Page 1

Preprint typeset in JHEP style - HYPER VERSION

MPP-2011-113

Transport in Anisotropic Superfluids: A Holographic

Description

Johanna Erdmenger, Patrick Kerner and Hansj¨ org Zeller

Max-Planck-Institut f¨ ur Physik (Werner-Heisenberg-Institut)

F¨ ohringer Ring 6, 80805 M¨ unchen, Germany

jke, pkerner, zeller@mppmu.mpg.de

Abstract: We study transport phenomena in p-wave superfluids in the context of

gauge/gravity duality. Due to the spacetime anisotropy of this system, the tensorial struc-

ture of the transport coefficients is non-trivial in contrast to the isotropic case. In partic-

ular, there is an additional shear mode which leads to a non-universal value of the shear

viscosity even in an Einstein gravity setup. In this paper, we present a complete study of

the helicity two and helicity one fluctuation modes. In addition to the non-universal shear

viscosity, we also investigate the thermoelectric effect, i.e. the mixing of electric and heat

current. Moreover, we also find an additional effect due to the anisotropy, the so-called

flexoelectric effect.

Keywords: Gauge-gravity correspondence, Black Holes.

arXiv:1110.0007v1 [hep-th] 30 Sep 2011

Page 2

Contents

1.Introduction2

2. Holographic Setup and Equilibrium

2.1Hairy Black Hole Solution

2.2 Thermodynamics

4

4

6

3.Perturbations about Equilibrium

3.1Characterization of Fluctuations and Gauge Fixing

3.2Equations of Motion, On-shell Action and Correlators

3.2.1Helicity two mode

3.2.2Helicity one modes

9

10

11

12

13

4.Transport Properties

4.1Universal Shear Viscosity

4.2Thermoelectric Effect perpendicular to the Condensate

4.3Non-Universal Shear Viscosity and Flexoelectric Effect

15

15

15

19

5.Conclusion25

A. Holographic Renormalization

A.1 Asymptotic Behavior

A.2 Counterterms

26

26

28

B. Constructing the Gauge Invariant Fields

B.1 Residual Gauge Transformations

B.1.1 Diffeomorphism Invariance

B.1.2 SU(2) Gauge Invariance

B.2 Physical Fields

29

29

30

31

31

C. Numerical Evaluation of Green’s Functions32

D. General Remarks on Viscosity in Anisotropic Fluids34

– 1 –

Page 3

1. Introduction

Hydrodynamics is a very powerful description of systems close to equilibrium. Its focus is

on slowly varying fluctuations with frequency ω and momentum k smaller than the typical

length scale, the mean free path. Hydrodynamics may be seen as the low-energy effective

description of interacting systems. Gauge/gravity duality is a very useful tool to further

develop the hydrodynamic description for various systems. New transport phenomena

have been uncovered by studying systems which violate parity by an anomaly [1–3]. The

transport in a system which shows the chiral magnetic effect induced by an axial anomaly

has been studied in [4–7]. Effects of anisotropy in strongly coupled systems have been

discussed in [8,9]. Recently the hydrodynamic description for s-wave superfluids which

may violate parity has been investigated in [10–12].

The hydrodynamical description of superfluids is interesting since an Abelian symme-

try is spontaneously broken. Due to the spontaneous breaking of a continuous symmetry, a

Nambu-Goldstone boson appears in the spectrum. Since it is massless, it behaves as hydro-

dynamic mode and has to be included into the hydrodynamical description. In this paper

we study p-wave superfluids where in addition to the Abelian symmetry, the rotational

symmetry is spontaneously broken and thus more Nambu-Goldstone bosons appear in the

spectrum. This leads to an anisotropic fluid in which the transport coefficients depend on

the direction, i.e. they are tensors. In the case we study here the fluid is transversely sym-

metric, i.e. the system has an SO(2) symmetry and we can use this symmetry to reduce

the tensors to the minimal amount of independent quantities. For instance, the viscosity

which relates the stress Tµνin a fluid with the strain ∇λuρ+ ∇ρuλgiven in terms of the

four velocity of the fluid uµis parametrized by a rank four tensor ηµνλρ(see e.g. [13,14]).

Using the symmetry we find two independent shear viscosities, in contrast to only one in

the isotropic case, i.e. SO(3) symmetry.

The shear mode is the transversely polarized fluctuation given for instance by ∇yuz+

∇zuyfor a momentum in x direction. In the isotropic case this is the unique shear mode

since any momentum can be rotated into the x direction by the SO(3) rotational symmetry.

In the transversely symmetric case, two momenta, one along and one perpendicular to the

favored direction, e.g. the x direction, must be considered. Thus there are two shear modes.

If the momentum is along the favored direction, the situation is similar to the isotropic

case and the strain is again ∇yuz+ ∇zuy. However if the momentum is perpendicular to

the favored direction say in y direction, the situation changes dramatically. Now the little

group is given by the discrete group Z2and the strain is given by ∇xuz+∇zux. Since the

shear viscosity can be evaluated at zero momentum, we can characterize the fluctuations

with respect to the full symmetry group which is in the transversely symmetric case SO(2).

