Holographic phase transitions of p-wave superconductors in Gauss-Bonnetgravity with backreaction

Rong-Gen Cai,* Zhang-Yu Nie,† and Hai-Qing Zhang‡

Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,P.O. Box 2735, Beijing 100190, China

(Received 14 January 2011; published 22 March 2011)

We investigate the phase transitions of holographic p-wave superconductors in (4þ 1)-dimensional

Einstein-Yang-Mills-Gauss-Bonnet theories, in a grand canonical ensemble. Turning on the backreaction

of the Yang-Mills field, it is found that the condensations of vector order parameter become harder if the

Gauss-Bonnet coefficient grows up or the backreaction becomes stronger. In particular, the vector order

parameter exhibits the features of first order and second order phase transitions, while only the second

order phase transition is observed in the probe limit. We discuss the roles that the Gauss-Bonnet term and

the backreaction play in changing the order of phase transition.

DOI: 10.1103/PhysRevD.83.066013 PACS numbers: 11.25.Tq, 74.20.�z

I. INTRODUCTION

The AdS/CFT correspondence [1–4] provides a novelapproach to study the strongly coupling systems at finitedensity. Therefore, it may have some useful applicationsin condensed matter physics. It has been applied to studythe holographic shear viscosity [5–8], holographic super-conductors [9,10], and holographic (non)fermi-liquids[11–13]. In this paper, we will focus on the holographicp-wave superconductors [14–16].

The phenomena of superconducting can be explained bythe spontaneously breaking of U(1) gauge symmetry [17].In the p-wave superconductor, the rotational symmetry isalso broken by a special direction of some vector field inaddition. This could be achieved by the condensation of acharged vector field. And the holographic modeling of thispicture could be simply realized by adding an SU(2) Yang-Mills field to an anti-de Sitter (AdS) black hole background[14]. In this case, after making some ansatz of the SU(2)field, one U(1) subgroup of SU(2) is considered as theelectromagnetic gauge group; in addition, a gauge bosongenerated by another SU(2) generator is charged underthis U(1) subgroup through the nonlinear coupling of thenon-Abellian gauge fields. In this setup, the superconduc-tor phase transition is studied, and conductivities showsome anisotropic behavior. Furthermore, the holographicp-wave superconductor with backreaction was investi-gated in Ref. [16]. The crucial point is that the backreactionwill dramatically change the order of the phase transition.More specifically, when the matter field coupling goesbeyond a critical value, the former second order phasetransition with a 1=2 mean-field theory critical exponentnear the critical temperature [14,16,18–21] will be changedto a first order phase transition.

In a previous paper [18], we studied the holographicGauss-Bonnet p-wave superconductors in the probelimit. The holographic Gauss-Bonnet superconductors arealso discussed in [22–27]. In this paper, we will investigatethe holographic p-wave superconductors with backreac-tion in order to find out how the matter couplings and theGauss-Bonnet coefficient affect the phase transition of thep-wave superconductor. We find that the bigger the Gauss-Bonnet coupling is, the bigger the condensation value ofthe order parameter is, and the lower the critical tempera-ture is. This reflects that the big Gauss-Bonnet coefficientwill make the superconducting phase transition hard,which is consistent with our previous conclusions [18].Besides, we also find that the stronger the matter fieldcouples to the background, the harder the condensationto be formed. In addition, we find that the phase transitionwill change from second order to first order when thebackreaction is strong, which is similar to the discussionsof [16]. In grand canonical ensemble, we study the freeenergy and entropy of the p-wave superconductorwhich also supports our claim of the change of phasetransitions.This paper is organized as follows: We will set up our

model of the holographic superconductors in Sec. II andstudy the condensation behavior of the vector order pa-rameter for different Gauss-Bonnet coefficients and differ-ent matter field couplings in Sec. III. In Sec. IV, we studythe thermodynamics of the p-wave superconductor byexploring the free energy and entropy. We draw our con-clusions in Sec. V.

