A holographic study on vector condensate induced by a magnetic field

Abstract

We study a holographic model with vector condensate by coupling the anti-de Sitter gravity to an Abelian gauge field and a charged vector field in (3+1) dimensional spacetime. In this model there exists a non-minimal coupling of the vector field to the gauge field. We find that there is a critical temperature below which the charged vector condenses via a second order phase transition. The DC conductivity becomes infinite and the AC conductivity develops a gap in the condensed phase. We study the effect of a background magnetic field on the system. It is found that the background magnetic field can induce the condensate of the vector field even in the case without chemical potential/charge density. In the case with non-vanishing charge density, the transition temperature raises with the applied magnetic field, and the condensate of the charged vector operator forms a vortex lattice structure in the spatial directions perpendicular to the magnetic field.

Published for SISSA by Springer

Received: September 17, 2013

Accepted: November 26, 2013

Published: December 6, 2013

Rong-Gen Cai,a Song He,a Li Lia and Li-Fang Lib

aState Key Laboratory of Theoretical Physics,

Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

bState Key Laboratory of Space Weather,

Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing 100190, China

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study a holographic model with vector condensate by coupling the anti-de Sitter gravity to an Abelian gauge eld and a charged vector eld in (3 + 1) dimensional spacetime. In this model there exists a non-minimal coupling of the vector eld to the gauge eld. We nd that there is a critical temperature below which the charged vector condenses via a second order phase transition. The DC conductivity becomes innite and the AC conductivity develops a gap in the condensed phase. We study the e ect of a background magnetic eld on the system. It is found that the background magnetic eld can induce the condensate of the vector eld even in the case without chemical potential/charge density. In the case with non-vanishing charge density, the transition temperature raises with the applied magnetic eld, and the condensate of the charged vector operator forms a vortex lattice structure in the spatial directions perpendicular to the magnetic eld.

Keywords: Gauge-gravity correspondence, Holography and condensed matter physics (AdS/CMT), Black Holes

ArXiv ePrint: 1309.2098

c

A holographic study on vector condensate induced by a magnetic eld

JHEP12(2013)036

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP12(2013)036

Web End =10.1007/JHEP12(2013)036

Contents

1 Introduction 1

2 The holographic model 4

3 Adding a constant magnetic eld 5

4 Phase diagram 84.1 Phase diagram at vanishing charge density 84.2 Phase diagram at nite charge density 10

5 Vortex lattice solution 11

6 Conclusion and discussion 12

A Condensate of vector eld and superconducting phase transition 15A.1 Condensate of vector eld 15A.2 Free energy 17A.3 Conductivity 18

1 Introduction

The gauge/gravity duality [13] turns out to be a useful and complimentary framework to study a strongly coupled system through an appropriate gravity theory living in a higher dimensional spacetime. There are two complementary approaches. In the top-down approach, one can obtain some low energy e ective theories in the bulk as consistent truncations of string/M theory by dimensional reduction on some compactied manifolds. The advantage of this approach is that one knows the origin of the e ective gravity theory in the bulk and the details of the dual eld theory. One of the well-known examples in this category is the duality between the IIB superstring theory on the AdS5 S5 and the N = 4

supersymmetric Yang-Mills theory on the AdS5 boundary. On the other hand, in the so-called bottom-up approach, the gravity theory in the hulk is usually constructed by some physical considerations, according to the AdS/CFT dictionary. The models constructed in this way are simple and universal, just like the Landau-Ginzburg theory describing superconductivity. The disadvantage is that the details of the dual eld theory are not very clear.

A well known example in the second category is the holographic superconductor model [4, 5]. More specically, to build a holographic superconductor model, one should rst introduce a U(1) gauge eld in the bulk, which corresponds to a global U(1) symmetry

1

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in the boundary side. To break this U(1) symmetry spontaneously, one needs a charged operator condensing at low temperature. Therefore, one includes a charged scalar eld in the bulk which is dual to the boundary scalar operator. For simplicity, the gauge eld and the charged scalar can be minimally coupled. This holographic toy model admits black holes with scalar hair at low temperatures (superconducting phase), but without scalar hair at high temperatures (normal phase). In this way, one mimics the superconductor/conductor phase transition in the eld theory side. Holographic superconductor models constructed in the top-down approach can also be found, for example, in refs. [69].

It is well known that superconductivity involves the formation of a quantum condensate state by pairing conduction electrons. The pair of electrons, called the Cooper pair, can be in a state of either spin singlet with total spin s = 0 or spin triplet with s = 1. The order parameter in a superconductor is expressed in terms of the gap function. Due to the anti-commuting properties of the electron wave function, the antisymmetric spin-singlet state is associated with a symmetric orbital wave function with orbital angular momentum l = 0 (s-wave), 2 (d-wave), etc, while the symmetric spin-triplet state is accompanied by an antisymmetric orbital wave function with orbital angular momentum l = 1 (p-wave), 3 (f-wave), etc. Since the condensed eld in the holographic setup [4, 5] is a scalar eld dual to a scalar operator in the eld theory side, it is therefore a holographic s-wave model. In the holographic setup, a d-wave order parameter is dual to a charged spin two eld propagating in the bulk [10, 11]. To mimic p-wave superconductor, one can consider a vector order parameter which is dual to a vector eld in the gravity side. A holographic p-wave model [12] was constructed by adding a SU(2) Yang-Mills eld into the bulk. A U(1) subgroup of the Yang-Mills eld is regarded as the gauge group of electromagnetism. A gauge boson generated by another SU(2) generator charged under this U(1) by the nonlinear coupling of the non-Abelian eld is dual to the vector order parameter. An alternative holographic realization of p-wave superconductivity emerges from the condensation of a 2-form eld in the bulk [13]. A holographic chiral px + ipy superconductor was discussed by generalized the SU(2) model introducing a Maxwell eld and a Chern-Simons term [14]. Note that in principle we can also build a holographic p-wave model by introducing a complex vector eld charged under a U(1) gauge eld in the bulk with possible couplings.

