#### Towards complete phase diagrams of a holographic p-wave superconductor model

Rong-Gen Cai
1
Li Li
1
Li-Fang Li
0
Run-Qiu Yang
1
0
State Key Laboratory of Space Weather, Center for Space Science and Applied Research, Chinese Academy of Sciences
,
Beijing 100190, China
1
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences
,
Beijing 100190, China
We study in detail the phase structure of a holographic p-wave superconductor model in a five dimensional Einstein-Maxwell-complex vector field theory with a negative cosmological constant. To construct complete phase diagrams of the model, we consider both the soliton and black hole backgrounds. In both two cases, there exist second order, first order and zeroth order phase transitions, and the so-called retrograde condensation also happens. In particular, in the soliton case with the mass of the vector field being beyond a certain critical value, we find a series of phase transitions happen such as insulator/superconductor/insulator/superconductor, as the chemical potential continuously increases. We construct complete phase diagrams in terms of temperature and chemical potential and find some new phase boundaries.
1 Introduction 2 3 4
The holographic model and no-hair solutions
AdS soliton with vector hair
3.1 Equations of motion and boundary conditions
3.2 Free energy
3.3 Phase transition in AdS soliton backgrounds
3.3.1 Phase transition for m2 > mc21
3.3.2 Phase transition for mc22 m2 mc21
3.3.3 Phase transition for m2 < mc22
AdS black hole with vector hair
4.1 Equations of motion and boundary conditions
4.2 Thermodynamics
4.3 Phase transition in black hole backgrounds
4.3.1 Phase transition for m2 mc2b
4.3.2 Phase transition for m2 < mc2b
The complete phase diagram
6 Summary and discussions
A The regular boundary conditions for the equations of motion
B The extremal hairy soliton solution
C The stress energy tensor
D The scaling relations between the critical chemical potential and charge 37
Introduction
The strong/weak duality characteristic of the anti-de Sitter/conformal field theory
correspondence (AdS/CFT) [13] enables us to study the properties of strong coupled systems
by a weak coupled AdS gravity with one higher dimension. There are a lot of strong
interacting phenomena in condensed matter physics, which are thought to be a good place
where the AdS/CFT correspondence is applicable (see refs. [47] for reviews). In order
to describe more general phenomena in this framework, recent efforts are devoted to
generalizing the correspondence to those systems with less symmetries (see refs. [814] for
example) and describing the far-from equilibrium problems quantitatively (see refs. [1519]
for example).
One of the most studied objects is the holographic superconductor (superfluid). In the
simplest model [20, 21], the onset of superconductivity is characterized by the instability
to form complex scalar hair in the bulk black hole spacetime, which corresponds to
condensation of composite charged scalar operator spontaneously breaking U(1) symmetry below
a critical temperature in the CFT side. This toy model describes a holographic s-wave
superconductor/conductor phase transition, i.e., a transition from black hole with no scalar
hair (normal phase/conductor) to the case with scalar hair at low temperatures
(superconducting phase). By using AdS soliton as the background in this Einstein-Maxwell-complex
scalar theory, the authors of ref. [22] constructed a holographic model mimicking a s-wave
superconductor/insulator transition at zero temperature. The normal insulating phase
is described by a pure AdS soliton which exhibits a mass gap and is used to describe a
confining phase in a dual theory [23]. Adding a chemical potential to the bulk, the
AdS soliton will become unstable to developing scalar hair for sufficiently large chemical
potential, which describes a superconducting phase. It was shown that as one changes
the temperature T and the chemical potential there are as many as four phases in this
model, including the AdS soliton, the AdS black hole and their superconducting phases.
The complete phase diagram in terms of T and was constructed in ref. [24].
Recently, the authors of ref. [25] proposed a holographic p-wave superconductor model
by introducing a complex vector field charged under a Maxwell field A in the bulk,
which is dual to a strong coupled system involving a charged vector operator with a global
U(1) symmetry. This setup meets the minimal requirement to construct a holographic
p-wave superconductor model, since the condensate of the dual vector operator may
spontaneously break the U(1) symmetry as well as spatial rotation symmetry. Actually, worked
in the probe limit neglecting the back reaction of the matter fields, the vector hair appears
at low temperatures and the condensed phase exhibits an infinite DC conductivity and a
gap in the AC conductivity. In this sense, this Einstein-Maxwell-complex vector model can
be regarded as a holographic p-wave model.1 The probe approximation is only justified
in the limit of large q with q and qA fixed. In the paper [28], the full back reaction
was taken into account in the four dimensional black hole background. Depending on the
charge q and mass square m2 of the complex vector field, the model presents a rich phase
structure. One can find zeroth order, first order and second order phase transitions. There
is also a retrograde condensation in which the hairy black hole solution exists only for
temperature above a critical value and is thermodynamically subdominant. In particular,
we can see in this model the conductor/superconductor/conductor phase transitions when
the temperature is continuously lowered. In addition, the behavior of entanglement entropy
in this model was discussed in ref. [29], and it can correctly manifest the phase structure
of the model.
The Einstein-Maxwell-complex vector model can be directly generalized to the AdS
soliton background to describe a holographic p-wave superconductor/insulator phase
transition. Actually, in the probe approximation, the vector condensate and instability induced
1The first holographic p-wave model was given in ref. [26] by introducing a SU(2) Yang-Mills field into
the bulk, where a gauge boson generated by one SU(2) generator is dual to the vector order parameter. An
alternative holographic realization of p-wave superconductivity comes from the condensation of a 2-form
field in the bulk [27].
by a magnetic field in a (4+1) dimensional AdS soliton background was studied in ref. [30],
where a vortex lattice structure can form in the spatial directions perpendicular to the
applied magnetic field.2 Note that in ref. [30] the vector condensation and AdS soliton
instability can be totally induced by an applied magnetic field because of the existence of
the coupling between the magnetic moment of the vector field and the background
magnetic field in this model. Furthermore, it was found that this toy model is a generalization
of the SU(2) p-wave model [26] in the sense that the vector field has a general mass and
gyromagnetic ratio.
As we have shown in the black hole case [28], it is worthwhile to consider the full
back reaction of the matter fields on the soliton geometry and to find all possible phase
behaviors. This is just one of the purposes of the present paper. On the other hand,
in order to construct complete phase diagrams of the model, we will also study phase
behaviors of the model in a (4+1) dimensional black hole background by generalizing the
study in ref. [28]. Since in this paper we do not turn on magnetic field, the model is
left with two parameters, i.e., the charge q and mass square m2 of the complex vector
field . More precisely, m2 determines the scaling dimension of the dual vector operator,
while q controls the strength of the back reaction. Depending on q and m2, the model
in the black hole case exhibits all known phase structure reported in ref. [28]. In the
soliton geometry, we can find second order transition, first order transition, zeroth order
transition as well as retrograde condensation in the sense that the hairy soliton solution
appears only for chemical potential below a critical value and has free energy larger than
the pure AdS soliton. Those phase behaviors are comparable to those in the black hole case.
Nevertheless, there is an additional interesting behavior in the soliton case. In some region
of model parameters, the condensate will be absent in some range of chemical potential,
where the condensate would be expected to appear naively. This leads to a series of phase
transitions. In a typical example, as one increases the chemical potential, we will encounter
for the insulator/superconductor/insulator/superconductor phase transitions.
Taking both the soliton geometry and black hole case into account, we obtain four
distinct solutions: AdS soliton, AdS black hole and their corresponding hairy solutions.
We construct the full phase diagram in terms of chemical potential and temperature for
various q and m2. Due to the rich phase behaviors of the model, the T - phase diagram is
much more complicated and interesting than the holographic s-wave model [24] and SU(2)
p-wave model [33, 34]. We can see some new phase boundaries that are never reported in
the literature in the framework of AdS/CFT correspondence.
The organization of this paper is as follows. In the next section, we introduce the
holographic model and give no-hair soliton and black hole solutions. In section 3, we give
our ansatz for the hairy soliton solution and specify the boundary conditions to be satisfied.
Then the phase transition for various parameters will be discussed in detail. In section 4,
a parallel discussion for the case of black hole geometry will be presented. The complete
phase diagrams in terms of temperature and chemical potential are constructed in section 5.
The conclusion and some discussions are included in section 6. Some calculation details
are shown in appendixes.
2In ref. [31], a similar phenomenon that the magnetic field leads to the instability was also found in the
SU(2) model.