II. HOLOGRAPHIC SET UP OF P-WAVESUPERCONDUCTORS

We consider the Einstein-Gauss-Bonnet gravity with anSU(2) Yang-Mills field in (4þ 1)-dimensional asymptoti-cally AdS space-time. The action is

*cairg@itp.ac.cn†niezy@itp.ac.cn‡hqzhang@itp.ac.cn

PHYSICAL REVIEW D 83, 066013 (2011)

1550-7998=2011=83(6)=066013(9) 066013-1 � 2011 American Physical Society

S ¼Z

d5xffiffiffiffiffiffiffi�g

p �1

2�25

�Rþ 12

L2þ �

2ðR2 � 4R��R��

þ R����R����Þ�� 1

4g2ðFa

��Fa��Þ

�þ Sbdy:; (1)

where �5 is the five-dimensional gravitational constantwith 2�2

5 ¼ 16�G5, and G5 a (4þ 1)-dimensional

Newton gravitational constant, g is the Yang-Millscoupling constant, and L is the AdS radius. The SU(2)Yang-Mills field strength is

Fa�� ¼ @�A

a� � @�A

a� þ �abcAb

�Ac�; (2)

where a, b, c ¼ ð1; 2; 3Þ are the indices of the generatorsof SU(2) algebra. �, � ¼ ðt; r; x; y; zÞ are the labels ofspace-time with r being the radial coordinate of AdS.The Aa

� are the components of the mixed-valued gauge

fields A ¼ Aa�

adx�, where a are the SU(2) generators

with commutation relation ½a; b� ¼ �abcc. �abc is thetotally antisymmetric tensor with �123 ¼ þ1. Thequadratic curvature term is the Gauss-Bonnet termwith � the Gauss-Bonnet coefficient and R�

��� ¼@��

��� � � � � . Sbdy includes boundary terms that do not

affect the equations of motion, namely, the Gibbons-Hawking surface term, as well as counterterms requiredfor the on shell action to be finite. We will write Sbdy term

explicitly in Sec. IV.The Einstein field equations can be derived from the

above action as

R�� � 1

2g��

�Rþ 12

L2

�þ �

2

�H�� � 1

2g��

H

2

�¼ �2

5T��

(3)

with

T�� ¼ 1

g2tr

�Fa��F

a�� � 1

4g��F

a��F

a��

�; (4)

H�� ¼ 2RR�� þ 2R���R��� � 4R��R

�� þ 4R�

�R����;

(5)

H ¼ H��; (6)

where ‘‘tr’’ takes the trace over the indices of SU(2) gen-erators. The Yang-Mills equations of motion are

r�Fa�� ¼ ��abcAb

�Fc��: (7)

Following Refs. [14–16], we choose the ansatz of thegauge fields as

AðrÞ ¼ ðrÞ3dtþ wðrÞ1dx: (8)

In this ansatz, we regard the U(1) symmetry generated by3 as the U(1) subgroup of SU(2). We call this U(1)subgroup as Uð1Þ3. The gauge boson with nonzero compo-nent wðrÞ along x direction is charged under A3

t ¼ ðrÞ.

According to AdS/CFT dictionary, ðrÞ is dual to thechemical potential in the boundary field theory, whilewðrÞ is dual to the x component of some charged vectoroperator J. The condensation of wðrÞ will spontaneouslybreak the Uð1Þ3 gauge symmetry and induce the phe-nomena of superconducting on the boundary field theory.Our metric ansatz following Ref. [16,28] is

ds2 ¼ �NðrÞ�ðrÞ2dt2 þ 1

NðrÞdr2 þ r2fðrÞ�4dx2

þ r2fðrÞ2ðdy2 þ dz2Þ; (9)

with

NðrÞ ¼ r2

2�

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4�

L2þ 4�mðrÞ

r4

s �; (10)

where mðrÞ is a function related to the mass and charge ofthe black hole. The reason for this metric ansatz is that thebackreaction of nonzero wðrÞ will change the backgroundof space-time. The condensation of wðrÞ will preserveonly SO(2) symmetry of the spatial direction, i.e., ðy; zÞdirection. The horizon of the black hole is located at rhwhile the boundary of the bulk is at rbdy ! 1. Note that

when r ! 1,

NðrÞ � r2

2�

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4�

L2

s �: (11)

So we can define an effective radius Lc of AdS space-time as

Lc � L

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þU

2

s; U ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4�

L2

s: (12)

