AdS 5 /CFT 4 correspondence
4.2. Holographic p-Wave Superconductor
4.2.2. Numerical solutions of the p-wave background equations of motion
Let us first consider the probe limit solutions withα“κ5{gYM“0. Since the Einstein equations decouple, they can be solved by the AdS-Schwarzschild solution discussed in the previous sec-tion. This is in agreement with the top-down solution of aD-brane wrappingP AdS5directions andQ S5 directions, where the background metric generated by the D3-branes is exactly the AdS5ˆS5-Schwarzschild metric (4.64) with temperatureT “1{πuH. As explained in the intro-duction theSUp2qI will be explicitly broken to aUp1q3 which in turn is subsequently broken by a spontaneously generated condensate. Thus the Ansatz for the gauge field is of the form
A“A1xpuqτ1dx`A3tpuqτ3dt . (4.106) On the field theory side this corresponds to the introduction of an isospin chemical potential by the boundary values of the time components of the gauge fieldA3t. This breaks theSUp2qI symmetry down to a diagonalUp1q3 which is generated byτ3. In order to study the transition to the superfluid state, we allow solutions with non-zero@
Jx1D
, such that we include the dual gauge fieldA1xin the gauge field Ansatz. Since we consider only isotropic and time-independent solutions in the field theory, the gauge fields exclusively depend on the radial coordinateu. With this Ansatz the Yang-Mills energy-momentum tensor defined in (4.105) is diagonal. The gauge field Ansatz (4.106) reduces the Yang-Mills equations to
A3t2puq `4´P
u A3t1puq ´
`A1x˘2
π2T2fpuqA3t “0, A1x2puq `
ˆf1puq
fpuq `4´P u
˙
A1x1puq `
`A3t˘2
π2T2fpuq2A1x“0,
(4.107)
where we work in dimensionless radial coordinates u ÝÑ uuH. Furthermore for numerical calculations the temperature can be set toT “1 by rescaling the gauge fieldsA3t ÝÑT A3t and A1xÝÑT A1x. The asymptotic expansions of the gauge fields at the horizon and the boundary are determined via the Frobenius method, implemented in Mathematica CodeD.1. The asymptotic expansions at the horizonuH“1read
A3tHpuq “Ca3tpu´1q ´Ca3t
˜ 2´1
2P`
`Ca1x˘2
8π2T2
¸
pu´1q2 (4.108)
`Ca3t
`Ca1x˘4
´12π2` Ca1x˘2
pP´5qT2`32π4`
P2´9P`20˘ T4
192π4T4 pu´1q3, (4.109)
`O`
pu´1q4˘
(4.110)
0 1 2 3 4 5 6 7 8 9
NcD3 ‚ ‚ ‚ ‚ ´ ´ ´ ´ ´ ´
NfD5 ‚ ‚ ‚ ´ ‚ ‚ ‚ ´ ´ ´
NfD7 ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ´ ´
xa x0 x1 x2 x3 u θ1 θ2 θ3 θ4 θ5
Table 4.2. The directions wrapped by theDp-branes are listed with a‚. The spatial directions are denoted by xµ, the radial AdS coordinate by u and the S5 angles by θi. As explained in the main text theD5-brane wrapsP “4AdS directions andQ“2S5 directions with ad“3field theory, while for thed“4field theory theNf “2D7 -branes yieldsP “5andQ“3. Note thatD3/Dp-brane embeddings withP´Q“2 are supersymmetric [225].