The first fluctuation ∇yuz+∇zuyis a helicity two state as the shear mode in the isotropic

case is. The second fluctuation ∇xuz+∇zuxhowever transform as helicity one state under

the rotational symmetry. This transformation property is due to the rotational symmetry

breaking and will be very important in this paper.

In the context of gauge/gravity duality, the spontaneous breaking of continuous sym-

metries by black holes developing hair was first achieved in [15] and later used to construct

– 2 –

Page 4

holographic superconductors/superfluids by breaking an Abelian symmetry [16,17]. Along

this line also p-wave superconductors/superfluids have been constructed [18] and gave rise

to the first string theory embeddings of holographic superconductors/superfluids [19–21].

In order to obtain the effects of spontaneous rotational symmetry breaking in the hydro-

dynamics of p-wave superfluids, we have to take the back-reaction of the superfluid density

into account, i.e. we consider the effect of the superfluid density on the energy-momentum

tensor. This was obtained e.g. in [22]. On the gravity side, the p-wave superfluid state

corresponds to an asymptotically AdS black hole which carries vector hair.

A very famous result in the context of gauge/gravity duality is that the ratio between

shear viscosity and the entropy density is universal [23–25]. The ratio is the same for all

field theories which have a Einstein gravity dual, i.e. the field theory is a large N gauge

theory at infinite ’t Hooft coupling λ. This universality can be proven as follows: The

shear mode is the only mode which transforms as a helicity two mode under the SO(2)

little group and thus decouple from all the other modes. In addition it can be shown that

the low energy dynamics of this mode is trivial such that the ratio is completely determined

by gravitational coupling constant which is universal. The universality is lost if finite N

and/or coupling is considered for instance by adding a Gauss-Bonnet term to the gravity

action (see e.g. [26,27]).

In the letter [28] we have shown that universality is also absent even at leading order

in N and λ if the fluid is anisotropic. In this case, the universality is lost since one of the

different shear modes transforms as a helicity one mode under the rotational symmetry

and can therefore couple to other helicity one modes present in the system. The coupling

generates non-trivial dynamics which lead to a non-universal behavior of the shear vis-

cosity. This result is valid for a field theory dual to Einstein gravity without additional

contributions to the gravity action. In this paper we present the detailed calculations for

this result. This calculation was suggested already in [29]. We study the complete set

of the helicity two and one modes in the SU(2) Einstein-Yang-Mills theory in the broken

phase at zero momentum.

Along this calculation we find some additional transport phenomena: the thermoelec-

tric effect in the transversal directions and the flexoelectric effect. The thermoelectric effect

is the phenomenon that the electric and heat current mix since charged object transport

charge as well as energy. This effect has been already studied for holographic s-wave su-

perfluids [17,30]. We find that the thermoelectric effect in the transversal directions agrees

with the result found for s-wave superfluids. The flexoelectric effect is known from nematic

liquids which consists of molecules with non-zero dipole moment (see e.g. [14]). A direction

can be preferred by the dipoles. In this anisotropic phase, a strain can lead to effective

polarization of the liquid and an electric field applied to the liquid can lead to a stress.

This is the first appearance of this effect in the context of gauge/gravity duality.

The paper is organized as follows: In section 2 we review the holographic setup in

which p-wave superfluids are constructed and describe their behavior in equilibrium. In

section 3 we study perturbations about equilibrium. We characterize the fluctuations in

terms of their transformation under the symmetry groups and determine their equations of

motion. In addition we calculate the on-shell action and read off the correlation functions.

– 3 –

Page 5

In section 4 we extract the transport properties out of the correlation functions and find

the non-universal shear viscosity, the thermoelectric effect and the flexoelectric effect. We

conclude in section 5. In the appendix A we discuss holographic renormalization. The

gauge covariant fields are constructed in appendix B. In appendix C we review the numer-

ical evaluation of correlator when operator mixing is present. Some general remarks on

anisotropic fluids are given in appendix D.

2. Holographic Setup and Equilibrium

We consider SU(2) Einstein-Yang-Mills theory in (4 + 1)-dimensional asymptotically AdS

space. The action is

?

where κ5is the five-dimensional gravitational constant, Λ = −12

stant, with L being the AdS radius, and ˆ g is the Yang-Mills coupling constant. The SU(2)

field strength Fa

MNis

S =d5x√−g

?

1

2κ2

5

(R − Λ) −

1

4ˆ g2Fa

MNFaMN

?

+ Sbdy,(2.1)

L2 is the cosmological con-

Fa

MN= ∂MAa

N− ∂NAa

M+ ?abcAb

MAc

N, (2.2)

where capital Latin letter as indices run over {t,x,y,z,r}, with r being the AdS radial

coordinate, and ?abcis the totally antisymmetric tensor with ?123= +1. The Aa

components of the matrix-valued gauge field, A = Aa

generators, which are related to the Pauli matrices by τa= σa/2i. Sbdyincludes boundary

terms that do not affect the equations of motion, namely the Gibbons-Hawking boundary

term as well as counterterms required for the on-shell action to be finite. Finally it is

convenient to define

α ≡κ5

Mare the

MτadxM, where the τaare the SU(2)

ˆ g,

(2.3)

which measures the strength of the backreaction.