From this relation, we can see that in order to have a well-defined vacuum for the gravity theory, there is an upperbound for � � L2=4. The saturation � ¼ L2=4 is calledChern-Simons limit. If we further consider the causalityconstraint of the boundary conformal field theory (CFT),there is an additional constraint on the Gauss-Bonnetcoefficient with �7L2=36 � � � 9L2=100 in five dimen-sions [29–35].The Hawking temperature of this black hole is

T ¼ �N0

4�

��������r¼rh

¼�

�

�L2� �2

g

02

12��

�r

��������r¼rh

; (13)

where �g � �5=g is regarded as the effective matter field

coupling and ‘‘ 0 ’’ denotes the derivative with respect to r.The Bekenstein-Hawking entropy of the black hole is

S ¼ A

4G5

¼ 2�A

�25

¼ 2�

�25

Vr3h; (14)

where A denotes the area of the horizon and V ¼ Rd3x.

The Einstein and Yang-Mills equations of motion withthe ansatz (8) and (9) can be explicitly written as

RONG-GEN CAI, ZHANG-YU NIE, AND HAI-QING ZHANG PHYSICAL REVIEW D 83, 066013 (2011)

066013-2

N0 ¼ � �2grf

4w22

3r2N�2 � 6�N2�2ðrf�2Þ0ðrfÞ02 ��2gr

302

3r2�2 � 6�N�2ðrf�2Þ0ðrfÞ02

� rðNL2ð6f2 þ 6r2f02 þ �2gf

6w02Þ � 12f2r2Þ þ 24L2r�N2f0ðrfÞ0ðf�1rf0Þ03L2r2f2 � 6L2�f2Nðrf�2Þ0ðrfÞ02 ; (15)

�0 ¼ �2grf

4w22

6r2N2�� 12�N3�ðrf�2Þ0ðrfÞ02 ��2gr

302

6r2N�� 12�N2�ðrf�2Þ0ðrfÞ02

þ 12r3�f2 þ L2�ð�2grNw02f6 � 3f2ðN0r2 þ 2Nðr� �ðrf�2Þ0ðrfÞ02N0ÞÞ þ 6r3Nf02Þ

6L2fNðr2 � 2�Nðrf�2Þ0ðrfÞ02Þ ; (16)

f00 ¼ f1 þ f2 þ f3; (17)

00 ¼ f4w2

r2Nþ

�� 3

rþ �0

�

�0; (18)

w00 ¼ � w2

N2�2� w0

�1

rþ 4

f0

fþ N0

Nþ �0

�

�; (19)

where

f1 ¼ ð��2gr

2f7w22 þ �2g�f

5Nw22ð2f� rf0ÞðrfÞ0Þ=ð3r4f2N2�2 þ 3�r2N2�ð2r2N�f02 � 2r2ff0ð�N0 þ 2N�0Þ� f2ðr�N0 þ 2Nðr�Þ0ÞÞ þ 6�2rN3�ðrfÞ02ð�N0 þ 2N�0ÞÞ (20)

f2 ¼ ð�2gr

2f7�w02 � �2g�f

5N�ð2f� rf0ÞðrfÞ0w02Þ=ð3r4f2�þ 3�r2ð2r2N�f02 � 2r2ff0ð�N0 þ 2N�0Þ� f2ðr�N0 þ 2Nðr�Þ0ÞÞ þ 6�2rNðrfÞ02ð�N0 þ 2N�0ÞÞ (21)

f3 ¼ ð�L2r3�f0ðfðr�N0 þ Nð3�þ r�0ÞÞ � rN�f0Þf3 þ r�ff0½ðL2�2N02r2 þ Nð�2g

02L2 þ 2�N0�0L2 þ 12�2Þr2þ 2L2N2�ð2�þ r�0ÞÞf3 þ rf0ð�4L2N2�2 þ L2r2N02�2 þ rNð�2

gr02L2 þ 2r�N0�0L2

þ 2�2ð6r� L2N0ÞÞÞf2 � 2L2r2N�f02ð2r�N0 þ 3Nfð2�þ r�0ÞÞ � 4L2r3N2�2f03�þ 4L2�2N2�f0ðfþ rf0Þ2ð�f2 þ rf0fþ r2f02Þð�N0 þ 2N�0ÞÞ=ðL2r4N�2f4

