Phase transitions in a holographic s+p model with backreaction

In a previous paper (arXiv:1309.2204, JHEP 1311 (2013) 087), we present a holographic s+p superconductor model with a scalar triplet charged under an SU(2) gauge field in the bulk. We also study the competition and coexistence of the s-wave and p-wave orders in the probe limit. In this work we continue to study the model by considering the full back-reaction The model shows a rich phase structure and various condensate behaviors such as the"n-type"and"u-type"ones, which are also known as reentrant phase transitions in condensed matter physics. The phase transitions to the p-wave phase or s+p coexisting phase become first order in strong back-reaction cases. In these first order phase transitions, the free energy curve always forms a swallow tail shape, in which the unstable s+p solution can also play an important role. The phase diagrams of this model are given in terms of the dimension of the scalar order and the temperature in the cases of eight different values of the back reaction parameter, which show that the region for the s+p coexisting phase is enlarged with a small or medium back reaction parameter, but is reduced in the strong back-reaction cases.

In a previous paper (arXiv:1309.2204, JHEP 1311 (2013) 087), we present a holographic s+p superconductor model with a scalar triplet charged under an SU (2) gauge field in the bulk and study the competition and coexistence of the s-wave and p-wave orders in the probe limit. In this work we continue to study the model by considering the full back reaction. The model shows a rich phase structure and various condensate behaviors such as the "n-type" and "u-type" ones. The phase transitions to the p-wave phase or s+p coexisting phase become first order in strongly back reacted cases. In these first order phase transitions, the free energy curve always forms a swallow tail shape, in which the unstable s+p solution can also play an important role. The phase diagrams of this system are given in terms of the dimension of the scalar order and the temperature in the cases of eight different values of the back reaction parameter, which show that the region for the s+p coexisting phase is enlarged with a small or medium back reaction parameter, but is reduced in the strongly back reacted case.

I. INTRODUCTION
Over the past years, AdS/CFT correspondence [1][2][3] has attracted a lot of attention as a tool to study strongly coupled systems. One of the most successful applications of the correspondence is the so-called holographic superconductors which was first proposed in Refs. [4,5]. In this model, the condensation of a charged field in asymptotically AdS spacetime can trigger the superconducting phase transition on the dual field theory. According to the different property especially the symmetry for the condensed fields, the holographic model has been extended to the p-wave and d-wave superconductor models [6][7][8][9].
Recently the competition and coexistence between different order parameters have been studied in some holographic framework. In Refs. [10][11][12], the authors studied the system with two s-wave orders. Refs. [13][14][15] presented the result of the holographic system with swave and p-wave orders simultaneously. The holographic systems with one s-wave order and one d-wave order have also been investigated in Refs. [16,17]. More works on holographic study of the multi-order systems can be found in Refs. [18][19][20][21][22][23][24][25][26][27][28]. These studies might shed light to the understanding of some universal properties of the competition and coexistence between different orders in real materials such as high T c cuprates and the 3 He superfluid.
In a previous work [13], we built a holographic model with a scalar triplet charged under an SU (2) gauge field in the bulk and studied the competition and coexistence between the s-wave and p-wave superconducting order parameters. The result showed that the system could be in an s-wave phase at higher temperature and then transfers to a p-wave phase at lower temperature via an s+p coexisting phase. In that work, we worked in the probe limit and found that the temperature region for the s+p coexisting phase is very narrow. It is interesting to consider the effect of the back reaction of the matter fields on the background geometry. Because the strength of the back reaction can affect the critical temperature of the s-wave and p-wave superconducting phase transitions, there might be rich phenomena in the holographic s+p model away from the probe limit. Indeed we see this issue by studying the condensation behavior for the different orders as well as the phase diagram in the case with finite back reaction strength.
Another interesting behavior when the back reaction is taken into account is that the phase transition for the p-wave condensation is first order when the back reaction parameter is large enough [29][30][31]. It would be interesting to study what would happen to the s+p coexisting phase when the p-wave phase transition becomes a first order type in our model.
In this paper, we turn on the back reaction of matter field on the background geometry in our holographic s+p model [13]. More precisely, the strength of the back reaction is controlled by a parameter b. For simplicity, we consider eight values of back reaction parameter b varying from 0.1 to 0.8, and build corresponding phase diagrams in terms of the dimension of the scalar order and the temperature.
This article is organized as follows. In the next section, we briefly introduce our holographic model and the definition for the parameters. In section III we calculate the free energy of the system. In section IV, we show the condensation behavior of some interesting phase transitions in different back reaction strength. We also give the free energy behavior for some special cases with unstable s+p solution to illustrate the important role played by the unstable s+p solution. In section V, we construct the phase diagrams with eight different values of back reaction parameter. In section VI, we give another setup of a holographic s+p model, from which with some special choice of model parameters we can get the same equations of motions and thus the same results in the metric ansatz made shortly. We summarize our results and make some outlooks in section VII.

