Entanglement entropy in a holographic p-wave superconductor model

In a recent paper, arXiv:1309.4877, a holographic p-wave model has been proposed in an Einstein-Maxwell-complex vector field theory with a negative cosmological constant. The model exhibits rich phase structure depending on the mass and the charge of the vector field. We investigate the behavior of the entanglement entropy of dual field theory in this model. When the above two model parameters change, we observe the second order, first order and zeroth order phase transitions from the behavior of the entanglement entropy at some intermediate temperatures. These imply that the entanglement entropy can indicate not only the occurrence of the phase transition, but also the order of the phase transition. The entanglement entropy is indeed a good probe to phase transition. Furthermore, the"retrograde condensation"which is a sub-dominated phase is also reflected on the entanglement entropy.


Introduction
The guage/gravity duality [1,2,3] provides us a novel method to study the strongly coupled field theories through a weakly coupled gravitational system in one higher dimension. In a recent paper [4], we have proposed a holographic model of p-wave superconductors by introducing a complex vector field ρ µ charged under a Maxwell gauge field A µ in the bulk, which is different from the previous holographic p-wave model constructed by adding a SU(2) Yang-Mill fields [5] or realized from the condensation of a two-form field in bulk [6]. According to the AdS/CFT dictionary, this charged ρ µ is dual to a charged vector operator with a global U(1) symmetry in a strongly coupled system. We have studied this model in the probe limit [4]. We have found the system undergoes a second order phase transition at a critical temperature and the critical exponent of this transition is one half coinciding exactly with the results in the Landau-Ginzburg theory. Our above computations were performed in the approximation scheme with neglecting the back reaction of the matter fields. As it is well known, the new phases may emerge [7,8] and the order of the phase transition can also be changed [9,10,11,12] in the fully back reacted geometry. We have further examined how the phase diagram changed due to the back reaction of the vector field on the above p-wave model in our following paper [13]. The results showed that depending on two parameters, the mass and charge of the vector field, such model exhibits a rich phase structure. The second order, first order and the zeroth order phase transitions happen when tuning the model parameters. We also found the "retrograde condensation", but it is sub-dominated in the sense that its free energy is much larger than the black hole without hair.
While several aspects of the new holographic p-wave model have been studied carefully, in this paper we will study the behavior of entanglement entropy in this model. For such a strongly coupled system, the entanglement entropy is looked as a robust tool for keeping track of the degrees of freedom when other traditional probes might not be available. And the previous studies have confirmed that the entanglement entropy is indeed a good probe to investigate the holographic phase transition. It can indicate not only the appearance, but also the order of the phase transition. Considering the new proposed model has a rich phase structure, it is quite interesting to see the behavior of entanglement entropy in this p-wave model, in particular to see how the entanglement entropy changes with the order of the phase transition.
The proposal of holographic entanglement entropy [14] in the context of AdS/CFT correspondence provides an effective way to calculate the entanglement entropy using bulk geometry (for reviews see refs. [15,16]). With this method, further works on the entanglement entropy have been widely studied. Refs. [17,18,19] studied the behaviors of entanglement entropy in holographic conductor/superconductor phase transition, including s-wave and p-wave cases. The results showed that the entanglement entropy decreases as one lowers temperature, indicating the degrees of freedom are reduced at lower temperature. And the behavior of the entanglement entropy changes dramatically when the order of the phase transition changes, which implies that the entanglement entropy is indeed a good probe to phase transition. Similarly, the behaviors of entanglement entropy for the holographic s-wave and p-wave superconductor/insulator model have also been investigated in refs. [20,21,22]. Unlike the conductor/superconductor phase transition, the entanglement entropy in the s-wave superconductor/insulator case [20,21] as a function of chemical potential is not monotonic: at the beginning of the transition, the entropy first increases and reaches its maximum at some chemical potential, and then decreases monotonically.
In this paper, we continue our study of the entanglement entropy in this new holographic p-wave superconductor model. The entanglement entropy is calculated for a straight strip geometry with the holographic proposal. We find that the behavior of entanglement entropy changes dramatically with the order of phase transition as we alter the mass square m 2 and the charge q of the vector field. These results suggest that the entanglement entropy is really a good probe to indicate the appearance as well as the order of phase transition. By comparing the entanglement entropy with the thermal entropy of the bulk black holes during the whole process of phase transition, we see the possibility for understanding the black hole entropy as the entanglement entropy [23,24,25,26]. Furthermore, the "retrograde condensation" which is a sub-dominated phase is also reflected on the entanglement entropy.
