Statical Structures of the BCS-like Holographic Superfluid in AdS4 Spacetime

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by  Shanquan Lan

Static Structures of the BCS-like Holographic Superfluid in AdS Spacetime
4
Shanquan Lan∗ and Wenbiao Liu†
Department of Physics, Beijing Normal University, Beijing, 100875, China
Yu Tian‡
School of Physics, University of Chinese Academy of Sciences, Beijing, 100049, China and
Shanghai Key Laboratory of High Temperature Superconductors, Shanghai, 200444, China
We investigate in detail the m2 = 0 Abelian Higgs model in AdS , which is considered as the
4
holographicdualofthemostBCS-likesuperfluid. Inhomogeneousandisotropicsuperfluidsolutions,
wecalculatethesoundspeeds,squareofwhichapproachesto1/2withincreasingchemicalpotential
(lowering temperature). Then we present the single dark soliton solutions, which becomes thinner
withincreasingchemicalpotential. Forthefirsttime,wealsofindtheinterestingdoubleandtriple
dark soliton solutions, which is unexpected and shows the possibility of more complicated static
7 configurations. Finally, we investigate vortex solutions. For winding number n = 1, the vortex
1 becomes thinner with increasing chemical potential. At a given chemical potential, with increasing
0 windingnumber,firstlythevortexbecomesbiggerandthechargedensitydepletionbecomeslarger,
2 then the charge density depletion settles down at a certain value and the growth of the vortex size
is found to obey a scaling symmetry.
n
a
J
I. INTRODUCTION AND MOTIVATION BEC superfluid to BCS superfluid. Thus our model cor-
1
responds to the BCS superfluid with the most loosely
1
InAdS/CFTcorrespondence, anAbelianHiggsmodel bounded cooper pairs of fermions. As we know the

] coupled with gravity in AdS [1] which exhibits sponta- BCS limit is currently not accessible to ultra-cold gas
h 4
neous symmetry breaking of U(1) symmetry has been experiments[18],soatheoreticalinvestigationofthis“ex-
t
- extensively studied. Firstly, this model was considered treme” BCS-like holographic superfluid is valuable.
p
e todescribeaholographicsuperconductor[2]. Inthiscase, Manycharacteristicsofsuperfluidareinteresting,such
h thespatialcomponentsoftheboundaryU(1)gaugefields asthe turbulencefeaturesthat have been investigated in
[ canbeturnedonwheredropletandvortexsolutionshave Refs.[12, 19–21] for different kinds of holographic super-

been discussed in Refs. [3–7]. Later on, interpretation of fluid. But before an investigation of this extreme BCS-
1
v this model as a holographic superfluid can be found in like holographic superfluid’s dynamic features, first we
1 Refs. [8–12]. In this case, the spatial components of shall know its static structures which is also interesting
2 the boundary U(1) gauge fields are turned off and yet and important by itself. Actually, some of the static fea-
9 consistentsolitonandvortexsolutionsarefoundinRefs. tures are already known. One can find the phase tran-
2 [13, 14]. All in all, the model with different boundary sition diagram in Ref.[22] and the single dark soliton,
0 conditions corresponds to superconductor or superfluid. vortex solutions in Ref.[17]. But a detailed and deeper
.
1 Inthispaper,wewillconsidertheholographicsuperfluid investigation are still necessary. In this paper, besides
0 case. thethreekindsofstaticstructuresmentionedabove(the
7 In the model, the complex scalar field is massive homogeneous superfluid, single dark soliton and vortex)
1
with a coupling constant m2. This parameter is re- constructed in the infalling Eddington Coordinates, we
:
v lated to the conformal dimension ∆ of condensate by also find some more complicated structures and investi-
Xi ∆± = 3/2±(cid:112)9/4+m2[15] in four dimensional space- gate their features.
time. Fromtheequation,onecanseem2 >−9/4whichis For the homogeneous and isotropic superfluid solu-
r
a known as the Breitenlohner-Freedman bound[16]. What tions, we present the sound velocity of the superfluid at
is more, if m2 > 0, we have ∆ < 0 and the scalar field different chemical potential. The square of sound speed
−
goestoinfinityattheboundaryifthesourceisturnedon approaches to 1/2 with increasing chemical potential.
which renders the model unstable. As a result, m2 ≤ 0 This is consistent with the argument in Ref.[23] where
and case m2 = 0 has a condensate with the maximum for the conformal fluids at zero temperature the square
conformal dimension ∆ = 3. In Refs.[13, 14, 17], after of sound speed is 1/2.