On the other hand, motivated by the possibility to create a very strong magnetic eld, for example, at RHIC and LHC, the interest to investigate the properties of QCD matter in a strong magnetic eld has been growing recently. Some new phenomena have been revealed such as the chiral magnetic e ect, a split between the deconnement phase transition and chiral symmetry restoration phase trasnition, see ref. [15] for a review. An interesting new phenomenon is the possibility that the QCD vacuum undergoes a phase transition to a new phase with charged -meson condensed in a su ciently strong magnetic eld [16, 17]. This exotic phase is a kind of anisotropic superconducting phase [18]. The author in ref. [16] adopted an e ective quantum electrodynamics action (DSGS model [19]) to discuss the condensate of -meson. A similar study based on Nambu-Jona-Lasinio model [20] can be found in ref. [17]. The uniform magnetic eld background is encoded in an Abelian gauge eld. As a vector boson, the possible condensate of the -meson in a uniform magnetic eld can be realized in the holographic framework by introducing a charged vector eld in

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the gravity side in the presence of a magnetic eld. Based on the Sakai-Sugimoto model, the holographic -meson was studied in the connement phase at zero temperature [21] where it is shown that the e ective mass of the -meson at strong magnetic eld becomes tachyonic. The inuence of the magnetic eld on the chiral transition temperature and deconnement transition temperature was discussed in ref. [22]. In the SU(2) model [12], a similar instability triggered by a non-Abelian magnetic eld has been found in refs. [23 25], which is reminiscent of the observation that non-Abelian magnetic eld induces the W-boson condensate exhibiting vortex lattices in at spacetime [2628].

In this paper we will construct a toy model by introducing a complex vector eld charged under an Abelian gauge eld in the bulk, which is dual to a strongly coupled system involving a charged vector operator with a global U(1) symmetry. In this model there exists a non-minimal coupling between the vector eld and the Abelian gauge eld. In this bottom-up approach, we do not know the details of the dual eld theory clearly. However, our setup meets the minimal requirement to construct a holographic p-wave superconductor model. This model has also potential to discuss the -meson condensate in the presence of magnetic eld.

We rst study the model in the case without a background magnetic eld, i.e., turn on the temporal component of the gauge eld only, which is the usual way of analyzing holographic superconductor at a rst step [4, 1012]. We nd a critical temperature below which a vector operator acquires a vacuum expectation value breaking the Abelian symmetry spontaneously. Furthermore, the condensate of this vector operator picks out a special spatial direction, thus the rotational symmetry is also broken in the condensed phase. The system undergoes a second order phase transition with the critical exponent one half which coincides with the result from mean eld theory. This condensed phase presents characteristics known from superconductivity, such as an innite DC conductivity and a gap in the optical conductivity.

An instability induced by a background non-Abelian magnetic eld has been reported, where the non-Abelian current operators obtain vacuum expectation values resulting in a vortex lattice structure [24, 25]. We are interested in how the applied Abelian magnetic eld inuences the instability of our model with Abelian gauge symmetry. For this, we turn on a uniform magnetic eld B in the bulk, which immerses the condensed phase into an external magnetic eld.

We nd that the increase of the magnetic eld induces the instability of the black hole background, which gives rise to a family of condensate induced by the applied magnetic eld in the dual strongly coupled system. This magnetic eld induced instability can happen even for the case with vanishing chemical potential or charge density. For nite magnetic eld, there is a tower of droplet solutions [29] in the sense that they are localized in a nite region. Further, the condensate shrinks in size as one increases the magnetic eld, which shares similarity with the Meissner e ect. But it is not the exact case with conventional superconductivity, where the superconductivity is suppressed by magnetic eld in the real materials. Of course, it is interesting to notice the fact that it has been reported recently that an applied magnetic eld may induce superconductivity [30, 31]. In addition, the emergence of charged vector operator condensate triggered by the applied

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magnetic eld is consistent with the appearance of electromagnetically superconducting phase in a strong magnetic eld studied from both eld theory method [16, 17] and the holographic setup [21, 22]. We also manage to construct the vortex lattice solution in the condensed phase near the phase transition, which is very reminiscent of the Abrikosov lattice in common type-II superconductors.

The plan of this paper is as follows. In section 2, we introduce the holographic model and deduce the general equations of motion of the system. In section 3, we turn on a uniform magnetic eld and give details about how to recover the Landau levels in this holographic model. Section 4 is devoted to discussing the phase diagram with/without chemical potential. The vortex lattice solutions are constructed in section 5. The conclusion and further discussions are included in section 6. We study, in appendix A, the condensate of the vector eld and the phase transition in the case without magnetic eld by calculating the conductivity to ensure the condensed phase to be a superconducting state.

