Analytical study on holographic superfluid in AdS soliton background

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by  Chuyu Lai

Analytical study on holographic superfluid in AdS soliton background
Chuyu Lai1, Qiyuan Pan1,2∗, Jiliang Jing1† and Yongjiu Wang1‡
1 Department of Physics, Key Laboratory of Low Dimensional
Quantum Structures and Quantum Control of Ministry of Education,
and Synergetic Innovation Center for Quantum Effects and Applications,
Hunan Normal University, Changsha, Hunan 410081, China and
6 2 Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo 05315-970, Brazil
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2 Abstract

r
p
A Weanalytically studytheholographic superfluidphasetransition in theAdSsoliton background
by using the variational method for the Sturm-Liouville eigenvalue problem. By investigating the
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holographic s-wave and p-wave superfluid models in the probe limit, we observe that the spatial
]
h component of the gauge field will hinder the phase transition. Moreover, we note that, different
t
-
p fromtheAdSblackholespacetime,intheAdSsolitonbackgroundtheholographicsuperfluidphase
e
transition always belongs to the second order and the critical exponent of the system takes the
h
[ mean-field valuein both s-waveand p-wavemodels. Ouranalytical results are found to bein good

2 agreement with thenumerical findings.
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PACSnumbers: 11.25.Tq,04.70.Bw,74.20.-z
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∗ [email protected]
† [email protected]
‡ [email protected]
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I. INTRODUCTION
As we know, the phenomenology of conventional superconductors is extremely well explained by Bardeen-
Cooper-Schrieffer (BCS) theory [1] and its extensions [2]. However, these theories fail to describe the core
mechanismgoverningthe high-temperaturesuperconductorsystems whichis one ofthe unsolvedmysteriesin
moderncondensedmatterphysics. Interestingly,theanti-deSitter/conformalfieldtheories(AdS/CFT)corre-
spondence [3–5], which can map strongly coupled non-gravitationalphysics to a weakly coupled perturbative
gravitational problem, might provide some meaningful theoretical insights to understand the physics of high
T superconductors from the gravitationaldual [6–9]. The main idea is that the spontaneous U(1) symmetry
c
breaking by bulk black holes can be used to construct gravitational duals of the transition from normal state
to superconducting state in the boundary theory, which exhibits the behavior of the superconductor [10, 11].
Inadditionaltothe bulkAdSblackholespacetime,itwasfoundthataholographicmodelcanbeconstructed
in the bulk AdS soliton backgroundto describe the insulator and superconductor phase transition [12].
In general, the studies on the gravitational dual models of the superconductorlike transition focus on the
vanishingspatialcomponentsoftheU(1)gaugefieldontheAdSboundary. Consideringthatthesupercurrent
insuperconducting materialsis awellstudied phenomenonin condensedmatter systems,the authorsofRefs.
[13, 14] constructed a holographic superfluid solution by performing a deformation of the superconducting
black hole, i.e., turning on a spatialcomponent of the gauge field that only depends on the radialcoordinate.
Itwasfoundthatthesecond-ordersuperfluidphasetransitioncanchangetothefirstorderwhenthevelocityof
thesuperfluidcomponentincreasesrelativetothenormalcomponent. Interestingly,theholographicsuperfluid
phasetransitionremainssecondorderforallallowedfractionsofsuperfluiddensityinthestrongly-backreacted
regime at low charge q [15]. However, in the case of the fixed supercurrent, the superfluid phase transition is
always of the first order for any nonzero supercurrent [16–18]. In Ref. [19], the effect of the scalar field mass
on the superfluid phase transition was investigated and it was observed that the Cave of Winds exists for
some special mass in the superfluid model. In order to explore the effect of the vector field on the superfluid
phasetransition,aholographicp-wavesuperfluidmodelinthe AdSblackholescoupledtoaMaxwellcomplex
vector field was introduced [20, 21] and it was revealed that the translating superfluid velocity from second
ordertofirstorderincreaseswiththeincreaseofthemasssquaredofthevectorfield. Ontheotherhand,from
the perspective ofthe QNManalysis,the questionofstabilityofholographicsuperfluids withfinite superfluid
3
velocitywasrevisitedanditwassuggestedthattheremightexistaspatiallymodulatedphaseslightlybeyond
the critical temperature [22, 23].