A1xHpuq “Ca1x´
`Ca3t˘2
Ca1x
64π2T2 pu´1q2
`
`Ca3t˘2
Ca1x´`
Ca1x˘2
`3p13´2Pqπ2T2
¯
576π4T4 pu´1q3`O`
pu´1q4˘
. (4.111) Due to the regularity conditionsA3tpuHq “ 0 we have only a two parameter family left at the horizon determined byCa3t andCa1x. For the numerical integration we actually used a asymp-totic horizon expansion up toO`
pu´1q8˘
. The boundary asymptotic solution atuB“0is given by
A3tBpuq “µ´ρ3t uP´3 pπTqP´3, A1xBpuq “ρ1x uP´3
pπTqP´3,
(4.112)
where we included directly the vanishing source constraint for the dual operatorJx1. For the numerical solution the temperatureT is set to one, so all dimensionful physical quantities are given in units ofT, e.g. the numerical values of the isospin chemical potential are in factµ{T. For ad“3dimensional field theory we may consider aD3/D5-brane setup where theD5-brane wrapsP “ 4 directions of the AdS5 space and Q “ 2 directions of the S5, whereas a d “ 4 dimensional field theory may be described by aD3/D7-brane setup withP “5andQ“3. The precise extensions of the branes are listed in Table4.2. Solving these equations numerically with the help of theMathematicaprogram outlined in Mathematica CodeD.2yields the map
pCa3t, Ca1xq ÞÑ pµ,0q. (4.113) Note that in general this map is not unique since there are higher excitation solutions with one or more nodes which are associated to higher energy states in analogy to higher modes of a harmonic oscillator in a box. However, we need to pick the ground state solution in order to apply the holographic dictionary. The isospin chemical potentialµI of theUp1q3 will be set as a free parameter, whereas the spontaneous symmetry breaking operator must not be sourcedi.e.
the coefficient of the leading term must be zero. The result of the shooting yields the inverse
00 5 10 15 20
20 25 30
40 60 80
Ca1x Ca3 t
MappCa3t, Ca1xq ÞÑ pµ,0q
Ground state 1stexcited state 2ndexcited state
0
´60
´40
´20 20
0.0 0.2 0.4 0.6 0.8 1.0
u A3 tpuq,A1 xpuq
Solution to Equations of Motion
Figure 4.2. The left plot shows the mapping of the numerically obtained solutions with bound-ary valuesA3tB“µandA1xB“0for aD7-brane setup. The critical temperature is read off atCa1x“0fromCa3t «24and hence according to (4.114)µ«12«3.18π.
Note that the curves corresponding to theground statesolutions and the1stand2nd excited state ofCa3t are approaching each other for large values of the free param-eterCa1xat the horizon. Thus, it is virtually impossible to numerically distinguish the sought ground state solutions from the higher excitation solutions since not only the first and second, but all higher excitations approaching each other. Phys-ically, this follows from the fact that forCa3t Ñ0,A3t “0, the system is described by the totally disordered high temperature phase. The right plots shows the re-spective ground state solutions and the2ndexcited statesolutions with two nodes in theA1xpuqprofile.
map displayed in Figure 4.2The critical temperature can be read off directly from Figure4.2 since forCa1x“0we find the trivial solutionA1x“0, so the initial conditionCa3t is identical to µaccording to the analytic solution ofA3t in the normal phase (4.48)9
A3tpuq “ ´Ca3t1´uP´3
P´3 ñ µ“ ´lim
uÑ0A3tpuq “ Ca3t
P´3. (4.114) The critical value isµ{Tc«3.81π. The expectation value of the operators dual to the gauge fields is determined via the holographic dictionary10
@JµAD
“δIDDBIp
δµ “TDpNf` 2πα1˘2
ρAµ “ p2πq´p`2 ˆ2π
g2YM
˙ ˆ2gYM2 Nc L4
˙pp´3q{4
NfρAµ
“ p2πq´p`22pp´3q{4L´p`3`
g2YMNc˘pp´7q{4
NcNfρAµ, (4.115)
where in the last line the expression is written entirely in terms of field theory quantities via gs´1 “ 2π{g2YM andα1 “a
2g2YMNc{L4´1. Plotting the subleading term of the numerical solution, which determines the expectation value of the dual current, we see again the typical mean-field behavior in Figure4.3. In the backreacted caseα‰0, the normal phase solution@
Jx1D
“0or A1x “0, respectively, are the AdS-Reissner-Nordström black brane solution discussed in Section
9Strictly speaking the two function differ by an overall sign of the chemical potential, but this is a matter of convention what we call aDp-brane or an antiDp-brane. In the following we will choose the convention thatµą0.
10P andpmust not be confused, the former denotes the number of AdS directions wrapped by the brane, whereas the latter numbers the spatial dimensionality of the brane.