II. HOLOGRAPHIC MODEL OF AN S+P SUPERCONDUCTOR
The action of our holographic s+p superconductor model takes the following form [13] Here Ψ a is an SU (2) scalar triplet and A a µ is the SU (2) gauge field with F a µν the field strength. a = (1, 2, 3) is the index of the generators of SU(2) algebra. From this action, we can get the equations of motion for the matter fields and that for the gravitational fields where b = κ g /g c characterizes the strength of the back reaction of the matter field on the background geometry, and T µν is the stress-energy tensor of the matter sector Here D µ is the covariant derivative with SU(2) connection on the curved spacetime background and ε abc is antisymmetrical with ε 123 = 1. The action of D µ on a spacetime tensor T a µ 1 ...µn with one group index a is We will work in d = 3 case as in Ref. [13]. In order to get both the s-wave and p-wave superconducting phases in this model, we can choose the A 1 µ field as the electro-magnetic U (1) gauge field, and Ψ 3 , A 3 x as the scalar and vector order parameters, respectively. Thus the ansatz for the matter fields is and all the other components of the scalar and gauge field are set to be zero. The background black hole geometry compatible with the ansatz of matter fields (9) is [31][32][33] with where L is the AdS radius. The function M (r) is related to the mass and charge of the black brane. The horizon of the black brane is located at r = r h where M (r h ) = r 3 h /2, and the temperature is The full equations of motion for this system under the above ansatz read Note that in the above equations, there exist four sets of scaling symmetries as follows These scaling symmetries are very useful in the numerics. For example, we can use the first scaling symmetry to set L = 1 and the second one to set r h = 1. After we get the numerical solution, we can use these scaling symmetries again to recover L and r h to any value. The last two scaling symmetries will be used to scale the solutions to asymptotically AdS type with lim r→∞ σ(r) → 1 and lim r→∞ f (r) → 1. In order to solve the equations of motion numerically, we need to specify the boundary conditions both on the horizon r = r h and on the boundary r = ∞. Without lose of generality, we set L = 1 in the rest of the paper. Then near the horizon, the functions can be expanded as One can check that only the coefficients {σ h0 , f h0 , φ h1 , Ψ xh0 , Ψ 3h0 } are independent. That means when we work at some fixed value of b and m 2 , we can get a group of solutions with these five independent parameters. However, we should further consider some constraints from the boundary r = ∞. Then the degrees of freedom of the solutions would be reduced. We can expand the functions near the AdS boundary as where ∆ = (3 + √ 9 + 4m 2 )/2 is the scaling dimension of the scalar order. Note that there is no mass term for the SU(2) fields, thus the dimension for the p-wave order is fixed to In order to make the boundary geometry to be asymptotically AdS(a-AdS), we should have σ b0 = f b0 = 1. These two conditions could be easily satisfied by a scale transformation from any known solution by using the last two scaling symmetries.
In the asymptotically AdS space-time, we can use the AdS/CFT correspondence to get information of the dual field theory. The AdS/CFT dictionary tells us that µ and ρ are related to the chemical potential and charge density respectively, while Ψ 3S , Ψ xb0 are related to the sources and Ψ 3E , Ψ xb1 are related to the expectation values of the dual s-wave and p-wave operators. To study the symmetry broken spontaneously phases, we further impose the source free condition with Ψ xb0 = Ψ 3S = 0. There is an alternative quantization with the expression of the source and expectation interpretation exchanged [34][35][36][37]. But here we only focus on the standard quantization case.
With these conditions from the boundary, we can account the degrees of freedom left for the equations. Besides the two parameters b and ∆ from the model, we have five free parameters at horizon to fix the solution. The solution we need should satisfy four conditions on the boundary. Thus the solutions satisfying the four boundary conditions would have only one free parameter besides b and ∆. In this paper, we would choose this free parameter as T /µ, and we further fix the value of µ because we work in the grand canonical ensemble. This is convenient for comparing the results in this paper to that in the probe limit in Ref. [13] .

III. FREE ENERGIES
In our model, there is one degree of freedom denoted by T for the solutions, once we have fixed the value of b and ∆. The model includes the AdS Reissner-Norsdtröm (RN) black brane solution describing the normal conductor phase, the s-wave superconductivity solution, the p-wave superconductivity solution, as well as the s+p superconductor solution with both the expectation values for the scalar and vector order parameters non zero. These four kinds of solutions are all possible solutions with one free parameter T at fixed value of (b, ∆), but these different solutions are in different branches. Thus at some fixed value of T , there may exist more than one solution, which means that the system may have more than one possible phase simultaneously. So we need to calculate the free energies of the different phases (solutions) to find out which one is the physically favored.
