Fluctuations about background fields

According to the holographic fluctuation-dissipation theorem explained in Section 3.5.1, the linear response functions are related to the linearized equations of motions in the fluctuations about the background solutions. Thus, in this section we will derive the full probe limit and backreaction linearized equations of motion of the field fluctuations.

Fluctuation in the probe limit

We now look at fluctuations of the scalar fieldΦpuqand the gauge field Apuqabout there fixed background values with the scalar potential VpΦq “ m²Φ². The metric will be fixed to the Schwarzschild solution since we do not consider any backreaction. The complex scalar field and the gauge field will be replaced by

ΦÝÑΦ`δφ, AaÝÑAa`aa, (4.70)

withΦ PRas stated in (4.23), butδφ PC. We allow a dependence on all coordinates for the fluctuations,i.e.δφ“δφpt,x, uqanda_a “a_apt,x, uq. Expanding the action (4.1) up to quadratic order in the fluctuations we find

SF“S_F^p0q`S<sub>F</sub><sup>p1q</sup>`S_F^p2q, (4.71) where

S_F^p0q“SM, (4.72)

S_F^p1q“ 1 e²

ż

d^d`1x?

´g

„

´1 4

`δFabF<sup>ab</sup>`FabδF^ab˘

´∇aΦ∇^apδφ`δφ^˚q

(4.73)

´iAa∇^apδφ´δφ^˚qΦ`i∇aΦA^apδφ´δφ^˚q

´AaA^aΦpδφ`δφ<sup>˚</sup>q ´2Aaa<sup>a</sup>Φ<sup>2</sup>´m<sup>2</sup>Φpδφ`δφ^˚q



, (4.74)

S_F^p2q“ 1 e²

ż

d^d`1x?

´g

„

´1

4δF_abδF^ab´∇_aδφ∇^aδφ^˚`iAa∇^apδφ^˚qδφ

´iAa∇^apδφqδφ^˚´iaa∇^apδφ´δφ^˚qΦ`i∇aΦa^apδφ´δφ^˚q

´AaA^aδφδφ^˚´2A^B_aa^aΦpδφ`δφ^˚q ´aaa^aΦ²´m²δφδφ^˚



. (4.75)

Variations with respect to the fluctuations ofS_F^p0qare vanishing, while the variation ofS_F^p1qgives rise to the equations of motions of the background fields as expected, since this is the usual variation procedure. The variations of the quadratic action in the fluctuations yield the linearized equations of motions for the fluctuations

p∇a´iAaq p∇^a´iA^aqδφ´m²δφ´i∇apa^aΦq ´iaa∇^aΦ´2Aaa^aΦ“0, (4.76)

∇^aδF_ab`i∇<sub>b</sub>Φpδφ´δφ<sup>˚</sup>q ´i∇<sub>b</sub>pδφ´δφ<sup>˚</sup>qΦ´2AbΦpδφ`δφ^˚q ´2abΦ²“0, (4.77) whereδF_abis given by

δFab“ BapAb`abq ´ BbpAa`aaq ´ BaAb` BbAa“ Baab´ Bbaa. (4.78) In order to work out the quasi-normal modes we apply a Fourier transformation and assume plain wave behavior for the spatial dependence

δφpt,x, uq “

ż dωd^d´1k

p2πq^d e^´iωt`ik¨xδφpuq, a_apt,x, uq “

ż dωd^d´1k

p2πq^d e^´iωt`ik¨xa_apuq.

(4.79)

Thus, we end up with the following equation of motion for the scalar fluctuations

δφ²puq ` ˆf¹puq

fpuq ´d´1 u

˙

δφ¹puq `

„pω`Atpuqq² fpuq² ´ k²

fpuq´ L²m² u²fpuq

 δφpuq

`

„ˆω`2Atpuq fpuq²

˙

a_tpuq ` 1

fpuqk¨apuq

 Φpuq

´iΦa¹_upuq ´i

„ˆf¹puq

fpuq ´d´1 u

˙

Φpuq `2Φ¹puq



a_upuq “0, (4.80)