Let us begin with a (4 + 1) dimensional Einstein-Maxwell-complex vector field theory with
a negative cosmological constant [25]
S = 212 Z d5xg R + L122 + Lm ,
Lm = 14 FF 12 m2 + iqF ,
where L is the AdS radius which will be set to be unity and 2 8G is related to the
gravitational constant in the bulk. g is the determinant of the bulk metric g and is
the complex vector field with mass m and charge q. We define F = A A and
= D D with the covariant derivative D = iqA. The last non-minimal
coupling term characterizes the magnetic moment of the vector field , which is crucial
in the presence of an applied magnetic field [25, 30]. However, in the present study, since
we only consider the case without external magnetic field, this term will not play any role.
So this model is left with two parameters m2 determining the scaling dimension of the
dual operator and q controlling the strength of the back reaction of matter fields on the
background geometry.
The full equations of motion deduced from the action (2.1) are the complex vector field
equations
D m2 + iqF = 0,
Maxwells equations
F = iq( ) + iq( ),
as well as the gravitational field equations
1 6 1 1
R 2 Rg L2 g = 2 FF + 2 Lmg
+ 21 {[ + m2 iq( )F] + }.
Since is charged under the U(1) gauge field, according to AdS/CFT correspondence,
its dual operator will carry the same charge under this symmetry and a vacuum
expectation value of this operator will then trigger the U(1) symmetry breaking spontaneously.
Thus, the condensate of the dual vector operator will break the U(1) symmetry as well as
the spatial rotation symmetry since the condensate will pick out one direction as special.
Therefore, viewing this vector field as an order parameter, the holographic model can be
used to mimic a p-wave superconductor (superfluid) phase transition. The gravity
background without vector hair ( = 0)/with vector hair ( 6= 0) is used to mimic the normal
phase/superconducting phase in the dual system.
First, let us consider solutions with = 0 and At = (r). One solution with planar
symmetry is the AdS Reissner-Nordstrom (RN-AdS) black hole, which reads [4]
ds2 = f (r)dt2 +
f (r) = r2 1
+ r2(dx2 + dy2 + dz2),
rh 4
r
2
rh 6
r
1
rh 2
r
The horizon locates at rh and is the chemical potential of the dual field theory. The
temperature T of the black hole is
2
1 6rh2
In the following, we will assume the coordinate z is compactified with period , whose
value will be given below, while x and y are in (, ). There is another trivial planar
solution, the so-called AdS soliton, which is given by
+ r2(dt2 + dx2 + dy2 + g(r)d2),
where r0 is the tip of the soliton. To avoid the potential conical singularity at r = r0, the
spatial direction must be made an identification with period
= . (2.8)
Thus, the AdS soliton is cigar shaped with the asymptotical geometry R1,2 S1 near
the AdS boundary r . Because the spacetime exists only for r > r0, the dual field
theory is in a confined phase and has a mass gap, Eg r0. Since the time component
of metric is regular at the tip, the soliton can be associated with any temperature in the
Euclidean sector.
In the holographic setup, the RN-AdS black hole (2.5) corresponds to a conductor while
the AdS soliton (2.7) describes an insulator [22]. In order to obtain a superconducting phase
in both soliton and black hole backgrounds, we need to find solutions with non-trivial
in the bulk. That is what we will do in the following.
AdS soliton with vector hair
Equations of motion and boundary conditions
To construct homogeneous charged solutions with vector hair in the soliton background,
we adopt the following ansatz
ds2 =
+ r2(f (r)dt2 + h(r)dx2 + dy2 + g(r)e(r)d2),
where g(r) vanishes at the tip r = r0 of the soliton. Further, in order to obtain a smooth
geometry at the tip r0, should be made with an identification
This gives a dual picture of the boundary theory with a mass gap, which is reminiscent of
an insulating phase.
One finds that the r component of (2.3) implies that the phase of x must be a constant.
Without loss of generality, we can take x to be real. Then, the independent equations of
motion in terms of the above ansatz are deduced as follows
2ff gg 2hh + 2 3r 2rq42gh2x = 0,
x + 2ff + gg 2hh 2 + 3r x + rq42fg2 x rm2g2 x = 0,
f 2ff gg 2hh + 2 5r f r22 2qr26g2xh2 = 0,
ff 2gg hh + 23rxh2 3r2f 23qr25f2xgh2 + r8g 8r = 0,
h + 2r2x2 2rq62f2xgh2 + 2rm4g2h2x = 0,
ffhh 2r2hx2 + r2f + 6rq62f2xgh2 2rm4g2h2x r224g + 2r42 = 0,
where the dots stand for the higher order terms in the expansion of 1/r and = 1
1 + m2.3 In general, in the above expansion we must impose x = 0, which meets
the requirement that the condensate appears spontaneously. According to the AdS/CFT
dictionary, up to a normalization, the coefficients , , and x+ are regarded as chemical
potential, charge density and the x component of the vacuum expectation of the vector
operator J in the dual field theory, respectively.
We impose the regularity conditions at the tip r = r0, which mean that all our functions
have finite values and admit a series expansion in terms of (r r0) as4
F = F (rh) + F (rh)(r r0) + .
By plugging the expansion (3.5) into (3.3), one can find that there are six independent
parameters at the tip {r0, x(r0), (r0), f (r0), h(r0), (r0)}. However, there exist four useful
scaling symmetries in the equations of motion, which read
r r, (t, x, y, ) 1(t, x, y, ), (, x) (, x),
where in each case is a real positive constant.
By using above four scaling symmetries, we can first set {r0 = 1, f (r0) = 1, h(r0) =
1, (r0) = 0} for performing numerics. After solving the coupled differential equations, one
should use the first three symmetries again to satisfy the asymptotic conditions f () = 1,
h() = 1 and () = 0.5 We choose (r0) as the shooting parameter to match the
source free condition, i.e., x = 0. Finally, for fixed m2 and q, we have a one-parameter
family of solutions labeled by x(r0). After solving the set of equations, we can read off
the condensate hJxi, chemical potential and charge density from the corresponding
coefficients in (3.4). It should be noticed that different solutions obtained in this way will
have different periods for direction. We should use the last scaling symmetry to set all
of the periods equal in order to obtain same boundary geometry. We shall fix to be
in this paper.
In the soliton background, there are two kinds of solutions: one is the pure AdS soliton
without the vector hair and the other is the hairy one. To determine which kind of solutions
is thermodynamically favored, we must calculate the grand potential of the system in the
grand canonical ensemble. In gauge/gravity duality the grand potential of the boundary
thermal state is identified with temperature times the on-shell bulk action in Euclidean
4In some extremal cases, the functions do not admit such Taylor series expansion. In appendix B, it is
shown that a type of solutions exists with a finite nonzero , but is expressed in terms of fractional order of
(r r0). This type of solutions corresponds to the limit case of the discontinuous points in subsection 3.3.1.
5Here we assume the difference of (r) between at the tip and at the boundary is finite. However, it
does not hold in the limit case of the discontinuous points discussed in subsection 3.3.1. The details will be
discussed in appendix B.
signature. One should also consider the Gibbons-Hawking boundary term for a well-defined
Dirichlet variational principle and further a surface counterterm for removing divergence.
Since we consider a stationary problem, the Euclidean action is related to the Minkowski
one by a minus sign as
d5xg(R + 12 + Lm) +
d4xph(2K 6),
where g is the determinant of the bulk metric (one should not confuse it with the
function g(r) appearing in the metric ansatz), h is the determinant of the induced metric on
the boundary, and K is the trace of the extrinsic curvature.6 By using the equations of
motion (3.3) and the asymptotical expansions of matter and metric functions near the
boundary, the grand potential turns out to be
where V3 = R dxdyd. There is a little subtle issue dealing with soliton case, since the
soliton background has no horizon and the associated Hawking temperature vanishes.
Nevertheless, one can introduce an arbitrary inverse temperature 1/T as the period of the
Euclidean time coordinate. The integration over the Euclidean time in the Euclidean
action just gives the factor 1/T , which cancels the temperature factor in the Euclidean action
and leads to a finite grand potential. In the case of m2 > 1, it is found that f4+h44 = 0
(see appendix C). Since we have scaled to be , g4 = 1 for the pure AdS soliton solution,
namely, in the normal insulating phase.