The free energy of the system is equal to the temperature T times the Euclidean on shell action of the bulk spacetime Ω = T S E .
In S E , we should consider both the bulk terms and some boundary terms. Remember that all the solutions are source free and we have fixed the chemical potential, which means we study the system in grand canonical ensemble. In this case, the action S E can be expressed as By substituting the solutions to the Euclidean action, the integrand can be written as a total derivative term, thus the on shell action and the free energy can be evaluated by the boundary value of the functions as where V 2 denotes the area of the two dimensional transverse space. For the normal phase which is dual to the AdS-RN black brane solution the temperature and free energy are while for the condensed phases, we have Using the above formulas, we can get the free energy of all the solutions and compare these free energies at the same temperature and (b, ∆). Thus we can draw the condensate of the s-wave and p-wave orders for the most stable phase at any temperature at fixed b and ∆ (Figs. 1, 2, 3, 4, 5). We can see from these figures that what kind of phase transitions would occur. At some values of b and ∆, the condensation behavior is very interesting, we show some typical cases in the next section.

IV. CONDENSATION AND PHASE TRANSITION
We study the model with eight discrete values of the back reaction strength b from 0.1 to 0.8. In each value of b, we can fix the value of ∆ to study the phase transitions of the system with a decreasing temperature. We should find all the possible solutions and calculate the relevant free energy to figure out the most stable phase at each temperature. Finally we can plot the condensation behavior of the s-wave and p-wave orders versus the temperature to show the phase transition at the fixed value of b and ∆.
We can also change the value of ∆ continuously to see what would happen to the stable phases and the phase transitions between them. With the information at different values of ∆, we can extract a phase diagram on the ∆ − T plane at the fixed value of b. We will show the eight phase diagrams with different values of b in the next section. In this section we show some interesting and typical phase transition behaviors by the figure of condensation values versus temperature at fixed b and ∆.
In each case of b, we vary the value of ∆ to find all the possible phase transitions. Below we pick some typical values of b and ∆, and draw figures for the condensation values of the s-wave and p-wave operators versus temperature in Figs. 1, 2, 3, 4, 5. In these figures, we use the solid red line to show the condensation value of the p-wave operator, and the solid blue line to show the condensation value of the s-wave operator. We also use dotted lines to show the condensation values in unstable phases. The dotted red lines denote the condensation value of the p-wave order in the unstable section of the p-wave phases, and the dotted blue lines denote the value of the s-wave order in the unstable section of the s-wave phases. The s+p coexisting phase might also become unstable in our study. When the s+p phase becomes unstable, we use dotted orange line to denote the condensation value of the p-wave order and use the dotted green line to denote that of the s-wave order. In Fig. 5 we also use vertical dashed black lines to denote critical temperatures of phase transitions between different condensed phases. In Fig. 1, the left figure is for the case b = 0.1 and ∆ = 1.9. In this figure, we can see a typical behavior of the s+p system, the s+p coexisting phase connects the p-wave phase in higher temperatures and the s-wave phase in lower temperatures. This behavior is similar to that in the probe limit [13], but the temperature region of the s+p coexisting phase is larger, compared to the case of the probe limit.
The right figure of Fig. 1 showes the case with b = 0.2 and ∆ = 1.93. We can see that the s+p coexisting phase emerges from a p-wave phase and is stable in the low temperature region ( the condensate of the p-wave order does not vanish even in low temperature). In the s+p coexisting phase, the condensation value of p-wave order is reduced while the value of the s-wave order is enhanced. This means that the s-wave order and the p-wave order still repel each other. Fig. 2 showes the results with b = 0.3 and ∆ = 1.885. The right figure is an enlarged version of the rectangle region in the left one. In this case, we find an interesting behavior for the s+p coexisting phase. The s+p coexisting phase begins from an s-wave phase at a higher temperature and finally goes back to the s-wave phase at a lower temperature. We can see from the picture that the condensation value of the p-wave order forms a shape of the letter "n", so we call this as an "n-type" condensation. In the n-type phase transition, the order parameter of the p-wave only gets non-zero values in a small region. Note that we can see from the right figure, that the value of s-wave order in the s+p coexisting phase is reduced, compared to the case of the pure s-wave phase.