TheaforementionedworksontheholographicsuperfluidmodelsconcentratedontheAdSblackholeconfig-
uration. More recently, the authors of Refs. [24, 25] extended the investigation to the soliton spacetime and
investigatednumericallytheholographics-wavesuperfluidmodelintheAdSsolitonbackground. Itwasfound
that, in the probe limit, the first-orderphase transition cannot be broughtby introducing the spatial compo-
nentofthevectorpotentialofthegaugefieldinthe AdSsolitonbackground,whichisdifferentfromthe black
holespacetime [25]. Inordertobackupnumericalresultsandfurther revealthe propertiesofthe holographic
superfluid model in the probe limit, in this work we will use the analytical Sturm-Liouville (S-L) method,
whichwasfirstproposedin[26,27]andlatergeneralizedtostudyholographicinsulator/superconductorphase
transitionin [28], to analyticallyinvestigatethe holographics-wavesuperfluid modelinthe AdS solitonback-
ground. Considering that the increasing interest in study of the Maxwell complex vector field model [29–40],
wewillalsoextendtheinvestigationtotheholographicp-wavesuperfluidmodelintheAdSsolitonbackground,
which has not been constructed as far as we know. Besides to be used to check numerical computation, the
analyticalstudycanclearlydisclosesomegeneralfeaturesfortheeffectsofthespatialcomponentofthegauge
field on the holographic superfluid model in the AdS soliton background.
The structure of this work is as follows. In Sec. II we will investigate the holographic s-wave superfluid
modelin the AdS solitonbackground. In particular,we calculatethe criticalchemicalpotentialofthe system
as well as the relations of condensed values of operators and the charge density with respect to (µ µ ). In
c
−
Sec. III we extend the discussion to the p-wave case which has not been constructed as far as we know. We
will conclude in the last section with our main results.
II. HOLOGRAPHIC S-WAVE SUPERFLUID MODEL
We start with the five-dimensional Schwarzschild-AdS soliton in the form
dr2
ds2 = r2dt2+ +f(r)dϕ2+r2(dx2+dy2), (1)
− f(r)
wheref(r)=r2(1 r4/r4)withthetipofthe solitonr whichisaconicalsingularityinthissolution. Wecan
− s s
remove the singularity by imposing a period β =π/r for the coordinate ϕ. As a matter of fact, this soliton
s
can be obtained from a five-dimensional AdS Schwarzschild black hole by making use of two Wick rotations.
In order to construct the holographic s-wave model of superfluidity in the AdS soliton background, we
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consider a Maxwell field and a charged complex scalar field coupled via the action
1
S = d5x√ g F Fµν ψ iqA ψ 2 m2 ψ 2 , (2)
µν µ µ
− −4 −|∇ − | − | |
Z (cid:18) (cid:19)
where q and m represent the charge and mass of the scalar field ψ respectively. Taking the ansatz of the
matter fields as
ψ =ψ(r), A dxµ =A (r)dt+A (r)dϕ, (3)
µ t ϕ
where both a time component A and a spatial component A of the vector potential have been introduced
t ϕ
in order to consider the possibility of DC supercurrent,we can get the equations of motionin the probe limit
3 f′ 1 q2A2 q2A2
ψ′′+ + ψ′ m2+ ϕ t ψ =0,
(cid:18)r f (cid:19) − f f − r2 !
1 f′ 2q2ψ2
A′′+ + A′ A =0,
t r f t− f t
(cid:18) (cid:19)
3 2q2ψ2
A′′ + A′ A =0, (4)
ϕ r ϕ− f ϕ
where the prime denotes the derivative with respect to r. From the equation of motion for ψ, we can obtain
the effective mass of the scalar field
q2A2 q2A2
m2 =m2+ ϕ t, (5)
eff f − r2
whichimpliesthattheincreasingm2andA ordecreasingA willhinderthes-wavesuperfluidphasetransition.
ϕ t
We will get the consistent result in the following calculation.
