Holographic phase transition probed by non-local observables

From the viewpoint of holography, the phase structure of a 5-dimensional Reissner-Nordstr\"{o}m-AdS black hole is probed by the two point correlation function, Wilson loop, and entanglement entropy. As the case of thermal entropy, we find for all the probes, the black hole undergos a Hawking-Page phase transition, a first order phase transition and a second order phase transition successively before it reaches to a stable phase. In addition, for these probes, we find the equal area law for the first order phase transition is valid always and the critical exponent of the heat capacity for the second order phase transition coincides with that of the mean field theory regardless of the size of the boundary region.


Introduction
Phase transition is a ubiquitous phenomenon for garden-variety thermodynamic systems. Due to the pioneering work by Hawking [1,2], a black hole is also a thermodynamic system. Such a fact is further supported by AdS/CFT correspondence [3,4,5], where a black hole in the AdS bulk is dual to a thermal system without gravity. So one can naturally expect a black hole can also undertake some interesting phase transitions as the general thermodynamic system. Actually it has been shown that a charged AdS black hole undergos a Hawking-Page phase transition [6,7], which is interpreted as the confinement/deconfinement phase transition in the dual gauge field theory [8], and a Van der Waals-like phase transition before it reaches the stable state [9]. The Hawking-Page phase transition implies that the thermal AdS is unstable and it will transit to the stable Schwarzschild AdS black hole lastly. The Van der Waals-like phase transition has been observed till now in many circumstantces. The first observation was contributed by [9] in the T − S plane. Specifically speaking, in a fixed charge ensemble, for a black hole endowed with small charge, there is an unstable black hole interpolating between the stable small hole and stable large hole, and the small stable hole will undertake a first order phase transition to the large stable hole as the temperature of the black hole reaches a critical temperature. As the charge increases to the critical charge, the small hole and the large hole merge into one and squeeze out the unstable phase so that an inflection point emerges and the phase transition is second order. When the charge exceeds the critical charge, the black hole is always stable. Recently in the extended phase space, where the negative cosmological constant is treated as the pressure while its conjugate acts as the thermodynamical volume, the Van der Waals-like phase transition has also been observed in the P − V plane [10,11,12,13,14,15,16]. In addition, it was shown in [17] that the Van der Waals-like phase transition also shows up in the Q − Φ plane. Particularly, in the Gauss-Bonnet gravity, it is found that the Gauss-Bonnet coupling parameter α also affects the phase structure of the space time, and in the T − α plane, a 5-dimensional neutral Gauss-Bonnet black hole also demonstrates the Van der Waals-like phase transition [18].
In this paper, we intend to probe the Hawking-Page phase transition and Van der Waals-like phase transition appeared in a 5-dimensional Reissner-Nordström-AdS black hole by the geodesic length, minimal area surface, and minimal surface area in the bulk, which are dual to the non-local observables on the boundary theory by holography, namely the two point correlation function, Wilson loop, and entanglement entropy, individually 1 . In fact, there have been some similar works to probe the phase structure by holographic entanglement entropy. In [26], the phase structure of entanglement entropy is studied in the T − S plane for both a fixed charge ensemble and a fixed chemical potential ensemble, and it is found that the phase structure of entanglement entropy is similar to that of the thermal entropy. In particular, the entanglement entropy is found to demonstrate the same second order phase transition at the critical point as the thermal entropy. Soon after, it is found that the entanglement entropy can also probe the Van der Waals-like phase transition in the P − V plane [27]. In [28], Nguyen has investigated exclusively the equal area law of holographic entanglement entropy and found that the equal area law holds regardless of the size of the entangling region. Very recently [29] investigated entanglement entropy for a quantum system with infinite volume, their result showed that the entanglement entropy also exhibits the same Van der Waals-like phase transition as the thermal entropy. They also checked the equal area law and obtained the critical exponent of the heat capacity near the critical point.
