 13. Along the coexistence curve, $R_{N}$ negatively diverges at the critical point for both the small and large black holes. However, different from the previous VdW-like phase transition, the scalar curvature of the small black hole gains a new divergence when the reduce temperature is around 0.48. While for the coexistence large black hole, there is no such phenomenon. The reason for this new divergence is that the leftmost of the coexistence small black hole curve is located on the spinodal curve, but the large black hole is not.

Near the critical point, the fitting results for the coexistence small and large black holes are given by

The numerical results and the fitting results are shown in Fig. 14 with highly consistent with each other. The fitting coefficients of the small and large black holes are respectively $\alpha_{LBH}$ =1.9536, $\alpha_{SBH}$ =2.0426. Taking into account the error of the calculation, we get the critical exponent $\alpha=2$. The dimensionless constant is

which is the same as that of the VdW-like phase transition.