þ 2L2r�2N2�ðfþ rf0Þ2ð�N0 þ 2N�0Þf2 þ L2r2�N�ð2N�f02r2 � 2ff0ð�N0 þ 2N�0Þr2� f2ðr�N0 þ 2Nð�þ r�0ÞÞÞf2Þ: (22)

There are four useful scaling symmetries in the aboveequations:

ðIÞ f ! �f; w ! ��2w; (23)

ðIIÞ � ! ��; ! �; (24)

ðIIIÞ r ! �r; m ! �4m; ! ! �!;

! �; N ! �2N(25)

ðIVÞ r ! �r; m ! �2m; L ! �L;

! ��1; �g ! ��g; � ! �2�:(26)

We can use the symmetries (25) and (26) to set rh ¼ 1and L ¼ 1 and use symmetries (23) and (24) to set

�ðr ! 1Þ ¼ fðr ! 1Þ ¼ 1 in order to make the solutionasymptotically approach to an AdS.

III. TWO BRANCHES OF SOLUTIONS

A. Analytic charged Gauss-Bonnet–AdS solutionwith wðrÞ ¼ 0

In the ansatz (8) with wðrÞ ¼ 0, there is an straightfor-ward analytic black hole solution which is a charged gen-eralization of Gauss-Bonnet (GB)-AdS black holes[36,37]. In this case,

fðrÞ ¼ �ðrÞ ¼ 1; ¼ �� Q

2r2; (27)

and

HOLOGRAPHIC PHASE TRANSITIONS OF p-WAVE . . . PHYSICAL REVIEW D 83, 066013 (2011)

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NðrÞ ¼ r2

2�

241�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4�

� ~M

r4� 1

L2� �2

g

Q2

6r6

�s 35; (28)

where ~M is related to the mass of the black hole, Q isthe charge of the black hole, and Q ¼ 2�r2h. Using the

formula (13), the temperature for this charged GB-AdSblack hole is

T ¼ rh�L2

� �2g

�2

3�rh: (29)

B. Superconducting solutions with wðrÞ � 0

In order to investigate the superconducting solutionswith wðrÞ � 0, we have to solve numerically the setof Eqs. (15)–(19).

First of all, we should impose boundary conditions onthe fields. The horizon is located at r ¼ rh withNðrhÞ ¼ 0.On the horizon, we should impose ðrhÞ ¼ 0 for the Uð1Þ3gauge field to have a finite norm, and �ðrÞ, fðrÞ, wðrÞshould be finite. We can expand the fields in the powersof (1� rh=r) near the horizon, they are

NearHorizon

8>>>>>>>>><>>>>>>>>>:

N¼0)m¼ r4h=L2þmð1Þ

H ð1�rh=rÞþ����¼�ð0Þ

H þ�ð1ÞH ð1�rh=rÞþ���

f¼fð0ÞH þfð1ÞH ð1�rh=rÞþ���¼ð1Þ

H ð1�rh=rÞþð2ÞH ð1�rh=rÞ2þ���

w¼wð0ÞH þwð1Þ

H ð1�rh=rÞþ��� ;(30)

where all the coefficients of the expansions are constants.At the boundary r ! 1, the asymptotical behavior of

these fields are

Near Boundary

8>>>>>>>>><>>>>>>>>>:

m ¼ mð0ÞB þmð2Þ

B =r2 þ � � �� ¼ �ð0Þ

B þ �ð4ÞB =r4 þ � � �

f ¼ fð0ÞB þ fð4ÞB =r4 þ � � � ¼ ð0Þ

B þð2ÞB =r2 þ � � �

w ¼ wð0ÞB þ wð2Þ

B =r2 þ � � � :

(31)

From the AdS/CFT dictionary, we know that ð0ÞB ¼ �,

ð2ÞB ¼ �, where � and � are, respectively, the chemical

potential and density of the charge at the boundary; wð0ÞB is

the source of the boundary operator J while wð2ÞB is the

expectation value of J and the nonzero wð2ÞB will induce

superconducting phase as we have mentioned.