Phase transition in AdS soliton backgrounds
In what follows we will look for hairy soliton solutions numerically. Our strategy is to fix
m2 and for each m2 we scan a wide range of q which determines the strength of the back
reaction of matter fields on the background. The precision of numerical calculation limits
us to investigate the entire range of q. Nevertheless, our numerical calculation reveals
that the system exhibits qualitatively different behavior depending on concrete value of
m2. More specially, there are two critical mass squares, mc21 = 0.218 0.001 and mc22.
Although it is difficult to determine the exact value of mc2 numerically, it is suggested to
be zero by numerical analysis. Depending on q and m2, we can find second order, first
order and zeroth order phase transitions. There also exists the retrograde condensation
in which the hairy soliton solution exists only for chemical potential below a critical value
and is thermodynamically subdominant. Whats more, a discontinuous condensed phase
appears when m2 > mc21 and q is less than some critical value, which is a new kind of
phase transition that does not exist in other models. We shall give more details in the
following. Since we would meet with more than one critical chemical potential at which a
phase transition happens as we increase the chemical potential, for brevity, we shall denote
the first critical chemical potential as c1, the second critical chemical potential as c2,
and so on.
6In principle, we should also consider the surface counterterm for the charged vector field , but one
can easily see that this term makes no contribution under the source free condition, i.e., x = 0.
Phase transition for m2 > mc21
For the case m2 > mc21, there exist three critical charges q1, q2 and q3 with q1 < q2 < q3.
Their values depend on m2 we choose. The three critical charges divide the parameter
space of q into four regions, shown in table 1. The phase behavior changes qualitatively
in each region. For the case q q3, the condensed phase will appear above c1 through
a second order transition and becomes thermodynamically preferred. As we decrease q to
a value smaller than q3, the order of transition from the normal (insulator) phase to the
superconducting phase becomes first order. Much more interesting thing happens when we
continue decreasing the value of q past q2. Starting from the small region, the system
undergos a first order transition from the normal phase to the condensed phase at c1, and
as increases to c2 there is a zeroth order transition back to the normal phase, then at
the larger chemical potential c3, it comes back to the condensed phase again by a zeroth
order transition. There is a region c2 < < c3 in which hairy solutions do not exist, so
the condensed phase should jump to the normal phase. As one can see clearly, there are
three phase transitions as we increase the chemical potential. The later two zeroth order
phase transitions have not been reported in any holographic model. For the case q q1,
the system undergoes a zeroth order transition from the normal phase to the condensed
phase at c1. The phase transition and their orders are summarized in table 1.
As a typical example, we consider the case m2 = 5/4. We find similar results for other
values of m2 in the region m2 > mc21. For m2 = 5/4, the three critical values of charge are
q3 1.5600, q2 1.5345, and q1 1.5020, respectively. The pure AdS soliton survives for
arbitrary value of q. Nevertheless, we can find additional solutions with non-vanishing x,
which are thermodynamically preferred for sufficiently larger chemical potential. That is
to say, for each value of q, there must be a phase transition from the normal phase to the
condensed phase occurring at a certain chemical potential. From the perspective of dual
system, it means that a charged vector operator obtains a non-zero vacuum expectation
value hJxi 6= 0, which breaks the U(1) symmetry as well as the rotation symmetry in x y
plane spontaneously.7
7The breaking of the rotation symmetry in x y plane is due to the fact that hJxi chooses x direction
as special. However, this anisotropy does not display in the stress energy tensor of the dual field theory. In
fact, the xx-component and yy-component of the stress energy tensor are equal. For more details one can
see in appendix C.
2
22/V3
3
For q q3, we take q = 1.600 as a typical example. Figure 1 shows the grand
potential and condensate as a function of chemical potential, from which one can see
that hJxi arises continuously from zero at c1. The grand potential in the left plot
of figure 1 indicates that above c1 the configuration with non-vanishing vector hair
is indeed thermodynamically preferred to the normal phase. It is a second order phase
transition with the critical behavior hJxi (/ c1 1)1/2 near the critical point.
For the case q2 q < q3, such as q = 1.540 in figure 2, the transition from the
normal phase to the condensed phase becomes first order. As one can see in figure 2, the
condensate with respect to chemical potential is multi-valued and the free energy develops
a characteristic swallow tail. The condensate has a jump from zero to the upper branch
of the hairy solution at c1 1.8054.
There is an additional complication when we decrease the value of q past q2. A concrete
example for q = 1.534 is presented in figure 3. As one increases the chemical potential,
there is a first order transition at c1 1.8094. However, if one continues increasing the
chemical potential, there will be two discontinuities at c2 1.9085 and c3 1.9499,
between which there does not exist any hairy solution. So there is a kind of zeroth order
phase transition from the condensed phase to the normal phase at c2 and a kind of zeroth
order phase transition from the normal phase to the condensed phase at c3. These kinds
of zeroth order phase transitions do not come from thermodynamics but from the kinetics
of the field equations (3.3). The details will be shown in appendixes A and B.
Recall that the pure AdS soliton resembles an insulating phase while the hairy
soliton mimics a superconducting phase, our result suggests that there is a kind of
insulator/superconductor/insulator/superconductor phase transitions by increasing chemical
potential. Here it looks strange that a zeroth order phase transition appears with a
discontinuity of free energy because it seems very rare to have such a phase transition in realistic
systems, if it is possible. On the other hand, if such a phase transition does not appear at
all in realistic systems, our results imply that something is missed in our analysis. We will
have more to say at the end of this paper.
As we continuously decrease q, the value of c1 decreases while the value of c2
increases. There exists a critical value of charge q1 1.5020 at which the first order phase
transition point c1 coincides with the first zeroth order point c2. In this case, the first
part of condensed phase such as in figure 3 disappears (see figures 4 and 5). When the value
of q is less than q1, the first order transition point disappears and the first discontinuous
1.17.8
Figure 5. The condensate of the vector operator in the case of m2 = 54 , q = q1 1.502 (left plot)
and q = 1.490 < q1 (right plot). For the condensed phase, only the blue line is thermodynamically
favored, while the dashed orange part is un-physical. Thus, the condensate has a sudden jump form
zero to a nonzero value. There exists a zeroth order phase transition from the normal phase to the
condensed phase.
point c2 appears above the line 22/V3 = 1, thus is un-physical. This case for q < q1
is shown in figure 4 with q = 1.490. There is only a zeroth order transition at critical point
c1 1.9499.
The ( -q) phase diagram for the case with m2 = 5/4 is drawn in figure 6. One should
note that the horizontal axis and vertical axis in the figure are labeled by 1/q and q ,
respectively. It is convenience to compare with the results in the probe limit, where we
take the limit q keeping q and qA fixed. Therefore, in order to compare our results
to the case in the probe analysis, we should make the scaling transformation q
and A qA. Under such transformation, the chemical potential becomes q and
Figure 6. The phase diagram for the case with m2 = 5/4. The red line stands for the second order
phase transition, the green line stands for the first order phase transition, and the black line stands
for the zeroth order phase transition. Numerical results show q 2.7852 when 1/q approaches
to zero.
Charge Phase transition and its order, mc22 m2 mc21
q qb < c1, S = c1, 2nd > c1, SC
qa q < qb < c1, S = c1, 1st > c1, SC
q < qa < c1, S = c1, 0th > c1, SC
the temperature T becomes T /q. Therefore, they are the quantities q and T /q that are
comparable to those in the probe limit. One can see from figure 6 that for sufficiently large
q, the critical value of q approaches to a constant, which quantitatively agrees with the
probe analysis in appendix D. Furthermore, let us mention that the zeroth order phase
transition can only appear in the strong back reaction case. Finally in some range of q,
one can first see one first order transition and then two zeroth order transitions with the
increase of chemical potential.
Phase transition for mc22 m2 mc21
In the case of mc22 m2 mc21, we find two critical values of q denoted as qa and
qb with qa < qb. For small back reaction with q > qb, there is a second order
insulator/superconductor transition at some critical chemical potential c1. As we decrease q
to the range qa < q < qb, the order of transition becomes first order. For larger strength
of back reaction with q < qa, the condensed phase appears through a zeroth order phase
transition. The main results are shown in table 2. The concrete values of qa and qb depend
on m2. To give more details, we shall consider a typical example with m2 = 0 in this
subsection. The two critical values of q for m2 = 0 is qa 1.172 and qb 1.317.