2 The holographic model

Let us introduce a charged vector eld into the (3+1) dimensional Einstein-Maxwell theory with a negative cosmological constant. The full action reads

S = 1

22

[integraldisplay] d4xg[bracketleftbigg]R +

6L2

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1

2 m2 + iqF [bracketrightbigg], (2.1)

where L is the AdS radius that we will set to be unity, 2 8G is related to the grav

itational constant in the bulk and m is the mass of the charged vector eld . The

strength of U(1) eld A is F = A A. The tensor in (2.1) is dened by

= D D. The covariant derivative D = iqA with q the charge of vector

. The last interacting term describes the non-minimal coupling of the charged vector eld to the U(1) gauge eld A. The parameter characterizes the magnetic moment of the vector eld and is assumed to be non-negative. The form of the action is reminiscent of the DSGS model which describes the quantum electrodynamics of the -meson proposed by Djukanovic, Schindler, Gegelia, and Scherer in ref. [19]. Compared to the action of the DSGS model, it is easy to nd that the action in (2.1) is just a simple generalization of the DSGS model to the anti-de Sitter space. Note that the part of neutral -meson in ref. [19] is neglected here since it is not relevant to our goal in this paper.

Varying the action (2.1) with respect to A yields the equation of motion for gauge eld

F = iq( ) + iq( ), (2.2)

while a variation of the action (2.1) with respect to gives the equation of motion for the charged vector eld

D m2 + iqF = 0. (2.3)

If one takes the limit q keeping q and qA xed, the back reaction of the matter

sources to the background can be ignored. This is the probe limit we will adopt in this

4

14FF

paper. The background is taken to be a (3 + 1) dimensional Schwarzschild-AdS black hole

ds2 = f(r)dt2 +

3 h

r3 ) and rh the horizon radius. The Hawking temperature for this black hole is T = f(rh)

4 =

3rh

4 , which sets the temperature of the boundary eld theory. In the probe limit, the matter elds and A can be treated as perturbations on the

Schwarzschild-AdS black hole background.

3 Adding a constant magnetic eld

In appendix A, we discussed the condensate of the vector eld induced by the bulk electric eld by turning on the scalar potential At associated with the Maxwell eld only. The normal phase corresponds to the black hole solution with a vanishing vector eld .

As one lowers the temperature, the normal phase becomes unstable to developing nontrivial conguration of the vector eld . It gives non-zero vacuum expectation value of the dual vector operator, which breaks the U(1) gauge symmetry and the rotational symmetry in x y plane. The calculation of the optical conductivity reveals that there

is a delta function at the origin for the real part of the conductivity, which means the condensed phase is indeed superconducting. For details, see appendix A.

We now turn on a magnetic eld to study how the applied magnetic eld inuences on the system. A consistent ansatz is as follows 1

dx = [x(r, x)eipy + O(3)]dx + [y(r, x)eipyei + O(3)]dy,

Adx = [(r) + O(2)]dt + [Bx + O(2)]dy,

rhr ). (3.3)

1One can also consider the dyonic black hole background with gauge eld xed [29]. is considered as a perturbation in such a background. In that case, the metric eld f(r) will depend on electric charge and magnetic eld B.

5

dr2f(r) + r2(dx2 + dy2), (2.4)

with f(r) = r2(1 r

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(3.1)

where x(r, x), y(r, x) and (r) are all real functions, p is a real constant, the constant is the phase of y and B > 0 is the constant magnetic eld perpendicular to the x y plane.

We have dened the deviation parameter from the critical point at which the condensate begins to appear.

The zeroth order of (2.2) gives the equation of motion for

(r) + 2r (r) = 0. (3.2)

The asymptotic value of gives the chemical potential = At(r ) of the dual eld theory. The boundary condition at the horizon is given by requiring that AA is nite

there. Thus we can obtain a unique solution

(r) = (1

The equations of motion for x and y can be deduced from (2.3) at order O(). We

further separate the variables as x(r, x) = x(r)X(x) and y(r, x) = y(r)Y (x). We nd that to satisfy the equations of motion of the model with the given ansatz, can only be chosen as + = 2 + 2n or = 2 + 2n with n an arbitrary integer. The equations of

motion for x(r), y(r), X(x) and Y (x) are divided into the following equations as

x X (qBx p)yY = 0, (3.4) x X (qBx p)yY = 0, (3.5)

x + f

f x +

q22

f2 x

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m2f x

+ x

r2f

[bracketleftbigg] (qBx p)

Y X

yx qB

Y X

yx (qBx p)2[bracketrightbigg] = 0, (3.6)

y + f

f y +

q22

f2 y

m2f y

+ y

r2f

[bracketleftbigg]

Y (qBx p)

X

Y

X [bracketrightbigg] = 0, (3.7)

where the prime denotes the derivative with respect to r and the dot denotes the derivative with respect to x. Here and below the upper signs correspond to the + case and the lower to the case. In order to satisfy (3.4), one should impose

y = cx, X c(qBx p)Y = 0, (3.8) where c is a real constant. This constraint is automatically satised by (3.5). We can see that only two of the four functions are independent. Substituting (3.8) into the remaining equations, we can nd the following three equations

x + f

f x +

q22

f2 x

xy (1 + )qB

x y

m2f x

Er2f x = 0, (3.9)

X c(1 + )qBY + (qBx p)2X = EX, (3.10)

(1 + )qBc X + (qBx p)2Y = EY. (3.11)

One can get the value of E for arbitrary constant c by solving the eigenvalue problem (3.10) and (3.11) with the constraint given in (3.8). There may exist the possibility that one can not obtain non-trivial solutions for some special values of c. Here we consider a simple case with c2 = 1, in which the equations of motion for X(x) and Y (x) can be solved exactly. Since c = 1/c, the c in the denominator in (3.11) is equivalent to the case in the numerator. Subtracting (3.10) from (3.11) and dening a new function as 2

(x) = X(x) Y (x), (3.12)

2To solve the above equations, one can also dene (x) = X(x)+Y (x). This case is equivalent to setting c ! c, which gives nothing new.

