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 –
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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
0.60.8 1.0
T
Tc
1.2 1.4
20
30
40
50
60
κ2
π4T4
5?Ttt?
c
(a)
0.6 0.81.0
T
Tc
1.21.4
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
0.00 0.02 0.04
T
µ
0.060.08
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 –
Page 16
Due to the breaking of the rotational symmetry we see a new coupling between the currents
Jy
1,2and the stress tensor Txyin this subset of the fluctuations. This new coupling generates
some interesting new physical effect: it induces a non-universal behavior of the ratio of shear
viscosity to entropy density and a flexoelectric effect known from nematic crystals. We will
study these effects in the next section.
4. Transport Properties
In this section we extract the transport properties of the holographic p-wave superfluid
from the correlation functions presented in the previous section. We split our analysis into
distinct transport phenomena.
4.1 Universal Shear Viscosity
Let us start by considering the helicity two mode hyz. It is well known that, in the isotropic
case, the corresponding component of the energy-momentum tensor may be written as2
?Tyz? = −(P + iωηyz)hyz.(4.1)
Using (D.4) we see that this result is still correct in the transversely symmetric case we
are studying here. The result also agrees with our gravity calculation (3.10)3. Thus the
shear viscosity is given by the well-known Kubo formula
ηyz= − lim
ω→0
1
ωIm(Gyz,yz) = − lim
ω→0
1
ω
2r4
κ2
h
5
Ξb
Ξb
2(ω)
0(ω).(4.2)
In the following we show that we can apply the proof for the universal result for the ratio
of the shear viscosity to entropy density described in [25]. In the ω → 0 limit the equation
of motion (3.6) corresponds to ∂rΠ = 0, with Π the conjugate momentum to the field
Ξ = gyyhyz. This is the decisive condition in order to apply the proof of [25]. Therefore we
conclude that here we obtain the universal result for the ratio of shear viscosity to entropy
density [23–25],
ηyz
s
In this subset we do not see any effect of the rotational symmetry breaking since the
fluctuation hyzis transverse to the condensate.
=
1
4π.
(4.3)
4.2 Thermoelectric Effect perpendicular to the Condensate
Now we relate the results of (3.15) to the thermoelectric effect on the field theory side. We
begin with the well known connection between electric ?J⊥
?Tt⊥? − µ?J⊥? transport perpendicular to the condensate direction (see e.g. [17,30,37] for
the same calculation for holographic s-wave superfluids), i.e.
?
?Q⊥?
2Note that gµν= ηµν− hµν.
3We do not see an ω4-term as in (3.10) in the linear hydrodynamic description since this term corresponds
to higher order term with four derivatives.
3? = ?J⊥? and thermal ?Q⊥? =
?J⊥?
?
=
?
σ⊥⊥Tα⊥⊥
Tα⊥⊥T¯ κ⊥⊥
??
E⊥
−(∇⊥T)/T
?
, (4.4)
– 15 –
Page 17
where the electric field E⊥and the thermal gradient −∇⊥T/T need to be related to the
background values of the gauge field
⊥
done in [30],
??a3
−∇⊥T
T
?a3
?b
0and the metric (Ψt)b
0. This identification is
E⊥= iω
⊥
?b
0.
0+ µ(Ψt)b
0
?
,
= iω (Ψt)b
(4.5)
Putting all together and comparing the relation of the electric and thermal transport to
the corresponding equations for ?Jy? and ?Tty? in (3.15), we can identify the transport
matrix of (4.4),
σ⊥⊥= −iG⊥,⊥
3,3
ω
?
?
= −α2rh
κ2
5
i
˜ ω
?
?
2?˜ a3
=i
?
⊥
?b
0
1
?˜ a3
⊥
ω?Jt
=i
ω?Ttt? + µ2σ⊥⊥.
?b
−˜ ω2
2
?
,
Tα⊥⊥= −i
T¯ κ⊥⊥= −i
ω
G⊥
3
t⊥− µG⊥,⊥
Gt⊥,t⊥+ µ2G⊥,⊥
3,3
3? − µσ⊥⊥,
ω
3,3
(4.6)
These results are in agreement with [30]. The coupling between thermal and electrical
transport is well known in condensed matter physics, since the charge carriers (electron or
0510
ω
1520
0
50
100
150
200
2πT
κ52Re(σ⊥⊥)
2α2T
01234567
0
2
4
6
8
Figure 3: Real part of the conductivity Re(σ⊥⊥) over the frequency ω/(2πT) for α = 0.032. The
color coding is as follows: blue T = ∞, red T = 1.34Tc, brown T = 1.00Tc, green T = 0.40Tc,
orange T = 0.19Tc. Note that the three curves with the highest temperature, blue, red and brown,
are nearly on top of each other. The agreement of the curves in the ω → 0 limit is due to the small
change in the strength of the Drude peak with temperature. Below Tc, the superfluid contribution
to the delta peak at ω = 0 is turned on and we obtain larger deviations from the T = ∞ curve,
since the area below the curves have to be the same for all temperatures (sum rule). Furthermore
the value for ω → 0 clearly asymptotes to 0 with decreasing temperature.
– 16 –
Page 18
051015
ω
20 2530
0
50
100
150
200
250
300
2πT
κ52Re(σ⊥⊥)
2α2T
0246810
0
2
4
6
8
10
Figure 4: Real part of the conductivity Re(σ⊥⊥) over the frequency ω/(2πT) for α = 0.316. The
color coding is as follows: blue T = ∞, red T = 1.00Tc, brown T = 0.88Tc, green T = 0.50Tc,
orange T = 0.19Tc. In this plot we see that the Drude peak has already a much stronger dependence
on the temperature than in the α = 0.032 case, since the blue and the red curve can be clearly
distinguished. Below Tcthe contributions of the superfluid phase to the delta peak leads again to
a tendency of the curve to vanish for frequencies in the gap region, since the area below the curves
have to be the same for all temperatures (sum rule).
holes) transport charge as well as heat. In this subset we do not observe any effect of the
breaking of the rotational symmetry since all the fields are in the transverse direction to
the condensate.