and for the gauge field fluctuations we have a²_tpuq ´d´3

u a¹_tpuq ´ ˆ k²

fpuq`2L²Φpuq² u²fpuq

˙

a_tpuq ´ ω

fpuqk¨apuq

´ L² u²fpuq

”`ω`2Atpuq˘

δφpuq ´`

ω´2Atpuq˘ δφ^˚ı

Φpuq

`iω ˆ

a¹_upuq ´d´3 u aupuq

˙

“0, (4.81)

a²puq ` ˆf¹puq

fpuq ´d´3 u

˙ a¹puq `

ˆ ω²

fpuq² ´ k²_K

fpuq´2L²Φpuq² u²fpuq

˙ apuq

` k fpuq

`k_K¨apuq˘

` ωk

fpuq²atpuq ` L²k u²fpuq

`δφpuq ´δφ^˚puq˘ Φpuq

´ik

„

a¹_upuq ` ˆf¹puq

fpuq ´d´3 u

˙ aupuq



“0, (4.82)

ω

fpuqa¹_tpuq `k¨a¹puq ´i ˆ

k²´ ω²

fpuq`2L²Φpuq² u²

˙ a_upuq

´L² u²

”`δφpuq ´δφ^˚puq˘

Φ¹puq ´`

δφ¹puq ´δφ^1˚puq˘ Φpuqı

“0, (4.83) where k_K denotes the transverse vector orthogonal to the directions given by the particular equations for the component ofapuq. For instance, we can look at the equation for one of the d´1spatial components,apuq “axpuqexsay, sok_K“kyey`kzez`¨ ¨ ¨ and hencek_K¨acouples the complementary components to the one chosen,i.e.aypuq, azpuq, . . .. Furthermore, assumingk“ kxex, we see that the equations of theaxpuqcomponent couples only with theatpuqcomponent (in this casek_K “ 0), whereas the equations for all other d´2 spatial components decouple.

Looking at the transverse directionse.g.k“kyey`kzez`¨ ¨ ¨, the equation for the fluctuations in thex-direction will decouple from all otherd´2fluctuation equations, but the equations for the remainingd´2fluctuations will couple with each other andatpuq. The background solutions are given by the AdS-Schwarzschild solutions forfpuq(4.65) and the numerical integrated solution for the coupled system (4.68) and (4.69).

Fluctuations with backreaction

Turning on the backreaction we need to consider the fluctuations of the metric field, too,

g_ab“G_ab`h_ab, (4.84)

Thus, we need to expand the determinant of the metric up to quadratic order δ^p2qg“detpG_ab`h_abq ´detG_ab

“ BpdetGq

BG_ab hab`1 2

B²pdetGq

BG_abG_cdhabhcd, (4.85)

and the square root of (4.85) δ^p2q?

´g“B? - detG BdetG

BdetG BGab

hab`1 4

˜ B BGab

B? - detG BGcd ` B

BGcd

B? - detG BGab

¸ habhcd

“?

´detG

„1

2G^abh_ab` ˆ1

8G^abG^cd´1 4G^bcG^ad

˙ h_abh_cd



, (4.86)

where we symmetrized the second partial derivative in its indices and inserted the well known relation

B? - detG BG_ab “1

2

?´GG^ab. (4.87)

Up to linear order in the fluctuations the indices of the metric fluctuationshabare raised and lowered by the background metricGab, so we can rewrite (4.86) as

adetpG`hq “? detG

„ 1`1

2trh´1 4tr`

h²˘

`1 8ptrhq²



, (4.88)

which can be compared to the expansion usingdetp1`Mq “ expptr lnp1`Mqq as given in AppendixA.1.2, (A.10). This will lead to additional terms in (4.74) and (4.75) coming form the quadratic expansion of?

´gin the fluctuations L^p1q

M “?

´G

„

´1

4G^ceG^dfpFcdδFef `δFcdFefq `G^cdL^p2q

φcd

´1 4

ˆ

G^csG^eth_stG^df`G<sup>ds</sup>G<sup>f t</sup>h<sub>st</sub>G<sup>ce</sup>`1

2G^sth_stG^ceG^df

˙ F_cdF_ef

` ˆ

G^csG^dth_st`1

2G^sth_stG^cd

˙ L^p0q

φcd



, (4.89)

L^p2q

M “?