In Fig. 3, we show the results with b = 0.4 and ∆ = 1.8544. In this figure, we can see that similar to the case in the right figure of Fig 1, the s+p coexisting phase is also stable in the low temperature region, but at this time, the s+p coexisting phase emerges from an s-wave phase instead. Fig. 4 showes two similar cases. The left figure is for b = 0.5, ∆ = 1.945 and the right one for b = 0.6, ∆ = 1.963. There are five regions of different phases in each of the figure, from the right to the left, they are the normal phase, the s-wave phase, the s+p coexisting phase, the p-wave phase and the second branch of s+p coexisting phase. The main difference between the two cases is that in the left figure, the phase transition from the s-wave phase to the s+p coexisting phase is second order, but in the right figure, this phase transition is first order. We can see from the two figures that the system goes into the p-wave phase from the s+p coexisting phase and goes back to the s+p coexisting phase at a lower temperature. The shape of the solid blue line forms a letter "u", so we call this as a "u-type" condensation. the s-wave phase and the p-wave phase. Besides this unstable s+p solution, there is another branch of stable s+p phase in the low temperature region. It is also interesting to notice that the two s+p phases overlap in some temperature region, that means one might find two different branches of solutions for the s+p coexisting phase at some temperatures. We should also notice that the condensation value of the p-wave order in the lower temperature branch of s+p phase differs from the value in the p-wave phase, although the two red lines (solid one for the s+p phase and dotted one for the p-wave phase) are very close to each other.
The third figure in Fig. 5 shows the case with ∆ = 2. We can see that the p-wave phase in this case is always unstable while the s+p phase is partly unstable, and the phase transition from the s-wave phase to the s+p phase is first order.
From the above figures we can see that this holographic system exhibits a rich phase structure. These different phase transition behaviors can be used to check the varies universalities in holographic systems. In particular, we have found that in this model, with some parameters, an s+p solution might be unstable, which is quite different from the previous studies, where once an s+p solution appears, it is usually stable. In the next section we will build the phase diagram of the system in terms of the dimension of the scalar order and the temperature. Before that, in the rest of this section, we will discuss the potential role of the unstable s+p solution in building the phase diagram and the feature of the free energy curve.
Note that to get the above condensation behaviors, we have calculated the free energies for all the phases to confirm the stability in each case. The critical temperatures for the first order phase transitions are also calculated from the free energy of the two related phases. From the free energy curves, we found that an s+p solution always exists when the two free energy curves for the s-wave phase and p-wave phase intersects. But this s+p solution is not always stable. We classify the three cases for the stability of this s+p solution as follows: 1. Stable: In this case, there exists a second order phase transition to and from the s+p phase instead of a first order phase transition between the s-wave and p-wave phases.
2. T otally U nstable: A first order phase transition between the p-wave phase and the s-wave phase occurs.
3. P artly Stable & P artly U nstalbe: A first order phase transition occurs between the s+p phase and the s-wave or p-wave phase.
In our previous work [13], we have shown the free energy curves for Case 1, where this s+p solution is stable. It is natural that in this case the s+p solution exists and is the most stable phase in its temperature region. But in the last two cases, especially in the second case, the s+p solution becomes totally unstable. Then we may ask what is the meaning for the existence of the unstable s+p solution?
To answer the question, we draw the free energy curves in Fig. 6 for the three condensation behaviors shown in We also use enlarged version in the third and fifth figure to show the detailed relation between the red and green lines when they are close to each other. In the three figures on the right side, we use thick black curves to show how the free energy curve of different phases form the shape of a swallow tail.
solid blue line to denote the free energy of the p-wave phase and s-wave phase respectively. The green line denotes the free energy for the s+p phase(solution) and the dashed black line denotes that for the normal phase. We also draw the enlarged version for the last two cases to see which one is lower when the green and red lines are very close to each other in some region.
We can see that when the s+p solution near the free energy intersection point becomes partly or totally unstable, a first order phase transition occurs. This first order phase transition can be from the s-wave phase to the p-wave phase as in the case when the s+p solution is totally unstable, or from the s-wave phase to the s+p phase as in the case when the s+p solution is partly unstable. In both the two cases, we can find a classical swallow tail in the free energy curves for the first order phase transition as is shown by the thick black curves in the right figures of Fig. 6. In the right figures, we can see that although the unstable part of the s+p solution is not experienced in the phase transition, it is still quite necessary in forming the swallow tail shape of the free energy curve. It seems that the existence of the unstable s+p solution is to bridge the s-wave phase to the p-wave phase and to form the standard swallow tail shape of free energy for the first order phase transitions. According to the precious results, the swallow tail shaped free energy curve seems always exist in any first order phase transition. If we assume the swallow tail shape of free energy curve as a universal property for the first order phase transition, we can explain the existence of s+p solution near the free energy intersection point as follows. When the free energy curves of the s-wave phase and the p-wave phase have an intersection point as in Fig. 6, the s+p solution must exist. Otherwise a first order phase transition would occur between the s-wave phase and p-wave phase, and without the s+p solution, the free energy curve can not form a swallow tail shape, which contradicts with the universality we assumed. This contradiction could be solved by an s+p solution near the intersection point of free energy curves. This s+p solution could either be (partly) stable to avoid the first order phase transition between the s-wave phase and p-wave phase, or be totally unstable but bridge the s-wave and p-wave phases and form the swallow tail shape of free energy curve for the first order phase transition between the s-wave and p-wave phases. We can also use the same logic to explain the existence of the s+d phase in the holographic s+d system studied in Ref. [16]. This method could be also useful in finding new solutions near some free energy intersection points in more general systems.