Near-horizon (30), mð1ÞH , �ð1Þ

H , fð1ÞH , ð2ÞH , and wð1Þ

H can be

evaluated as functions of �ð0ÞH , fð0ÞH , ð1Þ

H and wð0ÞH by sub-

stituting the expansions into the equations of motion.Therefore, there are four independent initial values left to

be specified, i.e., �ð0ÞH , fð0ÞH , ð1Þ

H , wð0ÞH .

For the boundary conditions at infinity, we impose�ðrÞ ¼ fðrÞ ¼ 1 to have an asymptotic AdS boundary,this could be reached by using the scaling symmetries

(23) and (24). We also impose wð0ÞB ¼ 0 to turn off

the source of J. In the numerical calculations, we will setrh ¼ L ¼ 1 by using the scaling symmetries (25) and (26).Armed with these equations of motion and boundary

conditions, we can numerically solve the set of equationsby the shooting method. Because we will work in thegrand canonical ensemble, we can fix the chemical poten-tial value � and then vary the four independent near-

horizon coefficients ð�ð0ÞH ; fð0ÞH ;ð1Þ

H ; wð0ÞH Þ until we find a

solution which produces the desired value of � and

�ð0ÞB ¼ fð0ÞB ¼ 1, wð0Þ

B ¼ 0.In the following, we will present our numerical results of

the condensation value of the order parameter J. We willscan through value of � from � ¼ �0:19 to � ¼ 0:09as well as �g from �g ¼ 0:0001 to �g ¼ 0:45. From the

AdS/CFT dictionary [4], we know that the conformaldimension of vector field in five dimension is � ¼ 3;

therefore, J1=3=Tc is the right dimensionless quantity.Figure 1 shows the condensation value of vector opera-

tor J for different Gauss-Bonnet coefficients and differentmatter field couplings. We can see that the condensationvalue grows if the Gauss-Bonnet coefficient grows or thematter field coupling grows. Table I lists all the explicitcondensation data of Fig. 1.Take the bottom-left plot in Fig. 1 as an example (i.e.,

the plot with � ¼ 0:01), when we decrease the tempera-ture, the condensation values for black (�g ¼ 0:0001) and

red (�g ¼ 0:2) curvewill emerge from zero at some critical

temperature Tc. When we keep on cooling down the sys-tem, those condensation values will continuously tend tosome constant values. Besides, when T � Tc the conden-sation will take a mean-field theory critical exponent 1=2 in

the form of J / ð1� T=TcÞ1=2. These are the second orderphase transitions which have been explored in our previouspaper [18]. However, for the blue (�g ¼ 0:4) and pink

(�g ¼ 0:45) lines the behavior is much different from the

above two curves. We see that these curves will bend to theright for T > Tc and there will be two condensation valuesfor T > Tc. However, when T � Tc they will also tend tosome constant values. The mean-field theory critical ex-ponent 1=2 never exists for these condensations. We willargue in Sec. IV that these peculiar condensation behaviorsare features of first order phase transitions and the realcritical temperature for these phase transitions in not ex-actly Tc.In the holographic p-wave superconductors, the normal

charge density �n is ð1ÞH near the horizon, while the total

charge density is �t ¼ 2ð2ÞB ¼ 2�, where the factor 2

appears due to the scaling behavior of on the infiniteboundary. The superconducting charge density is definedas �s ¼ �t � �n [14,18]. We plot the ratio �s=�t in Fig. 2.

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For the black and red curves which are of second orderphase transitions, the intersecting points where the curvesmeet the horizontal axis represent the critical temperaturesTc=� at which the superconducting phase occurs.However, for the blue and pink curves which are of firstorder phase transitions (see Sec. IV), the intersecting pointsare not the critical temperature for the phase transitions.But the real critical temperature TR

c for the first order phasetransition is a little above the intersecting points Tc. From

FIG. 1 (color online). The condensation values of vector operator J versus temperature for different � and different �g. The black,red, blue, and pink curves correspond to �g ¼ 0:0001, 0.2, 0.4, and 0.45, respectively.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.0

0.2

0.4

0.6

0.8

1.0

T

st

0.01

FIG. 2 (color online). The ratio of superconducting chargedensity and the total charge density versus the dimensionlesstemperature T=� at � ¼ 0:01. The black, red, blue, and pinkcurves correspond to �g ¼ 0:0001, 0.2, 0.4, and 0.45, respectively.