In figure 3, 4 and 5 we plot our numerical results for Re(σ⊥⊥) versus the frequency
ω/(2πT) for different values of α as defined in (2.3), namely α = 0.032 < αc, α = 0.316 ? αc
and α = 0.447 > αc, respectively. For large frequencies, i.e. ω ? 2πT, the conductivity
asymptotically has a linear dependence on the frequency (e.g. [38]),
Re(σ⊥⊥) →α2
κ52πωforω ? 2πT .(4.7)
For small temperatures, i.e. T < Tc, we see a gap opening up at small frequencies. The
size of the gap increases as the temperature is decreased. This is the expected energy gap
we know from superconductors. The gap ends at a frequency ωg with a sharp increase
of the conductivity. Beyond the gap the conductivity at small temperature, i.e. T < Tc,
is larger than the corresponding value at large temperature, i.e. T > Tc such that the
small temperature conductivities approach the asymptotic behavior (4.7) from above (cf.
[18,19,21]).
The value of Re(σ⊥⊥) at ω = 0 approaches zero with decreasing temperature. Below
Tcthe tendency for this part of the conductivity to vanish increases. Nevertheless, we still
find finite values even below Tc, i.e. these values seem to be suppressed but not identically
– 17 –
Page 19
01020
ω
30 40
0
100
200
300
400
2πT
κ52Re(σ⊥⊥)
2α2T
012345
0
2
4
6
8
10
Figure 5: Real part of the conductivity Re(σ⊥⊥) over the frequency ω/(2πT) for α = 0.447. The
color coding is as follows: blue T = ∞, brown T = 1.95Tc, green T = 1.00Tc, red T = 0.91Tc,
orange T = 0.34Tc. Again we see the same tendency as before for the curve to vanish at ω → 0
for decreasing temperatures. The strength of the Drude peak has a strong dependence on the
temperatures, since the blue and the brown curve are quite far apart (both curves were computed
for temperatures above Tc).
vanishing (c.f. [17]). In [34] it is shown that in the limit T → 0 there is a hard gap, i.e. the
value for the conductivity becomes zero. Finally, we observe that an increase in α leads to
a stronger suppression of the real part of the conductivity in the gap region.
Due to the sum rule for the conductivity, i.e. the frequency integral over the real
part of the conductivity is constant for all temperatures, a delta peak has to form at
zero frequency which contains the “missing area” of the gap region. The strength of the
delta peak has two contributions: the first is proportional to the superfluid density ns,
Re(σ⊥⊥) ∼ α2/κ2
contribution is a consequence of translation invariance of our system, the Drude peak, and
appears for all temperatures.
The delta peak is observed in the imaginary part of the conductivity by using the
Kramers-Kronig relation (see [17]),
5πnsδ(ω) and appears only for temperatures below Tc.The second
Im(σ⊥⊥) ?AD(α,T)
ω
+As(α)
ω
?
1 −T
Tc
?
,(4.8)
for T ? Tc, with As(α)
Drude peak. In figure 6 we present the imaginary part of the conductivity ωIm(σ⊥⊥) versus
the frequency ω/(2πT) for α = 0.316 and different temperatures. We see that ωIm(σ⊥⊥)
takes finite values for ω → 0 and T < ∞. The finite values above Tcare due to the Drude
peak, i.e. the ADpart of (4.8). Below Tcwe see a further contribution from the superfluid
?
1 −T
Tc
?
∝ nsand AD parametrizing the contribution from the
– 18 –
Page 20
012345
0
100
200
300
400
500
ω
2πT
ω
2πT
κ52Im(σ⊥⊥)
2α2T
Figure 6: Imaginary part of the conductivity ωIm(σ⊥⊥) over the frequency ω/(2πT) for α = 0.316.
The color coding is as follows: blue T = ∞, red T = 1.00Tc, brown T = 0.88Tc, green T = 0.50Tc,
orange T = 0.19Tc. The curves in this plot have a constant value for
by the δ-peak in the real part of the conductivity by the Kramers-Kronig relation. The values for
this constant are, in the same order as the temperatures above: 0, 8.0, 14.7, 50.3 and 362.4. Note
that we already see a finite value for T = 1.00Tc, this is due to the Drude peak. Below Tc the
values at ω = 0 are due to two contributions, first the Drude peak, as before, and second due to
the superfluid density.
ω
2πT→ 0, which is determined
density. By analyzing the temperature dependence of limω→0ωIm(σ⊥⊥), we get a smooth
curve, which is, however, not differentiable at Tcfor α ≤ αc, i.e. it behaves as equation
(4.8) anticipated. However, for α > αcwe see a jump at Tcas consequence of the jump in
the condensate. Furthermore ADand Ashave a non trivial dependence on α. Finally, as
expected, there is an increase in the superfluid density with decreasing temperature.
4.3 Non-Universal Shear Viscosity and Flexoelectric Effect
Now let us study the remaining three components of the helicity one modes, ?J⊥
and ?Tx⊥? as given by (3.17). We first focus on ?Tx⊥?, which for?a1
?Tx⊥? = −?Txx?hx⊥− iωηx⊥hx⊥,
1?, ?J⊥
0= 0 can be
2?
y
?b
0,?a2
y
?b
translated into the following dual field theory behavior
(4.9)
where ηx⊥is the second shear viscosity which is present in a transversal isotropic fluid (see
appendix D) and with ?Txx? as defined in (2.19). Here we see again that we can apply the
Kubo formula to determine the shear viscosity ηx⊥,
1
ωImηx⊥= − lim
ω→0
?
Gx⊥,x⊥?
.(4.10)
As described in [28], this shear viscosity has a non-trivial temperature dependence even
in the large N and large λ limit and is therefore not universal. In fig. 7 we compare our
– 19 –
Page 21
0.00.51.01.5
1.0
1.1
1.2
1.3
1.4
?
?
?
??
?
?
Figure 7: Ratio of shear viscosities ηyzand ηx⊥to entropy density s over the reduced temperature
T/Tcfor different values of the ratio of the gravitational coupling constant to the Yang-Mills coupling
constant α. The color coding is as follows: In yellow, ηyz/s for all values of α; while the curve
for ηx⊥/s is plotted in green for α = 0.032, red for α = 0.224 and blue for α = 0.316. The shear
viscosities coincide and are universal in the normal phase T ≥ Tc. However in the superfluid phase
T < Tc, the shear viscosity ηyz has the usual universal behavior while the shear viscosity ηx⊥is
non-universal.
numerical results for the ratio of the shear viscosity ηx⊥to the entropy density s with the
universal behavior of the shear viscosity ηyzfor different values of α. We see that in the
normal phase T ≥ Tc, the two shear viscosities coincide as required in an isotropic fluid.