Inorderto solveaboveequations,we haveto impose the appropriateboundaryconditions atthe tipr =r
s
and the boundary r . At the tip r =r , the fields behave as
s
→∞
ψ =ψ˜ +ψ˜ (r r )+ψ˜ (r r )2+ ,
0 1 s 2 s
− − ···
A =A˜ +A˜ (r r )+A˜ (r r )2+ ,
t t0 t1 s t2 s
− − ···
A =A˜ (r r )+A˜ (r r )2+ , (6)
ϕ ϕ1 s ϕ2 s
− − ···
where ψ˜, A˜ and A˜ (i = 0,1,2, and A˜ = 0) are the integration constants, and we have imposed
i ti ϕi ϕ0
···
the Neumann-like boundary conditions to render the physical quantities finite [12]. Obviously, we can find a
constantnonzerogaugefieldA (r )atr =r ,whichisinstrongcontrasttothatofthe holographicsuperfluid
t s s
model in the AdS black hole backgroundwhere A (r )=0 at the horizon [13, 14, 25].
t +
At the asymptotic AdS boundary r , we have asymptotic behaviors
→∞
ψ ψ ρ J
− + ϕ
ψ = + , A =µ , A =S , (7)
r∆− r∆+ t − r2 ϕ ϕ− r2
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where ∆ = 2 √4+m2 is the conformal dimension of the scalar operator dual to the bulk scalar field, µ
±
±
and S are the chemical potential and superfluid velocity, while ρ and J are the charge density and current
ϕ ϕ
inthedualfieldtheory,respectively. Notethat,provided∆ is largerthantheunitaritybound, bothψ and
− −
ψ can be normalizable and they will be used to define operators in the dual field theory according to the
+
AdS/CFT correspondence, ψ = O , ψ = O , respectively. We can impose boundary conditions that
− − + +
h i h i
either ψ or ψ vanishes [11, 41].
− +
Interestingly, from Eq. (4) we can get the useful scaling symmetries
1
r λr, (t,ϕ,x,y) (t,ϕ,x,y), (q,ψ) (q,ψ), (A ,A ) λ(A ,A ), (8)
t ϕ t ϕ
→ → λ → →
where λ is a real positive number. Using these symmetries, we can obtain the transformation of the relevant
quantities
(µ,S ) λ(µ,S ), (ρ,J ) λ3(ρ,J ), ψ λ∆iψ , (9)
ϕ ϕ ϕ ϕ i i
→ → →
with i = + or i = . We can use them to set q = 1 and r = 1 when performing numerical calculations and
s
−
check the analytical expressions in this section.
Applying the S-L method to analytically study the properties of the holographic s-wave model of superflu-
idity in AdS soliton background,we will introduce a new variable z =r /r and rewrite Eq. (4) into
s
f′ 1 1 qA 2 1 qA 2 m2
ψ′′+ ψ′+ t ϕ ψ =0, (10)
(cid:18)f − z(cid:19) "z2f (cid:18) rs (cid:19) − z4f2 (cid:18) rs (cid:19) − z4f#
1 f′ 2q2ψ2
A′′+ + A′ A =0, (11)
t z f t− z4f t
(cid:18) (cid:19)
1 2q2ψ2
A′′ A′ A =0, (12)
ϕ− z ϕ− z4f ϕ
with f =(1 z4)/z2. Here and hereafter in this section the prime denotes the derivative with respect to z.
−
A. Critical chemical potential
It has been shown numerically that [12, 42, 43], adding the chemical potential to the AdS soliton, the
solution is unstable to develop a hair for the chemical potential bigger than a critical value, i.e., µ>µ . For
c
lowerchemicalpotentialµ<µ ,thescalarfieldiszeroanditcanbeinterpretedastheinsulatorphasesincein
c
this modelthe normalphaseis describedbyanAdSsolitonwhere the systemexhibits a massgap. Therefore,
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thereisaphasetransitionwhenµ µ andthe AdSsolitonreachesthe superconductor(orsuperfluid)phase
c
→
for larger µ.
Before going further, we would like to discuss the phase transition between the AdS soliton and AdS
black holes at high chemical potential without the scalar (or vector) field since it is very important for us
to understand the phase structure of the holographic dual model in the backgrounds of AdS soliton [12, 42].