11.3.75
Figure 7. The grand potentials with respect to the chemical potential in the case of m2 = 0,
q = 1.500 (left) and q = 1.200 (right). In both figures, the grand potentials of the pure AdS soliton
and hairy soliton are described by the green dashed line and blue solid line, respectively. In the
left plot, the condensate begins at c1 1.5103, while in the right plot the condensate begins at
c1 1.8207.
For q > qb, the situation is very similar to the case with q > q3 in m2 = 5/4. There
exists a critical chemical potential c1 at which a second order transition happens. As we
go on decreasing q past qb, the transition from the normal phase to the condensed phase
is first order. The behaviors of the grand potential and condensate are shown in figures 7
and 8, respectively.
When q is slightly below qb, the curve of condensate (or grand potential) is continuous,
while as q is a little larger than qa, the curve becomes discontinuous. In the previous
subsection, this discontinuity leads to two zeroth order transitions for q1 < q < q2 (see
figure 2) and one zeroth order transition for q q1 (see figure 4). In the present case with
q qa, although there exist two discontinuous points in the grant potential curve denoted
as upper point with larger grand potential and lower point with smaller grand potential,
as shown in the left plot of figure 9, both points appear in the region where the grand
potential of the condensed solution is larger than the one in the normal phase, thus are
thermodynamically disfavored and will not appear in the physical phase space. So we are
left with only a first order transition. The lower point moves down as q decreases and
finally its grand potential becomes smaller than the one in the normal phase when q < qa
(see the right plot of figure 9). Take q = 1.150 < qa as an example, the grand potential
presented in figure 9 shows clearly that the value of has a sudden jump from the normal
phase to the condensed phase at c1 1.8597, indicating a zeroth order phase transition.
A phase diagram in terms of and q for the case with m2 = 0 is shown in figure 10.
Indeed, for large enough q, the critical value of q approaches to a constant, which is
consistent with the probe limit shown in appendix D. This phase diagram is reminiscent
of the case in the SU(2) holographic p-wave superconductor/insulator model presented in
figure 12 in ref. [32]. Comparing our model with the special case m2 = 0 to the SU(2)
model, we find that the equations of motion in our setup look very similar with each
other and the asymptotical expansions near the boundary are the same. Actually, we have
1
<Jx>1/2
Figure 8. The condensate of the vector operator in the case of m2 = 0, q = 1.500 (left plot) and
q = 1.200 (right plot). The left plot shows a second order phase transition, while the right plot
shows a first order phase transition from the normal phase to the condensed phase. In the right
plot, only the blue part is physical.
11.3.75
12.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1
carefully checked our numerical calculation in ref. [32] and found that similar discontinuous
points in the grand potential also appear for the back reaction larger than a certain value.
More precisely, the discontinuity first appears with grand potential larger than the normal
phase, thus can not change the phase structure. However, for very large back reaction,
i.e. larger value of in ref. [32], the lower discontinuous point enters the region which has
a lower grand potential than the one in the insulating phase. Thus we can also find a
zeroth order transition from the normal phase to the condensed phase in the SU(2) p-wave
model. If we add such a zeroth order transition region to the phase diagram in the SU(2)
p-wave model, the figure 12 in ref. [32] is quite similar to figure 10 in the present paper.
This result is consistent with our probe analysis in the presence of a magnetic field that
Figure 10. The phase diagram for the case of m2 = 0. The red, green and black lines stand for the
second order, first order and zeroth order phase transitions, respectively. Numerical results show
q 2.2653 when 1/q approaches to zero.
the Einstein-Maxwell-complex vector field model is a generalization of the SU(2) p-wave
model in the sense that the vector field has general m2 [30].
Phase transition for m2 < mc22
When m2 is less than mc22, there are two m2-dependent critical values of charge qc and qd
(qc < qd). The possible phase transitions and their orders are shown in table 3. For the
above two cases with m2 mc22, the condensed phase seems to survive even to the limit
.8 In contrast, in the case with m2 < mc22, the hairy solution can not exist above a
finite chemical potential. For the weak back reaction with q > qd, we first meet a second
order transition at c1 from the normal phase to the condensed phase, and then a zeroth
order transition from the condensed phase back to the normal phase at c2. In the range
qc < q < qd, the only difference is that the phase transition happening at c1 becomes first
order. When we decrease the value of q less than qc, the condensed phase only appears at
chemical potentials smaller than a critical one c1 and has grand potential larger than the
normal phase. The latter means the normal phase is thermodynamical preferred. We take
m2 = 3/4 as a typical case in the following.
In this case, the two critical q are qd 1.0801 and qc 0.9951. When q qd, there are
two critical chemical potentials denoted by c1 0.8690 and c2 1.8650, respectively.
As the chemical potential is increased to c1, a second order phase transition from the
normal phase to the condensed phased appears. If one continues scanning the shooting
parameters, there is a critical value c2 at which the curve of condensate turns back to
the small chemical potential region. We can see from figure 11 that above c2 the hairy
solution does not exist, hence, a zeroth order phase transition from the condensed phase
to the normal phase happens at c2.
8Due to the precision limit of numerical control, it becomes much difficult to solve the system numerically
for very large chemical potentials. Nevertheless, we have scanned a wide range of chemical potential up to
103 and found no evidence that the curves of grand potential and condensate would turn back to the small
chemical potential region.
q qd
Phase transition and its order, m2 < mc22
> c1, SC = c2,0th > c2,S
2
22/V3
3
With the value of q is decreased into the region qc q < qd, the first phase transition
at c1 becomes a first order one (see figures 11 and 12). The zeroth order transition at c2
still survives in this case. Our numerical calculation indicates that the difference between
c1 and c2 decreases as q decreases. Therefore, the region of chemical potential where the
condensed phase is thermodynamical preferred is reduced by increasing back reaction. At
the critical value q = qc, c1 is equal to c2, thus there is no thermodynamically favored
condensed phase.
For sufficiently large back reaction with q < qc, one can see from figure 13 that hairy
solutions only appear below a critical chemical potential and their grand potentials are
larger than those of the normal phase. Therefore in this case there is no physical phase
transition from the normal phase to the condensed phase when q < qc. This phenomenon is
an analog to the retrograde condensation appearing in the case of black hole background
in ref. [28].
3
<Jx>2/3
We construct the ( , q) phase diagram for the case with m2 = 3/4 in
figure 14. We can see from the phase diagram that when q > qc, there exist the
insulator/superconductor/insulator phase transitions as one simply increases the chemical
potential. For very large q, the back reaction of matter fields to the geometry is small, one
should return to the results obtained from the probe limit. Indeed, we find that the critical
chemical potential of the second order transition is very closed to a constant which is just
the one by the probe calculation in appendix D. For a very large q, the black curve
cor
q1
0.8 d
q1
1 c
Figure 14. The phase diagram for the case with m2 = 3/4. The red, green and black lines stand
for the second order, first order and zero order phase transitions, respectively. Numerical results
show c1q 1.7382 when 1/q approaches to zero.
responding to the zeroth order transition from the condensed phase to the normal phase
still exists. But we can not see this critical point in the probe limit. Actually, our
numerical results for q 1 suggest the following two scaling relations hold for two critical
chemical potentials
c1 1q , c2 q , (m2) = 0.4791m2 + 0.0492m4 + O(m6).
with negative m2. The first relation for c1 is consistent with the probe approximation by
using general Heun function analytically. The second relation for c2 is found by numerical
analysis. For more details please see appendix D.
This scaling relation on c2 shows that the value of c2 will be divergent for q
once the value of m2 is non-negative. That is to say, there does not exist similar turning
point to the normal phase for m2 0. This suggests that the value of mc22 is indeed
zero. Furthermore, for the case with m2 < mc22, one can see that q c2 q+1 is divergent
as q , which means that we can not see the turning point in the probe limit. The
retrograde condensation can only appear for strong back reaction case.
AdS black hole with vector hair
Equations of motion and boundary conditions
The hairy AdS charged black holes in the Einstein-Maxwell-complex vector field theory
have been well studied in four dimensions in the probe limit [25] as well as in the case with
full back reaction [28]. Especially for the latter case, depending on two model parameters
m2 and q, the model manifests a rich phase structure. In this section we will construct
asymptotically AdS charged black holes with non-trivial vector hair in (4+1) dimensions.