In this paper, we will further investigate whether one can probe the phase structure by two point correlation function and Wilson loop besides the entanglement entropy. We intend to explore whether they exhibit the similar Van der Waals-like phase transition as the entanglement entropy and thermal entropy. In addition, we also want to check whether these non-local observables can probe the Hawking-Page phase transition between the AdS black hole and thermal gas so that we can get a complete picture about the phase transition of the black holes in the framework of holography. This paper is organized as follows. In the next section, we will discuss the thermal entropy phase transition of a 5-dimensional Reissner-Nordström-AdS black hole in the T −S plane in a fixed charge ensemble. Then in Section 3, we will probe these phase transitions by geodesic length, Wilson loop, and holographic entanglement entropy individually. In each subsection, the equal area law is checked and the critical exponent of the heat capacity is obtained for different sizes of the boundary region. The last section is devoted to discussions and conclusions.

Thermodynamic phase transition of the 5-dimensional
Reissner-Nordström-AdS black hole Starting from the action where F µν = ∂ µ A ν − ∂ ν A µ , and l is the AdS radius. We shall focus on the case of n = 4, in which the charged Reissner-Nordström-AdS black hole can be written as [9] where φ ∈ (0, π), θ ∈ (0, π), ψ ∈ (0, 2π), are hyperspherical coordinates for the 3-sphere, and with M and Q the mass and charge of the black hole. Whence we can get the Hawking temperature of this space time as In addition, it follows from the Bekenstein-Hawking formula that the entropy of the black hole is given by where r + is the outer event horizon of the black hole, namely the largest root of the equation f (r + ) = 0. With this, the mass of the back hole can thus be expressed as the function of the event horizon M = 4l 2 Q 2 + 3l 2 π 2 r 4 + + 3π 2 r 6 + 8l 2 πr 2 Substituting (5) and (6) into (4), we can get the relation between the temperature T and entropy S of the 5-dimensional Reissner-Nordström-AdS black hole T = 12S 2 + l 2 −2π 2 Q 2 + 32 1/3 π 4/3 S 4/3 62 2/3 l 2 π 5/3 S 5/3 .
In addition, with the relation F = M − T S, the Helmholtz free energy can be expressed as Note that this formula for our free energy has implicitly chosen the pure AdS as the reference spacetime because the free energy vanishes for pure AdS by this formula. Now let us review the relevant phase transitions in the fixed charge ensemble by (7) and (8) in the T − S plane.
To achieve this, we should first find the critical charge by the following equations Inserting (7) into (9), we can get the values for the critical charge and critical entropy Substituting these critical values into (7), we can get the critical temperature We plot the isocharge curves for different charges in Figure 1. For the case Q = 0, There is a minimum temperature T 0 = √ 2 π [30], which is indicated by the red solid line in (a). When the temperature is lower than T 0 , we have only a thermal AdS. When the temperature is higher than T 0 , there are two additional black hole branches. The small branch is unstable while the large branch is stable. This can be justified by check the corresponding heat capacities, which is related to their slopes. The Hawking-Page phase transition occurs at the temperature given by T 1 = 3 2π [30], which is higher than T 0 and indicated by the red dashed line. This can be observed by the F − T relation in (a) of Figure 2, where T 0 is the horizontal coordinate of the cusp and T 1 is the horizontal coordinate for the intersection of the stable branch and the horizontal axis. Obviously, when the temperature is lower than T 1 , the thermal AdS is the most stable state. While when the temperature is higher than T 1 , the most stable state is taken over by the large black hole branch.
For the case Q = 0, the phase structure is similar to that of the Van der Waals phase transition. That is, for a small charge, there is an unstable black hole interpolating between the stable small hole and stable large hole. The small stable hole will jump to the large stable hole at the critical temperature temperature T ⋆ , which is labeled by the red dashed line in (b) of Figure 1. As the charge increases to the critical charge, the small hole and the large hole merge into one and squeeze out the unstable phase. So there is an inflection point in (c) of Figure 1. The heat capacity is divergent in this case, the phase transition is therefore second order. As the charge exceeds the critical charge, we simply have one stable black hole at each temperature, which can be justified by the slope of the curve in (d) of Figure 1. The Van der Waals-like phase transition can also be observed from the F − T relation. From (b) of Figure 2, we see a swallowtail structure, which corresponds to the unstable phase in (b) of Figure 1. The critical temperature T ⋆ = 0.4526 for the phase transition is apparently read off by the horizontal coordinate of the junction between the small black hole and the large black hole. As the temperature is lower than the critical temperature T ⋆ , the free energy of the small black hole is lowest, so the small hole is stable. As the temperature is higher than T ⋆ , the free energy of the large black hole is lowest, so the large hole dominates thereafter. The non-smoothness of the junction indicates that the phase transition is first order. When the charge is arriving at the critical charge Q c , the swallowtail structure in (b) of Figure 2 shrinks into a point as is shown in (c) of Figure 2.