V. PHASE DIAGRAMS
In the previous section, we have shown the condensation behaviors for some interesting cases. If we get all the condensation behaviors for different values of ∆ at fixed b, we can summarize the information at different ∆ to build a ∆ − T phase diagram at the fixed b. This method was used to get the ∆−T phase diagram in the probe limit [13]. In this section, we give the ∆ − T phase diagrams at eight different values of the back reaction parameter b from 0.1 to 0.8. With these phase diagrams, we can understand the rich phenomena shown in the previous section much more clearly. And we can also compare the phase diagrams to study the effect of back reaction concretely.
We show all the phase diagrams in Fig. 7 and Fig. 8, respectively. In the eight phase diagrams with b from 0.1 to 0.8, we can see that all the diagrams have four different phases, these are the normal phase, the s-wave phase, the p-wave phase and the s+p coexisting phase colored by light purple, light blue, light red and white, respectively. The thermodynamic stability of all these phases has been confirmed by our calculation of free energy.
We also use different lines in the figures to show different phase transition critical points  or emergent points of unstable phases. The solid lines divide the different phases, and the dashed lines near the zero temperature region are obtained by extrapolation from the solid lines with the same color. The dotted lines denote the points where an unstable solution emerges. We use different colors to mark the different critical lines. The second order s-wave and p-wave phase transitions from the normal phase have been well studied, we color the critical lines black. When the back-reaction is strong enough, the p-wave phase transition becomes first order, so we use purple line to denote the critical points. The s+p coexisting phase generally starts from the s-wave or the p-wave phase, so we use blue lines to denote the emergent points of the s+p coexisting phase on the s-wave phase, and use red curves to denote the emergent points of the s+p coexisting phase on the p-wave phase. When the red or the blue curve becomes dotted, the points denote the emergent points of unstable (part of) s+p solutions. In that case, a first order phase transition occurs. We use the solid  yellow curves to denote the critical points of first order phase transitions between the s-wave phases and the p-wave phases, and use the solid green curves to denote the critical points of first order phase transitions between the s-wave phases and the s+p coexisting phases.
In order to describe these phase diagrams more clearly, we define some special values of the dimension of the scalar order to mark vertical coordinates of some critical points. We list their definitions as follows. 5. ∆ pmin , ∆ smin : These two points denote the minimum value of the red line and the blue line respectively. But these two appear only in some limited cases, for example, ∆ pmin in b = 0.5, 0.6 and ∆ smin in b = 0.3.
Next we will discuss each phase diagram case by case. In each phase diagram, the vertical coordinate is the operator dimension of the s-wave order parameter, and the horizontal coordinate is the temperature over the fixed value of chemical potential T /µ. In order to explain the phase diagrams more clearly, we can fix the operator dimension to a special value ∆ = ∆ e.g. , that means we choose a special horizontal line in the phase diagram, with the vertical coordinate value equal to ∆ e.g. . We then force the system to be cooled slowly from a high temperature state in the normal phase region, so the point representing the phase of the system will go from the right side to the left along the line with ∆ = ∆ e.g. . We can see how many phase transitions would occur from the phase diagram, and we can also get the critical temperature as well as the order of each phase transition from the phase diagram. Notice that each condensation figure in the previous section shows the case of a horizontal line in one of the phase diagrams. Now let us begin with the first phase diagram with b = 0.1. The structure of this phase diagram is quite simple, it is qualitatively very similar to the phase diagram in the probe limit [13]. We can see that in this case ∆ p0 > ∆ s0 > ∆ t . We can analyze the system in four regions of the parameter ∆ e.g. . When ∆ e.g. < ∆ t , the system undergoes only one phase transition from the normal phase to the s-wave phase and stays in the s-wave phase in the low temperature region. When ∆ t < ∆ e.g. < ∆ s0 , there are three phase transitions. The system will first undergo a p-wave phase transition from the normal phase, then experience a second order phase transition form the p-wave phase to the s+p coexisting phase, and finally meet the final phase transition and change from the s+p coexisting phase to the s-wave phase. The left figure in Fig. 1 shows the typical condensation behavior of the system in this region. When ∆ s0 < ∆ e.g. < ∆ p0 , the system will also experience the p-wave phase transition and the phase transition from the p-wave phase to the s+p coexisting phase, but without the phase transition from the s+p coexisting phase to the s-wave phase, the system will finally stay in an s+p coexisting phase in the low temperature region. When ∆ e.g. > ∆ p0 , there is no s+p coexisting phase, and the the system will only undergo the p-wave phase transition and keep in the p-wave phase in the low temperature region.