Similar to ref. [28], we adopt the following metric ansatz
ds2 = a(r)eb(r)dt2 +
+ r2(c(r)dx2 + dy2 + dz2).
the value of x at the horizon. After solving the set of equations, we can read off the
condensate hJxi, chemical potential and charge density directly from the asymptotical
expansion (4.4).
We will discuss how to extract thermodynamic quantities from our solutions. We will work
in the grand canonical ensemble with chemical potential fixed. According to gauge/gravity
duality, the Hawking temperature of the black hole is identified with the temperature of
boundary thermal state, which is given by
4
and the thermal entropy S is given by the Bekenstein-Hawking entropy of the black hole
T =
a(rh)eb(rh)/2
S = 42 A = 4V23 rh3pc(rh),
d5xg(R + 12 + Lm) +
where h is the determinant of the induced metric on the boundary, and K is the trace of
the extrinsic curvature. We have introduced the Gibbons-Hawking boundary term for a
well-defined Dirichlet variational problem. Once again, the potential surface counterterm
for the matter fields has been neglected here since this term does not make any contribution
in our setup of the source free condition.
By using the equations of motion (4.3), the on-shell action reduces to
22SEonuclsidheealln = 2V3eb/2rac(Kr 2r a)|r,
= T SEonuclsidheealln = 2V32 (a4 + c4 b4).
Therefore, we can set the chemical potentials of all black hole solutions equal by using
above transformation rule directly. Without loss of generality, we shall fix the chemical
potential to be = 1. Finally let us mention that apart from the temperature T and
thermal entropy S, this scaling transformation (4.11) is also true for the soliton case.
Phase transition in black hole backgrounds
Our numerical calculation shows that the phase structure of the model in the (4+1)
dimensional black hole geometry looks precisely the same as the case in the (3+1) dimensional
black hole spacetime [28]. Therefore in this section we will just give main results and some
relevant details can be found in ref. [28]. The phase transition in the black hole background
depends on the values of the mass square m2 and charge q. There exists a particular value
of m2 denoted by mc2b. For m2 is above or below mc2b, our system exhibits distinguished
behaviors. Our numerical analysis suggests that mc2b = 0, up to a numerical error, which
is same as the value obtained in the (3+1) dimensional case [28]. For m2 mc2b, the
condensed phase exists even for sufficiently low temperatures, while for m2 < mc2b, the
condensed phase will be absent when the temperature is lowered to a critical value. All
possible phase transitions are summarized in table 4 and table 5.
In the case of m2 mc2b, there is a mass dependent critical charge qc. When the charge
of the vector field is larger than qc, the condensed phase will become thermodynamically
preferred when the temperature is less than a critical temperature at which a second
order phase transition occurs. When we decrease q to a certain value smaller than qc,
the transition from the normal phase/conductor to the condensed phase/superconductor
becomes a first order one. For large m2 mc2b, the condensed phase seems to survive even if
the temperature goes to zero, T 0. In contrast, in the case with m2 < mc2b, the condensed
phase cannot exist when the temperature is lower than a certain value. To determine the
precise value for mc2b , we need to solve the coupled equations of motion (4.3) at very low
temperatures to see whether the condensate would turn back to the higher temperature
region. The precision of numerical calculation prevents the investigation on the limit of
T 0. Nevertheless, our numerical calculation suggests that mc2b is very closed to zero.
We will consider one concrete example for both cases with m2 mc2b and m2 < mc2b,
respectively.
Phase transition for m2 mc2b
Phase transition and its order, mc2b m2
T > Tc, BH T = Tc, 2nd T < Tc, BC
T > Tc, BH T = Tc, 1st T < Tc, BC
Table 4. The phase transition and its order with respect to the charge and temperature in the case
2 2
of mcb m . In the table, BH=normal phase/conductor with the RN-AdS black hole solution,
BC=condensed phase/superconductor with hairy black hole solution. 1st and 2nd stand for the first
order and second order phase transitions, respectively.
Phase transition and its order, m2 < mc2b
q q T > Tc2, BH T = Tc2, 2nd Tc0 < T < Tc2, BC T = Tc0,0th T < Tc0,BH
q < q < q T > Tc1, BH T = Tc1, 1nd Tc0 < T < Tc1, BC T = Tc0,0th T < Tc0,BH
BH
Table 5. The phase transition and its order with respect to the charge and temperature in the case
2 2
of mcb > m . In the table, BH=normal phase/conductor with the RN-AdS black hole solution,
BC=condensed phase/superconductor with hairy black hole solution. 0th, 1st and 2nd stand for the
zeroth order, first order and second order phase transitions, respectively.
by q. For small back reaction, this transition is second order, while for large back reaction,
it becomes a first order one. The critical value of q denoted as qc depends on m2 we choose.
We here consider m2 = 5/4 as a concrete example, where qc 2.0972. One can find similar
results for other values of m2 mc2b.
Let us first consider, for example, the case with q = 2.500 > qc. The condensate
appears at Tc 0.0698 and arises gradually from zero for lower temperatures. Comparing
grand potential between the hairy solution and the RN-AdS solution, one can see from the
left plot of figure 15 that the condensed phase is thermodynamically favored. It is a second
order phase transition with the critical behavior hJxi (1 T /Tc)1/2 near Tc. The critical
exponent for the condensate is 1/2, the same as the one in mean field theory.
A qualitative change happens as q is decreased to below qc. The grand potential and
condensate as a function of temperature for q = 1.800 < qc are shown in figure 16. We can
see that a swallowtail appears in the grand potential and in this case, the condensate has
a jump from zero to a finite value at the critical temperature, which is a typical feature of
a first order phase transition. This tells us that when q < qc, the phase transition becomes
first order from the normal phase to the condensed phase as the temperature is lowered.
A phase diagram in terms of temperature and charge is shown in figure 17 in the case
with m2 = 5/4, where the red and blue curves stand for the second order and first order
phase transitions, respectively.
Phase transition for m2 < mc2b
The phase structure for m2 < mc2b is the same as the one in the (3+1) dimensional black
hole background [28]. The parameter space of q is divided into three regions by two special
values of q, denoted as q and q with q > q. The phase behavior changes qualitatively
in each region. To compare our results to the case with one dimension less, we take the
0.009.8004 0.006 0.008
0.012 0.014 0.016
same terminology, i.e., the critical temperatures for second order transition, first order
transition and zeroth order transition are denoted by Tc2, Tc1 and Tc0, respectively. In this
subsection, we consider m2 = 3/4 as a concrete example. In this case, q 1.766 and
q 1.700.
For the weak back reaction q > q, we show the condensate and grand potential versus
temperature in figure 18. We immediately see that the condensate arises below Tc2 and
then turns back to the high temperature region at T0 (see the dashed orange line, which
corresponds to the upper branch of the grand potential in the condensed phase). When
T < Tc0, there does not exist hairy black hole solution and thus in the region with T < Tc0
we have only the RN-AdS black hole solution. By comparing the grand potential for each
0.0135 0.14 0.145 0.15 T 0.155 0.16 0.165 0.17
solution, we find that the thermodynamically favored region of the condensed phase only
locates in the region Tc0 < T < Tc2. At Tc2, it is a second order phase transition, while there
is a zeroth order transition at Tc0, since at the critical temperature the grand potential has
a sudden jump from the condensed phase to the normal phase.
As we decrease the value of q to the region q < q < q, there is a little difference
compared to the previous case. As we lower the temperature, we will first see a first order
transition from the normal phase to the condensed phase at Tc1, and then a zeroth order
transition back to the normal phase at Tc0. A typical case with q = 1.719 is shown in
figure 19.
In the above two cases, as we lower the temperature, there exists a condensed phase
which is thermodynamically favored, although only in a narrow temperature range. The
00..13185
0.6
J1/2
x
0.4
0.405.1
0.209.13164
Figure 20. The grand potentials as a function of temperature in the case of m2 = 34 , q =
1.700(left) and q = 1.600(right). In both plots, the grand potential of the RN-AdS black hole
solution and hairy black hole solution are denoted by the green dashed line and blue solid line,
respectively.
story has a dramatic change as q is decreased past q. In this case, the hairy black hole
solution only appears in the high temperature region. In figure 20 we present the grand
potential as a function of temperature in the cases of q = 1.700 (left plot) and q = 1.600 for
3
m2 = 4 . One can clearly see that the grand potential of the hairy black hole solutions is
always larger than the one for the RN-AdS black hole at the same temperature. Thus these
hairy black hole solutions are thermodynamically disfavored. This is just the so called
retrograde condensation reported in ref. [28]. Let us notice that similar retrograde
condensation also happens in the soliton case in the sense that the hairy soliton solution
only appears below a critical chemical potential and is thermodynamically disfavored, as
shown in figure 13.