+
an investigation of this model for different m2 from var- For the single soliton solutions, we study the features
ious aspects, the authors suggest that the increasing of of their charge density and healing length varying with
conformal dimension corresponds to the crossover from respect to chemical potential. Then, for the first time,
wefindthemultiple(doubleandtriple)darksolitonsolu-
tions. Somewhat unexpectedly, those multiple dark soli-
tonsseemtobalanceatquitearbitrarypositions,atleast
∗ [email protected] to very high precision numerically. Nevertheless, the na-
† [email protected] ture of such multiple soliton solutions at finite tempera-
‡ [email protected] ture and their possible applications worth further inves-
2
tigation. here the overbar denotes complex conjugate. In what
For the winding number n = 1 vortex, we study the follows, we will take the units in which L=1,16πGq2 =
features of their charge density and healing length vary- 1, and z = 1. Working with the case of m2 = 0, the
h
ing with respect to chemical potential. Then we study asymptotic solution of A and Ψ near the AdS boundary
the vortices with different winding numbers at a certain can be expanded as
chemical potential. Interestingly, we find that there is a
scalingsymmetrywhenthewindingnumberisverylarge, A =a +b z+o(z),Ψ=ψ +ψz3+o(z). (6)
µ µ µ 0
which governs the growth of the vortex size with respect
to the winding number. Again, according to the holographic dictionary, the ex-
The paper is organized as follows. In the next section, pectation values of the boundary quantum field theory
we develop the m2 = 0 holographic superfluid model in operators jµ (currents) and O (condensate) can be ob-
the infalling Eddington Coordinates. Then we present tained explicitly as the variation of renormalized bulk
the homogeneous and isotropic superfluid solutions, the on-shell action with respect to the sources a and ψ re-
µ 0
darksolitonsolutionsandthevortexsolutionsinthenext spectively
three sections. In the end, we summarize what we have
found with some discussions. By the way, we have also (cid:104)jµ(cid:105)= δSonshell = lim√−gFzµ,
find the multiple soliton solutions for the m2 =−2 holo- δaµ z→0
graphic superfluid model and put it in the Appendix. δS
(cid:104)O(cid:105)= onshell =−3ψ. (7)
δψ
0
II. HOLOGRAPHIC SETUP
ChoosingtheaxialgaugeA =0tofixtheU(1)gauge
z
fields, the above equations of motion are rewritten as
AsarguedinRefs.[1,2],aholographicdualofatwodi-
mensionalsuperfluidisprovidedbyAbelianHiggsmodel 2(1−z∂ )∂ Ψ=(2+z3)∂ Ψ+2iA Ψ−iz∂ A Ψ
z t z t z t
in asymptotically AdS spacetime. The action is →− →− →− →−
4 −2izA ∂ Ψ−z(∂ −iA)·(∂ −iA)Ψ−zf∂2Ψ, (8)
t z z
1 (cid:90) √ 6 1
S = d4x −g(R+ + L ). (1)
16πG L2 q2 matter →− →−
M 0=z2(−∂2A +∂ ∂ · A)+i(Ψ∂ Ψ−Ψ∂ Ψ), (9)
z t z z z
Here G is the Newton’s constant, and the matter La-
grangian reads
2z2∂ ∂ A =z2(∂ ∂ A +∂2A −∂ ∂ A −3z2∂ A
z t x x z t y x x y y z x
1
Lmatter =−4FabFab−|DΨ|2−m2|Ψ|2, (2) +f∂z2Ax)−(iΨ∂xΨ−iΨ∂xΨ+2AxΨ2), (10)
where D =∇−iA with ∇ the covariant derivative com-
patible to the metric. 2z2∂ ∂ A =z2(∂ ∂ A +∂2A −∂ ∂ A −3z2∂ A
z t y y z t x y x y x z y
Inthispaper,wewillworkintheprobelimit(theback- +f∂2A )−(iΨ∂ Ψ−iΨ∂ Ψ+2A Ψ2), (11)
reaction of the matter fields onto gravity are ignored). z y y y y
According to the holographic dictionary, the tempera-
ture of a black hole in the above model corresponds to z2(∂ ∂ A +∂ ∂ A +∂ ∂ A )+i(Ψ∂ Ψ−Ψ∂ Ψ)
x t x y t y z t t t t
the temperature of the superfluid. Since we are consid- →− →−
=z2(∂2A +∂2A +f∂ ∂ · A)−2A Ψ2
eringthesuperfluidatfinitetemperature, sowetake the x t y t z t
Schwarzschildblackbranesolutionasourbackgroundge- +if(Ψ∂ Ψ−Ψ∂ Ψ). (12)
z z
ometry. IntheinfallingEddingtonCoordinates,themet-
ric is written as The first equation of motion will contribute two inde-
L2 pendent equations for the scalar field is complex. As a
ds2 = (−f(z)dt2−2dtdz+dx2+dy2), (3) result,therearesixequationsabove. Butwehavechosen
z2
A =0 gauge, so only five equations are independent.