6

one gets the equation

(x) + [E c(1 + )qB (qBx p)2](x) = 0. (3.13)

We further introduce a new variable =

p|qB|(x pqB ) and a constant = Ec(1+)qB|qB| ,

then the above equation becomes

d2d2 () + ( 2)() = 0. (3.14)

The regular and bounded solution of (3.14) is given by Hermite function Hn as

n() = Nne2/2Hn(), (3.15)

with the corresponding eigenvalue n = 2n + 1. Nn is a normalization constant and n is a non-negative integer. Thus we obtain the solution of equation (3.13) as

n(x) = Nne

1

2

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|qB|(x pqB )2Hn[parenleftbigg][radicalbig]|

qB|[parenleftbigg]x

p

qB

[parenrightbigg][parenrightbigg], (3.16)

with the corresponding eigenvalue

En = (2n + 1)|qB| c(1 + )qB. (3.17)

Combining (3.8), (3.12) and (3.16), one can obtain the exact congurations for X(x)

and Y (x), which read

Xn(x; p) = e

cqB

2 (x pqB )2

X(0)ecp2 2qB

cqBNn

[integraldisplay]

x t p qB

e|qB|cqB2 (t pqB )2Hn[parenleftbigg][radicalbig]|qB|[parenleftbigg]t pqB [parenrightbigg][parenrightbigg]dt[bracketrightbigg],

(3.18)

0

|qB|(x pqB )2Hn[parenleftbigg][radicalbig]|

qB|[parenleftbigg]x

[parenrightbigg][parenrightbigg], (3.19) where X(0) is a constant denoting the value of Xn(x; p) at the origin x = 0. The solutions of x and y corresponding to the eigenvalue En, denoted by xn and yn, can be obtained by solving the equation of motion (3.9) with En given in (3.17). So far, we have recovered the Landau levels. As one can see in appendix A, we have a second order phase transition from the normal phase with = 0 to the condensed phase with 6= 0. Therefore

we should encounter a marginally stable mode at the transition point. Theses solutions obtained from (3.9) just correspond to the marginally stable states.

We can see from (3.9) that the e ective mass of x is

m2e = m2 + En

r2

q22

f = m2 +

and

Yn(x; p) = Xn(x; p) Nne

1

2

p

qB

f , (3.20)

which is clearly shifted by the magnetic eld B. Depending on concrete parameters and Landau level, the appearance of magnetic eld can increase or decrease the e ective mass,

7

(2n + 1)|qB| c(1 + )qB

r2

q22

thus will hinder or enhance the transition from the normal phase to the condensed phase. In what follows, we consider the case with the lowest Landau level with n = 0, which reads

EL0 = |qB|,

XL0(x; p) = N0

2 e

|qB|

2 (x pqB )2 = Y L0(x; p).

(3.21)

4 Phase diagram

We are interested in how the applied magnetic eld inuences on the transition temperature from the normal phase to the condensed phase. As we can see from (3.20), the e ective mass of the charged vector eld in the lowest energy state, i.e., in the lowest Landau level n = 0 depends on the magnetic eld B and the non-minimal coupling parameter as

m2e = m2 |

qB|

r2

q22

f . (4.1)

It is clear that the increase of the magnetic eld B decreases the e ective mass and thus tends to raise the transition temperature.

4.1 Phase diagram at vanishing charge density

It is well known that the increase of the electric eld decreases the e ective mass of charged scalar or vector elds, inducing the transition from the normal phase to the condensed phase. We rst turn o the electric eld, which corresponds to the case with vanishing charge density = 0. We introduce a new coordinate z = rh/r. The equation of motion (3.9) can be rewritten as 3

x(z)

3z21 z3

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x(z) [bracketleftbigg]

m2z2(1 z3)

9 162(1 z3)[bracketrightbigg]

x(z) = 0, (4.2)

with = |qB|/T 2.

To solve such a second order equation by shooting method, we impose the regular condition at the horizon z = 1 as well as the source free condition at the boundary z = 0. More specically, we set x(1) = 1 in our numerical calculation due to the linearity of (4.2).

For a given m2, only for certain values of = |qB|/T 2 can the boundary conditions

be satised.

Figure 1 presents the three marginally stable curves of x(z) for m2 = 3/4. The three lowest-lying modes are in the sequence 0 < 1 < 2. The red line corresponding to the minimal value of has no intersecting points with horizontal axis at non-vanishing z. Such a mode with 0 72.84 is considered as a mode of node n = 0. Furthermore, the

green line to 1 339.24 and blue line to 2 780.10 are regarded as modes with nodes

n = 1 and n = 2, respectively. Since the radial oscillations in z-direction of x(z) will cost more energy, the later two curves are therefore thought to be unstable. Thus the lowest

3We apologize to the readers for here using a same notation to denote the derivative with respect to di erent variables for brevity. But the meaning of the derivative in the text is clear and will not be confused.