The Einstein and Yang-Mills equations derived from the above action are

RMN+

4

L2gMN= κ2

∇MFaMN= −?abcAb

5

?

TMN−1

MFcMN,

3TPPgMN

?

,(2.4)

(2.5)

where the Yang-Mills stress-energy tensor TMNis

TMN=

1

ˆ g2

?

Fa

PMFaPN−1

4gMNFa

PQFaPQ

?

.(2.6)

2.1 Hairy Black Hole Solution

Following ref. [18,22], to construct charged black hole solutions with vector hair we choose

the gauge field ansatz

A = φ(r)τ3dt + w(r)τ1dx.(2.7)

– 4 –

Page 6

The motivation for this ansatz is as follows. In the field theory we will introduce a chemical

potential for the U(1) symmetry generated by τ3. We will denote this U(1) as U(1)3. The

bulk operator dual to the U(1)3 density is A3

ansatz. We want to allow for states with a nonzero ?Jx

A1

x(r) ≡ w(r). With this ansatz for the gauge field, the Yang-Mills stress-energy tensor in

eq. (2.6) is diagonal. Solutions with nonzero w(r) will preserve only an SO(2) subgroup of

the SO(3) rotational symmetry, so our metric ansatz will respect only SO(2). In addition

the system is invariant under the Z2 parity transformation P?: x → −x and w → −w.

Furthermore, given that the Yang-Mills stress-energy tensor is diagonal, a diagonal metric

is consistent. Our metric ansatz is [22]

t, hence we include A3

1?, so in addition we introduce

t(r) ≡ φ(r) in our

ds2= −N(r)σ(r)2dt2+

1

N(r)dr2+ r2f(r)−4dx2+ r2f(r)2?dy2+ dz2?,

L2. For our black hole solutions we will denote the position of the

horizon as rh. The AdS boundary will be at r → ∞.

Inserting our ansatz into the Einstein and Yang-Mills equations yields five equations of

motion for m(r), σ(r), f(r), φ(r), w(r) and one constraint equation from the rr component

of the Einstein equations. The dynamical equations can be recast as (prime denotes

(2.8)

with N(r) = −2m(r)

r2

+r2

∂

∂r)

m?=α2rf4w2φ2

6Nσ2

+α2r3φ?2

6σ2

?

+ N

?

r3f?2

f2

+α2

6rf4w?2

?

,

σ?=α2f4w2φ2

3rN2σ

+ σ

2rf?2

f2

+α2f4w?2

3r

?3

?

,

f??= −α2f5w2φ2

φ??=f4w2φ

r2N

w??= −wφ2

3r2N2σ2+α2f5w?2

?3

N2σ2− w?

3r2

− f?

r−f?

f+N?

N+σ?

σ

?

,

− φ?

r+σ?

?1

σ

?

,

r+4f?

f

+N?

N+σ?

σ

?

.

(2.9)

The equations of motion are invariant under four scaling transformations (invariant

quantities are not shown),

(I)σ → λσ ,

f → λf ,

r → λr,

r → λr,

φ → λφ,

w → λ−2w,

m → λ4m,

m → λ2m,

(II)

(III)w → λw,

L → λL,

φ → λφ,

φ →φ

(IV )

λ,

α → λα,

where in each case λ is some real positive number. Using (I) and (II) we can set the

boundary values of both σ(r) and f(r) to one, so that the metric will be asymptotically

AdS. We are free to use (III) to set rhto be one, but we will retain rhas a bookkeeping

device. We will use (IV) to set the AdS radius L to one.

– 5 –

Page 7

A known analytic solution of the equations of motion is an asymptotically AdS Reissner-

Nordstr¨ om black hole, which has φ(r) = µ − q/r2, w(r) = 0, σ(r) = f(r) = 1, and

N(r) =r2−2m0

φ(r) at the boundary, which is the U(1)3chemical potential in dual field theory.

To find solutions with nonzero w(r) we resort to numerics. We will solve the equations

of motion using a shooting method. We will vary the values of functions at the horizon until

we find solutions with suitable values at the AdS boundary. We thus need the asymptotic

form of solutions both near the horizon r = rhand near the boundary r = ∞.

Near the horizon, we define ?h≡

of ?hwith some constant coefficients. Two of these we can fix as follows. We determine rh

by the condition N(rh) = 0, which gives that m(rh) = r4

A3

t(rh) = φ(rh) = 0 for A to be well-defined as a one-form (see for example ref. [31]). The

equations of motion then impose relations among all the coefficients. A straightforward

exercise shows that only four coefficients are independent,

?

where the subscript denotes the order of ?h(so σh

All other near-horizon coefficients are determined in terms of these four.

Near the boundary r = ∞ we define ?b≡?rh

relations among the coefficients. The independent coefficients are

?

where here the subscript denotes the power of ?b. All other near-boundary coefficients are

determined in terms of these.