z
wanhderzet=he0fathcteoArfd(Sz)bo=un1d−a(rzyzh.)A3 wsiatlhluzde=dzthotahbeohvoer,iztohne boAunsdaarnyoctoen,dwiteiocnosn,sider the situation with following
dualboundarysystemisplacedatHawkingtemperature,
which is given by ψ =0, a =µ. (13)
0 t
3
T = . (4) Here µ is chemical potential, source of particle density
4πz
h −b = ρ. When fixing the black hole temperature T
0
by rescaling, the only scale parameter is µ, which is in-
On top of this background geometry, the equations of
versely proportional to T. To solve the above equations,
motion for the matter fields can then be written as
we use the Chebyshev and the Fourier pseudo-spectral
D DaΨ−m2Ψ=0,∇ Fab =i(ΨDbΨ−ΨDbΨ), (5) methods[24].
a a
3
III. HOMOGENEOUS SUPERFLUID AND
SOUND SPEED 50 ρ
In this section, we reproduce the homogeneous and 40
isotropic superfluid solutions[22] in the infalling Edding-
ton Coordinates. Then we present the sound velocity 30
of the superfluid at different chemical potential. Same
20
as predicted in Ref.[23], the square of sound speed ap-
proachesto1/2withincreasingthechemicalpotentialin
10
our numerical results.
μ
Considering the homogeneous, isotropic and static
conditions, the matter fields only involve of Ψ(z) and 6 8 10 12 14 16 18 20
A (z). RewritethecomplexscalarfieldΨ(z)=φ(z)eiϕ(z)
t
0.4
(which is also done in the following sections), the inde-
3 <O>
pendent three equations of motion become
zf∂2φ−(2+z3)∂ φ−2zA φ∂ ϕ−zfφ(∂ ϕ)2 =0(1,4) 0.3 μ
z z t z z
At+f∂zϕ=0, (15) 0.2
z2∂2A +2φ2∂ ϕ=0. (16)
z t z 0.1
From the second equation, we find A (z = 1) = 0 and μ0
t
without loss of generality we set ϕ(z =1)=0. Together μ
with the boundary conditions in Section II, the above 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
equations of motion can be solved for different µ. The
numerical results show that, when µ is small, the solu-
tion is trivial φ = 0, while passing through the critical FIG. 1. The up panel shows the relation between chemi-
cal potential µ and charge density ρ, while the bottom panel
point µ = 7.56, φ begins to condense. This process de-
c
shows the corresponding condensation with respect to tem-
notes that when increasing µ (lowering the temperature
perature (inverse of the chemical potential µ). The critical
T), normal fluid condense to superfluid through a phase
chemicalpotentialisµ =7.56andthecriticalchargedensity
c
transition. This picture can be clearly seen in the phase
is ρ =7.56.
c
transition graph Fig.1.
With the above background solutions of Ψ(z) =
φ(z)eiϕ(z) = Ψr(z) + iΨi(z) and At(z), we are ready 0=−2ipΨ δΨ +2ipΨ δΨ +ipz2∂ δA
i r r i z t
to calculate the sound speed of the superfluid system by
+((2iwz2−3z4)∂ +z2f∂2−2(Ψ2+Ψ2))δA .(21)
thelinearresponsetheory[25,26]. Todothis,weassume z z r i x
the perturbation bulk fields take the following forms
The above linear perturbation equations are required to
δΨ =δΨ (z)e−iwt+ipx,δΨ =δΨ (z)e−iwt+ipx, have Dirichlet boundary conditions at the AdS bound-
r r i i
δA =δA (z)e−iwt+ipx,δA =δA (z)e−iwt+ipx.(17) ary and regular conditions at horizon. Then cast them
t t x x
intotheformL(w,p)v =0withv theperturbationfields
Thenfourindependentperturbationequations(theextra evaluated at the grid points by pseudo-spectral method.
fifth Eq.(9) is not considered here) can be written as The quasi-normal modes w are obtained by the con-
qnm
0=(2iw+p2z+(2+z3−2iwz)∂ −zf∂2)δΨ dition det[L(w,p)] = 0 (w is complex and its real part
z z r
is the quasi-normal mode). Thus the dispersion relation
+(z∂ A −2A +2zA ∂ )δΨ
z t t t z i can be obtained in Fig.2 and the sound speed υ can be
s
+(2z∂ Ψ −2Ψ +zΨ ∂ )δA
z i i i z t obtained by the fitting formula w = υ p. In Fig.3,
qnm s
−ipzΨ δA , (18) weshowthesquareofthesoundspeedwhichapproaches
i x
1/2 with respect to chemical potential. This maximum
0=(2At−z∂zAt−2zAt∂z)δΨr sound speed 1/2 feature is shared by conformal fluids at
+(2iw+p2z+(2+z3−2iwz)∂ −zf∂2)δΨ zero temperature[10, 23].