8

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

jx[LParen1]z[RParen1]

0.0 0.2 0.4 0.6 0.8 1.0

z

Figure 1. The marginally stable curves of the charged vector eld corresponding to various critical = |qB|/T 2. The three curves from top to down correspond to 0 72.84 (red), 1 339.24

(green) and 2 780.10 (blue), respectively. We choose m2 = 3/4.

100

80

60

40

20

Figure 2. The critical magnetic eld 0 versus m2 of the vector eld. The points are obtained by the shooting method to solve the equation (4.2).

value 0 just gives the critical magnetic eld above which the normal state is unstable to developing a vector hair. Figure 2 shows the critical magnetic eld in terms of 0 for various squared mass of the vector eld. It can be seen clearly that 0 increases as we increase the squared mass.

It should be pointed out that if we turn o the magnetic eld, the normal state will not become unstable to developing hairs with vanishing charge density. The interesting result here is that only the magnetic eld itself can trigger the phase transition. This result has an analogy to the QCD vacuum instability induced by a strong magnetic eld to spontaneously developing the -meson condensate. It is clear that the last term in (2.1) describing a non-minimal coupling of the vector eld to the gauge eld A plays a crucial role in the instability. Note that similar coupling can be found in many formalisms used to describe the coupling of magnetic moment to the background magnetic eld for charged particles of spin 1, i.e., vector particles [19, 32].

9

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0

0.0 0.5 1.0 1.5 2.0

m2

4.5

4.0

3.5

normal phase

3.0

TTc

2.5

condensed phase

2.0

1.5

1.0 0 2 4 6 8 10 12 14

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[VertBar1]qB[VertBar1]

Figure 3. The transition temperature from the normal phase to the condensed phase as a function of magnetic eld. It corresponds to the case with En = |qB|. Tc is the critical temperature in

the case without magnetic eld. The magnetic eld raises the transition temperature. We choose m2 = 3/4.

4.2 Phase diagram at nite charge density

Let us now consider the system with xed charge density . Here we do not limit ourselves to the lowest energy state. The equation of motion (3.9) can be written in terms of and z as

x(z)

3z21 z3

x(z) [bracketleftbigg]

m2 z2(1 z3)

+ (En/)

1 z3

q22

(1 + z + z2)2

x(z) = 0, (4.3)

where =

r2h and En is given in (3.17). The lowest energy state corresponds to En = EL0 = |qB|. For numerical convenience and to match the behavior at the boundary, we

further denex(z) = ( zrh )F (z). (4.4)

Then we can obtain

F (z)

1 z

[parenleftbigg]

3z31 z3

2[parenrightbigg]F (z) [bracketleftbigg]

m2 z2(1 z3)

1 z3

( 1) z2

+ 3z

F (z)

[bracketleftbigg]

q22

+ (1 + z + z2)2

(En/)

1 z3 [bracketrightbigg]

F (z) =0.

(4.5)

34 and En . To solve such second order equation, we impose the regular condition at the horizon z = 1 as well as the source free condition F (0) = 0 at the boundary z = 0. It has a non-trivial solution only when there is a relation between such two parameters, which just gives the transition temperature as a function of the magnetic eld.

The (T, B) phase diagram with the lowest Landau level is drawn in gure 3. To determine which side of the phase transition line is the condensed phase, we can consider

10

This equation depends on two physical parameters T

=

1.0

0.9

0.8

0.7

0.6

normal phase

condensed phase

0.5 0 2 4 6 8 10

[VertBar1]qB[VertBar1]

Figure 4. The transition temperature from the normal phase to the condensed phase versus magnetic eld. It corresponds to the case with En = |qB|. Tc is the critical temperature in the

case without magnetic eld. The magnetic eld leads to lowering the transition temperature. We choose m2 = 3/4.

the equation (4.1). It suggests that the magnetic eld decreases the e ective mass. So if we increase the magnetic eld at a xed temperature, the normal state will become unstable for su ciently large magnetic eld. Figure 3 looks very similar to gure 9 in ref. [22] where the chiral transition temperature rises with magnetic eld, indicating chiral magnetic catalysis. Furthermore, to compare with the lowest Landau level case, we present an example with positive En = |qB| in (4.3) in gure 4. One can see clearly in this case

that the transition temperature lowers with the increase of the applied magnetic eld. It is the well known property of the ordinary superconductor, which has been rst discussed in a holographic setup in ref. [33].

5 Vortex lattice solution

Let us now construct the vortex lattice solution. It is enough to consider the n = 0 solution only, i.e.,

0(x; p) = N0e

+

y(r, x, y) = ceix0(r)

i a2a2

TTc

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|qB|(x pqB )2. (5.1)

Due to the fact that the eigenvalue En is independent of p, a linear superposition of the solutions eipyxn(r)Xn(x; p) and eipyyn(r)Yn(x; p) with di erent p is also a solution of the

model at O(). We introduce two functions

x(r, x, y) = x0(r)

1

2

X=ceipyXL0(x; p),

+

(5.2)

X=ceipyY L0(x; p),

c = e

1 2, p = 2[radicalbig]|qB|

a1 ,

11

which satisfy the full equations of motion. a1 and a2 are arbitrary constants. Following ref. [33], we can obtain the vortex lattice solution

(r, x, y) x(r, x, y) ceiy(r, x, y) = x0(r)

+

X=ceipy0(x; p). (5.3)

(r, x, y) has a pseudo-periodicity

(r, x, y) = [parenleftbigg]r, x, y +

a1

p|qB|[parenrightbigg] ,

r, x + 2

a1

p|qB|, y + a2 a1

p|qB|[parenrightbigg]= e

i2a1 (|qB|y+

a2 2a1 )

(r, x, y),

(5.4)

m + 12[parenrightbigg]b1 + [parenleftbigg]n +1 2

b2, (5.5)

with two vectors b1 = a1

|qB|

as well as a zero at

xm,n =

y. m and n are two integers.