Consideringthatthe GibbsEuclideanactionofAdSsolitoncoincideswiththatofthe AdSchargedblackhole
in the grand canonical ensemble, we find that the phase boundary between the AdS black hole and the AdS
soliton at zero temperature will be at a chemical potential µ = 21/231/4 1.861 assuming r = 1, which
d s
≃
has been discussed in Refs. [12, 42]. Obviously, the AdS soliton solution should be replaced with the AdS
black hole at µ = µ and the superconductor (or superfluid) phase transition gets unphysical if µ > µ .
c c d
Employingthe analysisofthe string theoryembedding foundin [44], the authorsof[12] avoidedthis problem
in an explicit string theory setup. In the following discussion, we will accept this way if we were in a similar
situation.
At the criticalchemicalpotential µ , the scalarfield ψ =0. Thus,below the criticalpoint Eq. (11) reduces
c
to
1 f′
A′′+ + A′ =0, (13)
t z f t
(cid:18) (cid:19)
which leads to a general solution
1+z2
A =µ+c ln , (14)
t 1 1 z2
(cid:18) − (cid:19)
where c is an integration constant. Obviously, the second term is divergent at the tip z = 1 if c = 0.
1 1
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Considering the Neumann-like boundary condition (6) for the gauge field A at the tip z = 1, we have to
t
set c = 0 to keep A finite, i.e., in this case A will be a constant. Thus, we can get the physical solution
1 t t
A (z) = µ to Eq. (13) if µ < µ . This is consistent with the numerical results in Figs. 1 and 2 which plot
t c
the condensates of the operator O = ψ and charge density ρ with respect to the chemical potential µ for
i i
h i
different values of the dimensionless parameter k=S /µ.
ϕ
Similarly, from Eq. (12) we have
1
A′′ A′ =0, (15)
ϕ− z ϕ
which results in a solution
A =S (1 z2), (16)
ϕ ϕ
−
7
5
1.2
Sj(cid:144)Μ=0.00 Sj(cid:144)Μ=0.00
4 SSjj(cid:144)(cid:144)ΜΜ==00..2550 1.0 SSjj(cid:144)(cid:144)ΜΜ==00..2550
3 SSjj(cid:144)(cid:144)ΜΜ==01..7050 0.8 SSjj(cid:144)(cid:144)ΜΜ==01..7050
<O+> Ρ 0.6
2
0.4
1
0.2
0 0.0
2.0 2.2 2.4 2.6 2.8 3.0 3.2 1.5 2.0 2.5 3.0
Μ Μ
FIG.1: (Coloronline)ThecondensateoftheoperatorhO+iandchargedensityρwithrespecttothechemicalpotential
µ for different values of the dimensionless parameter k = S /µ in the holographic s-wave model of superfluidity by
ϕ
using the numerical shooting method. In each panel, the five lines from left to right correspond to increasing S /µ,
ϕ
i.e., S /µ=0.00 (orange), 0.25 (blue), 0.50 (red), 0.75 (green) and 1.00 (black) respectively. We choose m2 =−15/4
ϕ
and scale q=1 and r =1 in thenumerical computation.
s
4
Sj(cid:144)Μ=0.00 1.2 Sj(cid:144)Μ=0.00
Sj(cid:144)Μ=0.25 Sj(cid:144)Μ=0.25
3 Sj(cid:144)Μ=0.50 1.0 Sj(cid:144)Μ=0.50
Sj(cid:144)Μ=0.75 Sj(cid:144)Μ=0.75
Sj(cid:144)Μ=1.00 0.8 Sj(cid:144)Μ=1.00
<O-> 2 Ρ 0.6
0.4
1
0.2
0 0.0
1.0 1.5 2.0 2.5 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Μ Μ
FIG.2: (Coloronline)ThecondensateoftheoperatorhO−iandchargedensityρwithrespecttothechemicalpotential
µ for different values of the dimensionless parameter k = S /µ in the holographic s-wave model of superfluidity by
ϕ
using the numerical shooting method. In each panel, the five lines from left to right correspond to increasing S /µ,
ϕ
i.e., S /µ=0.00 (orange), 0.25 (blue), 0.50 (red), 0.75 (green) and 1.00 (black) respectively. We choose m2 =−15/4
ϕ
and scale q=1 and r =1 in thenumerical computation.
s
which is consistent with the boundary condition A (1)=0 given in (6).