z z i
+(2Ψ −2z∂ Ψ −zΨ ∂ )δA
r z r r z t
+ipzΨ δA , (19)
r x IV. INHOMOGENEOUS STRUCTURES: DARK
SOLITON
0=(2iwΨ +2fΨ ∂ −2f∂ Ψ −4A Ψ )δΨ
i i z z i t r r
+(2f∂ Ψ −2iwΨ −2fΨ ∂ −4A Ψ )δΨ
z r r r z t i i Following the work in Refs.[11, 13] where dark soli-
+(iwz2∂z−p2z2At−2(Ψ2r+Ψ2i))δAt ton solutions are constructed in the m2 = −2 Abelian
+(ipz2f∂ −wpz2)δA , (20) Higgs holographic model, here we reproduce the single
z x
4
2∂ ϕφ2+z2(6z∂ ϕ+6z2∂2ϕ
0.14 z z z
wqnm −f∂3ϕ−∂ ∂2ϕ)=0. (25)
0.12 z z x
Ideally, the size of x direction should be infinite. But
0.10
in numerical method, a cut off is needed. We will con-
0.08 -0.00013+0.6077p sider the system in a box of size 1 × 2L in the z and
x directions respectively. The boundary conditions are
0.06
introduced as follows. At z = 0, φ(z = 0) = 0 (the con-
0.04 densate source is turned off), ∂ ϕ(z = 0) = −µ (setting
z
the chemical potential which is introduced as Eq.(23)).
0.02
p At z = 1, we put regular boundary condition for φ, and
ϕ(z =1)=0(whichisintroducedasEq.(23)bythesame
0.00 0.05 0.10 0.15 0.20
argument as that in the above section). At x=−L and
x=L, we will introduce Neumann boundary conditions
FIG. 2. The dispersion relation for the Goldstone mode in for both φ and ϕ fields. With the equations of motion
superfluid phase at µ=10. and boundary conditions set up, we can use relaxation
methods ( Newton iteration method, or Gauss-Seidel it-
0.5 eration method) to find solutions. First, we choose seed
vs2 configurations of φ(x,z) and ϕ(x,z). Then relax these
configurations towards the solutions.
0.4
0.3
IV.1. single dark soliton solutions
0.2
In this subsection, we study in detail the single dark
soliton solutions for different chemical potential. The
0.1
seedconfigurationsarechosenasφ(x,z)=atanh(x)and
μ ϕ(x,z) = b, where a,b are constants. Then relax these
configurations towards the solutions.
8 10 12 14 16 18 20
For chemical potential µ = 8, the numerical results
of the bulk fields configurations are shown in Fig.4.
FIG. 3. The sound speed square υ2 varies with respect to Condensate and charge density as functions of x are
s
chemical potential µ. υ2 approaches 1/2 with increasing µ shown in Fig.5. The condensate is well fitted by func-
s
(lowering temperature). tion
(cid:104)O(cid:105)(x)=15.0tanh(x/1.25), (26)
dark soliton solutions for m2 =0 case[17] in the infalling
Eddington Coordinates. Then we study the charge den- where 15.0 is condensate for homogeneous and isotropic
sity and healing length of single dark soliton solutions, solution with the same chemical potential and 1.25
and present multiple (double and triple) dark soliton so- should be thought of as healing length of the soliton.
lutions. The charge density is well fitted by function
Single dark soliton solutions and multiple dark soliton
solutions with several single dark soliton paralleling to ρ=−0.45sech2(x/1.404)+8.51, (27)
each other have one dimensional translation invariance
symmetry. Thus we can assume the fields are indepen- where 8.51 is charge density for homogeneous and
dent of y coordinate, and A = 0. As we are looking isotropicsolutionwiththesamechemicalpotential,1.404
y
for static solutions, all the fields are independent of t co- should also be thought of as healing length of the soli-
ordinate and the currents in the gravity system should ton and 0.45 is the charge density depletion. Thus
be zero. As a result, rewrite the complex scalar field there are two healing lengths for a holographic soliton.
Ψ(x,z)=φ(x,z)eiϕ(x,z), the currents in Eq.(5) will con- Compare the results with the soliton solutions of the
tribute two constraint equations Gross-Pitaevskii equation (GPE)[13] in condensed mat-
ter physics, it is also found that the forms of the fitting
A (x,z)−∂ ϕ(x,z)=0, (22)
x x functions are the same, but the charge density in the
soliton core is zero and there is only one soliton healing
At(x,z)+f(z)∂zϕ(x,z)=0. (23) length for the GPE case.