Since the expectation value of the operator dual to is given by the coe cient at boundary r , the quantity J = h

x i

yi indeed exhibits the vortex struc

ture with the cores of vortices located at xm,n. In particular, the triangular lattice with three adjoining vortices forming an equilateral triangle can be obtained by choosing the following parameters

a1 = 2

4

JHEP12(2013)036

y and b2 = 2

a1

|qB|

x + a2

a1

|qB|

23. (5.6)It should be stressed that it is the special combinations J which exhibit the vortex lattice

structure. In particular, for q > 0, it is J that corresponds to the lowest Landau level,

while for q < 0, it is J+. The form of operator presenting vortex lattice structure is the same as the one in eld theory study without gravity [16]. This also provides the evidence for the correctness of choosing c2 = 1 in section 3.

Figure 5 shows the conguration of the norm of condensate J in the x y plane for

the triangular lattice. Obviously, to obtain the true ground state, we should calculate the free energy of the solutions with di erent lattice structures from the action to nd which conguration minimizes the free energy. It turns out that the linear analysis presented here is not su cient to determine the most stable solution. We should include higher order contributions just as done in refs. [24, 33]. The calculation is much more complicated and is not very relevant to our purpose of this paper. We leave it for our further study.

6 Conclusion and discussion

In this paper we studied a holographic model with a complex vector eld charged under a U(1) gauge eld in a (3 + 1) dimensional AdS black hole background, aiming to shed some light on the real strongly coupled systems which are of gravity duals. In this model, there is a non-minimal coupling of the vector eld to the U(1) gauge eld, which describes the interaction between the magnetic dipole moment of the vector eld to the background

12

3 , a2 =

JHEP12(2013)036

Figure 5. The vortex lattice structure for the triangular lattice in x y plane. The contour plot

is also drawn in the bottom. In particular, the condensate vanishes in the core of each vortex.

magnetic eld. This model includes the minimal ingredients to build a holographic p-wave superconductor model. We found a critical temperature at which the system undergoes a second order phase transition. The critical exponent of this transition is one half which coincides with the case in the Landau-Ginzburg theory. In the condensed phase, a vector operator acquires a vacuum expectation value breaking the Abelian symmetry as well as rotational symmetry spontaneously. Our calculation showed that this condensed phase exhibits an innite DC conductivity and a gap in the frequency-dependent conductivity, which is quite similar to properties of the ordinary superconductivity.

We paid more attention on the response of this system to an applied magnetic eld. We obtained the Landau level, from which we can nd the contribution to the e ective mass of the vector eld by the magnetic eld (see (3.20)). Due to the non-minimal coupling given in the last term of (2.1), the applied magnetic eld can reduce the e ective mass of the vector eld, thus inducing the instability of the black hole background even when the chemical potential/charged density is absent. That is, for the case with vanishing chemical potential or charged density, the black hole background becomes no longer stable when the magnetic eld is beyond a certain critical value. For the case with non-vanishing chemical potential, the phase boundary is determined by a relation between the transition temperature and the magnetic eld, which is presented in gure 3. The transition temperature increases with the applied magnetic eld. The response of this system to the magnetic eld is quite di erent from the behavior of ordinary superconductor where the magnetic eld makes the transition more di cult as drawn in gure 4. But our result is quite similar to the case of QCD vacuum instability induced by strong magnetic eld to spontaneously developing the

13

-meson condensate [16, 17]. Although so, it was shown that in our model, the condensate of the vector operator forms a vortex lattice structure in the spatial directions perpendicular to the magnetic eld. Of course, the non-minimal coupling term in the action plays a crucial role in both cases. Therefore in some sense, our model is a holographic setup of the study of -meson condensate in refs. [16, 17].

In ordinary superconductors an external magnetic eld suppresses superconductivity via diamagnetic and Pauli pair breaking e ects. However, it has also been proposed that the magnetic eld induced superconductivity can also be realized in type-II superconductors [34, 35], in which the Abrikosov ux lattice may enter a quantum limit of the low Landau level dominance with a spin-triplet pairing. And possible experimental evidence for the strong magnetic induced superconductivity can be found, for example, in refs. [30, 31]. It was also shown in Gross-Neveu type model that applied magnetic eld might induce superconductivity in the planar system with 4-fermion interaction [36].

We mention here that similar studies can also be generalized other gravitational backgrounds, such as the AdS soliton background which has been adopted to mimic superconductor/insulator phase transition [37]. The study of magnetic eld e ect in the superconductor/insulator case can be found in ref. [38]. In a forthcoming paper [39], we generalize the present study to the case with the AdS soliton as the background. It is found that the magnetic eld can induce the instabilities of the AdS soliton background. Comparing our model with the SU(2) model with a constant non-Abelian magnetic eld [23, 25], we nd that our complex vector eld model in some sense is a generalization of the SU(2) model to the case with a general mass squared m2 and magnetic moment characterized by . In the setup of the present paper, the SU(2) model corresponds to our model with m2 = 0

and = 1.