ϕ
As µ µ from below the critical point, the scalar field equation (10) becomes
c
→
f′ 1 1 qµ 2 (1 z2)2 qS 2 m2
ψ′′+ ψ′+ − ϕ ψ =0. (17)
(cid:18)f − z(cid:19) "z2f (cid:18)rs(cid:19) − z4f2 (cid:18) rs (cid:19) − z4f#
With the boundary condition (7), we assume ψ takes the form
O
ψ(z) h iiz∆iF(z), (18)
∼ r∆i
s
where the trial function F(z) obeys the boundary conditions F(0) = 1 and F′(0) = 0. From Eq. (17), we
arrive at
2 2
qµ qS
(TF′)′+T U +V W ϕ F =0, (19)
" (cid:18)rs(cid:19) − (cid:18) rs (cid:19) #
8
where we have defined
∆ ∆ 2 f′ m2 1 (1 z2)2
T =z2∆i−1f, U = i i− + , V = , W = − . (20)
z z f − z4f z2f z4f2
(cid:18) (cid:19)
According to the Sturm-Liouville eigenvalue problem [45], the minimum eigenvalue of Λ = qµ/r can be
s
obtained from variation of the following functional
qµ 2 1T F′2 UF2 dz
Λ2 = = 0 − , (21)
(cid:18)rs(cid:19) R01T((cid:0)V −k2W)F(cid:1)2dz
R
wherewewillassumethetrialfunctiontobeF(z)=1 az2 withaconstanta. Whenk =0,Eq. (21)reduces
−
tothe caseconsideredin[28]forthe holographics-waveinsulator/superconductorphasetransition,wherethe
spatial component A has been turned off.
ϕ
Fordifferentvaluesofkandm2withthefixedoperator O or O ,wecanobtaintheminimumeigenvalue
+ −
h i h i
of Λ2 and the corresponding value of a. As an example, we have Λ2 = 3.650 and a = 0.3214 for k = 0.25
min
with m2 = 15/4, which gives the critical chemical potential Λ = Λ = 1.911 for the operator O . In
c min +
− h i
Table I, we presentthe critical chemicalpotential Λ =qµ /r for chosenk with fixed mass of the scalar field
c c s
bym2 = 15/4intheholographics-wavesuperfluidmodel. Obviously,theagreementoftheanalyticalresults
−
derived from the S-L method with the numerical calculation shown in Table I is impressive.
TABLE I: The critical chemical potential Λ =qµ /r obtained by the analytical S-L method (left column) and from
c c s
numericalcalculation(rightcolumn)withthechosenvaluesofk=Sϕ/µforthescalaroperators<O− >and<O+ >
in theholographic s-wave superfluidmodel. Herewe fix themass of the scalar field by m2=−15/4.
<O− > <O+ >
k=0.00 0.8368 0.8362 1.890 1.888
k=0.25 0.8534 0.8528 1.911 1.909
k=0.50 0.9096 0.9092 1.975 1.973
k=0.75 1.032(7) 1.032(8) 2.094 2.067
k=1.00 1.320(5) 1.320(3) 2.291 2.290
We see that, from Table I and Figs. 1 and 2, the critical chemical potential Λ = qµ /r increases as the
c c s
dimensionless parameter k = S /µ increases for the fixed mass of the scalar field, i.e., the critical chemical
ϕ
potentialbecomeslargerwiththeincreaseofthesuperfluidvelocity,whichindicatesthatthespatialcomponent
of the gauge field to modeling the superfluid hinders the phase transition. This result is consistent with the
observation obtained from the effective mass of the scalar field in Eq. (5), which implies that the increasing
A will hinder the s-wave superfluid phase transition.
ϕ
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B. Critical phenomena
Now we are in a position to study the critical phenomena of this holographic s-wave superfluid system.
