Background equations of motion for scalar hair black branes

Let us first derive the complete set of equations for the holographic s-wave superconductor back-ground in arbitrary dimensionsd. To my knowledge, this calculation has not been done in full generality, so allow me to be a little more detailed in the derivation. Starting with the Einstein equations in AdS_d`1

R_ab´1

2Rg_ab´dpd´1q

2L² g_ab“α²L²T_ab, (4.16) we have inserted the AdSd`1cosmological constant whereLdenotes the AdS-radius, related to the curvature of the AdS-space. The main ingredients of a holographic superconductor are a charged black brane that may “grow scalar hair”,i.e. a condensate described by a charged scalar fieldΦand a respectiveUp1qgauge field. Due to Lorentz symmetry the fields can only depend on the radial AdS coordinateuand it is sufficient to have a non-zero time component of the gauge field, acting as a chemical potential at the boundary. Thus, we may take the Ansatz

Φ“Φpuq, A“A_tpuqdt, ds²“ L² u²

ˆ

´fpuqe^´χpuq dt²`dx<sup>2</sup>`du² fpuq

˙

, (4.17) and for simplicity, we will assume a special scalar potential namely,

Vp|Φ|q “m²|Φ|², (4.18)

where the mass of the potential needs to be fixed to a value above the Breitenlohner-Freedman boundm²L² ě ´^d2{⁴. Note that it is sufficient to assume that only quadratic terms are present in the potentialVp|Φ|q “m²|Φ|², since we are only interested in the behavior near the critical point where higher order interactions do not contribute. Deep in the condensed phase,i.e. close to zero temperature, the ground state of the holographic superconductor depends heavily on the from of the scalar field potential. Inserting the Ansatz (4.17) into the equations of motions of

the scalar field (4.8) yields Φ²puq `

ˆf¹puq

fpuq ´χ¹puq

2 ´d´1 u

˙

Φ¹puq `

ˆA_tpuq²e^χpuq

fpuq² ´ L²m² u²fpuq

˙

Φpuq “0. (4.19) TheUp1qgauge field Ansatz in (4.17) gives rise to the field strength

F “dA“ BuAtpuqdu^dt“ ´BuAtpuqdt^du ñ Ftu“ ´A¹_tpuq, (4.20) and the gauge field equation of motions (4.9) forb“t, read

A²_tpuq ` ˆχ¹puq

2 ´d´3 u

˙

A¹_tpuq ´2L²|Φpuq|²

u²fpuq Atpuq “0. (4.21) The second non trivial Maxwell equation forb“u, yields a reality condition for the scalar field

Φ^˚BuΦ´ΦBuΦ^˚“0. (4.22)

For arbitraryΦpuq “|Φpuq|e^iϕpuq this reduces to

2iϕpuqϕ¹puq “0. (4.23)

Therefore we can conclude that ϕ¹ “ 0 ñ ϕ “ const. and we can choose without loss of generalityϕ“0ôΦPR. Hence, the current can be reduced toj_b “2Φ²A_b.

For the Einstein tensorG_abwe need to calculate the Christoffel symbolsΓ^a_bc, the Riemann tensor R_abcd, the Ricci tensorR_aband the Ricci scalarR. The non-vanishing Christoffel symbols defined by

Γ^a_bc“ 1 2g_ab`

Bbg_dc` Bcg_ab´ Bdg_bc˘

, (4.24)

read

Γ^t_ut“Γ^t_tu“ ´1

u` f¹puq

2fpuq´χ¹puq

2 , Γⁱ_iu “Γⁱ_ui“ ´1 u, Γ^u_tt“1

2fpuq² ˆ

´2

u`f¹puq

fpuq ´χ¹puq

˙

, Γ^u_ii “fpuq

u , Γ^u_uu“ ´1

u´ f¹puq 2fpuq.

(4.25)

The Riemann tensor Rabcd “1

2

`BdBagbc´ BdBbgac` BcBbgad´ BcBagbd

˘´gef

´

Γ^e_acΓ^f_bd´Γ^e_adΓ^f_bc¯

, (4.26)

with all indices lowered, has the following symmetries

• R_rabsrcds,i.e. antisymmetric in its first two and last two indices

• R_abcd “R_cdab,i.e. symmetric under the interchange of the first and the last pair of indices