In addition, the ratio of shear viscosity to entropy density is universal. In the superfluid
phase T < Tc, the two shear viscosities deviate from each other and ηx⊥is non-universal.
However it is exciting that ηx⊥/s ≥ 1/4π, such that the KSS bound on the ratio of shear
viscosity to entropy density [23] is still valid.
The difference between the two viscosities in the superfluid phase is controlled by α
as defined in (2.3). In the probe limit where α = 0, the shear viscosities also coincide in
the superfluid phase. By increasing the back-reaction of the gauge fields, i.e. rising α, the
deviation between the shear viscosities becomes larger in the superfluid phase as shown
in fig. 7. If α is larger than the critical value αc= 0.365 found in [22] (see fig. 2) where
the phase transition to the superfluid phase becomes first order, the shear viscosities are
also multivalued close to the phase transition as seen in fig. 8. Since there is a maximal α
denoted by αmax= 0.395 for which the superfluid phase exists (see fig. 2), we expect that
the deviation of the shear viscosity ηx⊥from its universal value is maximal for this αmax.
Unfortunately numerical calculations for large values of α are very challenging such that
we cannot present satisfying numerical data for this region. It is interesting that also the
– 20 –
Page 22
0.00.51.01.5
1.0
1.2
1.4
1.6
1.8
2.0
?
?
?
??
?
?
Figure 8: Ratio of shear viscosities ηyz(blue) and ηx⊥(red) to entropy density s over the reduced
temperature T/Tcfor α = 0.447, which is larger than the critical value where the phase transition
becomes first order: The shear viscosities coincide in the normal phase T ≥ Tcand are universal.
In the superfluid phase ηx⊥ is non-universal. Close to the phase transition, it is multivalued as
expected for a first order phase transition.
deviations due to λ and Nccorrections are bounded. In this case the bound is determined
by causality [39].
As described in our letter [28], we also have found numerically that for α < αc
1 − 4πηx⊥
s
∝
?
1 −T
Tc
?β
withβ = 1.00 ± 3%.(4.11)
Interestingly, the value of β appears to be independent of α. This result has recently been
confirmed by an analytic calculation in [40].
The non-universality of the shear viscosity can be understood in the following way.
For the ηx⊥component, the equation of motion (3.12a) in the ω → 0 limit includes also
non-vanishing source terms besides the derivative of the conjugate momentum Πxof Ψx,
i.e. ∂rΠx = source. This is in contrast to equation of motion which leads to the ηyz
component. Note that the source term depends on the condensate w and the fluctuation
a1
⊥and vanishes if the condensate w vanishes. Hence, as we confirm numerically in fig. 7,
when the condensate is absent (i.e. for the T > Tc case) we obtain again the universal
result, since in this case the same proof as described above for the helicity two mode applies.
For hx⊥ = 0, we obtain flavor charge transport, i.e. a flavor field a1,2
flavor current ?J⊥
way a±
⊥since they transform in the fundamental representation of the U(1)3
⊥
generates a
1,2?. In the unbroken phase it is useful to combine the fields a1,2
⊥± ia2
⊥in the
⊥= a1
– 21 –
Page 23
symmetry and are complex conjugate to each other. To make contact to the unbroken
phase, we also use this definition in the broken phase.
We also use the definition for the currents ?J±? = 1/2(?J1? ± i?J2?), such that the full
transport matrix becomes
+
−
where the flavor conductivities are given by
?
=α2r2
2κ2
5
⊥
G⊥,⊥
4
=α2r2
2κ2
5
⊥
Gx⊥⊥
2
?J⊥
?J⊥
?Tx⊥?
+?
−?
=
G⊥,⊥
+,+
G⊥,⊥
−,+
Gx⊥⊥
G⊥,⊥
+,−
G⊥,⊥
−,−
Gx⊥⊥
G⊥
+
x⊥
G⊥
−
x⊥
−?Txx? − iωηx⊥
a+
⊥
a−
⊥
hx⊥
,(4.12)
G⊥,⊥
±,±(ω) =1
4
G⊥,⊥
??˜ a1
1,1(ω) + G⊥,⊥
?b
2,2(ω) ∓ i
?˜ a2
?
G⊥,⊥
1,2(ω) − G⊥,⊥
−(˜ µ ∓ ˜ ω)2
2
2,1(ω)
??
?˜ a2
h
⊥
1(ω)
?b
?˜ a1
0(ω)
+
⊥
?b
1(ω)
?b
?˜ a2
⊥
0(ω)
?
?b
?
?
∓ i
??˜ a1
y
?b
1(ω)
?b
y
0(ω)
−
?˜ a2
y
?b
1(ω)
?b
?˜ a1
y
0(ω)
??
,
±,∓(ω) =1
?
G⊥,⊥
??˜ a1
1,1(ω) − G⊥,⊥
?b
2,2(ω) ± i
?˜ a2
G⊥,⊥
1,2(ω) + G⊥,⊥
??˜ a1
2,1(ω)
??
?˜ a2
?b
y
h
⊥
1(ω)
?b
?˜ a1
1(ω) ∓ iGx⊥⊥
0(ω)
−
⊥
1(ω)
?b
?˜ a2
2(ω)
⊥
0(ω)
= −?Jx
± i
y
?b
1(ω)
?b
+r3
κ2
?˜ a2
y
0(ω)
+
y
?b
1(ω)
?b
2(ω)
?b
y
?Ψx
?˜ a1
y
0(ω)
??
?Ψx
,
±(ω) =1
?
?
Gx⊥⊥
1?
4
h
5
??Ψx
? ?˜ a1
?˜ a1
0(ω)
?b
∓ i
?b
2(ω)
?b
y
?Ψx
?˜ a2
± i
y
?˜ a2
0(ω)
?b
?
,
G⊥
±
x⊥(ω) =1
2
G⊥
1
x⊥(ω) ± iG⊥
2
x⊥(ω)= −?Jx
1?