Considering that the condensation of the scalar operator O is so small near the critical point, we can
i
h i
expand A (z) in O as
t i
h i
A (z) µ + O χ(z)+ , (22)
t c i
∼ h i ···
where we have introduced the boundary condition χ(1)=0 at the tip. Defining a function ξ(z) as
2q2µ
c
χ(z)= O ξ(z), (23)
r2∆i h ii
s
we obtain the equation of motion for ξ(z)
(Qξ′)′ z2∆i−3F2 =0, (24)
−
with
Q(z)=zf(z). (25)
According to the asymptotic behavior in Eq. (7) and Eq. (23), we will expand A when z 0 as
t
→
2
ρ q O 1
A (z) µ z2 µ +2µ h ii ξ(0)+ξ′(0)z+ ξ′′(0)z2+ . (26)
t ≃ − r2 ≃ c c r∆i 2 ···
s (cid:18) s (cid:19) (cid:20) (cid:21)
From the coefficients of the z0 term in both sides of the above formula, we have
q Oi 1 1
rhs∆ii = [2µcξ(0)]12 (µ−µc)2 , (27)
with
1 1 z
ξ(0)=c c + x2∆i−3F(x)2dx dz, (28)
2 3
− Q(z)
Z0 (cid:20) Z1 (cid:21)
where c and c are the integration constants which can be determined by the boundary condition of χ(z).
2 3
For example, for the case of k = 0.25 with m2 = 15/4, we have O 1.776(µ µ )1/2 when a = 0.3214
+ c
− h i ≈ −
(we have scaled q =1 and r =1 for simplicity), which is in good agreementwith the numerical result shown
s
in the left panel of Fig. 1. Note that our expression (27) is valid for all cases considered here, so near the
critical point, both of the scalar operators O and O satisfy O (µ µ )1/2. This analytical result
+ − i c
h i h i h i ∼ −
shows that the phase transition of the holographic s-wave superfluid model belongs to the second order and
thecriticalexponentofthesystemtakesthemean-fieldvalue1/2,whichcanbeusedtobackupthenumerical
findings as shown in Figs. 1 and 2.
10
Comparing the coefficients of the z1 term in Eq. (26), we observe that ξ′(0) 0, which agrees well with
→
the following relation by making integration of both sides of Eq. (24)
ξ′(z) 1
= z2∆i−3F2dz. (29)
z −
(cid:20) (cid:21)(cid:12)z→0 Z0
(cid:12)
(cid:12)
Considering the coefficients of the z2 term(cid:12)in Eq. (26), we get
ρ q O 2
= h ii µ ξ′′(0)=Γ(k,m)(µ µ ), (30)
r2 − r∆i c − c
s (cid:18) s (cid:19)
with a prefactor
1 1
Γ(k,m)= z2∆i−3F2dz, (31)
2ξ(0)
Z0
whichisafunctionoftheparameterkandscalarfieldmassm2. Forthecaseofk=0.25andm2 = 15/4with
−
the operator O , as an example, we can obtain ρ = 1.323(µ µ ) when a = 0.3214 (we have scaled q = 1
+ c
h i −
andr =1forsimplicity),whichisconsistentwiththeresultgivenintherightpanelofFig. 1. Obviously,the
s
parameterk andmassofthescalarfieldm2 willnotchangethelinearrelationbetweenthechargedensityand
chemical potential near µ , i.e., ρ (µ µ ), which is in good agreement with the numerical results plotted
c c
∼ −
in Figs. 1 and 2.
On the other hand, near the critical point Eq. (12) becomes
1 2S (1 z2) q O z∆iF 2
A′′ A′ ϕ − h ii =0. (32)
ϕ− z ϕ− z4f r∆i
(cid:18) s (cid:19)
Thus, we finally arrive at
q O 2 2x2∆i−5(1 x2)F(x)2
A =S (1 z2)+S h ii z − dx dz, (33)
ϕ ϕ − ϕ r∆i f(x)
(cid:18) s (cid:19) Z (cid:20)Z (cid:21)
which obeys the boundary condition A (1) = 0 presented in (6) at the critical point. For example, for the
ϕ
case of k =0.25with m2 = 15/4,we obtainA =S [(1 z2) 0.07382 O 2z2+ ] when a=0.3214(we
ϕ ϕ +
− − − h i ···
have scaled q =1 and r =1 for simplicity) for the operator O . Obviously,Eq. (33) is consistent with the
s +
h i
behavior of A in Eq. (16) at the critical point.
ϕ
III. HOLOGRAPHIC P-WAVE SUPERFLUID MODEL
Since the S-L method is effective to obtain the properties of the holographic s-wave model of superfluidity
in the AdS soliton background, we will use it to investigate analytically the holographic p-wave model of
superfluidity in the AdS soliton backgroundwhich has not been constructed as far as we know.