• R_arbcds“0, the first or algebraic Bianchi identity

The only non-vanishing components thus read

R_ijij“R_jiji“ ´R_ijji“ ´R_jiij“ ´g_uupΓ^u_iiq²“ ´L² u⁴fpuq, R_titi“R_itit “ ´R_itti “ ´R_tiit “ ´g_uuΓ^u_ttΓ^u_ii“ L²fpuq

u⁴

„ 1´u

2

ˆf¹puq

fpuq ´χ¹puq

˙

e^´χpuq,

Riuui“Ruiiu“ ´Riuiu“ ´Ruiui“ 1

2Bu²gii´gii

`Γⁱ_iu˘2

`guuΓ^u_iiΓ^u_uu“³L²2fpuq ´uf¹puq 2u⁴fpuq Rutut“Rtutu“ ´Ruttu“ ´Rtuut“1

2B²ugtt´gtt

`Γ^t_tu˘2

`guuΓ^u_uuΓ^u_tt,

“ L² u⁴

„

fpuq ´uf¹puq `u²

2 f²puq `u 2

ˆ

fpuq ´3

2uf¹puq `u 2f χ¹puq

˙

χ¹puq ´u²

2 fpuqχ²puq

 . The Ricci tensor can be derived from the Riemann tensor by Rac “ g^bdRabcd which yield a symmetric tensor. The non-zero components are

R_tt“g^bdR_tbtd “

d´ÿ1 i“1

gⁱⁱR_titi`g^uuR_tutu

“ e^´χpuq u²

„

dfpuq²´d`1

2 ufpuqf¹puq `u²

2 fpuqf²puq

`u 2

ˆ

dfpuq²´3

2ufpuqf¹puq `1

2ufpuq²χ¹puq

˙

χ¹puq ´u²

2 fpuq²χ²puq

 ,

R_ii“g^bdR_ibid“g^ttR_itit`

dÿ´1 j“1

g^jjR_ijij⁴`g^uuR_iuiu“ fpuq u²

„

´d`u ˆf¹puq

fpuq ´χ¹puq 2

˙

,

R_uu“g^bdR_ubud“g^ttR_utut`

d´ÿ1 i“1

gⁱⁱR_uiui

“ 1 u²

"

´d` pd`1qu 2

f¹puq fpuq ´u²

2 f²puq

fpuq ´u 2

„ 1´u

2 ˆ

3f¹puq

fpuq ´χ¹puq

˙

χ¹puq `u² 2 χ²puq

* , and we find a diagonal Ricci tensor reflecting our spherical symmetric Ansatz and matching the diagonal form of the energy-momentum tensorTab. Finally, the Ricci scalar is the trace of the Ricci tensor

R“g^abR_ab“g^ttR_tt`

d´ÿ1 i“1

gⁱⁱR_ii`g^uuR_uu

3No summation overi; original term of the formgefΓ^e_iuΓ^f_uiallows only one term,i.e. wheni“e“f.

4řd´1

j“1g^jjRijijhas one zero summand, namelygⁱⁱRiiii“0, so this sum yields a factor ofpd´2q.

“ ´ 1 L²

„

dpd`1qfpuq ´2duf<sup>1</sup>puq `u²f²puq

`u ˆ

dfpuq ´3

2uf¹puq `u

2fpuqχ¹puq

˙

χ¹puq ´u²fpuqχ²puq



. (4.27)

We are now able to write the three different Einstein equations as dpd´1q`

fpuq ´fpuq²˘

` pd´1qufpuqf¹puq

2u²e^χpuq “α²L²Ttt,

´ 1 2u²

„

dpd´1q ´dpd´1qfpuq `2pd´1quf¹puq ´u²f²puq

`ufpuq ˆ

´d`1`3 2uf¹puq

fpuq ´u 2χ¹puq

˙

χ¹puq `u²fpuqχ²puq



“α²L²T_ii,

´dpd´1q ´dpd´1qfpuq ` pd´1qu“

f¹puq ´fpuqχ¹puq‰

2u²fpuq “α²L²T_uu.

(4.28)

The right-hand side of the Einstein equations are the full energy-momentum tensor defined in (4.13)–(4.15). The gauge field energy-momentum tensor reads

T_ab^em“g^ttF_atF_bt`

d´ÿ1 i“1

gⁱⁱF_aiF_bi`g^uuF_auF_bu´1

2g_abg^ttg^uupF_tuq². (4.29) The only non-vanishing components are

T_tt^em“ ˆ

g^uu´1

2g_ttg^ttg^uu

˙

pF_tuq²“1

2g^uupF_tuq²“ 1 2

u²fpuq

L² e^χpuqA¹_tpuq², T_jj^em“ ´1

2g_jjg^ttg^uupF_tuq²“1 2

u² L²A¹_tpuq², T_uu^em“

ˆ g^tt´1

2guug^ttg^uu

˙

pFtuq²“1

2g^ttpFtuq²“ ´1 2

u²

L²fpuqe^χpuqA¹_tpuq².