4
+α2r3
h
κ2
5
1(ω)
?b
0(ω)
1(ω)
?b
0(ω)
?
(4.13)
.
First note that for µ = 0 where the SU(2) symmetry is restored, i.e. a1≡ a2, the Green’s
function is diagonal and G⊥,⊥
is still valid for µ ?= 0, since the a±do not couple to each other, while G⊥,⊥
µ ?= 0. In the unbroken as well as in the broken phase we find
G⊥,⊥
+,+= G⊥,⊥
−,−= G⊥,⊥
3,3. In the unbroken phase, G⊥,⊥
+,−= G⊥,⊥
+,+?= G⊥,⊥
−,+≡ 0
−,−for
−,−(ω) = G⊥,⊥
G⊥
+
+,+(−ω)∗,
x⊥(−ω)∗
G⊥,⊥
Gx⊥⊥
+,−(ω) = G⊥,⊥
+(ω) = Gx⊥⊥
−,+(−ω)∗,
−(−ω)∗,
x⊥(ω) = G⊥
−
and
(4.14)
as expected since a1(ω) =?a1(−ω)?∗, a2(ω) =?a2(−ω)?∗and Ψx(ω) = (Ψx(−ω))∗.
and 9(d), showing Im(G⊥,⊥
±,±), that for temperatures T > Tc, the quasinormal modes tend
towards the origin (in these plots we see their projection on the real axis) (see e.g. [18,41]).
For T ≤ Tc we see a pole at the origin which is due to the massless Nambu-Goldstone
modes. These Nambu-Goldstone modes are related to rotations of the director ?J1
real space which are generated by the fluctuations a1
In figure 9 we plot the real and imaginary parts of G⊥,⊥
±,±(ω). We see in fig. 9(c)
x? in
⊥
4. Furthermore, as expected for
4The other Nambu-Goldstone mode is related to the change of the phase of the condensate and corre-
spond to the fluctuation a2
xwhich shows up in the helicity zero sector.
– 22 –
Page 24
2468101214
?40000
?30000
?20000
?10000
0
ω
2πT
4κ52Re(G⊥,⊥
α2T2
−,−)
(a)
2468101214
?10000
?8000
?6000
?4000
?2000
0
2000
4000
ω
2πT
4κ52Re(G⊥,⊥
α2T2
+,+)
(b)
246810 1214
0
5000
10000
15000
20000
ω
2πT
4κ52Im(G⊥,⊥
α2T2
−,−)
(c)
2468101214
0
5000
10000
15000
20000
ω
2πT
4κ52Im(G⊥,⊥
α2T2
+,+)
(d)
Figure 9: These plots show the real and imaginary part of the correlators G⊥,⊥
frequency ω/(2πT) for α = 0.316 at different temperatures: T = ∞ blue line, T = 3.02Tcred line,
T = 1.00Tcbrown line, T = 0.88Tcgreen line and T = 0.50Tcorange line.
±,±versus the reduced
2468101214
?14000
?12000
?10000
?8000
?6000
?4000
?2000
0
ω
2πT
4κ52Re(G⊥,⊥
α2T2
+,−)
(a)
2468101214
?500
0
500
1000
1500
2000
2500
3000
ω
2πT
4κ52Im(G⊥,⊥
α2T2
+,−)
(b)
Figure 10: These plots show the real and imaginary part of the correlator G⊥,⊥
frequency ω/(2πT) for α = 0.316 at different temperatures: T = ∞ blue line, T = 3.02Tcred line,
T = 1.00Tc brown line, T = 0.88Tc green line and T = 0.50Tc orange line. The curves for the
temperatures above Tcare exactly zero for all frequencies.
+,−versus the reduced
large frequencies, the Green’s function grows proportional to ω2in the ±± components
as for the correlator G⊥,⊥
3,3. In figure 9 the correlators for the different temperatures do
not seem to have the same asymptotic behavior. However, in the present case we have
contribution from terms such as ωµ, i.e. of first order in ω, which are not existent in the
G⊥,⊥
3,3
component. Hence to see that all correlators have the same limit, larger values of
ω have to be considered. Even not present in our figures we verified numerically that the
asymptotics of the correlators at different temperatures agree. A more detailed study of
– 23 –
Page 25
02468 1012 14
0
5000
10000
15000
20000
25000
ω
2πT
2κ52Re(G⊥
α2T2
+
x⊥)
(a)
02468 10 1214
?5000
0
5000
10000
15000
20000
ω
2πT
2κ52Im(G⊥
α2T2
+
x⊥)
(b)
02468101214
0
5000
10000
15000
20000
25000
30000
ω
2πT
2κ52Re(G⊥
α2T2
−
x⊥)
(c)
02468101214
?10000
?5000
0
5000
ω
2πT
2κ52Im(G⊥
α2T2
−
x⊥)
(d)
Figure 11: These plots show the real and imaginary part of the correlators G⊥
reduced frequency ω/(2πT) for α = 0.316 at different temperatures: T = ∞ blue line, T = 3.02Tc
red line, T = 1.00Tcbrown line, T = 0.88Tcgreen line and T = 0.50Tcorange line. The curves for
the temperatures above Tcare exactly zero for all frequencies.
±
x⊥versus the
this sector in the probe approximation is in preparation [42].
In figure 10 we plot G⊥,⊥
not couple in the unbroken phase. Furthermore, below Tc, a pole at ω = 0 due to the
Nambu-Goldstone mode appears. We do not show G⊥,⊥
alike. Nevertheless, there is a difference between them in the broken phase. The difference
arises from the contributions to the correlators due to the mixed terms, G⊥,⊥
the corresponding equation in (4.13). However, these are suppressed in relation to G⊥,⊥
which contains the Goldstone mode in the broken phase.
Finally, in figure 11 we show G⊥
±
parts of the correlators is not included in the corresponding plots since it just shifts the
curves by a constant. Furthermore, we have that Im(G⊥
const = Re(Gx⊥⊥
±), i.e. there is a constant offset between the real parts of these correlators.