We used scaling symmetries to set σb

since we do not want to source the operator Jx

spontaneously broken). In our shooting method we choose a value of µ and then vary

the four independent near-horizon coefficients until we find a solution which produces the

desired value of µ and has σb

In what follows we will often work with dimensionless coefficients by scaling out factors

of rh. We thus define the dimensionless functions ˜ m(r) ≡ m(r)/r4

˜ w(r) ≡ w(r)/rh, while f(r) and σ(r) are already dimensionless.

?

r2 +2α2q2

3r4

?

, where m0=

r4

h

2+α2q2

3r2

h

and q = µr2

h. Here µ is the value of

r

rh−1 ? 1 and then expand every function in powers

h/2. Additionally, we must impose

φh

1,σh

0,fh

0,wh

0

?

, (2.10)

0is the value of σ(r) at the horizon, etc.).

r

?2? 1 and then expand every function

in powers of ?bwith some constant coefficients. The equations of motion again impose

mb

0,µ,φb

1,wb

1,fb

2

?

, (2.11)

0= fb

0= 1. Our solutions will also have wb

1in the dual field theory (U(1)3 will be

0= 0

0= fb

0= 1 and wb

0= 0.

h,˜φ(r) ≡ φ(r)/rhand

2.2 Thermodynamics

Next we will describe how to extract thermodynamic information from our solutions [22].

Our solutions describe thermal equilibrium states in the dual field theory. We will work in

the grand canonical ensemble, with fixed chemical potential µ.

We can obtain the temperature and entropy from horizon data. The temperature T is

given by the Hawking temperature of the black hole,

?

T =

κ

2π=

σh

12π

0

12 − α2(˜φh

1)2

σh

0

2

?

rh.(2.12)

– 6 –

Page 8

Here κ =

?

∂Mξ∂Mξ

???rh

is the surface gravity of the black hole, with ξ being the norm of

the timelike Killing vector, and in the second equality we write T in terms of near-horizon

coefficients. In what follows we will often convert from rhto T simply by inverting the

above equation. The entropy S is given by the Bekenstein-Hawking entropy of the black

hole,

S =2π

κ2

5

Ah=2πV

κ2

5

r3

h=2π4

κ2

5

V T3

123σh

0

3

1)2α2?3,

?

12σh

0

2− (˜φh

(2.13)

where Ahdenotes the area of the horizon and V =?d3x.

The central quantity in the grand canonical ensemble is the grand potential Ω. In

AdS/CFT we identify Ω with T times the on-shell bulk action in Euclidean signature. We

thus analytically continue to Euclidean signature and compactify the time direction with

period 1/T. We denote the Euclidean bulk action as I and Ion-shellas its on-shell value (and

similarly for other on-shell quantities). Our solutions will always be static, hence Ion-shell

will always include an integration over the time direction, producing a factor of 1/T. To

simplify expressions, we will define I ≡˜I/T. Starting now, we will refer to˜I as the action.

˜I includes a bulk term, a Gibbons-Hawking boundary term, and counterterms,

˜I =˜Ibulk+˜IGH+˜ICT.(2.14)

˜Ion-shell

bulk

To regulate these divergencies we introduce a hypersurface r = rbdywith some large but

finite rbdy. We will ultimately remove the regulator by taking rbdy→ ∞. For our ansatz,

the explicit form of the three terms may be found in [22]. Finally, Ω is related to the

on-shell action,˜Ion-shell, as

Ω =˜Ion-shell.

and˜Ion-shell

GH

exhibit divergences, which are canceled by the counterterms in˜ICT.

(2.15)

The chemical potential µ is simply the boundary value of A3

density ?Jt

t(r) = φ(r). The charge

3? of the dual field theory can be extracted from˜Ion-shellby

?Jt

3? =lim

rbdy→∞

δ˜Ion-shell

δA3

t(rbdy)= −2π3α2

κ2

5

T3

123σh

0

3

1)2α2?3˜φb

?

12σh

0

2− (˜φh

1.(2.16)

Similarly, the current density ?Jx

1? is

?Jx

1? = lim

rbdy→∞

δ˜Ion-shell

δA1

x(rbdy)= +2π3α2

κ2

5

T3

123σh

0

3

1)2α2?3˜ wb

?

12σh

0

2− (˜φh

1. (2.17)

The expectation value of the stress-energy tensor of the CFT is [32,33]

?Tµν? = lim

rbdy→∞

2

√γ

δ˜Ion-shell

δγµν

=lim

rbdy→∞

?r2

κ2

5

(−Kµν+ Kρργµν− 3γµν)

?

r=rbdy

,(2.18)

– 7 –

Page 9

where small Greek letter as indices run over the dual field theory directions {t,x,y,z} and

Kµν=1

2

?N(r)∂rγµνis the extrinsic curvature. We find

?Ttt? = 3π4

κ2

5

T4

124σh

0

4

1)2α2?4˜ mb

1)2α2?4

124σh

0

2− (˜φh

?