This paper is organized as follows. In the next section, we briefly review the holographic p-wave superconductor model proposed recently. In section 3, the fully back-reacted system is solved by shooting method and the main phase structure of the model is briefly summarized. Section 4 is devoted to exploring the behaviors of the entanglement entropy in the p-wave superconductor model. We present numerical results in this section and for each given m 2 , we scan a widely range of q to find all possible behaviors of the entanglement entropy. The conclusions and some discussions are included in section 5.

The holographic model
Let us start from the holographic model of p-wave superconductors proposed in refs. [4,13] where κ 2 ≡ 8πG is the gravitational constant, L is the radius of AdS spactime (we will take L = 1 in the following numerical calculation), the Maxwell field strength The last non-minimal coupling term in the matter sector plays an important role when including a background magnetic field [4]. Following ref. [13], we only consider the case without turning on a magnetic field, thus this model is left with only two independent parameters, i.e., the mass m of the vector field ρ µ giving the dimension of the dual vector operator and its charge q controlling the strength of the back reaction on the background geometry.
Varying the action, we have the equations of motion for gauge field A µ and charged vector field ρ µ as and the corresponding Einstein's field equations The system admits an analytical solution with vanishing ρ µ , corresponding to the normal phase. This solution with planar symmetry is the AdS Reissner-Nordström black hole given by with r h is the horizon radius and µ is the chemical potential of the dual field theory. The ). When we tune the temperature, the system exhibits an instability which triggers the condensation of the charged vector field ρ µ . According to AdS/CFT correspondence, this hairy black hole with appropriate boundary conditions can be explained as a condensed phase of the dual field theory. The process from black hole without hair to black hole with non-trivial vector hair mimics the conductor/superconductor phase transition. To study the behavior of the entanglement entropy during this process, we need to consider the back reaction of the matter fields on the background geometry. Therefore we take the following ansatz [13] ds 2 = −f (r)e −χ(r) dt 2 + dr 2 f (r) + r 2 h(r)dx 2 + r 2 dy 2 , The temperature T of the black hole is given by And the thermal entropy of the black hole is given by the Bekenstein-Hawking area formula: where V 2 is the area spanned by coordinates x and y. With the above ansatz, the independent equations of motion turn out to be where the prime denotes the derivative with respect to r. We will use shooting method to solve the above equations (7). To do this, we have to first specify the boundary conditions for this system. Near the AdS boundary r → ∞, these fields have the following expansion behaviors: . According to the AdS/CFT dictionary, the constants µ and ρ can be interpreted as the chemical potential and the charge density in the dual field theory, respectively. ρ x − is the source of the dual operator and ρ x + gives its expectation value. To require the U(1) symmetry being broken spontaneously, we will take ρ x − = 0 in the numerical calculation.
We impose regular conditions on the horizon r = r h . Particularly, one has f (r h ) = 0 and φ(r h ) = 0. Then we are left with five independent parameters {r h , ρ can be used to set r h = 1 for performing numerics. Similarly, the scaling symmetries and lead us to set {χ(r h ) = 0, h(r h ) = 1}. Thus we finally have two independent parameters {ρ x (r h ), φ ′ (r h )} at hand. Given φ ′ (r h ) as the shooting parameter to match the source free condition, i.e, ρ x− = 0, we can solve the fully coupled differential equations. For the details of numerical calculation, please refer to ref. [13].

Various phase transitions
In order to determine which phase is thermodynamically favored, we should calculate the free energy of the system for both normal phase and condensed phase. We will work in grand canonical ensemble with fixed chemical potential. Following ref. [13], the free energy Ω can be expressed as For the normal phase shown in (4), one has f 3 = −r 3 h − µ 2 r h 4 , and h 3 = χ 3 = 0. As stated before, we only have two independent parameters m 2 and q left. We take different m 2 into consideration and then for each m 2 we scan a wide range of q. Our numerical results revealed that the system exhibits diverse behaviors depending on concrete m 2 . There exists a critical mass square m 2 c . For m 2 > m 2 c , the condensed phase seems to survive even at a low temperature shown in figure 1, while in figure 2 with the case m 2 < m 2 c , the condensed phase cannot survive below a finite temperature. Our numerical result suggests that m 2 c ≃ −0.004 ± 0.005. The complete diagrams are drawn in ref. [13]. Figure 1 presents the grand potential Ω as a function of temperature in the case m 2 > m 2 c . It is clear that below the critical temperature T c , the state with non-vanishing vector "hair" is indeed thermodynamically favored over the normal phase. The q = 3/2 plot describes a second order phase transition. When we decrease q past q c , taking q = 6/5 as an example, the free energy versus temperature develops a characteristic "swallow tail" which characterizes a first order phase transition. The critical value of q c is 1.3575 [13].