TABLE I. The condensation values of J for different � and �g,numbers in boldface represent first order phase transition.

�gn� �0:19 0.0001 0.01 0.09

0.0001 5.891 03 6.019 42 6.027 64 6.103 52

0.2 6.862 88 7.353 03 7.388 71 7.744 25

0.4 12.283 75 16:424 25 16:779 13 20:719 940.45 16:170 33 24:767 84 25:656 23 35:200 14

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Sec. IV, we can find the values of TRc =� for � ¼ 0:01 and

varying �gs, which are listed in Table II. Despite of the first

and second order phase transitions, we still see that thedimensionless critical temperature TR

c =� decreases when�g grows, which reflects that the stronger the matter field

couples to the gravity, the harder the phase transitionoccurs. For sufficiently low temperature, all the curveswill tend to one, which reflects the fact that the super-conducting charge is dominated in the total charge.

IV. THERMODYNAMICS

A. Euclidean action and counterterm methods

From the AdS/CFT correspondence, a nonextremalblack hole corresponds to a thermal equilibrium state atthe boundary. The Hawking temperature of black hole isthe temperature of the boundary field. In the following, wewill work in the grand canonical ensemble with the fixedvalue of chemical potential�. The partition function of thebulk theory is

Z ¼ e�IE½g��; (32)

where IE½g�� is the Euclidean action evaluated on the onshell value of g. Because of the Euclidean action, thecompactified time direction has a period 1=T. The freeenergy now is

� ¼ �T logZ ¼ TIE½g��: (33)

In the path integral, the Euclidean on shell action shouldadditionally include the Gibbons-Hawking surface term togive the correct Dirichlet variational problem and someother boundary counterterms to render the action finite[38]. In the computation, we introduce a hypersurface atlarge but finite r ¼ rbdy as the boundary and then calculate

the on shell action by putting rbdy ! 1.

The bulk action evaluated on the on shell values is

Ibulkon-shell ¼V

T�25

�r2N�ðrfÞ0

f��NðrfÞ0

f3ð2f2ððr�NÞ0 þ r�0NÞ

� rf0ðr�N0 þ 2Nðr�Þ0Þf� 4r2N�f02Þ���������r¼rbdy

;

(34)

where V=T ¼ Rdtd3x is the volume of the (3þ 1)-

dimensional hypersurface. The usual Gibbons-Hawkingsurface term is

Ið1ÞGH ¼ � 1

�25

Zd4x

ffiffiffiffiffiffiffiffi��p

K

¼ � V

T�25

�3r2N�þ r3�N0

2þ r3N�0

���������r¼rbdy

; (35)

where � is the induced metric on the r ¼ rbdy hypersur-

face, n� ¼ ffiffiffiffiffiffiffiffiffiffiNðrÞp

�;r is the outward-pointing normalvector to the hypersurface, and K ¼ K�

� is the trace ofthe extrinsic curvature K�� ¼ rð�n�Þ.For the GB term there is also a generalized Gibbons-

Hawking term [39,40]

Ið2ÞGH ¼ � �

�25

Zd4x

ffiffiffiffiffiffiffiffi��p ðJ � 2GijK

ijÞ

¼ �VN

T�25f

3ðð3r�N0 þ 2Nð�þ 3r�0ÞÞf3

� 3r2f02ðr�N0 þ 2Nð�þ r�0ÞÞf� 4r3N�f03Þjr¼rbdy ; (36)

where Gij is the Einstein tensor of the metric �ij and

J ¼ �ijJij with

Jij ¼ 1

3ð2KKikK

kj þ KklK

klKij � 2KikKklKlj � K2KijÞ:

(37)

No additional counterterms for matter fields are neces-sary because the fields fall off sufficiently near the bound-ary [41].1 Besides, in our metric ansatz the scalar curvatureR for the hypersurface is zero. So the simplest countertermis [43–46]