In this paper, we restricted ourselves to the probe approximation, neglecting the e ect of matter elds on the background geometry. This can indeed reveal some signicant properties of the model, but something might be lost in this approximation, see ref. [40] as an example. It is therefore helpful to understand full properties of the model by considering the back reaction of matter elds on the background geometry. In a recent paper [41], going beyond the probe approximation, we found a rich phase structure in this model without magnetic eld. Depending on mass square m2 and charge q of the vector eld, not only second order but also rst order and zeroth order phase transition can appear. Interestingly, there also exists a so-called retrograde condensation in which the hairy solution exists only for temperatures above a critical value and is thermodynamically sub-dominant. Particularly, the zeroth order transition and retrograde condensation can be observed for the case with small m2. Indeed this model has much more phase behaviors than the SU(2) model, thus it can be used to mimic much richer phenomena in dual strongly coupled systems.

In our model, it is the magnetic eld itself that can induce the condensate. A natural question arises how about the Meissner e ect known as that superconductors expel weak external magnetic eld. This e ect is due to the large superconducting currents induced in the superconductor by the external magnetic eld, which generates a back-reacting magnetic eld screening the external magnetic eld. In the holographic SU(2) p-wave

14

JHEP12(2013)036

model, it has been pointed out [25] that the vortex currents ow in the opposite direction to the one in conventional superconductors, thus can enhance the applied magnetic eld in the regions between the vortices. It seems that the model presented here is similar to the SU(2) p-wave model. It is interesting to ask whether a similar phenomenon also appears in our model. To answer this question, we need to consider the contribution of higher order terms in (3.1), which is also required to nd the true vortex ux lattice. We leave it for further study.

In this paper we only calculated the conductivity in the case without magnetic eld in one spatial direction. Although it is enough to see the superconductivity feature of the condensed phase, in order to study the model in a more realistic manner, it is desirable to calculate the conductivity in another direction and further to study the transport properties of the lattice state. As a phenomenological approach, this toy model would have potential application to mimic strongly coupled systems with a vector like order parameter. It should be applicable in a wide variety of condensed matter systems, heavy ion physics and beyond. We hope to report further progresses in future.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (No.10821504, No.11035008, No.11205226,No.11305235, and No.11375247), and in part by the Ministry of Science and Technology of China under Grant No.2010CB833004.

A Condensate of vector eld and superconducting phase transition

As a toy model with a charged vector eld in the bulk dual to a vector operator in the eld theory, there exists the possibility that the condensate of this vector eld can serve as an order parameter to mimic a holographic p-wave superconductor phase transition, like the s-wave case [4, 5]. More precisely, we hope that this system has stable black hole solutions with vector hair at low temperatures, but without vector hair at high temperatures. If this is true, in the condensed phase, the condensate of the dual vector operator will break not only the U(1) symmetry but also the rotational symmetry since the condensate of vector eld picks out one special direction. This situation is very similar to the one in the holographic p-wave superconductor model with SU(2) gauge eld in ref. [12]. In this sense, our model can also be regarded as a holographic p-wave model.

A.1 Condensate of vector eld

We adopt the following ansatz

dx = x(r)dx + y(r)dy, Adx = (r)dt. (A.1)

One can use the U(1) gauge symmetry to set x to be real. Then one nds that the r component of (2.2) implies that the phase of y must be constant. Without loss of

15

JHEP12(2013)036

generality, we take y to be real. Then, the independent equations of motion in terms of the above ansatz are deduced as follows

x + f

f x +

q22x

f2

m2x

f = 0,

y + f

f y +

q22y

f2

m2y

f = 0,

(A.2)

+ 2r

2q2r2f (2x + 2y) = 0,

where the prime denotes the derivative with respect to r.

In order to nd the solutions for all the three functions F = {x, y, } one must impose suitable boundary conditions at the AdS boundary r and at the horizon

r = rh. In addition to f(rh) = 0, one must require (rh) = 0 in order for gAA to be nite at the horizon.

In order to match the asymptotical AdS boundary, the general fallo of the matter elds near the boundary r should behave as

=

r + . . . , x =

JHEP12(2013)036

x r +

x+r + + . . . , y =

y r +

y+r + + . . . , (A.3)

where =

11+4m2

2 . 4 We impose x = 0 and y = 0, since we want the U(1)

symmetry to be broken spontaneously. According to the AdS/CFT dictionary, up to a normalization, the coe cients , , x+ and y+ are interpreted as chemical potential, charge density and the x and y components of the vacuum expectation value of the vector operator in the dual eld theory, respectively.

There is a useful scaling symmetry in the equations of motion

r r, (t, x, y) 1(t, x, y), (, x, y) (, x, y), (A.4) where is an arbitrary positive constant. Under this symmetry, the revelent quantities transform as

T T, , 2, (x+, y+) ++1(x+, y+). (A.5) We assume the condensate to pick out the x direction as special, so we can consistently set y = 0. The condensate as a function of temperature is presented in gure 6. It is clear that as we lower the temperature, the normal phase with vanishing charged vector becomes unstable to developing vector hair which breaks the U(1) symmetry as well as rotational symmetry in the dual eld theory. Fitting the curve near the critical temperature Tc, we

nd that for small condensate there is a critical behavior with critical exponent 1/2, which precisely meets the result given by mean eld theory and it is typically a second order phase transition. In the case with m2 = 3/4 we obtain

h

xi 339T 5/2c(1 T/Tc)1/2, as T Tc. (A.6)

4The m2 has a lower bound as m2 = 1/4 with + = = 1/2, in which case there is a logarithmic term in the asymptotical expansion. Such a term should be considered as the source set to be vanishing [42].