A phase diagram in terms of temperature and charge is shown in figure 21 in the case
with m2 = 3/4. There the red, blue and black curves represent second order, first order
and zeroth order phase transitions, respectively.
The complete phase diagram
The above studies tell us that, depending on two model parameters m2 and q, the
EinsteinMaxwell-complex vector field theory exhibits rich phase structures. From the perspective
of dual field theory, the system admits four phases including an insulating phase described
by the pure AdS soliton, a soliton superconducting phase described by the hairy soliton, a
conducting phase described by the RN-AdS black hole, and a black hole superconducting
phase described by the hairy black hole. For given m2 and q, the complete phase diagram
can be constructed in terms of temperature and chemical potential. At each point in T -
plane, one should find the phase which has the lowest grand potential. Since the dual
system in spatial directions is homogeneous and infinite, the total grand potential is
divergent. So we consider the grand potential density defined by = /V3 for both the
soliton and black hole cases. Remember that we fix the size of the compact directions
and z to be = .
As a warmup, let us first show some properties of the grand potential density (T, ).
We note that one may associate an arbitrary temperature to the Euclidean soliton for a
given chemical potential. So the grand potential density of the soliton case is independent
of temperature, i.e., soliton(T, ) = soliton( ). Therefore, the phase boundary between
insulator and soliton superconductor should be a line parallel to the T axis. According
to the scaling law (4.11), the grand potential density of black hole for different chemical
potential is given by BH(T, ) = BH(T, = 1) 4. Taking advantage of above properties,
one can easily find a first order phase transition between the AdS soliton and RN-AdS
black hole,9 and the phase boundary between the RN-AdS black hole and the AdS soliton
9This first order phase transition is the planar analogous of the Hawking-Page transition for black holes
in global AdS coordinates [36].
at zero temperature is at a chemical potential = 121/4 1.8612, while the boundary
at zero chemical potential locates at T = 1/. In addition, note that since the critical
temperature between the RN-AdS black hole and the hairy black hole is proportional to
, this phase boundary between them is a straight line passing through the original point
(T, ) = (0, 0).
Now we begin to construct the T - phase diagram. To find the thermodynamically
favored phase among four phases, we adopt a method similar to the one in ref. [24]. In the
first step, we scale all solutions to the ones with the same value of chemical potential by
using above scaling rules. Then we scan temperature to find the physical phase boundary
at this by comparing grand potential densities for all solutions. Repeating this procedure
with different values of , we can assemble the full T - phase diagram.
Remember that for different choice of model parameters m2 and q, the phase behaviors
can qualitatively change. More precisely, for regions m2 > mc21, 0 m2 mc21 and m2 < 0,
our system exhibits distinguished thermodynamics. It is a very hard work to construct the
T phase diagram for all values of m2. Instead, we consider some typical examples for
each parameter range. We hope that the phase diagram will not have a dramatic change
for other choice of m2 in each region. We draw some typical examples in figure 22.
In the region with m2 > mc21, let us consider m2 = 5/4 as an example. The phase
structure depends on the strength of back reaction. For weak back reaction with q > q2 1.5345,
the phase diagram looks like the top-left plot in figure 22. In this range of q, as we lower q,
the phase boundary between the insulator and soliton superconductor goes toward larger ,
and the slope of the line separating black hole superconductor from conductor is reduced.
This is consistent with the fact that the back reaction makes the phase transition more
difficult. Furthermore, the boundary between the two kinds of superconducting phases moves
to a lower temperature for smaller q. As q is lowered past q2, there will be two discontinuous
points in grand potential indicating zeroth order phase transition (see figure 3). Between
two discontinuous points, there is no soliton superconducting phase, thus it changes the
phase diagram a lot. As shown in the top-right plot of figure 22, the soliton superconducting
phase is broken into two parts and there is a particular range of chemical potential where
the thermodynamically favored phase is replaced by black hole superconductor for lower T
and conductor for higher T . When the value of q is decreased to the case that the transition
from the pure AdS soliton to the hairy soliton becomes zeroth order (see figure 4), for some
suitable values of charge in this region, the insulating phase and soliton superconducting
phase separate from each other, between them there are phases coming from the black hole
background. A typical example for this is shown in the middle-left plot in figure 22.
In the region of mc22 m2 mc21, we take m2 = 0 as an example. We find that in this
case the T - phase diagram is qualitatively similar to the one numerically constructed for
the holographic s-wave model [24] as well as the one schematically plotted in ref. [33] for the
SU(2) p-wave model. The four phases typically meet in two triple points. As we decrease
q, the two triple points gradually approach to each other, merge into one quadruple point
where four phases can coexist and then pass through each other. There exists a phase
boundary in which the black hole superconductor becomes an insulator via a first order
phase transition.
Phase diagram(m2=5/4, q=1.5)
Phase diagram(m2=3/4, q=1.15)
8x 103
0.35
0.3
0.25
T 0.2
0.15
0.1
0.05
0
0
Figure 22. The complete phase diagram for the model with different mass square and charge of
the vector field. In the figure, S=pure AdS soliton, BH=RN-AdS black hole, SC=hairy soliton,
BC=hairy black hole.
In contrast to above two cases, when m2 is negative, the hairy solution can exist only
in a certain chemical potential region for soliton background and temperature region for
black hole background. As one can see from figure 13 and figure 20 that the hairy solution
becomes thermodynamically irrelevant for sufficiently strong back reaction, even more in
the near zero temperature region. Due to this, the T - phase diagram has a dramatic
change compared to the cases with m2 0. Let us consider the case m2 = 3/4 as an
SC
BH
BH
example. Indeed, the structure of the phase diagram depends on the value of q we choose.
A typical diagram for small back reaction is drawn in the middle-right plot of figure 22.
One can see that the lower-right region of the phase diagram is replaced by the conducting
phase. As we lower q, the region in which the two superconducting phases dominate is
deduced. In the case with q = 1.150 shown in the bottom-left plot of figure 22, the soliton
superconducting phase shrinks to a smaller region, while the black hole superconducting
phase disappears. For sufficiently small q, both the black hole superconducting phase and
the soliton superconducting phase become thermodynamically disfavored compared to the
uncondensed phases. In this case, we are left with only the conducting phase and insulating
phase at hand. There is a first order phase transition between them shown in the
bottomright plot in figure 22, which is the well-known confining/deconfining phase transition in
gauge field theory [23].
Summary and discussions
In this paper we give a detailed study on the holographic p-wave superconductor model
based on the Einstein-Maxwell-complex vector field theory with a negative cosmological
constant. We work in five dimensional asymptotically AdS backgrounds with full back
reaction of matter field on the background geometries. In this model, we find four kinds
of bulk solutions given by the pure AdS soliton, RN-AdS black hole and their vector
hairy counterparts. According to the AdS/CFT dictionary, the hairy solution is dual to
a system with a non-zero vacuum expectation value of the charged vector operator which
breaks the U(1) symmetry and the spatial rotation symmetry spontaneously. The above
four solutions in the bulk correspond to an insulating phase, a conducting phase, a soliton
superconducting phase and a black hole superconducting phase, respectively.
In this model there exist two model parameters, i.e., the charge q and the mass square
m2 of the charged vector filed . The phase structure of the model heavily depends on
the two parameters. In the black hole background case, the phase behavior is qualitatively
same as the one in the (3+1) dimensional case studied in [28]. One can see second order
transition, first order transition, zeroth order transition as well as the retrograde
condensation. The latter two cases can only appear for the case m2 < mc2 with mc2 = 0 suggested
by numerical analysis.
For the soliton case, our system exhibits distinguished behavior depending on concrete
value of m2. Qualitatively, the parameter space for m2 is divided into three regions with
m2 mc21 = 0.218 0.001, 0 m2 mc21 and m2 < 0, respectively. In each region,
the phase behavior depends on the strength of back reaction q. There also exist second
order, first order and zero order phase transitions as well as the retrograde condensation.