The b = 0.2 phase diagram looks also very similar to the case in the probe limit. The only difference is that the region for the s+p coexisting phase is larger. There are also four regions for the parameter ∆, in these regions, the phase transition properties are qualitatively the same as that in the case b = 0.1. We plot the condensation behavior of the system with ∆ s0 < ∆ e.g. < ∆ p0 in the right figure of Fig. 1.
In the b = 0.3 case, the phase diagram still looks similar to the previous two cases. But it seems that the slope of the blue line changes to be positive in some region. We draw an enlarged version of the phase diagram to show the case clearly. From the enlarged version we can see that the blue line has a minimum value of ∆ = ∆ smin at a non zero temperature, below this temperature the blue line has a negative slope. We have four special values of ∆ with ∆ smin < ∆ s0 < ∆ t < ∆ p0 , and thus we have five regions for ∆ e.g. . The most interesting is the region with ∆ smin < ∆ e.g. < ∆ s0 , the condensation behavior for this case is shown in Fig. 2. In this region, there would be three phase transitions. The system first goes from the normal phase to the s-wave phase via an s-wave phase transition, then it goes from the s-wave phase to the s+p coexisting phase by the second phase transition. In the third phase transition, the system goes back to the s-wave phase from the s+p coexisting phase. In this case, the behavior of the p-wave order is rather interesting, the condensation value of the p-wave order emerges from zero at some critical temperature. Once reaching its maximal value, it would decrease and finally vanish when the system goes back to the s-wave phase. We plot the condensate of the s-wave and the p-wave orders for this kind of phase transition in Fig. 2 and call this condensation as a "n-type" one. From the phase diagram we can see that the "n-type" condensation occurs because the blue line has a minimum value of ∆ at a non-zero temperature.
In the b = 0.4 case, the phase diagram is qualitatively the same as to the case with b = 0.3, so the above analysis of the b = 0.3 phase diagram is still valid here. It should be pointed out, however, that the lowest point of the blue line seems to be in the dashed part of the line. Because the dashed part is obtained by extrapolation, the lowest point needs to be confirmed by more precise numerical calculations.
In the b = 0.5 phase diagram, the blue line always has a positive slope, but the red line has a minimum value at ∆ = ∆ pmin at a non-zero temperature in this case. Then there are four special values of ∆ that will divide the phase diagram into five different regions. The interesting part is the region ∆ pmin < ∆ e.g. < ∆ t . In this region, the s-wave phase transition occurs first, then the system goes into the s+p coexisting phase via the second phase transition. In the third and forth phase transitions, the system goes into the p-wave phase and then goes back to the s+p coexisting phase. We plot the condensation for this case in Fig. 4, and we call this kind of condensation as "u-type". We can see easily from the phase diagram that the "u-type" condensation can occur because the red line has a minimum value of ∆ at a non-zero temperature.
In the above five phase diagrams, there is always a quadruple intersection point connecting to all the four phases, and all the phase transitions are second order. In the next three phase diagrams, there does not exist any quadruple intersection point, but there are two triple intersection points. And we can see that in this case, some phase transitions become first order.
In the phase diagram with b = 0.6, the p-wave phase transition from the normal phase is still second order, but in the region ∆ I < ∆ e.g. < ∆ t− , the s+p coexisting phase transition is first order. When ∆ t− < ∆ e.g. < ∆ t+ , the s+p coexisting phase connecting the s-wave phase and p-wave phase still exists, but it always has a higher free energy than the s-wave and p-wave phases. Thus a first order phase transition occurs between the s-wave phase and the p-wave phase. But we should point out that the unstable s+p coexisting phase is still essential to form the swallow tail shape of the free energy curve for the first order phase transition. In this phase diagram, the red line still has a minimum point as in the b = 0.5 phase diagram, then the system would also go into the p-wave phase from an s+p coexisting phase and goes back to the s+p coexisting phase later when ∆ pmin < ∆ e.g. < ∆ t− . We plot the condensation behavior for this case in this region in the right figure of Fig. 4. We draw an enlarged version of this phase diagram in Fig. 9 to show the details near the triple intersection points.