16

0.8

0.6

<Jx>25

0.4

0.2

0.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

JHEP12(2013)036

TTc

Figure 6. The condensate as a function of temperature. We choose q = 1 and m2 = 3/4. The

condensate begins to appear at Tc 0.102.

Thus we have obtained two black hole solutions in the system. When T > Tc, we

have the black hole solution without the vector eld, while we have the black hole solution with non-trivial vector eld as T < Tc. This behavior is in complete agreement with the holographic superconducting phase transition in the literature. Therefore we expect that the black hole solution with non-trivial vector eld can describe a superconducting phase. To prove this, it is helpful to calculate the optical conductivity. Before dong this, we should rst ensure that the black hole solution with non-trivial vector eld is more thermodynamically stable than the one without the vector eld as T < Tc.

A.2 Free energy

In order to determine which solution is thermodynamically favored, we should calculate the free energy of the system for both black hole solutions. We will work in canonical ensemble in this paper, where the charge density is xed. In gauge/gravity duality Helmholtz free energy F of the boundary thermal state is identied with temperature times the on-shell bulk action with Euclidean signature. Since we work in the probe approximation, we can ignore the gravity part. Given that the system is stationary, the Euclidean action is related to the Minkowski case by a total minus. Employing the equations of motion (2.2) and (2.3), we have

22SEuclidean =[integraldisplay] d4xg[parenleftbigg]

14FF

1

2 m2 + iqF [parenrightbigg]

+ [integraldisplay] d3xhn

AF + Sct

= [integraldisplay] d4xg

1

2AF + [integraldisplay] d3xhn

12AF [parenrightbigg] + Sct,

(A.7)

where h is the determinant of the induced metric h on the boundary r and n is

the outward pointing unit normal vector to the boundary. Sct denotes the surface counter term for removing divergence.

17

0

-2

-4

-6

-8

-10

-12

2 2 DF

V

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TTc

Figure 7. The di erence of Helmholtz free energy between in the condensed phase and in the normal phase as a function of temperature. We choose q = 1 and m2 = 3/4 for which the critical temperature Tc 0.102.

Substituting the asymptotically expansion (A.3) into (A.7) and introducing a counter term Sct = [integraltext]

dx3hh, 5 we nd the free energy F as 22F

V =

1

2 (2 + 1)(xx+ + yy+) [integraldisplay]

dxdy. Regarding x and y as sources, Helmholtz free energy tells us that

sub-leading terms x+ and y+ are the expectation values of the dual operator in the eld theory side. The di erence of Helmholtz free energy between the condensed phase and the normal phase as a function of temperature is presented in gure 7. It is clear that below the critical temperature Tc, the state with non-vanishing vector hair is indeed thermodynamically favored, compared to the normal phase. The phase transition is second order, which can be seen, for example, from the derivative of the free energy with respect to the temperature.

A.3 Conductivity

We now calculate the conductivity in the dual eld theory side as a function of frequency . We need to turn on uctuations of the matter contents in the bulk. We assume perturbations have a time dependence of the form eit with zero spatial momentum. It turns out that one can consistently turn on the perturbations of Ay only. We can obtain the equation of motion for Ay by linearizing the equations of motion (2.2), which reads

Ay + f

f Ay + [parenleftbigg]

18

drg

1

2AF , (A.8)

JHEP12(2013)036

with V =

[integraltext]

Ay = 0. (A.9)

Since the conductivity is related to the retarded two-point function of the U(1) current, we impose the ingoing boundary condition near the horizon. The gauge eld Ay near the boundary r falls o as

Ay = A(0) + A(1)r + . (A.10)

5We are not sure whether the counter term works or not in a general case. But we nd this counter term works well for the ansatz (A.1) with the asymptotically expansion (A.3).

2 f2

2q22x r2f

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0 10 20 30 40 50 60

T

Figure 8. The optical conductivity as a function of frequence. The solid lines in the left plot are the real part of the conductivity, while the dashed lines in the right plot are the imaginary part of the conductivity. We choose q = 1 and m2 = 3/4. The horizontal lines correspond to the temperature above Tc. Other curves from left to right correspond to T/Tc 0.830 (purple),

T/Tc 0.519 (green), T/Tc 0.388 (blue), and T/Tc 0.290 (red), respectively. There is a delta

function at the origin for the real part of the conductivity in the condensed phase.

According to the AdS/CFT dictionary, one can obtain the conductivity as

yy() = A(1)iA(0) . (A.11)

The AC conductivity as a function of frequency is presented in gure 8. We can see clearly that the optical conductivity along the y direction in this model behaves qualitatively similar to the case in the p-wave model with SU(2) gauge symmetry [12]. In particular, from the Kramers-Kronig relation, one can conclude that the real part of the conductivity has a Dirac delta function at = 0 since the imaginary part has a pole, i.e., Im[yy()]

1 . Furthermore, it is clear that the optical conductivity develops a gap at some special frequency g known as gap frequency. As suggested in ref. [42], it can be identied with the one at the minimum of the imaginary part of the AC conductivity. Re[yy] is

very small in the infrared and rises quickly at g. There also exists a small bump slightly above g which is reminiscent of the behavior due to fermionic pairing [12]. For our chosen parameter, we have g 8Tc. Compared to the corresponding BCS value g 3.5Tc, the

result shown here is consistent with the fact that our holographic model describes a system at strong coupling.

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