An interesting observation is that when m2 < 0, the hairy black hole solution does not
exist as the temperature is lower than a certain value (see figure 18), which leads to the
zeroth phase transition at the critical point. We draw the phase diagrams (see figures 6, 10
and 14) in each region of mass parameter space. Compared to the black hole background
case, an additional complication appears in the soliton background case.
Recall that our one parameter family of hairy solutions are labeled by the value of
x at the tip of the soliton. For m2 0, as we lower q to a critical value, there exists a
region of chemical potential, in which the hairy soliton solution does not exist. As a result,
the curve of the condensate (or grand potential) parameterized by x(r0) breaks down,
which results in two discontinuous points (see figures 3, 4, 5, and 9). For the existence
of this discontinuous point and the hairy soliton solution at this point, we give a detailed
analysis in appendix A and appendix B. Some physical meanings of this discontinuous point
both at the gravity side and the dual field theory should be further studied. For the case
with m2 < 0, the hairy soliton solution terminates at a finite chemical potential for weak
back reaction, while for sufficiently large back reaction, it becomes thermodynamically
subdominant compared to the pure AdS soliton. See figures 11 and 13 for details.
With all four phases at hand, we construct some complete phase diagrams in terms
of chemical potential and temperature. As one can see, there are many types of phase
transitions in both soliton and black hole backgrounds, which depend on the values of
parameters m2 and q. The T - phase diagrams are expected to be much more complicated
than the ones for the holographic s-wave model and SU(2) p-wave model [24, 33]. Some
typical examples are shown in figure 22. It has been suggested that the
Einstein-Maxwellcomplex vector field model is a generalization of the SU(2) p-wave model in the sense that
the vector field has a general mass and gyromagnetic ratio . Comparing our complex
vector field model to the SU(2) model with a constant non-Abelian magnetic field, we find
that the SU(2) p-wave model can be recovered by the restriction m2 = 0 and = 1 at
least in our setup in ref. [30]. Similar statement holds for the black hole case. Indeed, for
the special case m2 = 0, our model exhibits very similar phase structures as the SU(2)
model [32, 35]. In particular, the complete phase diagrams in our model with m2 = 0 are
qualitatively same as those drawn schematically in ref. [33]. We have freedom to choose
other values of m2 in our model, thus it can be used to describe much rich phenomena
in the dual strong coupled systems. As one can see, depending on the values of m2 and
q, we have indeed many other types of phase diagrams which do not appear in the SU(2)
p-wave model [33] as well as in holographic s-wave model [24]. In addition, let us mention
that in the soliton background case, we have checked that there also exists a zeroth order
phase transition in the SU(2) p-wave model as the back reaction is strong enough, which
is missed in the previous studies.
Finally some remarks are in order. (1) There exist two critical masses of the vector
field in our model, mc21 = 0.218 0.001 and mc22 = 0, in the soliton background case,
a critical mass, mc2b = 0, in the black hole background case. Note that the latter also
appears in the (3+1) dimensional black hole case [28], there some possible implications of
the critical mass have been discussed and therefore we will not repeat here. While the
critical mass mc21 = 0.218 0.001 looks spacetime dimension dependent, its appearance is
interesting and is worthy to further investigate. Unfortunately at the moment we have no
any physical interpretation for this. (2) In both the soliton and black hole backgrounds, our
model exhibits zeroth order phase transition, which has a discontinuity of grand potential
(free energy) at the critical point. In ref. [38] the author argues that a zeroth order phase
transition could appear in the theory of superfluidity and superconductivity and presents
an exactly solvable model for such a phase transition. The authors of ref. [39] show that a
zeroth order phase transition exists in an exactly solvable pairing model for superconductors
with px + ipy-wave symmetry. (3) In the soliton background case, our model shows the
insulator/superconductor/insulator/superconductor phase transitions when the chemical
potential continuously increases. Indeed such a series of phase transitions is expected to
appear from theoretical perspective in some superconducting materials, for example, see
the phase diagram in the figure 10 of ref. [40]. (4) We can see from the complete phase
diagrams in terms of temperature and chemical potential that in some cases, more than
one superconducting phase appear in a phase diagram in our model. The phase diagram
for some realistic superconducting materials is usually complicated, and indeed, more than
one superconducting phases can occur, for example, see ref. [4146]. Definitely, it is of
great interest to see whether our model is relevant to these superconducting materials. (5)
It should be stressed here that although the zeroth order phase transition is argued to exist
theoretically in [38, 39], it seems not yet be observed experimentally. If the zeroth order
phase transition is not allowed in realistic systems, the existence of the zeroth order phase
transition might imply that either the gauge/gravity duality is invalid in some cases, or
we miss some solutions in our analysis, for example, the solutions without the translation
invariance and the solutions with more components of matter fields, which have not been
considered in the present setup. Anyway, certainly it is required to further study this issue.
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.
The regular boundary conditions for the equations of motion
In this appendix, some details for solving the coupled equations of motion (3.3) will be
given. In particular, we will show why in the soliton background case, the hairy soliton
solution does not exist in some region of chemical potential, shown in subsection 3.3.1. The
black hole case will be compared in the end of this appendix.
In order to find the solutions for all the six functions F = {g(r), f (r), h(r), (r), x(r),
(r)}, one must impose suitable boundary conditions at boundary r and the tip
r = r0. The asymptotic behaviors for these functions near the infinite boundary can be
easily obtained from the equations (3.3), which are shown in equations (3.4). The condition
of x = 0 leads the problem to be a boundary value problem. In this paper, the method to
solve them is shooting method, i.e, by finding some suitable initial values at the tip for each
function in F in order to give result of x = 0 near the boundary when we integrate the
equations (3.3) from the tip to the boundary. To solve the equations (3.3), one needs ten
initial conditions, i.e, {g(r0), f (r0), h(r0), (r0), x(r0), (r0), f (r0), h(r0), x(r0), (r0)}.
By the scaling symmetries in (3.6)(3.9), one can first set {r0 = 1, f (r0) = 1, h(r0) =
1, (r0) = 0} . The first order derivatives of f (r), h(r), x(r) and (r) at the tip can be
obtained by imposing the requirement that each term in equations (3.3) behaves regular
at the tip. By some calculations, we have
h(r0) = 2x00 = 22x0(m2 q220)/g(r0),
where x0 = x(r0) and 0 = (r0), which are two free parameters at hand. We can choose
fixing x0 using (r0) as the shooting parameter or vice versa to match the source free
condition, i.e., x = 0. After solving the coupled differential equations, we should use
the scaling symmetries (3.6)(3.9) again to satisfy the asymptotic conditions f () = 1,
h() = 1 and () = 0 and the fixed compacted length = of the coordinate . Then
we can obtain the condensate hJxi, chemical potential , charge density , and grand
potential by reading off the corresponding coefficients in (3.4), respectively.
The equation (A.3) should be considered carefully, as it gives a restriction on the
solution for equations (3.3). By definition, the tip r0 is the maximal zero point of g(r).
Therefore, g(r0) must be non negative. For the non-extremal case, we should have a
positive value of g(r0), which gives a restriction for the initial values of x0 and 0,
It is precisely this restriction so that we cannot get the hairy soliton solution within some
region of chemical potential in the condensed phase, which is shown in figures 3, 4 and 9,
respectively. In order to show this discontinuous phenomenon more visualizable, in figure 23
we show two kinds of orbits for the initial values x0 and 0 for the case with m2 = 5/4
and q = 1.900 and 1.520, respectively. In the figure, The blue solid line is the orbit for the
initial values x0 and 0, while the red dashed lines stands for the curve g(r0) = 0. The
region below the red line has g(r0) > 0.
From figure 23, one can see that the orbit of 0 and x0 behaves well and has no
any intersection with the dashed red line when q = 1.900. In this case, the solution
orbit leads to the continuous grand potential and condensate in the condensed phase, as
shown, for example, in figure 1. However, when q = 1.520 as shown in the right plot of
figure 23, the solution orbit is broken into two parts by the curve g(r0) = 0. It leads to the
discontinuity of the grand potential in the condensed phase, just as shown in figures 3, 4
and 9, respectovely. Between these two intersection points, there do not exist any hairy
soliton solutions and thus in that region we have to replace hairy soliton solutions by the
pure AdS soliton because the latter can be associated with any chemical potential.