When b = 0.7, the p-wave phase transition from the normal phase becomes first order, so the vertical line separating the p-wave phase and normal phase is marked brown. In this case, the s+p phase transition from the s-wave phase is still first order in the region ∆ I < ∆ e.g. < ∆ t− , and the s+p coexisting phase connecting the s-wave and p-wave phases is always unstable in the region ∆ t− < ∆ e.g. < ∆ t+ .
The b = 0.8 phase diagram is very similar to the one with b = 0.7, but notice that the orange line seems not to intersect with the blue line, that means the phase transition from the s+p coexisting phase to the s-wave phase will be alway first order. We show the condensate behaviors in Fig. 5 for three typical regions for ∆ e.g. . In Fig. 5, the first figure is for ∆ p0 < ∆ e.g. < ∆ t+ , the second figure is for the case ∆ t− < ∆ e.g. < ∆ p0 , and the third one is for ∆ s0 < ∆ e.g. < ∆t − . We also plot the free energy for the three cases in Fig. 6.
From the above analysis, we can see that the qualitative condensation behaviors at fix b and ∆ can be read off easily along a horizontal line in each phase diagram. The different shape of the curves separating the different phases leads to different phase transition behaviors. For example, the "n-type" behavior in Fig. 2 is due to the existence of the minimum of the blue line in the b = 0.3 phase diagram, while the "u-type" behavior in Fig. 4 is due to the existence of the minimum of the red lines in the b = 0.5 and b = 0.6 phase diagrams.
We can also see the influence of the back reaction from the phase diagrams. The most obvious one is that the region for the s+p coexisting phase is enlarged by the back reaction when the back reaction strength b is not too large. This is in accordance with the intuition that the back reaction on the metric adds an additional attraction between the s-wave and p-wave orders. This also happens in the s+s system in Ref. [11].
But we here have found that for a very large value of b, the region for the s+p coexisting phase is reduced. This is mainly caused by the low critical temperatures of the s-wave and p-wave phase transitions in the strongly back reacted case. We should also notice that the s+p solution near the triple intersection points becomes unstable in the strong back reaction case, this also makes the region for the s+p coexisting phase smaller.
In the parameter region we have tested, the s-wave phase transition from the normal phase is alway second order, but the p-wave phase transition becomes first order when the back reaction is large enough. The critical value of the back reaction at which the p-wave phase transition becomes first order can be read off from Ref. [30], and the critical value is b c = 0.62. In this work, we see that the phase transition from the s-wave phase to the s+p coexisting phase can also be first order when the back reaction is large. And even when both the p-wave and s-wave phase transitions are still second order, the s+p coexisting phase transition can be first order, as exhibited in the phase diagram of b = 0.6.

VI. AN ALTERNATIVE SETUP OF THE HOLOGRAPHIC S+P MODEL
Before we conclude our work, we give another setup for a holographic s+p model in tis section. From this new setup, we can get the same equations of motion as that presented in Sec. II. Thus we can get the same condensation behavior and the same phase diagrams as in the previous sections.
It is well known that the origin of superconductivity involves the formation of a quantum condensate state by pairing conduction electrons. Nevertheless, from a symmetry point of view, the phenomenon of superconductivity can be thought of as the spontaneously breaking of U(1) gauge symmetry. In the previous setup, the p-wave and s-wave order parameters are dual to the gauge field pointing in the third direction inside the SU(2), i.e., A 3 µ and the third component of the scalar triplet Ψ 3 , respectively. By fixing a gauge, we choose the SU(2) gauge field in the first direction as the U(1) gauge symmetry which we call U (1) 1 . A non-vanishing value of the time component of U (1) 1 gauge field induces a chemical potential on the dual theory. The non-trivial profile of the gauge field A 3 x would induce a vacuum expectation value of the dual operator with no source and thus breaks the U (1) 1 symmetry spontaneously. Similar story is also true for the scalar field Ψ 3 .