Ict ¼ 1

�25

Zd4x

ffiffiffiffiffiffiffiffi��p 3

Lc

�2þU

3

�

¼ V

T�25

ffiffiffiffiN

p�r3

2þU

Lc

��������r¼rbdy

: (38)

Thus the total on shell Euclidean action IE½g�� isIE½g�� ¼ Ibulkon-shell þ Ið1ÞGH þ Ið2ÞGH þ Ict: (39)

Then the free energy is

� ¼ TIE½g�� ¼ TðIbulkon-shell þ Ið1ÞGH þ Ið2ÞGH þ IctÞ: (40)

B. Phase diagrams of free energy and entropy

In this subsection, we will discuss the numerical resultsof the free energy � and the entropy S. In Fig. 3, we plotthe free energy and entropy for some typical values of� and �g.

TABLE II. The quantity of TRc =� for different �g while fixing

� ¼ 0:01. Boldfaces represent first order phase transition.

�g TRc =�

0.0001 0.078 85

0.2 0.064 37

0.4 0:028 800.45 0:020 63

1If one works in the canonical ensemble, the charge density �should be fixed, and we should add an additional boundary termto the Euclidean action as �IE / R

d4xffiffiffiffiffiffiffiffi��

ptrðn�Fa

��Aa�Þ [42].

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0.6 0.7 0.8 0.9 1.0 1.1 1.230

28

26

24

22

20

18

16

T Tc

52

Tc4 V

0.19, g 0.4

0.6 0.7 0.8 0.9 1.0 1.1 1.20

2

4

6

8

10

T Tc

52 S

2 Tc3 V

0.19, g 0.4

0.6 0.7 0.8 0.9 1.0 1.1 1.212

10

8

6

4

T Tc

52

Tc4 V

0.01, g 0.3

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.20

1

2

3

4

5

6

T Tc

52 S

2 Tc3 V

0.01, g 0.3

0.80 0.85 0.90 0.95 1.00 1.05 1.10125

120

115

110

105

T Tc

52

Tc4 V

0.01, g 0.4

0.5 0.6 0.7 0.8 0.9 1.0 1.10

5

10

15

20

25

30

T Tc

52 S

2 Tc3 V

0.01, g 0.4

0.8 0.9 1.0 1.1 1.2 1.3900

880

860

840

820

800

780

760

T Tc

52

Tc4 V

0.01, g 0.45

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.30

20

40

60

80

100

120

T Tc

52 S

2 Tc3 V

0.01, g 0.45

FIG. 3 (color online). The free energy and entropy versus temperature for different � and �g. The purple line is for the chargedGB-AdS black hole (i.e., wðrÞ ¼ 0), while the blue line is for the superconducting solutions (i.e., wðrÞ � 0).

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From the first two up-left plots in Fig. 3, we find that

for � ¼ �0:19, �g ¼ 0:4 and � ¼ 0:01, �g ¼ 0:3, the

charged GB-AdS solution (purple line) (28) exists for all

the temperature T > Tc and T � Tc. On the contrary,

the superconducting solution (blue line) exists only for

T � Tc. But for T < Tc, the free energy of superconduct-

ing solution is smaller than that of the charged GB-AdS

solution, which means that the superconducting solution is

thermodynamically favored, compared to the charged

GB-AdS solution. This represents that when T decreases

across Tc, a phase transition occurs when the bulk solution

goes from the charged GB-AdS solution to a solution with

nonzero vector hair, which induces an order parameter of

the p-wave holographic superconductor. In addition, we

see that at T ¼ Tc the purple line and blue line are not only

continuous but also C1 differentiable, which can be seen

from the plots of entropy S because S ¼ �@�=@T, see thefirst two up-right plots in Fig. 3. The entropy for these two

solutions at T ¼ Tc is continuous but not differentiable. So

according to Ehrenfest’s classification of phase transitions,

these phase transitions are second order for � ¼ �0:19,�g ¼ 0:4 and � ¼ 0:01, �g ¼ 0:3. This is consistent with

our remarks in the previous section.From the two down-left plots of Fig. 3, i.e., plots with