´G

"

´1

4G^ceG^dfδFcdδFef `G^cdL^p2q

φcd

´1 4

ˆ

G^csG^eth_stG^df `G<sup>ds</sup>G<sup>f t</sup>h<sub>st</sub>G<sup>ce</sup>`1

2G^sth_stG^ceG^df

˙

pF_cdδF_ef `δF_cdF_efq

` ˆ

G^csG^dth_st`1

2G^sth_stG^cd

˙ L^p1q

φcd

´1 4

„

G^csG^eth_stG^dmg^{f n}h_mn`1

2G^mnh_mn`

G^csG^eth_stG^df`G^dsG^{f t}h_stg^ce˘

` ˆ1

8g^stG^mn´1

4G^tmG^sn

˙

h_sth_mnG^ceG^df

 F_ceF_df

`

„1

2G^sth_stG^cmG^dnh_mn` ˆ1

8G^stG^mnh_sth_mn´1

4G^tmG^sn

˙

h_sth_mnG^cd

 L^p0q

φcd

*

. (4.90)

Note that here we write the total action in the form S“ 1

2κ² ż

d<sup>d`1</sup>x`

L_EH`2α²L²L_M˘

“SEH`SM, (4.91)

and the scalar field LagrangianL_φcdis given by L^p0q

φcd“ ´ p∇cΦ`iAcΦq p∇dΦ´iAdΦq ´m²Φ², L^p1q

φcd“ ´∇cδφ^˚∇dΦ´∇dδφ∇cΦ`i∇cΦAdδφ´i∇dΦAcδφ^˚´iAc∇dδφΦ

`iAd∇cδφ<sup>˚</sup>Φ´AcAdΦpδφ`δφ^˚q ´ pAcad`AdacqΦ<sup>2</sup>´m<sup>2</sup>Φpδφ`δφ^˚q, L^p2q

φcd“ ´∇cδφ∇dδφ^˚´iAc∇dδφδφ^˚`iAd∇cδφ<sup>˚</sup>δφ´iac∇dδφΦ`iad∇cδφ^˚Φ

`i∇<sub>c</sub>Φadδφ´i∇<sub>d</sub>Φacδφ<sup>˚</sup>´A<sub>c</sub>A<sub>d</sub>δφδφ<sup>˚</sup>´ pA<sub>c</sub>a<sub>d</sub>`A_da_cq pδφ`δφ^˚q

´aaabφ²´m²δφ^˚δφ.

(4.92)

The Einstein-Hilbert term expanded up to quadratic order in the fluctuations can be casted in the following form

L^p1q

EH “?

´Gδ^p1qR`δ^p1q?

´gpR´2Λq, (4.93)

L^p2q

EH “?

´Gδ^p2qR`δ^p1q?

´gδ^p1qR`δ^p2q?

´gpR´2Λq, (4.94) where

δ^p2qR“ ´h^ab∇^p_c^Gq∇^pGqch_ab´3

4G^abBah^cdBbh_cd´g^abBaln?

´dethBbln?

´deth, (4.95) with ∇^pGq denoting the background curved space covariant derivative constructed only out of the background metricG_ab. The full backreacted equations of motions are listed in AppendixC, where we applied the Fourier transformation for all fluctuationsi.e.

δφpt,x, uq “

ż dωd^d´1 k

p2πq^d e^´iωt`ik¨xδφpuq, aapt,x, uq “

ż dωd^d´1 k

p2πq^d e^´iωt`ik¨xaapuq, habpt,x, uq “

ż dωd^d´1 k

p2πq^d e^´iωt`ik¨xhabpuq.

(4.96)

For most of the computations, conducted in Section 4.3 and 4.4, involved in applications to linear response and Homes’ law, the normal phase equations of motions are sufficient. The normal phaseΦ“0andχ“0equations of motion for the scalar field fluctuations are given by

δφ²puq ` ˆf¹puq

fpuq ´d´1 u

˙

δφ¹puq `

„pω`Atq² fpuq² ´ k²

fpuq´ L²m² u²fpuq



δφpuq “0. (4.97)