However, there are still some limitations in the previous setup of the holographic s+p model. One can change the scaling dimension of the dual scalar operator for the s-wave order by adjusting the mass square of Ψ a . But dual to the constraint from the SU(2) gauge symmetry, the dimension of the p-wave order has to be fixed to be ∆ p = d − 1 where d is the spatial dimension in the bulk. And furthermore, the SU(2) gauge invariance also makes the "charge" of the p-wave order equal to the one of the s-wave order. Therefore in our previous setup, we only have two free parameters b and ∆ s = ∆ at hand. In principle, the dimension and the charge for the p-wave order can also have different values from its s-wave cousin, so we also need to find the way to vary these parameters in the holographic setup. One way is to use the holographic p-wave model [38] in terms of a charged complex vector field. In this new setup, we can use a complex scalar and a complex vector minimally coupled with a U (1) gauge field as the matter part of the holographic action. The matter part of the action can be written as where the superscript " † " means complex conjugate, F µν = ∇ µ A ν −∇ ν A µ is the field strength for the U(1) gauge field. Here Ψ is now a complex scalar charged under the U(1) gauge field withD µ = ∇ µ − iq s A µ , where q s is the U(1) charge of Ψ. ρ µ is a complex vector field charged under the same U(1) field but with a different charge q p . The field strength of ρ µ is ρ µν =D µ ρ ν −D ν ρ µ with covariant derivativeD µ = ∇ µ − iq p A µ . The equations of motion from the previous model under the ansatz (9) can also be obtained in the particular case with q p = q s = g c and m p = 0 in this new model under the following ansatz with all other matter field components being set to zero. Thus all results in the previous sections can also be recognized as that of the q s = q p = g c and m p = 0 case in the new s+p model in this section. It should be worthy pointing out that although those two s+p models show same aspects in particular cases, they are distinct from each other, for example, the transport coefficients might exhibit essential differences [30]. And with this new holographic s+p model, we can consider the effect of different charges of s-wave and p-wave orders. Furthermore, we can also introduce the mass term of ρ µ to deform the scaling dimension of the p-wave vector operator. So we can study the s+p system holographically in a more complete parameter space. It has been shown that depending on q p and m 2 p , only the p-wave part of action (36) can exhibit a rich phase structure [7,39,40]. Thus the new model (36) is expected to show more fruitful phase behaviors in a larger parameter space. This would be left for further study.

VII. CONCLUSIONS AND DISCUSSIONS
In this paper, we have continued our study on the holographic s+p model with a scalar triplet charged in an SU(2) gauge field in the bulk [13], by considering the back reaction of the matter fields on the background black brane geometry. We have shown that the model exhibits a rich phase structure depending on the model parameters. Taking eight different values of the back reaction parameter b from 0.1 to 0.8, we have plotted the condensate behaviors of the s-wave and p-wave orders and built corresponding phase diagrams in terms of the dimension of the s-wave order and the temperature of the system. We have also given another setup for a holographic s+p model, which with special model parameters, can lead to the same equation of motion as the model proposed in [13].
In this holographic s+p superconductor model, we have discovered some new condensate behaviors such as the "n-type" and "u-type", where the condensations for the p-wave or s-wave orders are non-monotonic. These interesting behaviors can only occur when the blue or red curve in the phase diagram has a minimum at non-zero temperature. These new phase transition behaviors should be helpful in further understanding strongly coupled systems.
We also have found that the s+p solution might be totally unstable when the back reaction is strong enough, and a first order phase transition between the s-wave and p-wave phases occurs at that time. Although the s+p solution in that case is totally unstable, it is still essential in forming the swallow tail in free energy curves for the first order phase transition.
According to this property, if we assume the swallow tail of free energy curve as a universal feature for the first order phase transitions, we can explain why the s+p solution can always exist when the free energy curves for the s-wave and p-wave phases have an intersection point.
The resulting phase diagrams show that the back reaction of the matter sector on the black brane background generally enlarges the region for the s+p phases when the back reaction is not too large, which is similar to the case of the s+s system [11]. But in the very strong back reaction case, the region for the s+p phase becomes smaller. This is mainly caused by the decreasing of the critical temperatures for the s-wave and p-wave phase transitions when the back reaction becomes strong. The fact that the s+p phase becomes unstable near the triple intersection points also contributes to the decreasing of the region for the s+p phase in the phase diagram.
In the strong back reaction case, the p-wave phase transition from the normal phase becomes first order. With the back reaction getting stronger, there is a growing parameter space where the transition from the s-wave phase to the s+p phase is of first order. The first order phase transition from the s-wave phase to the s+p phase even exists when both the s-wave and p-wave phase transitions are still second order.
There are many further directions to explore based on this work. With the new setup proposed in section VI, we can study the s+p system more completely with various charges and dimensions of the p-wave order. Since our s+p system has shown many new phase transition behaviors, one can use them to test some universal properties in holographic systems. In addition, these new s+p models provide the setups, in which we can study time dependent or inhomogeneous solutions in order to see the non-equilibrium physics or the lattice effects. Finally if the swallow tail shape of the free energy curve is indeed a universal property for the first order phase transitions, it can be used to predict new phases(solutions) near the intersection point of two different phases.