We expect that this constant offset is generated by the term?Ψx
term of Ψxis probably not sourced. A analytic calculation similar to [40] may confirm this
claim. In the unbroken phase these correlators vanish since the differential equations of the
corresponding fields decouple. In the broken phase the correlators present a rich structure,
which we cannot fully address at present. However, it seems that the coupling between the
a±
⊥flavor fields and the strain hx⊥generates new quasiparticles which appear as bumps in
+,−(ω). We see that this correlator vanishes since the a±do
−,+(ω) since G⊥,⊥
+,−and G⊥,⊥
−,+look
1,2and G⊥,⊥
2,1, in
1,1,
x⊥(ω). Note that the contribution of ?Jx
1? to the real
±
x⊥) = Im(Gx⊥⊥
±) and Re(G⊥
?b
±
x⊥)−
?b
2(ω)/?˜ a1
y
0(ω) which may
be constant in the limit ω → 0 since a1
yhas a normal mode at ω = 0 and the subleading
– 24 –
Page 26
the curves.
In addition to the flavor conductivity and the shear viscosity we obtain a coupling
between the stress ?Tx⊥? and the flavor fields a±
hx⊥described in (4.12). This coupling introduces an effect which is called flexoelectric effect
in nematic crystals [14] and only appears in fluids with broken rotational symmetry. We
have a current ?Jx
flavor fields a±
⊥. This interaction induces a force on the current which pushes the current
in its perpendicular direction generating the stress ?Tx⊥?. In the similar way, a strain
hx⊥introduces an inhomogeneity in the current ?Jx
generates the currents ?J⊥
⊥as well as the currents ?J⊥
±? and the strain
1? in a favored direction in the background which interacts with the
1? resulting in a flavor field a±
⊥which
±?.
5. Conclusion
In this paper we have studied transport phenomena in holographic p-wave superfluids con-
structed in the SU(2) Einstein-Yang-Mills theory. We classify the perturbations about
equilibrium according to their transformation properties under the symmetry group. At
zero momentum, there is an SO(2) symmetry left which allows us to divide the perturba-
tions into different helicity sectors: helicity two, one and zero states. While the helicity
two state is trivial and leads to the universal ratio of shear viscosity to entropy density, the
helicity one states are non-trivial. Due to a Z2parity, this sector splits into two blocks. In
the first block we find the thermoelectric effect transversal to the direction favored by the
condensate. In the second block we obtain two interesting new phenomena: a non-universal
shear viscosity and a flexoelectric effect. These two effects are due to the anisotropy of our
system.
Anisotropic fluids have been studied in particular in the context of nematic crystals
whose hydrodynamic description is given in [14, 43].
the connection of the hydrodynamic description of anisotropic fluids with gauge/gravity
duality. The results we obtain in this paper are in agreement with this description, i.e. the
transport coefficients found here can be related to the ones in [43]. However since we have
not studied the helicity zero modes in much detail, we have not yet described all transport
properties. In particular, the thermoelectric effect along the condensate as well as the
coefficients ζx, ζyand λ described in appendix D are still missing. In the future we plan to
study these coefficients in detail. This study may also lead to a covariant hydrodynamic
description of anisotropic superfluids.
In this paper, we have initiated
Furthermore, analytic results close to the phase transition can be found for small values
of α [40], which determines the ratio of the gravitational to Yang-Mills coupling. On the
one hand this analytic approach allows for a detailed study of the transport coefficients
close to the phase transition. On the other hand it also permits us to use the fluid/gravity
correspondence in order to obtain the complete hydrodynamic description of the system
directly from gravity. Similar analyzes for holographic s-wave superfluids can be found
in [10–12]. We intend to follow this line of thought further.
– 25 –
Page 27
Acknowledgements
We are grateful to A. Buchel, M. Haack, J. Mas, M. Natsuume and G. Policastro for
discussions. J.E. is grateful to the KITP Santa Barbara for hospitality during the final
stages of this work. This work was supported in part by The Cluster of Excellence for
Fundamental Physics - Origin and Structure of the Universe.
A. Holographic Renormalization
The goal we are pursuing in this section is to find covariant counterterms which can be
subtracted from the action (3.7) and (3.13) in order to make it finite. We follow the lead
of the references [44,45] to perform the holographic renormalization.
A.1 Asymptotic Behavior
In this section we look at the behavior of the fields at the horizon and at the boundary. We
want to calculate real-time retarded Green’s functions [36,46], i.e. at the horizon, besides
regularity5, we have to fulfill the incoming boundary condition. For this purpose we plug
in the ansatz,
F(r)??r→rH= ?β
for the behavior of the fields near the horizon, into the equations of motion (3.6), (3.11)
and (3.12). It turns out that, as expected, we obtain two possibilities for β, namely
h
?
i≥0
?i
hFh
i,with?h=
r
rh
− 1,(A.1)
β = ±i
ω
4πT,
(A.2)
with T being the temperature defined in equation (2.12). As said before, we choose the
solution with the “−“ sign which corresponds to the incoming boundary condition. Note
that the other solution represents the outgoing boundary condition.
Our ansatz at the boundary is similar to the one used for the background calculation
in section 2. However, here we have to add a logarithmic term to get a consistent solution
(c.f. [44]). Therefore we use
?
F(r)??r→rbdy=
?
i≥0
?i
b
Fb
i+1
2
ˆFb
iln?b
?
,with?b=
?rh
r
?2
.(A.3)
Let us now use the above expansions for the helicity one states. In the case of the
equations (3.11) we have 3 independent expansion coefficients at the boundary (4 free
parameters from the 2 second order differential equations minus 1 free parameter due to
the constraint). We choose them to be?a3
the remaining two parameters we can get rid of one by using the constraint equation
(3.11b). Therefore we are left with one free parameter at the horizon, we choose
y
?b
0,?a3
y
?b
1and?Ψt
?b
0. At the horizon we already
halved the independent parameters by choosing the incoming boundary condition. From
?a3
y
?h
0.
5Even with all fluctuations switched on, there is no need for a further constraint besides φ(rH) = 0 at
the horizon to guarantee regularity.
– 26 –
Page 28
When solving these equations numerically we set
values of ω.