For different chemical potential, the parameters which
By using these two constraint equations, the equations
characterize soliton are listed in Table.I. As we increase
of motion in Section II will be dramatically simplified as
the chemical potential (lower the temperature), (1)the
zf(∂ ϕ)2φ−(2+z3)∂ φ+zf∂2φ+z∂2φ=0, (24) condensate increases dramatically, (2)the depletion of
z z z x
5
15
<O>
10 x
15.0Tanh 
1.25
5
x
-10 -5 5 10
-5
-10
-15
8.5 ρ
8.4
x 2
8.3 -0.45 Sech  +8.51
1.404
8.2
8.1
x
-10 -5 0 5 10
FIG.5. Condensateandchargedensityofsolitonasfunctions
ofxatµ=8. Thesolidlinesarefittinggraphesofrespective
functions.
µ (cid:104)O(cid:105) ρ δρ δρ/ρ (cid:15) (cid:15)
max max max c ρ
8 15.0 8.51 0.45 0.053 1.250 1.404
9 34.2 10.9 1.43 0.131 0.630 0.806
10 54.5 13.6 2.27 0.167 0.456 0.638
11 77.9 16.5 3.17 0.192 0.367 0.541
12 105.4 19.7 4.07 0.207 0.313 0.475
13 137.5 23.1 5.06 0.219 0.272 0.428
14 174.7 26.9 6.06 0.225 0.244 0.388
15 217.3 30.9 7.07 0.229 0.227 0.361
16 266.0 35.2 8.16 0.232 0.211 0.333
17 321.2 39.7 9.35 0.236 0.194 0.310
18 383.2 44.5 10.64 0.239 0.179 0.291
19 452.6 49.6 12.0 0.242 0.165 0.274
20 530.0 55.0 13.5 0.246 0.152 0.260
TABLEI.Therelationbetweenchemicalpotentialµandmax
condensate(cid:104)O(cid:105) ,maxchargedensityρ ,chargedensity
max max
depletionδρ,healinglength(cid:15) forcondensate,healinglength
c
(cid:15) for charge density.
ρ
charge density increases and the two healing lengths
(width of the soliton) decreases, and as a result, the soli-
tonsbecomethinnerandsmallerbutdeeper,(3)δρ/ρ
max
FIG. 4. The bulk configurations of the fields increasesfastnearthecriticalpointbutslowsdownlater,
φ(x,z),ϕ(x,z),A (x,z),A (x,z) from up to down re- for all µ, this ratio is small which means the depletion of
t x
spectively for µ=8. soliton is shallow in charge density and it is not likely to
increaseto1(thechargedensityatthesolitoncoreisnot
likely to be 0).
6
IV.2. multiple dark soliton solutions
ItisknownthattheGross-Pitaevskiiequationalsohas
multipledarksolitonsolutions,butsuchsolutionscannot
be static in general, due to the force of interaction be-
tween these solitons (see, e.g. Ref.[27]). Since the dark
soliton is a localized structure, static configurations of
multiple dark solitons with large enough distance be-
tween them act as important approximate solutions of
the theory, which are in principle unstable but can have
arbitrarily long lifetime. Therefore, it is interesting to
explore similar solutions at finite temperature by holog-
raphy,approximateorexact. However,asfarasweknow,
there is no such exploration up to now.
Inthissubsection,weinitiatesuchexplorationbyfind-
ingthedoubleandtripleparalleldarksolitonsolutionsin
the holographic superfluids. Interestingly, it seems that
these solutions are exact, at least in the conditions that
we consider. Actually, these solutions can be found both
in the m2 =−2 case and the m2 =0 case (and probably
in other cases as well), but we will focus on the m2 = 0
case here and leave the m2 =−2 case in the Appendix.
For the double parallel dark soliton solution, the seed
configurationsarechosenasφ(x,z)=a[tanh(x+b)−1−
tanh(x−b)] and ϕ(x,z) = c, where a,b,c are constants.
In this situation, the fields can have periodic symmetry
in the x direction(We have checked that a solution also
exists for non-periodic boundary conditions in the x di-
rection. See also the following triple dark soliton case.).
So we can choose the Fourier pseudo-spectral method as FIG. 6. The bulk configurations of the double soliton fields
theChebyshevcollocationpointsaresparsearoundx=0 φ(x,z),ϕ(x,z) from up to down respectively for µ=9.
in the x direction, and still use the Chebyshev pseudo-
spectral method in the z direction. Setting a=6, b=4,
c = 1 for chemical potential µ = 9, we obtain the so-
lution of φ(x,z) and ϕ(x,z) as shown in Fig.6 and the 30
<O>
condensateandchargedensityinFig.7. Whenbisbig,in
20
a wide range of a and c, there is always a double soliton
10
solution. But when b is small enough, e.g. b = 0.5, the x
resultreducestoahomogeneousandisotropicsuperfluid -10 -5 5 10
solution. -10
For the triple parallel dark soliton solution, the seed -20
configurations are chosen as φ(x,z) = a[tanh(x+b)−
-30
tanh(x)+tanh(x−b)] and ϕ(x,z) = c, where a,b,c are
constants. Inthiscase,thexdirectioncannotbeperiodic
and we then use the Chebyshev pseudo-spectral method 10.8 ρ
inboththexandzdirections. Settinga=6,b=6,c=1 10.6
for chemical potential µ = 9, we obtain the solution of
10.4
φ(x,z)and ϕ(x,z) asshown inFig.8 andthe condensate
10.2
and charge density in Fig.9. When b is big, in a wide
range of a and c, there is always a triple soliton solution. 10.0
Butwhenbissmallenough,theresultreducestoadouble
9.8
soliton solution.