For the black hole case, we can also construct similar boundary conditions at the black
hole horizon and AdS boundary. In this case, we can use the values of x(rh) and (rh)
as the shooting parameters. There is also a restriction at the horizon as
One can see this restriction only depends on (rh), which is very different from the
one (A.4) in the soliton case. The numerical results in the region where we have scanned
Figure 24. The asymptotic behavior of . From the figure, one can see that when g(r0) 0,
there is an asymptotic behavior = r1r0 . Here the values of g(r0) are chosen from 2 102 to
4 104.
show this restriction does not cause any discontinuity in the condensed phase in the black
hole case.
The extremal hairy soliton solution
In the main text we have assumed g(r0) > 0 in seeking for the solutions with the regular
condition at the tip. In this appendix we will give the extremal solution with g(r0) = 0. In
order to find the extremal solution, it is helpful to observe the behaviors of these functions
in F when g(r0) approaches to zero.
In figure 24 we plot the behavior of ln near the tip when g(r0) tends to zero. It
shows that when g(r0) goes to zero, the behavior of behaves as = r1r0 . Taking this
behavior into account, we propose the following expansion near the tip
g(r) = gj (r r0)1+j + ,
f = 1 + f10(r r0)1j + ,
h(r) = 1 + h10(r r0)1j + .
Here the scaling symmetry has been used to scale f (r) and h(r) to be unity at the tip.
Substituting these assumptions into the equations (3.3), we find only j = 12 can give a
self-consistent expansion with the coefficients
022x0q2 12 = 0, gj = 4 q48q220 24m2 4m2q220 72q2 + 3q440 + m4.
q0
In this case with j = 1/2, a remarkable fact is that although g(r0) = 0, the solution with
the expansion (B.1) still has a finite compacted length = 8/(gj r0) for the coordinate .
In addition, the solution looks singular at the tip because (r) is divergent there. In fact,
this is a coordinate singularity and the solution is regular at the tip. To see this, one can
introduce = (r r0)1/4, near the tip r = r0, the metric then has the following form
ds2 = 16 d2 + r02(dt2 + dx2 + dy2 + gj 2d2). (B.3)
gj
One can use the expansion (B.1) with the coefficients (B.2) to solve the equations (3.3)
with the source free condition by scanning the shooting parameter 0.10 In the case of
m2 = 5/4 and q = 1.520, we find the first two values of 0 are 1.302900 and 1.537516,
which are very closed to the corresponding values in figure 23, the latter are obtained by
setting g(r0) = 4 104.
The stress energy tensor
K = 21 (nb + nb). (C.2)
In our case, the boundary spacetime is flat, so the Einstein tensor for the boundary metric
vanishes. In what follows we discuss the stress energy tensor for the soliton and black hole
backgrounds separately.
10To avoid the divergence of , f , h, , at the tip in the numerical calculations, it is better to
change the coordinate from r to and make a suitable variable substitution on such as e/4.
The soliton case. By the metric ansatz (3.1) and the asymptotic expansion for the
metric in (3.4), the leading terms of the Brown-York tensor turn out to be
In the following, we omit the factor 22 for convenience. The nonvanishing components of
the stress energy tensor t for the dual field theory on the boundary are
22T tt = r6(4h4 + g4 44),
22T xx = r6(44 g4 4f4),
22T yy = r6(44 g4 4f4 4h4),
22T = r6(3g4 4h4 4f4).
T tt = r6(4c4 3a4),
T xx = r6(4b4 a4),
T yy = r6[4(b4 c4) a4],
T zz = r6[4(b4 c4) a4],
The trace of the boundary stress energy tensor is
ff 2gg hh + 23rxh2 3r2f 23qr25f2xgh2 + r8g 8r = 0.
It is clear from the above equation that the dual field theory is traceless as > 0. When
= 0, the trace looks nonvanishing with T r(t) = 22x+. But this is not true. The reason
is as follows. Note that the case = 0 is equivalent to the lower bound with m2 = 1.
As mentioned above, in this case, the expansion in (3.4) is no longer valid, instead some
logarithmic terms should appear in the asymptotic expansion. In fact, it can be shown
that with these additional terms the dual stress energy tensor is still traceless.
The black hole case. In this case, by the metric ansatz (4.1) and the asymptotic
form of the metric in (4.4), we have the nontrivial terms of the Brown-York tensor near
the boundary
where we have omitted the factor 22 for convenience. The stress energy tensor t for the
dual field theory on the boundary reads
with its trace
ttt = 4c4 3a4,
txx = 4b4 a4,
tyy = [4(b4 c4) a4],
tzz = [4(b4 c4) a4],
In this case, let us consider the equation of motion for the metric function b(r)
+ 8ar 4r = 0.
Substituting the expansion (4.4) into the above equation, and taking the limit r , we
have the following asymptotic form
4(c4 b4) =
As the soliton case, one can see that the stress energy tensor is also traceless in the black
hole background case.
Finally let us mention an interesting fact on the stress energy tensor of the dual field
theory. Naively thinking, one may expect that the stress energy tensor should be anisotropic
with txx 6= tyy in both hairy soliton and black hole cases, for which the metric functions
h(r) and c(r) are nontrivial and from the point of view of dual field theory, the x-component
of the vacuum expectation value of the vector operator is non zero, while the y-component
vanishes. But this anisotropy does not appear in the stress energy tensor. Our numerical
results suggest that h4 and c4 are zero,11 which leads to txx = tyy in both cases.
A similar phenomenon has been reported in the paper [48], the authors have given an
expression for the variation of the free energy of periodic AdS black holes with respect to
the periods of transverse coordinates, which gives an explanation for this phenomenon in
the black hole and soliton backgrounds.
The scaling relations between the critical chemical potential and
charge
In this appendix, we will give some discussions on the scaling relation (3.12), which holds
for the large q case. We will focus on the soliton case. The relation between c1 and q
can be obtained by semi-analytical method through the general Heun function, while the
11In the soliton case with m2 = 3/4, q = 2.000 , the numerical results show that f4, g4 and 4 are of
order one, but h4 is zero up to a numerical error 1012, while in the black hole case with the same model
parameters, the numerical results show that a4 is of order one, but c4 and b4 vanish up to a numerical
error 1012.
This equation has a general solution as
F (z) = C1HeunG
1, 41 q2 c21, +2 3 , +2 1 , + 1, 1; z2
1, 14 q2 c21, 3 , 1 , 1 , 1; z2 .
2 2
relation between c2 and q can only be obtained by numerical method. We first study the
former case.
In the case with large q, the probe limit is a good description near the phase transition
point. In that case, the chemical potential takes the critical value, while the vector field is
very small, whose influence on the background and electromagnetic field can be neglected.
This means that we can just treat the vector field as a perturbation and take the background
solution as
g(r) = 1 r14 , f (r) = h(r) = 1, (r) = 0, (r) = c1.
Then the equation for the vector reads
x(r) = F (z)z1+, = p1 + m2,
Here C1 and C2 are two constants and HeunG stands for the general Heun function with
the feature HeunG( , , , , , ; 0) = 1 . To satisfy the source free condition at the boundary, it
is easy to see that one has to take C2 = 0. On the other hand, in order to have a regularity
for the vector field x at the tip, there exists a constraint condition at z = 1 as
Substituting the solution (D.5) with C2 = 0 into (D.6), we can obtain a nonlinear equation
for c21q2. The equation can be easily solved by numerical method. The solution of course
depends on or m2. As a result we can write the solution as
This is just the first scaling relation in (3.12). For the cases with m2 = 5/4, 0 and 3/4,
we have c1q 2.784531829, 2.265193164 and 1.737772548, respectively. It is easy to see
that they are very close to corresponding ones obtained by using the shooting method
in section 3. This agreement also provides a piece of evidence to support the results we
obtained by shooting method to solve the equations (3.3).
0.5
ln(c2)
It seems impossible to get the second scaling relation in (3.12) by an analytical method
since it happens in the condensed phase with large condensate. In this case, the back
reaction of the vector field can not be neglected, and we have to solve the equations (3.3)
without any approximation. By numerical analysis, we find that when q is very large, the
critical chemical potential for the phase transition from the condensed phase to the normal
phase has a power law form as
A typical case with m2 = 3/4 is shown in the left plot of figure 25. The fitting of for
different m2 is shown in the right plot of figure 25. One can see that the relation between
and m2 is fitted well by a second order polynomial with a very small constant term. This
constant term in fact is less than our numerical error. Thus if neglect this constant term,
we can obtain the following fitting relation
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