The phase structures are more complicated for the case m 2 < m 2 c shown in figure 2. With the different back reaction strength q, the phase diagram is divided into three regimes. The upper two plots tell us that the superconducting phase emerges at a critical temperature with T 2 representing a second order phase transition and T 1 a first order phase transition. When one continues to lower the temperature, the free energy will have a jump to the normal phase at T 0 indicating a zeroth phase transition in both cases. More explicitly, the q = 2 plot disposes a second order phase transition at temperature T 2 , while the q = 39/40 plot exhibits a first order phase transition at T 1 . The critical value between first order and second order phase transition is q α ≃ 1.0175 in the case with m 2 = −3/16. When we continue decreasing q, making q less than q β ≃ 0.9537, we see that the superconducting phase is not thermodynamically favored in that the free energy for condensed phase is much larger than the one for the normal phase shown in the q = 19/20 plot and q = 9/10 plot.
In summary, depending on the model parameters m 2 and q, the model exhibits a rich phase structure. We will further study these phase transitions through the holographic

Entanglement entropy
In the framework of AdS/CFT correspondence, a holographic method to calculate the entanglement entropy has been proposed in ref. [14]. Following ref. [14], for a conformal field theory (CFT) which has a dual gravitational configuration living in one higher dimension, the entanglement entropy of the CFT in a subsystem A with its complement can be obtained by searching the minimal area surface γ A extended into the bulk with the same boundary ∂A of A. That is, the entanglement entropy of A with its complement is given by the "area law" In this section we will study the behavior of entanglement entropy in this holographic p-wave superconductor model. We will consider a belt geometry with a finite width l along the x direction and extends in y direction. The holographic dual surface γ A is defined as a two-dimensional surface R is the regularized length in y direction. To avoid the UV divergence, we consider the subsystem A sits on the slice r = 1 ǫ with ǫ → 0 the UV cutoff. More specifically, the holographic surface γ A starts from x = ℓ 2 at r = 1 ǫ , extends into the bulk until it reaches r = r * , then returns back to the AdS boundary r = 1 ǫ at x = − ℓ 2 . For such embedding, The q = 2 plot shows the second order phase transition but the condensed phase terminates at finite low temperature T 0 . The q = 39/40 plot exhibits a first order phase transition but also the superconducing phase terminates at finite low temperature T 0 . The q = 19/20 and q = 9/10 plots demonstrate that the condensed phase has free energy much larger than the normal phase and thus is not thermodynamical favored.
the entanglement entropy of the subsystem is given by where the UV divergence part 1/ǫ has been separated from the total entropy. Thus S E is the finite part of physical relevance. The width l of the subsystem A and r * are connected by the relation l 2 = In addition, note that under the transformation (11), the scale invariants are S E /µ and ℓµ.
Once solved the metric functions numerically, we can calculate the entanglement entropy with the (15) and (16). We will give the entanglement entropy for typical m 2 and characteristic q. We find that the entanglement entropy S E with respect to strip width l behaves quite similar for different choice of parameters m 2 and q.
An example is shown in figure 3 with m 2 = 3/4 and q = 3/2. Different colorful lines reveal how the entanglement entropy versus belt width behaves when changing the temperature by fixing m 2 and q. From the top to bottom, the temperature decreases. The curve at the top is for the critical temperature T c , which is identical with the pure AdS Reissner-Nordström case. From figure 3, we observe that the slope of the curve decreases when the temperature lowers in superconductor situation as is expected that the lower the temperature is, the more the degrees of freedom will condense. This phenomenon can be seen more clearly later. The entanglement entropy evolving with temperature by fixing the belt width will be given in the following. As stressed in the last section, the system exhibits distinct behavior depending on m 2 and q of the vector field. There is a critical mass square m 2 c , above which the condensed phase survives even down to sufficiently low temperature while for the case m 2 < m 2 c , the condensed phase cannot exist below a finite temperature. For each m 2 , there are some special values of q, which divide the order of the phase transition. We will see how the entanglement entropy evolves with temperature in all the above cases.