0 5 10 15 20
0.0 0.2 0.4 0.6 0.8 1.0
T{Tc
ρ1 x
D5condensate withP“4
0 20 40 60 80 100
0.6 0.7 0.8 0.9 1.0 1.1
T{Tc
ρ1 x
D7condensate withP“5
Figure 4.3. The condensate arising from the non-zero expectation value of the dual operator@ Jx1D
is plotted in the D5 andD7 case. In the probe limit, we find again a typi-cal mean-field behavior withβ “1{2for the second order superconducting phase transition.
4.1.3. The gauge field solution is given by (4.48) A“µ
ˆ 1´ u2
u2H
˙
τ3dt, (4.116)
whereτ3 “ σ3{2iwithσ3 the third Pauli matrix. We consider the diagonal representations of the gauge group since we may rotate the flavor coordinates until the chemical potential lies in the third isospin direction. Furthermore, the condensed phase is described by the solution to the full set of Einstein and Yang-Mills equations regarding the Yang-Mills energy-momentum tensor (4.105). This is the main difference to the background solutions of the holographic s-wave superconductors listed in Table4.1: the AdS-Reissner-Nordström black brane solution with scalar hair is replaced by a vector hair solution. The numerical calculation seems to be more convenient and simpler inr-coordinates given byr “L2{u, so in the following we will work in these coordinates. Given that the Yang-Mills energy-momentum tensor is diagonal, a diagonal metric is consistent
ds2“ ´Nprqσprq2dt2` 1
Nprqdr2`r2fprq´4dx2`r2fprq2`
dy2`dz2˘
, (4.117)
with
Nprq “ ´2mprq r2 ` r2
L2. (4.118)
Solutions with@ Jx1D
‰ 0 also break the spatial rotational symmetry SOp3q down toSOp2q11 such that our metric Ansatz will respect onlySOp2q. In addition, the system is invariant under theZ2 parity transformationPk : x ÝÑ ´xandA1x ÝÑ ´A1x. Inserting our Ansatz into the Einstein and Yang-Mills equations leads to six equations of motion for mprq, σprq, fprq, φprq, wprqand one constraint equation from therrcomponent of the Einstein equations, where the
10In principle, equations of motions can be converted to different coordinates related by arbitrary bijective coordinate transformations (c.f. AppendixA.3). However, for a set of coupled equations of motions it is advisable to start again from the coordinate free formulation.
11Note that the finite temperature and chemical potential already break the Lorentz group down toSOp3q.
gauge fieldsA3t “φandA1x“whave been renamed in order to comply with the general naming scheme commonly used in the p-wave literature. The dynamical equations may be written as
m1“α2rf4w2φ2
6N σ2 `r3α2φ12 6σ2 `N
ˆr3f12 f2 `α2
6 rf4w12
˙ ,
σ1“α2f4w2φ2 3rN2σ `σ
ˆ2rf12
f2 `α2f4w12 3r
˙ ,
f2“ ´α2f5w2φ2
3r2N2σ2 `α2f5w12 3r2 ´f1
ˆ3 r´f1
f `N1 N `σ1
σ
˙ ,
φ2“f4w2φ r2N ´φ1
ˆ3 r ´σ1
σ
˙ ,
w2“ ´ wφ2 N2σ2 ´w1
ˆ1 r`4f1
f `N1 N `σ1
σ
˙ .
(4.119)
The equations of motion are invariant under scaling transformations analogous to the s-wave superconductor. As illustrated for the probe limit solution where we rescaled the gauge fields to removeuHwe can apply this rescaling to the complete set of equations (4.119) via
rÝÑcr, mÝÑc4m, wÝÑcw, φÝÑcφ. (4.120)
to setrhto one. By a similar scaling symmetry
rÝÑcr, mÝÑc2m, LÝÑcL, φÝÑc´1φ, αÝÑcα, (4.121) the AdS radius may be set toL“1. More importantly, the metric at the boundary needs to be asymptotically AdS spacetime, so we use the scaling
f ÝÑcf, wÝÑc´2w, (4.122)
to set the boundary valuefprBq “1and
σÝÑcσ, φÝÑcφ, (4.123)
in order to fix the boundary valueσprBq “1.