12σh

0

2− (˜φh

124σh

0,

?Txx? =π4

κ2

5

T4

0

4

?

?

12σh

0

2− (˜φh

?

?

˜ mb

0− 8fb

2

?

?

,

?Tyy? = ?Tzz? =π4

κ2

5

T4

4

12σh

01)2α2?4

˜ mb

0+ 4fb

2

.

(2.19)

Notice that ?Ttx? = ?Tty? = ?Ttz? = 0. Even in phases where the current ?Jx

the fluid will have zero net momentum. Indeed, this result is guaranteed by our ansatz

for the gauge field which implies a diagonal Yang-Mills stress-energy tensor and a diagonal

metric. The spacetime is static.

Tracelessness of the stress-energy tensor (in Lorentzian signature) implies ?Ttt? =

?Txx? + ?Tyy? + ?Tzz?, which is indeed true for eq.

we always have a conformal fluid. The only physical parameter in the dual field theory is

thus the ratio µ/T.

For ˜ mb

thermodynamic properties of the Reissner-Nordstr¨ om black hole, which preserves the SO(3)

rotational symmetry. For example, we find that ?Txx? = ?Tyy? = ?Tzz? and Ω = −V ?Tyy?,

i.e. ?Tyy? is the pressure P. For solutions with nonzero ?Jx

SO(2). In these cases, we find that ?Txx? ?= ?Tyy? = ?Tzz?. In the superfluid phase, both

the nonzero ?Jx

equations above, we also find

Ω = −V ?Tyy?.

This again suggest the identification of ?Tyy? as the pressure P.

breaking of the SO(3) symmetry ?Txx? is not the pressure P but most also contain terms

which are non-zero in the broken phase, i.e. terms which contain the order parameter ?Jx

For instance it may be written as

1? is nonzero,

(2.19), so in the dual field theory

0=1

2+α2˜ µ2

3, σh

0= 1,˜φh

1= 2˜ µ, fb

2= 0, and˜φb

1= −˜ µ we recover the correct

1?, the SO(3) is broken to

1? and the stress-energy tensor indicate breaking of SO(3). Just using the

(2.20)

However due to the

1?.

?Txx? = P + ∆?Jx

1??Jx

1?,(2.21)

where ∆ is a measure for the breaking of the rotational symmetry and is given by

?

∆ = −

3κ2

5

α2π2T2

12σh

0

2− (˜φh

122σh

1)2α2?2

2

0

fb

2

?˜ wb

1

?2. (2.22)

Using this identification we can write down the stress-energy tensor for the dual field

theory in equilibrium in a covariant form

?Tµν? = ?uµuν+ P Pµν+ ∆PµλPνρ?Jλ

a??Jρ

a?,(2.23)

– 8 –

Page 10

0.60.81.0

T

Tc

1.21.4

20

30

40

50

60

κ2

π4T4

5?Ttt?

c

(a)

0.60.81.0

T

Tc

1.2 1.4

6

8

10

12

14

16

18

20

κ2

2π3T3

5?Jt

3?

cα2

(b)

Figure 1: The energy density ?Ttt? (a) and the charge density ?Jt

ature T/Tcfor α = 0.316. The red line is the solution without a condensate and the blue line the

solution with ?J1

3? (b) over the reduced temper-

x? ?= 0 below Tc.

where ? = ?Ttt? is the energy density and Pµν= uµuν+ ηµνis the projector to the space

perpendicular to the velocity uµ.

In figure 1 we plot ?Ttt? and ?Jt

that in both cases there is one solution for temperatures above Tcand two for temperatures

below Tc. From considerations in [22] we know that the solution with condensate (blue

line) is the thermodynamically preferred one. For further plots see [22].

3? versus the reduced temperature, respectively. We see

In addition in [22] it was found that the order of the phase transition depends on the

ratio of the coupling constants α. For α ≤ αc= 0.365, the phase transition is second order

while for larger values of α the transition becomes first order. The critical temperature

decreases as we increase the parameter α. The quantitative dependence of the critical

temperature on the parameter α is given in figure 2. The broken phase is thermodynami-

cally preferred in the blue and red region while in the white region the Reissner-Nordstr¨ om

black hole is favored. The Reissner-Nordstr¨ om black hole is unstable in the blue region

and the phase transition from the white to the blue region is second order. In the red

region, the Reissner-Nordstr¨ om black hole is still stable however the state with non-zero

condensate is preferred. The transition from the white to the red region is first order. In

the green region we cannot trust our numerics. At zero temperature, the data is obtained

as described in [34,35].