?a3
y
?h
0= 1 and scan through different
We can perform similar considerations for the second part (equations (3.12)). In this
case it is even simpler. We do not have any constraint, just three fields and their corre-
sponding equations of motion. Therefore at the boundary we have six independent pa-
rameters, namely?a1
yy
horizon by choosing the incoming boundary condition. Again by choosing the values for
all fields at the horizon the system is fully determined.
y
?b
0,?a1
y
?b
1,?a2
y
?b
0,?a2
y
?b
1,?Ψx
?b
0and?Ψx
?b
2. At the horizon we have
?Ψx
?h
0,?a1
?h
0,?a2
?h
0. Note that, as before, we already fixed three free parameters at the
Notice that the same is true for the helicity 2 state. We have again 2 independent
components at the boundary, namely
0and
boundary condition and
0at the horizon. Therefore as before, the equation is fully
determined.
?Ξ?b
?Ξ?b
2which are fixed by the incoming
?Ξ?h
In the following we state the first few non-vanishing coefficients of the expansion at
the boundary of the different fields. We need them later on to determine divergences in the
on-shell action and to calculate the Green’s function. The explicit form of the dependent
coefficients are
?ˆ a1
?ˆ a2
?ˆ a3
(Ξ)b
y
?b
?b
?b
1=1
1=1
1=1
2
?
?
˜ ω2?a1
˜ ω2?a2
2˜ ω2?a3
4˜ ω2?Ξ?b
2= −α2˜φb
(Ψx)b
y
?b
?b
0,
0− 2i˜ ω˜ µ?a2
y
?b
?b
0+ ˜ µ2?a1
y
?b
?b
0
?
?
, (A.4a)
y
2
y
?b
0,
?˜ a3
?b
0+ 2i˜ ω˜ µ?a1
y
0+ ˜ µ2?a2
y
0
, (A.4b)
y
y
(A.4c)
1=1
?ˆΞ
?b
?ˆΨt
?ˆΨx
2=
1
16˜ ω4?Ξ?b
?b
2=
0,(A.4d)
?Ψt
?b
2y
?b
0,
3= −α2˜ ω2
1
16˜ ω4?Ψx
3
˜φb
2
?˜ a3
y
?b
0, (A.4e)
1=1
4˜ ω2?Ψx
0,
?b
?b
0. (A.4f)
Note that µ ≡ φb
0and φb
1are the expansion coefficients of φ(r) at the boundary.
We do not state the expansion at the horizon, since there is no additional information
to equation (A.1), and the explicit form of the non-independent coefficients is very long.
– 27 –
Page 29
A.2 Counterterms
By plugging the expansions (A.3) into (3.7) and (3.13), resulting in
?
−3
−1
0
+ iα2˜ ω˜ µ?˜ a1
+
32˜ ω4ln?b
?1
+1
4α2˜ µ2ln?b
0
− iα2˜ ω˜ µ?˜ a1
Note that Son-shell=
κ2
5
expansion coefficient is always a field of −k and the second of k, e.g.
is obvious from the last few lines of (A.5) that we have to add counter terms to the on-shell
action Son-shellto take the divergences in r into account.
The terms that have to be considered are the ones in (A.5) with explicit r dependence,
?1
1
4α2ln?b
0
0
− iα2˜ ω˜ µ?˜ a1
First we need the induced metric γ on the r = rbdyplane. The induced metric is defined
by
γµν=∂xM
∂˜ xµ
Lrb
r4
h
=?Ξ?b
0
?Ξ?b
2˜ mb
2+
−2fb
2+ α2??˜ a1
y
?b
64˜ ω4+1
2−˜ mb
0
2
??Ξ?b
?b
?b
0
2+?Ψx
1+?˜ a2
− ˜ wb
?b
?˜ a2
?˜ a1
0
?Ψx
y
?b
?b
2+
?
?b
4fb
2−˜ mb
?˜ a3
0+ 2˜φb
0
2
??Ψx
?b
0
2
0
?Ψt
?b
0
y
0
2?
?˜ a1
y
+ α2?
?b
y
?b
2
0
?b
0
1+?˜ a3
?Ψx
y
0
y
?b
?˜ a3
1
?
y
4α2˜ µ2
??˜ a1
y
?b
?˜ a2
2+?˜ a2
y
0
y
0
y
?b
2
?b
0
?Ψt
?b
0
?
0
?b
?1
8?b˜ ω2+1
4α2˜ ω2−1
???Ψ?b
y
0
2?
.
0
2+?Ψx
y
0
?b
0
2?
?b
−
4α2˜ ω2ln?b
??˜ a1
?˜ a2
?
???˜ a1
y
????r=rbdy
?b
2+?˜ a2
?b
2+?˜ a3
y
0
2?
y
?b
0ln?b
2+?˜ a2
?b
0
y
?b
1
0
y
?b
d4k
(2π)4 Lrbwith r = rbdy? 1. Moreover, as before, the first field
(A.5)
?Ξ?b
0(−k)?Ξ?b
2(k). It
8?b˜ ω2+1
32˜ ω4ln?b
??˜ a1
y
0
???Ξ?b
2+?˜ a2
0ln?b
0
2+?Ψx
?b
?b
?b
0
2?
2?
,
?
˜ ω2
y
?b
?b
y
2+?˜ a3
y
0
+ ˜ µ2
??˜ a1
y
?b
0
2+?˜ a2
y
?b
0
2??
and
?b
?˜ a2
y
∂xN
∂˜ xνgMN(r)
????r=rbdy
, (A.6)
resulting in
ds2
rbdy= −N(rbdy)σ(rbdy)2dt2+
r2
bdy
f(rbdy)4dx2+ r2
bdyf(rbdy)2(dy2+ dz2).(A.7)
We do not literally derive the covariant counter terms here in this work. However, by
looking at the counter terms of B. Sahoo and H.-U. Yee calculated in [45] we get an idea
how they should look like, namely some combinations of R[γ], Rµν[γ] and Fa
two are the Ricci scalar and Ricci tensor on the induced surface respectively, the latter
µν. The first
– 28 –
Page 30
is the field strength tensor on that surface. Possible covariant combinations of the three
terms are√−γR[γ],√−γRµν[γ]Rµν[γ] and√−γFa
expansion for r ? 1.
√−γR[γ]
2
√−γRµν[γ]Rµν[γ]
2
√−γFa
??a1
Therefore by adding the real space action
?
to the action Son-shell(3.7) and (3.13) we get a divergence-free theory (up to the second order
in the fluctuations) for rbdy? 1, i.e. also the real time Green’s functions are divergence
free. The renormalized rbdy? 1 Lagrangian is then given in (3.8) and (3.14).