The gauge field equations are extended by metric fluctuations as well, whereh_tpuq,h_upuqdenotes the spatial entries of the corresponding row/column

a²_tpuq ´d´3 u a¹_tpuq ´

ˆ k² fpuq

˙

atpuq ´ ω

fpuqk¨apuq `iω ˆ

a¹_upuq ´d´3 u aupuq

˙

´3u²A¹_tpuq

2L²fpuqh¹_ttpuq `u²A¹_tpuq 2L²

dÿ´1 i“1

h¹_iipuq `3u²fpuqA¹_tpuq 2L² h¹_uupuq

` u L²A¹_tpuq

dÿ´1 i“1

hiipuq `iuA¹_tpuq L²

dÿ´1 i“1

huipuqki`3 2

u²fpuq L²

ˆf¹puq fpuq ` 2

u

˙

A¹_tpuqhuupuq “0, (4.98)

a²puq ` ˆf¹puq

fpuq ´d´3 u

˙ a¹puq `

ˆ ω²

fpuq²´ k²_K fpuq

˙

apuq ` k fpuq

`k_K¨apuq˘

` ωk

fpuq²atpuq ´ik

„

a¹_upuq ` ˆf¹puq

fpuq ´d´3 u

˙ aupuq



´u²A¹_tpuq

L²fpuqh¹_tpuq ´2uA¹_tpuq

L²fpuqhtpuq ´iu²ωA¹_tpuq

L²fpuq hupuq “0, (4.99)

ω

fpuqa¹_tpuq `k¨a¹puq ´i ˆ

k²´ ω² fpuq

˙ aupuq

´3 2

u²ωA¹_tpuq

L²fpuq² httpuq ´u²A¹_tpuq L²fpuq

dÿ´1 i“1

htiki`u²ωA¹_tpuq 2L²fpuq

dÿ´1 i“1

hiipuq `3 2

u²ωA¹_tpuq

L² huupuq “0.

(4.100) Note that in the normal phase the background solution for the metric is given by the AdS-Reissner-Nordström black brane, sofpuqis given by (4.49), as well asA_tpuqby (4.48).

Numerical solution

Numerical solutions are integrated by adapting the Mathematica CodeD.2for fluctuations. In particular, we need to modify the asymptotic expansion at the horizon to use an index associated with infalling boundary conditions, as explained in detail in Section4.3.6, in order to extract the retarded response function according to the holographic fluctuation-dissipation theorem outlined in

i

_to

iv

_{on page}111. The numerical response function is defined as the ratio of the subleading coefficient to the leading coefficient in the boundary expansion. The numerically obtained key features of holographic superconductors are shown in Figure4.1

10

0 2 4 6 8

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Superconducting Order Parameter

T{^Tc xOyrTcs

axO2y xO¹y

0 50 100 150 200 250

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Optical Conductivity

ω{^T ReσrTcs

Figure 4.1. In the left panel, the expectation value of the dual operator is shown which serves as a order parameter for the superconducting phase transition for α “ 0, d “ 3 andm²L² “ ´2. In this case we may define two dual operators,

O

∆ and

O

d´∆, since ´⁹{⁴ ă m²L² ă ´⁵{⁴ as explained below (3.77). In order to compare the curves for the dimension one and two operator we need to take the square root of

O

∆“2. In the following we will only work with the dimension two operator

O

∆with RG scaling fieldd´∆“1. The continuous phase transition is of mean-field type with the critical exponentβ “¹{²,c.f. Table2.6. The right panel shows the response function of the spatial gauge field fluctuation corresponding to the optical conductivity σpωq. The real part measures the dissipation of the system, c.f. Section4.3.2, which is exponentially suppressed in the superconducting phase.

The normal phase solutions for d “ 3 is given by a frequency independent opti-cal conductivity. This is a general properties of conformal theories [181]. For IR deformation induced by the non-trivial profile of the scalar field solution in the limitω !T, the conformal solution is lost and a superconducting condensate al-lows for nearly dissipationless transport. The curves are color coded as folal-lows:

T{^Tc “ 1,0.5274,0.2464,0.165,0.1212,0.0974. Generically, upon approaching the conformal limit in the UV,ω "T, the optical conductivity is proportional toω^d´3 fordą2as derived in (4.190). Deeper implications of the gap in the real part of the optical conductivity and its connection to the superfluid strength is discussed extensively in Section4.4.1.

Im Dokument Gauge/Gravity duality (Seite 151-157)