3. Perturbations about Equilibrium

In this section we study the response of the holographic p-wave superfluid under small

perturbations. On the gravity side these perturbations are given by fluctuations of the

metric hMN(xµ,r) and the gauge field aa

modes: 5 from the massless graviton in 5 dimensions and 3×3 from the massless vectors in

five dimensions. Due to time and spatial translation invariance in the Minkowski directions,

M(xµ,r). Thus we study in total 14 physical

– 9 –

Page 11

0.000.020.04

T

µ

0.06 0.08

0.0

0.1

0.2

0.3

0.4

α2

Figure 2: The phase structure of the theory: In the blue and red region the broken phase is the

thermodynamically preferred phase while in the white region the Reissner-Nordstr¨ om black hole is

the ground state. In the blue region the Reissner-Nordstr¨ om black hole is unstable and the transition

from the white to the blue region is second order. In the red region the Reissner-Nordstr¨ om black

hole is still stable. The transition form the white to the red region is first order. The black dot

determines the critical point where the order of the phase transition changes. In the green region

we cannot trust our numerics.

the fluctuations can be decomposed in a Fourier decomposition

hMN(xµ,r) =

?

?

d4k

(2π)4eikµxµˆhMN(kµ,r),

d4k

(2π)4eikµxµˆ aa

aa

M(xµ,r) =

M(kµ,r).

(3.1)

To simplify notations we drop the hat on the transformed fields which we use from now on

if not stated otherwise.

3.1 Characterization of Fluctuations and Gauge Fixing

In general we have to introduce two spatial momenta: one longitudinal to the condensate

k?and one perpendicular to the condensate k⊥, i.e. kµ= (ω,k?,k⊥,0). Introducing the

momentum perpendicular to condensate breaks the remaining rotational symmetry SO(2)

down the discrete Z2 parity transformation P⊥: k⊥→ −k⊥and x⊥→ −x⊥. Thus in-

troducing this momentum forbids the usual classification of the fluctuations in different

helicity states of the little group since the symmetry group just consists of discrete groups

at best P?× P⊥. We do not study this case further in this paper. However a momentum

exclusively in the direction longitudinal to the condensate or zero spatial momentum pre-

serves the SO(2) rotational symmetry such that we can classify the fluctuations according

to their transformation under this SO(2) symmetry (see table 1). The modes of different

– 10 –

Page 12

dynamical fields

hyz,hyy− hzz

hty,hxy;aa

htz,hxz;aa

htt,hxx,hyy+ hzz,hxt;aa

constraints

none

hyr

hzr

htr,hxr,hrr;aa

# physical modes

2

4

4

4

helicity 2

helicity 1

y

z

helicity 0

t,aa

xr

Table 1: Classifications of the fluctuations according to their transformation under the little group

SO(2). The constraints are given by the equations of motion for the fields which are set to zero due

the fixing of the gauge freedom: aa

r≡ 0 and hrM≡ 0. The number of physical modes is obtained

by the number of dynamical fields minus the number of constraints. Due to SO(2) invariance the

fields in the first and second line of the helicity one fields can be identified.

helicity decouple from each other. The momentum longitudinal to the condensate, however,

breaks the longitudinal parity invariance P?.

In order to obtain the physical modes of the system we have to fix the gauge freedom.

We choose a gauge where aa

r≡ 0 and hMr ≡ 0 such that the equations of motion for

these fields become constraints. These constraints fix the unphysical fluctuations in each

helicity sector and allow only the physical modes to fluctuate. The physical modes may be

constructed by enforcing them to be invariant under the residual gauge transformations,

δaa

r= 0 and δhMr= 0 (see appendix B),

helicity two:Ξ = gyyhyz,hyy− hzz,

Ψ = gyy(ωhxy+ k?hty);aa

helicity one:

y,

(3.2)

and helicity zero:

Φ1=ξy,

Φ2=a1

t+iω

φa2

t+ik?ω2− φ2?

ξt+2k

(k2− w2)φa2

ωξtx,

t−1

− iω2ww?+ k2φφ?

x+w?ω2− φ2?

(k2− w2)φa3

x,

Φ3=ξx−k2Nσ2f4

x+k

ω2r2

Φ4=a1

ωa1

2wξx−w?

φ?a3

t+φw?

2φ?ξt−k?ω2w?+ wφφ??

ω (k2− w2)φ?a3

x

ωφ?(k2− w2)a2

x,

(3.3)

with

ξy= gyyhyy,ξx= gxxhxx,ξt= gtthtt,ξtx= gxxhtx.(3.4)

3.2 Equations of Motion, On-shell Action and Correlators

In the following we will focus on the response exclusively due to time dependent pertur-

bations, i.e. kµ= (ω,0,0,0). In this case in addition to the SO(2) symmetry, P?parity is

conserved which allows us to decouple some of the physical modes in the different helicity

blocks. In this section we write down the equations of motion for the fluctuations, deter-

mine the on-shell action and vary the on-shell action with respect to the fluctuations to

– 11 –

Page 13

obtain the retarded Green’s functions G of the stress-energy tensor Tµνand the currents

Jµ

a,

?

Gµ,ν

a,b(ω,0) = −i

?

Gρ

a

dtd3xeiωtθ(t)?[Jρ

Tµνand Jµ

a are the full stress-energy tensor and current, respectively. Thus they include

the equilibrium parts, ?Tµν? and ?Jµ

arise due to the inclusion of fluctuations in our model. In the following we split the analysis

into the different helicity blocks.