m 2 =3/4
First, we focus on the case m 2 > m 2 c . We take m 2 = 3/4 as before [13]. The entanglement entropy as a function of temperature with fixed belt width for q = 3/2 is listed in figure 4 and q = 6/5 in figure 5. The critical temperatures T c are determined by the grand potential presented in figure 1. Figure 4 shows that the entanglement entropy is continuous at critical temperature T c , but its slope is not. This discontinuity may signal a significant reorganization of the degrees of freedom of the system. Since there is a condensate generated at the transition point, it is expected that there is a reduction of degrees of freedom. The behavior of the entanglement entropy versus temperature indicates a typical second order phase transition. As we decrease q to q = 6/5 which is less than q c = 1.3575, the entanglement entropy as a function of temperature is presented in figure 5. The blue curve from normal phase is physical as T > T c , while the curve with the lowest entanglement entropy is physically preferred below T c . There is an obvious jump in the entanglement entropy itself as well as its slope at critical temperature T c . Once again, the entanglement entropy versus temperature indicates this conductor/superconductor phase transition is first order.
Note that the behavior of the entanglement entropy derived here is qualitatively the same with the thermal entropy obtained in our recent paper [13]. Considering the thermal entropy is equivalent to the black hole entropy in the holographic setup, our results indicate that the black hole entropy could be due to the entanglement entropy [23,24,25,26].
All in all, we see the qualitative behavior of entanglement entropy versus temperature is dramatically different for small and large q. For the case q > q c , the entanglement entropy is continuous at the critical temperature T c , while it has a jump for q < q c . That is, the behavior of the entanglement entropy with respect to temperature shows that the phase transition is second order when q > q c , while it is first order as q < q c . The entanglement entropy indeed indicates the occurrence and the order of the phase transition.

m 2 =-3/16
In this subsection, we will explore the entanglement entropy with respect to temperature for different q in the case m 2 < m 2 c . Just like in ref. [13] we take m 2 = −3/16 as an example. According to thermodynamic behavior, the parameter space of q is divided into three regions, q > q α , q α < q < q β and q < q β , where q α ≃ 1.0175 and q β ≃ 0.9537. Here we concentrate on the concrete cases q = 2, q = 39/40, q = 19/20 and q = 9/10 which belong to the above three regions, respectively. The numerical results are shown in figures 6-9, respectively. For the case q > q α , we focus on q = 2. The corresponding results are shown in figure 6. The entanglement entropy for the superconducting phase is divided into two branches, i.e., the upper-branch with large S E and a down-branch with small S E . Comparing the free energy (the top left plot in figure 2) for each solution, we find that the entanglement entropy for condensed phase in upper-branch is physical, which only exists in a small range T 0 < T < T 2 . At other temperatures, including T < T 0 and T > T 2 , it is the entanglement entropy for normal phase physically relevant. From figure 6, we see that S E is continuous but has a kink at T 2 , indicating a second order transition. And at a finite lower temperature T 0 , a sudden jump from the condensed phase to the normal phase in the entanglement entropy, which may correspond to the destruction of the Cooper pair and thus increasing the degrees of the freedom, signals a zeroth order transition.
For the case q β < q < q α , we consider q = 39/40. The behavior of the entanglement entropy as a function of temperature is shown in figure 7, which is much more complicated than the case in figure 6. The entanglement entropy S E is also multi-valued above T 0 . According to the value of S E , we denote them as the upper-branch for large S E , the middle-branch for middle S E and the down-branch for small S E , respectively. Based on the thermodynamical analysis (the top right plot in figure 2), when lowers the temperature, at the beginning the entanglement entropy of the normal phase is physical till T 1 , and then the middle-branch of the condensed phase dominates, finally at the temperature T 0 the entanglement entropy jumps back to the normal phase. From the behavior of the entanglement entropy versus temperature, we observe a first order transition from the The dashed blue curves are from the AdS Reissner-Nordström solutions, while the solid curves are from superconducting solutions. The physical entropy is selected by choosing the dashed curve above T 2 ≃ 0.1487µ and below T 0 ≃ 0.0597µ. As T 0 < T < T 2 , the upper branch of condensed phase with higher entanglement entropy is physically preferred. The S E is continuous but not differentiable at T 2 , characterizing a second order phase transition.