9.6 x
We have also investigated the convergence of our nu-
-10 -5 5 10
merical solutions of multiple dark solitons, as shown in
Fig.10. It can be seen that these solutions satisfy the
equations of motion at a very high precision, but it is FIG. 7. Condensate and charge density of the double soliton
impossible at the numerical level to determine whether as a function of x at µ=9.
a solution is exact or not. Nevertheless, the multiple
7
40 50 60 70 80 90
m
-1
-2
-3
-4
-5
Log10[Q(m+8)-Q(m)]
-6
FIG. 10. m is collocation points in x direction and Q(m)
is the total charge calculated for m collocation points(see,
Fig.9). Quantity|Q(m+8)−Q(8)|hasalogarithmdecrease
with respect to m which denotes a very high precision of the
numerical results.
darksolitonsolutionsconstructedaboveareatleastvery
wellapproximatesolutionsoftheholographicsuperfluids,
which may play an important role in superfluid physics
at finite temperature.
For the periodic (double dark soliton) case, the solu-
tion can actually be viewed as an infinite sequence of
soliton-anti-soliton pairs. It is easy to see by symme-
try arguments that there should be an exact solution if
the (anti-)solitons are equally spaced, since the forces of
interaction on an (anti-)soliton are balanced, similar to
the GPE case. But there is no obvious reason for the
existence of an exact solution in the unequally spaced
case or under non-periodic boundary conditions. So it
remainsaninterestingopenquestiontoseethenatureof
FIG. 8. The bulk configurations of the triple soliton fields
the multiple dark soliton solutions we have constructed
φ(x,z),ϕ(x,z) from up to down respectively for µ=9.
numerically. To end this section, we would like to point
out that it is reasonable to conjecture that one can con-
structmultiplesolitonsolutionswitharbitrarynumberof
30 <O>
singlesolitonandwithdifferentdistancesbetweenthem.
20
10
x
V. INHOMOGENEOUS STRUCTURES:
-10 -5 5 10
VORTEX
-10
-20
The holographic superconductor vortex solutions (the
-30 sourcesfortheflowsareturnedon)arestudiedinRefs.[3–
7] for different m2 models in the Schwarzschild coordi-
10.8 ρ nate. The holographic superfluid vortex solutions (the
sources for the flows are turned off) are studied in
10.6
Refs.[14,17]fordifferentm2 modelsintheSchwarzschild
10.4 coordinate. Here we reproduce the superfluid vortex so-
10.2 lutions for m2 = 0 model in the infalling Eddington co-
ordinates. Thenwestudythechargedensityandhealing
10.0
lengthofvortexsolutionswithdifferentwindingnumbers
9.8
at different chemical potential.
9.6 x Single vortex solutions have cylindrical symmetry.
-10 -5 0 5 10 When we rewrite the metric as
1
ds2 = (−f(z)dt2−2dtdz+dr2+r2dθ2), (28)
FIG. 9. Condensate and charge density of the triple soliton z2
as a function of x at µ=9.
all the gauge invariant fields will be independent of θ
8
coordinate. Thenwecanrewritethecomplexscalarfield
asΨ(r,θ,z)=φ(r,z)eiϕ˜(r,θ,z) whereφisgaugeinvariant.
Whatismore,thecurrentsinEq.(5)arealsoindependent
of θ. The currents in r and z directions should be zero.