µνFaµν. Now lets have a look at their
????r?1
????r?1
=r2ω2
??Ξ?b
??Ξ?b
?
0
2+?Ψx
2+?Ψx
??a1
?b
?b
0
2?
,
????r?1
=ω4
0
?b
0
2?
and
µνFaµν
= −2ω2
y
?b
0
2+?a2
?b
y
?b
− 4iωµ?a1
0
2+?a3
y
?b
?b
0
2?
?a2
+ µ2
y
0
2+?a2
y
0
2?
y
0
y
?b
?????r=rbdy
0
?
.
(A.8)
Sct= −
d4x√−γ
?1
4R[γ] +1
16Rµν[γ]Rµν[γ]ln?b−α2
8Fa
µνFaµνln?b
(A.9)
B. Constructing the Gauge Invariant Fields
B.1 Residual Gauge Transformations
The transformations we have to look at are diffeomorphisms and SU(2) gauge transforma-
tions. On the one hand, we demand that the fields be diffeomorphism invariant, i.e.
δΣΦ = LΣΦ = 0.(B.1)
LΣis the Lie derivative along Σ, i.e.
LΣgMN= ∇MΣN+ ∇NΣM= ∂MΣN+ ∂NΣM− 2ΓP
LΣAa
with ΓP
MNbeing the Christoffel symbol.
On the other hand they have to be invariant under the SU(2),
MNΣP,
P∂MΣP,
M= ΣP∇PAa
M+ Aa
P∇MΣP= ΣP∂PAa
M+ Aa
(B.2)
Φ → M(Λ)Φ = Φ, (B.3)
with M(Λ) being the SU(2) transformation matrices, this is equivalent to
δΛΦ = 0.(B.4)
Φ stands for the physical modes in our system and is composed of the helicity 0 fields
ξtx, ξt, ξx, ξy,aa
t, with a = 1,2,3. The invariance of Φ under the above transforma-
tions translates into
?
with τibeing the r dependent coefficients.
xand aa
δΦ = (δΣ+ δΛ)Φ =
3
a=1
(τaδaa
x+ τ3+aδaa
t) + τ7δξtx+ τ8δξt+ τ9δξx+ τ10δξy= 0,(B.5)
– 29 –
Page 31
B.1.1 Diffeomorphism Invariance
Let us look at the invariance under diffeomorphisms. We begin by defining
ˆ gMN= gMN+ hMN,
ˆAa
M= Aa
M+ aa
M.
(B.6)
Furthermore note that ΣM(and later on Λ) are of the same order as the fluctuations.
Through the gauge choice hMr= 0 we can determine the form of ΣMup to some constants,
because
LΣˆ gMr= 0
⇒∂MΣr+ ∂rΣM− 2ΓP
MrΣP= 0.
(B.7)
Note that we just need the Christoffel symbols to zeroth order in fluctuations, i.e. the
background Christoffel symbols. They are
Γr
rr=c4?
c4,Γt
tr=c1?
c1,Γx
xr=c2?
c2,Γy
yr=c3?
c3,
Γr
tt=c1c1?
c42,Γr
xx= −c2c2?
c42
andΓr
yy= −c3c3?
c42,
(B.8)
with
ds2= gMNdxMdyN= −c1(r)2dt2+ c2(r)2dx2+ c3(r)2(dy2+ dz2) + c4(r)2dr2.
We get 4 equations (+1 for the z component which is exactly the same as the one for the
y component), which read
(B.9)
− iωΣr+ Σt?− 2c1?
ikΣr+ Σx?− 2c2?
Σy?− 2c3?
2Σr?− 2c4?
c1Σt= 0,(B.10a)
c2Σx= 0,(B.10b)
c3Σy= 0, (B.10c)
c4Σr= 0.(B.10d)
We work in momentum space, i.e. the ansatz used is
ΣM(t,x,r) =
?
d4x eikµxµΣ(ω,k,r),(B.11)
with kµ= (ω,k,0,0). The solutions to the equations above are
Σt(ω,k,r) = Ktc2
1+ iωKrc12A, with A =
?
?
drc4
c12,
Σx(ω,k,r) = Kxc2
2− ikKrc22B, with B =drc4
c22,
Σy(ω,k,r) = Kyc32,
Σr(ω,k,r) = Krc4,
(B.12)
– 30 –
Page 32
with Kibeing constants. Using these solutions in the remaining equations δΣξi, we get
δΣξt=2iω
c12Σt,
δΣξx=2ik
c22Σx,
δΣξy= 0,
δΣξtx= −iω
c22Σx+ik
c22Σt.
(B.13)
Here we see already that ξyis a physical mode!
Applying the same procedure to the gauge fields, i.e. δΣaa
µ= LΣˆAa
µ, it results in
δΣa1
x=w?
c42Σr+ikw
φ?
c42Σr+iωφ
c22Σx,
δΣa3
t=
c12Σt.
(B.14)
The Lie derivatives of the remaining components vanish.
B.1.2 SU(2) Gauge Invariance
This transformation only affects the gauge fields, therefore we do not have to care about
the metric fluctuations here. A field in the adjoint SU(2) representation transform under
the SU(2) as
δΛaa
M= ∇MΛa(t,x,r) + ?abcAb
MΛc. (B.15)
Again we constrain the possible Λaby using the gauge choice aa
r= 0, i.e.
0 = δΛaa
⇒ Λa(t,x,r) = Λa(t,x).
?d4x eikµxµΛa(ω,k) with kµ= (ω,k,0,0) to calculate
with the rest and we can forget about them. We end up with
r= ∇rΛa(t,x,r) = ∂rΛa(t,x,r)
(B.16)
We choose the ansatz Λa(t,x) =
δΛaa
µ. Note that by using the definition above of kµthe y and z components do not mix
δΛa1
δΛa2
δΛa3
x= ikΛ1,
x= ikΛ2− wΛ3,
x= ikΛ3+ wΛ2,
δΛa1
δΛa2
δΛa3
t= −iωΛ1− φΛ2,
t= −iωΛ2+ φΛ1,
t= −iωΛ3.