The horizontal coordinate of the inflection point corresponds to the critical temperature T c of the second order phase transition, which is consistent with the analytical result in (12). For the first order phase transition in (b) of Figure 1, we would like to check whether Maxwell's equal area law holds with the following formula in which T (S, Q) is defined in (7), S 1 and S 3 are the smallest and largest roots of the equation T (S, Q) = T ⋆ . After a simple calculation, we find S 1 = 0.186987 and S 3 = 2.60575.
With these values, we find A 1 and A 3 in (13) equal 1.09481 and 1.09482 respectively. So the equal area law in the T − S plane holds within our numerical accuracy. For the second order phase transition in (c) of Figure 1, we are interested in the critical exponent associated with the heat capacity Near the critical point, writing the entropy as S = S c + δ and expanding the temperature in terms of small δ, we find in which we have used (9). In this case, (14) further implies C Q ∼ (T − T c ) −2/3 , namely the critical exponent is −2/3, which is the same as the one from the mean field theory. In addition, taking logarithm to (15), we have a linear relation with 3 the slope. In what follows, we will use this logarithm to check the critical exponent for the analogous heat capacities in the framework of holography. It it noteworthy that by holography the whole phase structure described above is not only for the bulk black hole but also for the dual boundary system, where the thermal entropy is simply given by the black hole entropy, and so on so forth.

Phase transition in the framework of holography
In this section we shall investigate the phase structures of some non-local observables such as two point correlation function, Wilson loop, and entanglement entropy in the dual fled theory by holography to see whether they have the same phase structure as the thermal entropy.

Phase transition of two point correlation function
According to the AdS/CFT correspondence, if the conformal dimension ∆ of scalar operator O of dual field theory is large enough, the equal time two point correlation function can be holographically approximated as [31] where L is the length of the bulk geodesic between the points (t 0 , x i ) and (t 0 , x j ) on the AdS boundary. Taking into account the spherical symmetry of the 5-dimensional Reissner-Nordström-AdS black hole, we can simply choose (φ = π 2 , θ = θ 0 , ψ = 0) and (φ = π 2 , θ = θ 0 , ψ = π) as the two boundary points. Then with θ to parameterize the trajectory, the proper length is given by in whichṙ = dr/dθ. Imagining θ as time, and treating L as the Lagrangian, one can get the equation of motion for r(θ) by making use of the Euler-Lagrange equation, that is which can be solved by imposing the following boundary conditionṡ To explore whether the size of the boundary region affects the later phase structure, we here choose θ 0 = 0.14, 0.2 as two examples. Note that for a fixed θ 0 , the geodesic length is divergent, so it should be regularized by subtracting off the geodesic length in pure AdS with the same boundary region, denoted by L 0 . To achieve this, we are required to set a UV cutoff for each case, which is chosen to be r(0.139) and r(0.199), respectively for our two examples. In this paper, we obtain L 0 also by numerics though there is an analytical result for r(θ 0 ) for pure AdS in Einstein gravity. We label the regularized geodesic length as δL ≡ L − L 0 . We plot the relation between T and δL for different θ 0 in Figure 3 and Figure 4. As shown in Figure 3 and Figure 4, δL demonstrates a similar phase structure as the thermal entropy. Moreover, we find that the minimum temperature T 0 as well as Hawking-Page phase transition temperature T 1 in (a), the first order phase transition temperature T ⋆ in (b), and second order phase transition temperature T c in (c) are exactly the same as those in T − S plane, which justifies our notation. To be more specific, it is easy to check T 0 by locating the position of local minimum. But in order to confirm T ⋆ and T c , we are required to examine the equal area law for the first order phase transition and obtain −2/3 as the critical exponent for the second order phase transition, which are documented as follows.