Thus we have three constraint equations as
A (r,θ,z)−∂ ϕ˜(r,θ,z)=0, (29)
r r
A (r,θ,z)+f(z)∂ ϕ˜(r,θ,z)=0, (30)
t z
∂ (A (r,θ,z)−∂ ϕ˜(r,θ,z))=0. (31)
θ θ θ
In order to have a single valued function Ψ, its phase
term has to be ϕ˜(r,θ,z) = nθ+ϕ(r,z). Constant n is
an integer which denotes the quantized winding number
of vortex. As a result, the three constraint equations
become
A (r,z)−∂ ϕ(r,z)=0, (32)
r r
A (r,z)+f(z)∂ ϕ(r,z)=0, (33)
t z
∂ A (r,θ,z)=0. (34)
θ θ
Byusingtheseconstraintequations,theequationsofmo-
tion will be dramatically simplified,
r2zf∂2φ+r2z∂2φ−(3−f)r2∂ φ+rz∂ φ
z r z r
+r2zfφ(∂ ϕ)2−zφ(A −n)2 =0, (35)
z θ
z2(∂ ∂ ϕ+r∂2∂ ϕ−6rz∂ ϕ−6rz2∂2ϕ
r z r z z z
+rf∂3ϕ)−2rφ2∂ ϕ=0, (36)
z z
rz2(∂2A +f∂2A −3z2∂ A )−z2∂ A
r θ z θ z θ r θ
−2r(A −n)φ2 =0. (37)
θ
Now all the above field functions are independent of
θ. So we only need to consider the system in a box of
size 1×L in the z and r directions respectively. The
boundaryconditionsareintroducedasfollows. Atz =0,
φ(z = 0) = A = 0 (sources turned off) and ∂ ϕ(z =
θ z
0) = −µ (choose a chemical potential). At z = 1, we
put regular boundary conditions for φ and A functions,
θ
and ϕ(z = 1) = 0 (introduced as Eq.(33) with the same
argument as before). At r =0, we put regular boundary
conditions for ϕ and A functions, and φ(r =0)=0(the
θ
condensate at vortex core is zero). At r = L, we will
introduceNeumannboundaryconditionsforallthefields
φ,ϕandA . Withtheequationsofmotionandboundary
θ
conditions set up, we can use relaxation methods to find
solutions. First, we choose seed configurations φ(r,z) =
a tanh(br), ϕ(r,z) = c and A (r,z) = d where a,b,c,d
θ FIG. 11. The bulk configurations of the fields
are constants. Then relax these configurations towards
φ(r,z),ϕ(r,z),A (r,z),A (r,z),A (r,z) from up to down
the solutions. t r θ
respectively for µ=9.
For vortex with winding number n = 1 and chemi-
cal potential µ = 9, the numerical results of the bulk
9
35 <O> µ (cid:104)O(cid:105)max ρmax δρ δρ/ρmax (cid:15)c (cid:15)ρ
30 8 15.0 8.51 0.41 0.048 1.515 1.704
r
25 34.2tanh 9 34.2 10.9 1.18 0.108 0.782 0.968
0.782
10 54.5 13.6 1.84 0.136 0.591 0.762
20
11 77.9 16.5 2.49 0.151 0.493 0.649
15
12 105.4 19.7 3.17 0.161 0.431 0.573
10
13 137.5 23.1 3.87 0.167 0.386 0.517
5
r 14 174.7 26.9 4.61 0.172 0.350 0.473
2 4 6 8 10 15 217.3 30.9 5.37 0.174 0.321 0.436
ρ 16 266.0 35.1 6.15 0.175 0.296 0.405
10.8
17 321.2 39.7 6.95 0.175 0.274 0.378
10.6 r
-1.18sech2 +10.9 18 383.2 44.5 7.78 0.175 0.255 0.354
0.968
10.4 19 452.6 49.6 8.63 0.174 0.239 0.333
10.2 20 530.0 54.9 9.53 0.174 0.225 0.315
10.0
TABLE II. The relation between chemical potential µ and
9.8 r
max condensate (cid:104)O(cid:105) , max charge density ρ , charge
max max
2 4 6 8 10 densitydepletionδρ,healinglength(cid:15) forcondensate,healing
c
length (cid:15) for charge density at winding number n=1.
ρ
FIG.12. Condensateandchargedensityofvortexasfunctions
ofr atµ=9. Thesolidlinesarefittinggraphesofrespective
V.1. vortex with winding number n=1 at different
functions.
chemical potential
For vortices with winding number n = 1 at different
fields configurations are showed in Fig.11. Condensate
chemicalpotential,theparameterswhichcharacterisethe
andchargedensityasfunctionsofrareshowedinFig.12.
vorticesarelistedinTable.II.Asweincreasethechemical
The condensate is well fitted by function
potential(lower the temperature), (1) the condensate in-
creases dramatically, (2) the depletion of charge density
r
(cid:104)O(cid:105)(r)=34.2tanh( ), (38) increases and the two healing lengths (width of the vor-
0.782 tices) decreases, as a result, the vortices become thinner
and smaller, (3) δρ/ρ increases fast near the critical
max
where34.2isexactlythecondensateofhomogeneousand point but slows down later, this ratio is small for all µ
isotropic solution with the same chemical potential and which means the depletion of vortices is shallow in the
0.782 should be thought of as healing length of the vor- charge density and it is not likely to increase to 1 (the
tices. The charge density is well fitted by function charge density at the vortex core is not likely to be 0).