Gµν,ρσ(ω,0) = −idtd3xeiωtθ(t)?[Tµν(t,? x),Tρσ(0,0)]?,

?

dtd3xeiωtθ(t)?[Jµ

a(t,? x),Jν

b(0,0)]?,

Gµνρ

a(ω,0) = −idtd3xeiωtθ(t)?[Tµν(t,? x),Jρ

a(0,0)]?,

µν(ω,0) = −i

?

a(t,? x),Tµν(0,0)]?.

(3.5)

a?, as well as the corresponding dissipative parts which

3.2.1 Helicity two mode

First we look at the non-trivial helicity two mode displayed in table 1. If we expand the

action (2.1) up to second order in the fluctuations, this mode decouples from every other

field. Therefore it can be written as a minimal coupled scalar with the equation of motion

?

3Nσ2

Ξ??+

1

r+4r

N−rα2φ?2

?

Ξ?+

ω2

N2σ2Ξ = 0.(3.6)

The contribution from this mode to the on-shell action is

?

˜Son-shell

helicity 2=

1

κ2

5

d4k

(2π)4

?

r3Nσ

??

3

2√N

−1

r+f?

2f−N?

4N−σ?

2σ

?

Ξ2−1

4ΞΞ?

??

r=rbdy

,

(3.7)

which is divergent as we send rbdy→ ∞. The divergence can be cured by holographic

renormalization (see appendix A). The renormalized on-shell action is

?

where ˜ ω = ω/rhis the dimensionless frequency, Ξb

quantities in (2.11) and Ξb

Son and Starinets [36] to compute the Green’s function of this component. The response

due to the perturbation hyzis given by

Son-shell

helicity 2=r4

h

κ2

5

d4k

(2π)4

?

Ξb

0Ξb

2−1

2

?

˜ mb

0+ 4fb

2−1

32˜ ω4

??

Ξb

0

?2?

,(3.8)

0and Ξb

0(ω)Ξb

2are defined similarly to the

2(−ω). Now we use the recipe by

0Ξb

2is a short form for Ξb

?Tyz?(ω) =

δSon-shell

helicity 2

δΞb

0(−ω)

?2r4

= Gyz,yz(ω)Ξb

0(ω),(3.9)

with

Gyz,yz(ω) =

h

κ2

5

Ξb

Ξb

2(ω)

0(ω)− ?Tyy? +1

32ω4

?

, (3.10)

where ?Tyy? is the equilibrium contribution given by the pressure P. As we will see in section

4.1, the Green’s function of this helicity mode will lead to a shear viscosity component with

universal behavior, i.e. ηyz/s = 1/4π.

– 12 –

Page 14

3.2.2 Helicity one modes

Now we look at the helicity one modes displayed in table 1. Again we obtain their equations

of motion by expanding the action (2.1) up to second order in the fluctuations and varying

it with respect to the corresponding fields. The equations of motion are

0 = a3

y

??+

?1

r2f2a3

r−2f?

f

+N?

N+σ?

σ

?

a3

y

?+

?

ω2

N2σ2−f4w2

r2N

−2α2φ?2

Nσ2

?

a3

y, (3.11a)

0 = Ψ?

t+2α2φ?

y,(3.11b)

and

0 =Ψ??

x+

?1

r+4r

N+6f?

f

−rα2φ?2

3Nσ2

2α2wφ2

r2f2N2σ2a1

+N?

?

Ψ?

x+2α2w?

r2f2a1

y

?+

ω2

N2σ2Ψx

+2iα2ωwφ

r2f2N2σ2a2

?1

−2iωφ

?1

+2iωφ

N2σ2a1

y−

y,(3.12a)

0 =a1

y

??+

r−2f?

fN+σ?

σ

?

a1

y

?− f6w?Ψ?

x+

?

ω2

N2σ2+

φ2

N2σ2

?

a1

y

N2σ2a2

y,(3.12b)

0 =a2

y

??+

r−2f?

y−iωf6wφ

f

+N?

N+σ?

σ

?

a2

y

?+

?

ω2

N2σ2+

φ2

N2σ2−f4w2

r2N

?

a2

y

N2σ2Ψx.(3.12c)

where Ψt = gyyhty and Ψx = gyyhxy. Note that due to the parity P?, the helicity one

modes split into two blocks where the modes of the first block are even while the modes

of the second block are odd under P?. In the first block there is only one physical mode

a3

ywhile the value of the other field Ψtis given by the constraint (3.11b). This can also

be seen in the gauge invariant fields (3.2) since htydrop out for k?= 0. The other three

physical modes appear in the second block where Ψx= Ψ for k?= 0.

The contribution from these modes to the on-shell action is

˜Son-shell

helicity 1=1

κ2

+3r4f2

5

?

d4k

(2π)4

?r5f2

4σΨtΨt?−1

r

√N

yΨx−r3α2φ?

4r3f6NσΨxΨx?−rα2Nσ

Ψt2+r3f6Nσ

2

?????r=rbdy

2f2

?

−N?

a1

ya1

y

?+ a2

ya2

?

y

?+ a3

ya3

y

??