� ¼ 0:01, �g ¼ 0:4 and � ¼ 0:01, �g ¼ 0:45, the behav-

ior of free energy is dramatically different from the one we

mentioned above, there is a characteristic ‘‘swallowtail’’

shape of the free energy indicating a first order phase

transition. Consider the � ¼ 0:01, �g ¼ 0:45 case, for

example. When we decrease the temperature, entering

the figure along the purple line from the right, we reach

the temperature T 1:2Tc where new solutions appear

(the blue curves represent the new solutions). However,

for now the charged GB-AdS solution is thermodynami-

cally favored because it has a lower free energy. If we keep

on cooling down the system, we will still remain in the

charged GB-AdS solution until T 1:12Tc. Then for

T < 1:12Tc, the superconducting solution will have a

lower free energy than the charged GB-AdS solution.

Therefore, superconducting solution will be thermody-

namically favored, compared to the charged GB-AdS

solution in the range T < 1:12Tc. So the exact critical

temperature for this kind of phase transition is TRc ¼

1:12Tc. This is the reason why the quantities for

�g ¼ 0:4 and �g ¼ 0:45 in Table II are not the exact

values which the intersecting points denote. This is a first

order phase transition due to the nondifferentiable free

energy at T ¼ 1:12Tc. From the point view of entropy,

we see that the entropy will jump from the purple curve to

the lowest part of the blue curve at T ¼ 1:12Tc (see the

down-right plot in Fig. 3. The entropy is not continuous,

which also reveals the feature of the first order phase

transition. Therefore, from the plots in Fig. 3, we can

read the real critical temperature for the first order phase

transition as T ¼ 1:02Tc when � ¼ 0:01, �g ¼ 0:4 and

T ¼ 1:12Tc when � ¼ 0:01, �g ¼ 0:45, respectively.

V. CONCLUSIONS

In a previous paper [18], we studied holographic p-wavesuperconductors within Gauss-Bonnet gravity in the probelimit. In this paper, we continued this study by includingbackreaction of Yang-Mills field. We found that both theGauss-Bonnet coefficient and backreaction will make thesuperconducting condensation difficult. This difficulty canbe seen both from the growing condensation values and thedecreasing critical temperatures. By studying the thermo-dynamics of the system in grand canonical ensemble, wefound two kinds of phase transitions of the holographicp-wave superconductors. Note that in the probe limit,the superconducting phase transition is always secondorder. With backreaction, we found that when fixing theGauss-Bonnet coefficient, there was a critical value forthe matter field couplings �gðcÞ (see Fig. 4). If �g < �gðcÞ,the phase transition is second order (yellow region); how-ever, if �g > �gðcÞ, the phase transition becomes first order

(orange region). It was found that the stronger backreactionwill not only make the condensation value bigger but alsowill change the order of the phase transition. This is con-sistent with the conclusions of [16]. On the contrary, if wefixed the matter field couplings �g, the Gauss-Bonnet

couplings would change the order of the phase transitionjust for a small range of �g, i.e., 0:366 � �g � 0:427.

However, out of this range, the Gauss-Bonnet term wouldnot change the order of the phase transition. Therefore, wemay conclude that although both the Gauss-Bonnet coef-ficient and backreaction will make the p-wave condensa-tion hard, the backreaction plays a major role in changingthe order of the phase transition.

2nd order phase transition

1st order phase transition

0.20 0.15 0.10 0.05 0.00 0.05 0.100.0

0.1

0.2

0.3

0.4

0.5

g

FIG. 4 (color online). Classification of phase transition versus�ð�0:19 � � � 0:09Þ and �gð�g 0Þ. The order of phase

transition depends on the coefficients � and �g. The yellow

region below the dashed curve is of the second order phasetransition, while the orange region is of the first order phasetransition.

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ACKNOWLEDGMENTS

Z.Y.N. and H.Q. Z. would like to thank Bin Hu,Ya-Peng Hu, Huai-Fan Li, Zhi-Yuan Xie, and Yun-LongZhang for helpful discussions. This work was supportedin part by a grant from Chinese Academy of Sciences

and in part by the National Natural ScienceFoundation of China under Grant Nos.10821504,10975168, and 11035008, and by the Ministry ofScience and Technology of China under GrantNo. 2010CB833004.

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