(4.30)

The energy-momentum tensor for the scalar field (4.15) can be reduced to T_tt^Φ“Atpuq²|Φpuq|²´gtt

´

g^uuBuΦ^˚puqBuΦpuq ´m²|Φ|²¯ , T_ii^Φ“ ´gii

´

g^ttAtpuq²|Φpuq|²`g^uuBuΦ^˚puqBuΦpuq ´m²|Φ|²¯ , T_uu^Φ “ BuΦ^˚puqBuΦpuq ´guu

´

g^ttAtpuq²|Φpuq|²´m²|Φ|²¯ .

(4.31)

Inserting the metric Ansatz (4.17) and the reality condition (4.22) simplifies the full Einstein equations to

0“ dpd´1q

u² `d´1

u f¹puq ´dpd´1q u² fpuq

´2α²L²

„

fpuqΦ¹puq²`

ˆe^χpuqA_tpuq²

fpuq ´m²L² u²

˙ Φpuq²



´α²u²e^χpuqA¹_tpuq², (4.32)

0“dpd´1q

u² ´f²puq `

ˆ2pd´1q u `3

2χ¹puq

˙

f¹puq ´dpd´1q u² fpuq

`

„

χ²puq ´ ˆd´1

u `1 2χ¹puq

˙ χ¹puq

 fpuq

`2α²L² ˆ

e^χpuqAtpuq²

fpuq `m²L² u²

˙

Φpuq²´2fpuqΦ¹puq²`α²u²e^χpuqA¹_tpuq², (4.33)

0“ ´dpd´1q

u² ´d´1

u f¹puq `

ˆpd´1qχ¹puq

u `dpd´1q u²

˙ fpuq

´2α²L²

„

fpuqΦ¹puq²`

ˆe^χpuqAtpuq²

fpuq `m²L² u²

˙ Φpuq²



`α²u²e^χpuqA¹_tpuq². (4.34) The horizon of the scalar hair AdS-Reissner-Nordström black brane defines a temperature ac-cording to (3.7)

TH“ 1 2π

„ 1

?g_uu d du

?´gtt



u“uH

“ e^´χpuq{2 4πu

„

uf¹puq ´fpuq`

uχ¹puq `2˘ˇˇˇˇ

u“uH

. (4.35)

We may use either thett(4.32) oruu(4.34) Einstein equation to eliminatef¹puqand the reg-ularity conditions at the horizon, i.e.fpuHq “ 0 andAtpuHq “ 0, which reduces the Hawking temperature to

TH“ e^´χH{2dpd´1q ´2α²m²L⁴Φ²_H`α²u⁴_He^χHA¹_tH² 4πpd´1quH

, (4.36)

where we have a set of four parameters, the horizon radius uH set by fpuHq “ 0, the scalar field at the horizonΦpuHq “ΦH, the electrical field perpendicular to the horizonA¹_tpuHq “A¹_tH and the metric fieldχH “χpuHq. In order to identify the black brane temperatureTHwith the temperature of the boundary (Euclidean) field theory, we need the proper normalization of the gravitational redshift and thus the emblackening factor has to approach one,f Ñ1foruÑ 0, at the boundary. Therefore the additional constraint χ Ñ0 asuÑ 0must be imposed on the solutions of the Einstein equations. The equations of motion, the metric and the one form are invariant under the scaling symmetry

tÝÑct, AtÝÑc^´1At, e^χ ÝÑc²e^χ, (4.37)

which we can use to set χ “ 0 at the boundary. Furthermore, there are two more scaling symmetries⁵

tÝÑct, xÝÑx, uÝÑcu, ds²ÝÑc²ds², mÝÑc^´1m, LÝÑcL, A_tÝÑc^´1A_t, ΦÝÑc^´1Φ,

(4.38)

5The rescaling of the metric can be absorb in a Weyl rescaling.

and

pt,xq ÝÑcpt,xq, uÝÑcu, AtÝÑc^´1At, (4.39) which can be used to setL“1anduH“1, respectively. This will be done in numerical calcula-tions in order to work with purely dimensionless equacalcula-tions of motion, but we will keep the factors in all expressions which allows for easy identification of their correct dimensionality. Effectively, the temperature depends only on the remaining set of parameterspΦH, A¹_tHqand the additional external parametersα,mandd, that changes the theory of the holographic superconductor.

Numerical solution

The solution to the equations of motion (4.19), (4.21) and (4.32)–(4.34) can be obtained only by numerical integration. We will employ the following procedure:

i

Determine the asymptotic solution to the equations of motion in a power series about the regular singular point at the horizonuH “ 1 under the regularity conditions fpuHq “ 0 andA_tpuHq “0. This approach is known as the Frobenius method. The numerical value of the asymptotic solution close to the horizonuÀ1is used as the initial data depending on the initial conditions set bypΦH, A¹_tH, χHq. From a naïve counting we would expect six initial conditions for a set of three coupled, second order, ordinary differential equations.