2σ

?

1 −

??

3

√N

−2

r−2f?

f2N−σ?

σ

Ψx2

+rα2f4Nσw?

2

a1

2σ

a3

yΨt

,

(3.13)

– 13 –

Page 15

which is again divergent1. The renormalized on-shell action is given by

?

−1

2

−1

+ α2?

Son-shell

helicity 1=r4

h

κ2

5

d4k

(2π)4

?

4α2(˜ µ2+ ˜ ω2)

??Ψx

4−1

?b

32˜ ω4

??˜ a1

?Ψt

0

?Ψx

?b

2+ α2??˜ a1

y

?b

−3

+ iα2˜ ω˜ µ?˜ a1

?Ψx

0

?˜ a1

2˜ mB

y

?b

0

1+?˜ a2

??Ψt

y

y

?b

?b

?˜ a2

0

?2

?˜ a2

y

?b

1+?˜ a3

−1

?b

y

?b

0

?˜ a3

y

y

?b

2

1

?

˜ mb

0− 8fb

???Ψx

2+?˜ a2

0− ˜ wb

?b

?b

?b

0

2?

?2

0

4α2˜ ω2?˜ a3

?b

0

y

?b

0

y

0

?b

0

y

0

2˜φb

1

?˜ a3

y

?b

0

?b

1

?˜ a1

y

0

?b

0

??

,

(3.14)

where ˜ aa

fluctuations a3

µ= rhaa

µis dimensionless.

yand htyby variation of the on-shell action,

We obtain the response of the system due to the

??Jy

?Tty?(ω)

3?(ω)

?

=

δSon-shell

helicity 1

δ(a3

δSon-shell

helicity 1

δ?

ty(ω)

y)

b

0(−ω)

?b

Ψt

0(−ω)

=

?

?Gy,y

Gtyy

3,3(ω) Gy

3

ty(ω)

3(ω) Gty,ty(ω)

?

?

?a3

y

?b

0(ω)

?b

?Ψt

0(ω)

,(3.15)

with

?Gy,y

Gtyy

3,3(ω) Gy

3

3(ω) Gty,ty(ω)

=

α2r2

κ2

h

5

?

2(˜ a3

(˜ a3

y)

b

1(ω)

b

y)

−?Jt

0(ω)−˜ ω2

3?

2

−?Jt

−?Ttt?

3?

.(3.16)

This result agrees with the result obtain in the holographic s-wave superfluids [17,30] and

thus the breaking of the rotational symmetry has no effect on this subset of fluctuations.

The coupling between the current Jy

3and the momentum Ttyis known as the thermoelectric

effect which we will study in the next section.

The response due to the fluctuations a1

Ψx

y, a2

yand hxyis given by

?Jy

?Jy

?Txy?(ω)

1?(ω)

2?(ω)

=

δSon-shell

helicity 1

δ(a1

δSon-shell

helicity 1

δ(a2

δSon-shell

helicity 1

δ?

y)

b

0(−ω)

y)

b

0(−ω)

?b

?

?

2(ω)

b

0(ω)

0(−ω)

=

Gy,y

1,1(ω) Gy,y

Gy,y

1,2(ω) Gy

2,2(ω) Gy

1

xy(ω)

2,1(ω) Gy,y

Gxyy

2

xy(ω)

1(ω) Gxyy

2(ω) Gxy,xy(ω)

?a1

?Ψx

y

?b

?b

0(ω)

?b

0(ω)

?a2

y

0(ω)

, (3.17)

where the matrix of the Green’s functions is given by

2

κ2

5

(˜ a1

α2r2

κ2

h

5

?

2(˜ a1

(˜ a1

?

y)

b

1(ω)

b

y)

0(ω)−˜ µ2+˜ ω2

2(˜ a2

(˜ a1

?

2

α2r2

κ2

h

5

?

2(˜ a2

(˜ a2

2(˜ a1

(˜ a2

y)

b

1(ω)

b

y)

b

1(ω)

y)

?

(˜ a2

0(ω)+ i˜ ω˜ µ

?

−?Jx

1?

2

+ 2α2r3

h

κ2

5

(˜ a1

?

?b

1

32˜ ω4

y)

b

1(ω)

?b

?

Ψx

0(ω)

α2r2

κ2

h

5

y)

b

1(ω)

b

y)

0(ω)− i˜ ω˜ µ

α2r2

κ2

h

5

?

y)

b

0(ω)−˜ µ2+˜ ω2

?b

2

?

2α2r3

κ2

?b

h

5

(˜ a2

?

y)

b

1(ω)

Ψx

0(ω)

−?Jx

1?

+ 2r3

h

Ψx

?b

y)

2r3

κ2

h

5

Ψx

2(ω)

b

0(ω)

y)

r4

h

κ2

5

?

2

?

Ψx

2(ω)

?b

?

Ψx

0(ω)+

− ?Txx?

.

(3.18)

1Note that the contribution of the on-shell action is zero at the horizon since we can set Ψt to zero there.

– 14 –