In the δL − T plane, we define the equal area law as in which T (δL) is an Interpolating Function obtained from the numeric result, and δL 1 , δL 3 are the smallest and largest roots of the equation T (δL) = T ⋆ . For the case θ 0 = 0.14, we find δL 1 = 0.0000556147, δL 3 = 0.000194497. Substituting these values into (21), we find A 1 = 0.0000628401, A 3 = 0.0000628583. For the case θ 0 = 0.2, after simple calculation, we find A 1 = 0.0000108924, A 3 = 0.0000108875. It is obvious that for different θ 0 , A 1 and A 3 are equal within our numeric accuracy. Thus the equal area law also holds in the δL − T plane. In addition, in order to investigate the critical exponent for the analogous heat capacity of the geodesic length. we are interested in the logarithm of the quantities T −T c , δL−δL c , in which T c is the critical temperature defined in (12), and L c is obtained numerically by the equation T (δL) = T c . We plot the relation between log | T − T c | and log | δL − δL c | for different θ 0 in Figure 5, where these straight lines can be fitted as log | T − T c |= 23.2318 + 3.06832 log | δL − δL c |, for θ 0 = 0.14, 31.9841 + 3.00077 log | δL − δL c |, for θ 0 = 0.2.
It is obvious that the slope is about 3, which indicates that the critical exponent is −2/3 for the analogous heat capacity and the phase transition is also second order at T c for the geodesic length.

Phase transition of Wilson loop
In this subsection, we are going to study the phase structure of the Wilson loop, which in the bulk corresponds to the minimal area surface by holography. Wilson loop operator is defined as a path ordered integral of gauge field over a closed contour, and its expectation value is approximated geometrically by the AdS/CFT correspondence as [32] W where C is the closed contour, Σ is the minimal bulk surface ending on C with A its minimal area, and α ′ is the Regge slope parameter. Next we choose the line with φ = π 2 and θ = θ 0 as our loop. Then we can employ (θ, ψ) to parameterize the minimal area surface, which is invariant under the ψ-direction by our rotational symmetry. Thus the corresponding minimal area surface can be expressed as in whichṙ = dr/dθ. Making use of the Euler-Lagrange equation, one can get the equation of motion for r(θ). Then with the boundary conditions r ′ (0) = 0, r(0) = r 0 , we can further get the numeric result of r(θ). Similar to the case of geodesic length, we choose θ 0 = 0.14, 0.2 as two examples and the corresponding UV cutoffs are set to be r(0.139), r(0.199). We label the regularized minimal area surface as δA ≡ A − A 0 , where A 0 is the minimal area in pure AdS with the same boundary region. We plot the relation between δA and T for different θ 0 in Figure 6 and Figure 7. Comparing Figure 6 with Figure 7, we find they are the same nearly besides the scale of the horizonal coordinate. In other words, θ 0 affects only the value but not the phase structure of minimal area surface in the T − δA plane. The result tells us that the similar phase structure also shows up for the minimal surface area. Here we concentrate only on scrutinizing the equal area law for the first order phase transition and the critical exponent of the analogous heat capacity for the second order phase transition. First, in the δA − T plane, the equal area law can be similarly defined as in which T (δA) is an Interpolating Function obtained from our numeric result, and δA 1 , δA 3 are the smallest and largest roots of the equation T (δA) = T ⋆ , respectively. As the same as that of the geodesic length, for a fixed θ 0 , we first obtain δA 1 , δA 3 , and then substitute these values into (25) to produce A 1 , A 3 . The concrete values are listed in Table  1. Obviously, for both θ 0 , A 1 and A 3 are equal within the reasonable numeric accuracy. The equal area law thus holds in the δA − T plane, which reinforces the fact that the minimal surface area has the same first order phase transition behavior as that of the thermal entropy.  Table 1: Check of the equal area law in the T − δA plane for different θ 0 . Second, in order to check whether the minimal surface area also demonstrates the same second order phase transition as the thermal entropy, we would like to evaluate the critical exponent of the analogous heat capacity at the critical point in the δA − T plane. To this end, we plot the relations between log | T − T c | and log | δA − δA c | in Figure 8. The numerical results for these curves can be fitted as log | T − T c |= 27.5226 + 3.04698 log | δA − δA c |, for θ 0 = 0.14, 24.692 + 3.00462 log | δA − δA c |, With 3 the slope, we can conclude that the minimal surface area also has the same second order phase transition as the thermal entropy.