However, the Einstein equations are reduced to two first order differential equations and imposing the regularity conditionfpuHq “ 0 reduces the number of initial conditions to one, namely χH. The four initial conditions of the scalar and Maxwell equations are re-duced to the remaining twoΦHandA¹_tHby imposing the regularity conditionAtpuHq “0.

ii

Determine the boundary asymptotic near the asymptotic AdS-spacetime boundaryuÑ0.

Again a power series expansion will yield the asymptotic solution. In order to have a well-defined boundary variational problem, we need to use a holographic renormalization scheme to regularize the boundary on-shell action. Apart from the standard gravitational counterterms,i.e. the Gibbons-Hawking term and a boundary cosmological constant term, we need counterterms for the scalar field. The total counterterm reads

Scounter“ ż

d^dx?

´γ

„

2γ^µν∇µnν´2pd´1q

L `2|Φ|

L ` pΦ<sup>˚</sup>n<sup>µ</sup>BµΦ`Φn^µBµΦ^˚qˇˇˇˇ

u“0^`

“ 2 ż

d^dx?

´γ ˆ

γ^µν∇µnν´pd´1q L `Φ

L `Φn^µBµΦ˙ˇˇˇˇ

u“0^`

, (4.40)

whereγdescribes the induced metric on a shell close to the boundary andn^νis the outward pointing normal vector on the boundary shell.

iii

Finally, integrate the equations of motion from the horizonuH“1to the boundaryuB“0.

After stabilizing the numerical integration by choosing suitable values close touH“1and uB“0, the solution is fitted to the boundary expansion in order to obtain the coefficients of the leading and subleading terms. Formally, the integration defines the map

pΦH, A¹_tH, χHq ÞÑ pµ,ΦB, χBq, (4.41) whereAtB“µis the chemical potential of the dual field theory, fixing the charge density.

For non-zeroΦBthe backgroundUp1qsymmetry is explicitly broken, whereas spontaneous

symmetry breaking is induced by a non-zero vacuum expectation value of the dual field operator. Thus, we need to impose the boundary conditionsΦB “ 0 and as mentioned above for a well-define temperature of the field theoryT “ TH, the condition χB “ 0.

Using a so-called “shooting method” where we fix the boundary values in (4.41) topµ,0,0q we determine the horizon initial conditions by searching for roots of the difference function between the integrated solution and the desired values. Operationally, we fit the numerical solution to the boundary expansion

ΦB“Φ^p_B^1qu^d´∆`Φ<sup>p</sup><sub>B</sub><sup>2q</sup>u<sup>∆</sup>“Φ<sup>p</sup><sub>B</sub><sup>1q</sup>u<sup>12</sup>p<sup>d´?d2`4m2</sup>q `Φ<sup>p</sup><sub>B</sub><sup>2q</sup>u<sup>12</sup>p<sup>?d2`4m2`d</sup>q, (4.42)

A_tB“µ`ρu^d´2. (4.43)

This method allows us to invert the map (4.41) and express the horizon initial conditions in terms of the dimensionless⁶boundary chemical potentialµ,¯ i.e.ΦHpµ¯q,A¹_tHpµ¯qandχHpµ¯q. We are thus left with a one parameter family for each set of the external parametersd, mandαof solutions for differentµ. The remaining physical quantities of the dual field¯ theory are then completely determined by inserting the inverted map,e.g. the temperature Tp¯µqvia (4.36), the charge densityn, the order parameter of the superconducting phase transitionx

O

∆yand the energy densityǫas

x

O

∆y “ lim

uÑ0

?1γ

δSon-shell

δΦB

,

n“@ J^tD

“lim

uÑ0

?1γ

δSon-shell

δAtB

,

ǫ“@ T⁰⁰D

“lim

uÑ0

?1γ

δSon-shell

δg_ttB

.

(4.44)

Note that working with fixed chemical potential relates to the grand canonical ensemble and the grand canonical thermodynamic potentialΩpT, V, µq. At the critical value of µc

orTc the boundary conditionspΦpuHq, A¹_tpuHqqare such that a non-zero vacuum value of O leads to a condensation of the charged scalar field, hovering over the charged black brane, at the critical chemical potential/temperatureµ¯c. Due to the geometry of the AdS-space there is a stable solution where the electrostatic repulsion cancels the gravitational attraction.

The numerical integration is executed byMathematica. For simplicity, only the probe limit Math-ematica code is presented inD.1, since all the key features are outlined and the extension to include the backreaction is straightforward, whereas the pedagogical value would be drastically reduced by the increased complexity of the more complicated equations of motion. In the fol-lowing let us discuss the remaining special cases listed in Table4.1.

Im Dokument Gauge/Gravity duality (Seite 141-147)