normal phase to the condensed phase at higher temperature T 1 and then a zeroth order transition from the condensed phase to the normal phase at lower temperature T 0 . As q decreases past q β , we see a dramatic change in the behavior of the entanglement entropy in figure 8 and figure 9 for the cases q = 19/20 and q = 9/10. The entanglement entropy S E is multi-valued in figure 8. As q diminishes, this multi-valued behavior disappears in figure 9. Furthermore, especially in the case q = 9/10, the entanglement entropy always decreases with the temperature increases, i.e., the slope is negative in figure 9. This behavior is opposite to the usual idea that increasing the temperature should promote the entanglement entropy. This strange behavior is due to the condensed phase emerging at high temperatures rather than at low temperatures. Comparing free energy Ω between the condensed phase and normal phase (bottom left and bottom right plots in figure 2), we clearly see that the condensed phase has free energy much larger than the normal phase and thus is not thermodynamically favored. Therefore, in each case only the entanglement entropy of the normal phase is physical.
Through the above analysis, as expected, we observe a second order transition, first order transition and zeroth order transition in the entanglement entropy, showing its utility as an independent probe of the phase structure of the superconductor. Especially for the case m 2 < m 2 c , as we low the temperature to a critical value, the conductor/superconductor phase transition is second order for sufficiently large q, while it is first order for smaller q. When the temperature lowers further to T 0 , an additional zeroth order transition exists. It may be caused by the limitation of a simple case with the charged vector ρ x non-vanishing only. We may turn on the temporal component ρ t , and search for the hairy black hole configuration with the least free energy among all possible configurations. The dashed blue curves are from the AdS Reissner-Nordström solutions, while the solid curves are from superconductor solutions. The physical entropy is selected by choosing the dashed curve above T 1 ≃ 0.04102µ and below T 0 ≃ 0.03992µ. As T 0 < T < T 1 , the middle branch of condensed phase entropy is thermodynamically preferred. The S E and its slope is not differentiable at T 1 , characterizing a first order phase transition.

Conclusions and discussions
In a recent paper [4] we constructed a holographic p-wave superconductor model in a four dimensional Einstein-Maxwell-complex vector field theory with a negative cosmological constant. Taking the back reaction of the vector fields into consideration, such model exhibits a rich phase structure [13]. When the mass m 2 and charge q of the vector field ρ µ change, the second order phase transition, first order transition and zeroth order transition show up. Considering the entanglement entropy as an independent probe of superconductor, we studied the behavior of entanglement entropy for a straight strip geometry in this model. The behaviors of the entanglement entropy have a dramatic change from large m 2 to small m 2 . For large m 2 with weak strength of back reaction, the entanglement entropy evolving with the temperature is continuous at the critical temperature T c but its slope changes discontinuously at T c . This discontinuity signals a second order phase transition from the normal phase to the condensed phase shown in figure 4. However, as we enhance the strength of the back reaction, both the entanglement entropy versus temperature and its slope at the critical temperature in figure 5 have an obvious jump. The phase transition becomes first order. In contrast to the previous case, for the case with small m 2 , no matter the value of the back reaction is, there is a particular temperature below which the entanglement entropy turns back to the one at the high temperature as a characteristic of a zeroth order transition shown in figure 6 and figure 7. In figure 6, the back reaction is weak, the entanglement entropy is continuous at the critical temperature T 2 but the slope at T 2 is discontinuous indicating a second order transition, then at a lower temperature T 0 , the entanglement entropy jumps to the normal phase, which implies a zeroth order phase transition. As we strengthen the back reaction, the entanglement entropy first encounters a  first order transition at critical temperature T 1 , the following is a zeroth order transition at lower temperature T 0 . The entanglement entropy undergoes two jumps at temperatures T 1 and T 0 respectively in figure 7. For sufficiently strong back reaction, the condensed phase only occurs at high temperature rather than low temperature, which means that only the entanglement entropy of the normal phase is physically relevant. The entanglement entropy is really a good probe to phase transition. Note that in this paper we only study the entanglement entropy with finite width along the x direction. In such model, the spatial rotational symmetry is broken in the superconducting phase. To capture the anisotropy on x direction and y direction, we should also study the entanglement entropy along y direction. We do not calculate it here since the behavior of the entanglement entropy along y direction is expected to be qualitatively similar to the case in x direction. As a non-local quantity, the entanglement entropy describes the new degrees of freedom emerging in the superconducting phase. Since the degree of anisotropy of the superconducting phase is characterized by the value of condensate, therefore, it is not expected to have qualitatively difference between x direction and y direction [22].
There are other aspects about the model to be studied, such as competition and coexistence of the multiple superconduction order parameters, implication of the zeroth order phase transition, and the effect of external magnetic field, etc. We expect to report further progress on these issues.