Phase transition of entanglement entropy
Holographic entanglement entropy is another non-local observable, and it has been used extensively to probe the superconductivity phase transition besides the thermalization process recently [33,34,35,36,37,38,39,40]. In this subsection, we intend to employ it to probe the phase structure of a 5-dimensional Reissner-Nordström-AdS black hole.
According to the formula in [41,42], holographic entanglement entropy can be given by the area A Σ of a minimal surface Σ anchored on the boundary entangling surface ∂Σ, namely For simplicity, we choose φ = φ 0 as our entangling surface and employ (φ, θ, ψ) to parameterize the minimal surface. But with the symmetry of (2), (27) can be rewritten as φ 0 = 0.14  δS and T for φ 0 = 0.14, 0.2 in Figure 9 and Figure 10, respectively. As one can see, it exhibits a similar behavior as the thermal entropy.
To be more precise, we would like to check the equal area law with the following equation in which T (δS) is an Interpolating Function obtained from the numeric result and δS 1 , δS 3 are the smallest and largest roots of the equation T (δS) = T ⋆ . For different φ 0 s, the results of δS 1 , δS 3 and A 1 , A 3 are listed in Table 2. It is obvious that A 1 nearly equals A 3 regardless of the choice of φ 0 . That is, the equal area law is also valid for the entanglement entropy.
To get the critical exponent of second order phase transition of entanglement entropy, we should find the slope of a linear function represented by log | T −T c | and log | δS −δS c |, in which S c is the critical entropy obtained numerically by the equation T (δS) = T c . The numeric results for different φ 0 are plotted in Figure 11. The results for these curves can be further fitted as log | T − T c |= 26.653 + 3.00107 log | δS − δS c |, for φ 0 = 0.14 21.2674 + 2.92789 log | δS − δS c |, for φ 0 = 0.2 (30) One can see that the slope is always about 3 for different φ 0 . So we can conclude that the entanglement entropy also has the same second order phase transition as the thermal entropy.

Concluding remarks
Investigation on the phase transition of the black holes is important and necessary. On one hand, it is helpful for us to understand the structure and nature of the space time.
On the other hand, it may uncover some phase transitions of the realistic physics in the conformal field theory according to the AdS/CFT correspondence. It is well-known now that the Hawking-Page phase transition in the gravity system is dual to the confinement/deconfinement phase transition, and the phase transition of a scalar field is dual to the superconductivity phase transition in the dual conformal field theory.
In this paper, we investigated the van der Waals-like phase transition in the framework of holography so that we can explore whether there is a realistic similar phase transition in physics. Taking the 5-dimensional Reissner-Nordström-AdS black hole as the gravity background, we investigated the phase structure of the two point correlation function, Wilson loop, and holographic entanglement entropy. For all the non-local observables, we observed that the black hole undergos a van der Waals-like phase transition. This conclusion is reinforced by the investigation of the equal area law and critical exponent of the analogous heat capacity in which we found that the equal area law is valid always and the critical exponent of the heat capacity coincides with that of the mean field theory regardless of the size of the boundary region. In addition, we found the black hole undergos a Hawking-Page phase transition before the van der Waals-like phase transition for all the non-local observables. We also obtained the minimum temperature and Hawking-Page phase transition temperature. Our investigation thus provides a complete picture depicting the phase transition of charged AdS black hole in the framework of holography.