Actually, it seems to set down at 0.175.
x
ρ(r)=−1.18sech2( )+10.9, (39)
0.968
V.2. vortex with different winding number at
certain chemical potential
where 10.9 is exactly the charge density of homogeneous
and isotropic solution with the same chemical potential,
0.968 should also be thought of as healing length of the Forvorticeswith different windingnumber n at chem-
vortex and 1.18 is the charge density depletion. Thus ical potential µ = 10, the condensates are well fitted by
therearetwohealinglengths(oneforcondensateandthe function
other for charge density) for holographic vortex. From r
(cid:104)O(cid:105)(r)=(cid:104)O(cid:105) tanhn( ), (40)
Table.I and Table.II, comparing the results with that max (cid:15)
c
of the holographic soliton solutions, holographic vortices
have the same forms of fitting functions and two heal- where (cid:104)O(cid:105)max is the max condensate of its profile and
ing lengths. But the healing lengths are bigger and the (cid:15)c should be thought of as healing length of the vortices.
charge density depletion is smaller at the same chemical The charge density is well fitted by function
potential. Comparing the results with that of the vortex r
ρ(r)=δρ tanhn+1( )+ρ , (41)
solutions of the GPE, the difference are that the charge (cid:15) 0
ρ
density in the vortex core is zero and there are two heal-
ing lengthes (but one for the vortex core and the other where ρ = ρ +δρ is the max charge density of its
max 0
for the vortex tail) for the GPE solutions. profileand(cid:15) shouldalsobethoughtofashealinglength
ρ
10
2.5 <O>
50
2.0
40
ϵρ
1.5 30
ϵc
1.0 20
10
0.5 r
n
2 4 6 8 10
2 4 6 8 10 12 13.5 ρ
13.0
12.5
FIG.13. Therelationbetweenwindingnumbernandhealing
length (cid:15) for condensate, healing length (cid:15) for charge density 12.0
c ρ
whichareobtainedbyEqs.(40)and(41)atchemicalpotential 11.5
µ=10.
11.0
10.5
0.26 r
δρ/ρmax 10.0
0.24 2 4 6 8 10
0.22
FIG. 15. Condensate and charge density of vortex as a func-
0.20
tion of r at µ=10.
0.18
0.16 equations of motion become
n
0.14
r˜2(zf∂2φ−(3−f)∂ φ+zfφ(∂ ϕ)2)
2 4 6 8 10 12 z z z
−zφ(A˜ −1)2 =0, (43)
θ
FIG.14. Therelationbetweenwindingnumbernandthera-
z2(f∂3ϕ−6z∂ ϕ−6z2∂2ϕ)−2φ2∂ ϕ=0, (44)
tioofchargedensitydepletiontomaxchargedensityδρ/ρmax z z z z
at chemical potential µ=10.
z2(f∂2A˜ −3z2∂ A˜ )−2(A˜ −1)φ2 =0. (45)
z θ z θ θ
of the vortices and δρ is the charge density depletion. It Surprisingly, these equations are independent of wind-
is easy to check that when n = 1, the fitting formulas ing number n. As a result, the system has a scaling
for charge density (Eq.(39) and Eq.(41)) are equivalent. symmetry Eq.(42) for large n. The scaling symmetry
The two healing lengthes and the ratio of charge density includes linear change of r coordinate and leaves z co-
depletiontomaxchargedensityareplottedinFig.13and ordinate unchanged with respect to n, so the two heal-
inFig.14. ThesetwoEqs.(40)and(41)arewellfittingat ing lengthes which are in r direction increase linearly
least for n≤12. andthechargedensitydepletionwhichisindependentof
As we increase the winding number, (1) the two heal- r coordinate maintains the same value with respect to
inglengthesincreaselinearly,(2)theratioofchargeden- large n. The condensate and charge density profiles of
sitydepletiontomaxchargedensityδρ/ρ (orsaythe the numerical vortex solution of the above equations at
max
charge density depletion δρ as the max charge density is chemical potential µ = 10 are shown in Fig.15. The ra-
that of homogeneous and isotropic superfluid) increases tio of charge density depletion to max charge density is
fast at small winding number but sets down at a certain δρ/ρ =3.6/13.6=0.265. Comparingtheresultwith
max
value(around0.258−0.259)atlargewindingnumber. As themaxchargedensitydepletionratioinFig.14,onecan
a result, the vortices will become fatter (bigger, but not see that they are consistent.
deeper) if we continue to increase the winding number.
Tounderstandthisphenomenon, wemakesuchascal-
ing VI. SUMMARY AND DISCUSSION
r =nr˜, For the m2 = 0 Abelian Higgs Model in AdS , in
4
A =nA˜ , (42) theinfallingEddingtoncoordinates,wehavenumerically
θ θ
solved the equations of motion and investigated in detail
and plug them into Eqs.(35-37). Setting n → ∞, the threetypesofstaticholographicsuperfluidsolutions: the