(B.17)
B.2 Physical Fields
Plugging everything into equation (B.5) results in 6 equations, due to the fact that Σt, Σx, Σr, Λ1, Λ2
– 31 –
Page 33
and Λ3are independent. The equations are
0 =ikτ1− iωτ4+ φτ5,
0 =ikτ2+ wτ3− φτ4− iωτ5,
0 = − wτ2+ ikτ3− iωτ6,
0 = − iωτ7+ 2ikτ9+ ikwτ1,
0 =2iωτ8+ ikc12
c22τ7+ iωφτ6,
0 =w?τ1+ φ?τ6.
(B.18)
The four physical fields, Φi=?10
i=1τi·(Helicity 0 fields), we get by solving above equations,
are
Φ1=ξy,
Φ2=a1
t+iω
φa2
ω2c22ξt+2k
x+k
ωa1
− iω2ww?+ k2φφ?
t+ik?ω2− φ2?
(k2− w2)φa2
ωξtx,
2wξx−w?
x+w?ω2− φ2?
(k2− w2)φa3
x,
Φ3=ξx−k2c12
Φ4=a1
t−1
φ?a3
t+φw?
2φ?ξt−k?ω2w?+ wφφ??
ω (k2− w2)φ?a3
x
ωφ?(k2− w2)a2
x.
(B.19)
C. Numerical Evaluation of Green’s Functions
Here we review the algorithm to evaluate Green’s function when there is operator mixing
[47]. For concreteness we present the algorithm for the second block of the helicity one
fields where we have the mixing of the fluctuations a1
action (3.13) and have a look at the components we need,
y,a2
y,Ψx. Let us first go back to the
Son-shell⊃
1
κ2
?3
5
2r3f6√Nσ − r2f6Nσ − r3f5Nσf?−1
+rα2f4Nσw?
2
?
?
d4k
(2π)4
?
−1
4r3f6NσΨxΨx?−rα2Nσ
2f2
4r3f6σN?−1
a1
ya1
y
?−rα2Nσ
2f2
a2
ya2
y
?
?
+
2r3f6Nσ?
Ψx2
a1
yΨx
?ΛT(−k,r)A(k,r)∂rΛ(k,r) + ΛT(−k,r)B(k,r)Λ(k,r)?????r=rbdy
y(k,r),Ψx(k,r)?, A(k,r) and B(k,r) are the matrices containing the
?????r=rbdy
=
=
1
κ2
5
d4k
(2π)4
.
(C.1)
ΛT(k,r) =?a1
y(k,r),a2
– 32 –
Page 34
coefficients of the field’s bilinear forms. Their explicit form for r = rbdy? 1 is
A(k,rbdy) =
−
α2r3
bdy
2
0
00
−
α2r3
bdy
2
0
0
r5
bdy
4
0
−
,B(k,rbdy) =
0 0 −α2wb
0 0
0 0
1r2
h
0
2r4
4fb
h
.
(C.2)
Note that all components of B(k,rbdy) are contact terms.
The next step is to define the boundary condition at the horizon for the vector Λ(k,rh).
This includes the incoming boundary condition we already introduced before (c.f. eq.
(A.1)). We define
(a)(k,r → rH) ? ?−iω
ΛI
4πT
h
?
eI
(a)+ O(?h)
?
,(C.3)
the index I refers to the three fields a1
second order differential equations we need 6 independent boundary conditions. We already
halved them by demanding incoming boundary conditions. We determine the remaining 3
conditions by choosing 3 linear independent vectors e(1), e(2)and e(3), with
y, a2
yand Ψx. Due to the fact that we have 3 coupled
eT
(1)= (1,0,0),eT
(2)= (0,1,0),eT
(3)= (0,0,1).(C.4)
Note that alternate choices are possible, but we get the best numerical result (with least
noise) using these values. We are now able to generate three linearly independent solutions
for the equations of motion by solving them numerically using the three linearly inde-
pendent boundary conditions. We put them together in a 3 by 3 matrix H(k,r) defined
by
HI
a(k,z) = ΛI
(a)(k,r),(C.5)
with I = 1,2,3 and a = 1,2,3. Now we can define a further matrix
H(k,r) = H(k,r)H(k,rbdy)−1.(C.6)
This matrix is basically the fields divided by the field’s value at the boundary, i.e. if we had
decoupled differential equations we would get H = diag?Λ1(r)/Λ1(rbdy),Λ2(r)/Λ2(rbdy),...?.
H?(k,r) · H(k,rbdy)−1.
H?(r) = diag
derivative of the field and the field into account. This is the know result [36,48]. What is
missing for the Green’s function is the r dependent prefactor, which is exactly the content
of A(k,r). Therefore we finally get
Next we take the derivative with respect to r of this matrix, leading us to H?(k,r) =
Again looking at a possible decoupled case we would obtain
?
Λ1?(r)/Λ1(rbdy),Λ2?(r)/Λ2(rbdy),...
?
which took the ratio between the
GR(ω) = 2 lim
rB→∞A(ω,rbdy)H?(ω,rbdy) + contact- and counter terms,(C.7)
which allows us to calculate the retarded Green’s function in systems where the operators
may mix.
– 33 –
Page 35
D. General Remarks on Viscosity in Anisotropic Fluids
In general, viscosity refers to the dissipation of energy due to any internal motion [49].
For an internal motion which describes a general translation or a general rotation, the
dissipation is zero.Thus the dissipation depends on the gradient of the velocities uµ
only in the combination uµν=1
function Ξ =1
are given by
ηµνλρ= ηνµλρ= ηµνρλ= ηλρµν.
2(∇µuν+ ∇νuµ), and we may define a general dissipation
2ηµνλρuµνuλρ, where ηµνλρdefines the viscosity tensor [13]. Its symmetries
(D.1)
The part of the stress tensor which is dissipative due to viscosity is defined by
Πµν= −∂Ξ
∂uµν
= −ηµνλρuλρ.(D.2)
We consider a fluid in the rest frame of the normal fluid ut= 1. To satisfy the condition of
the Landau frame uµΠµν= 0, the stress energy tensor and thus the viscosity has non-zero
components only in the spatial directions i,j = {x,y,z}. In general only 21 independent
components of ηijklappear in the expressions above.
For an isotropic fluid, there are only two independent components which are usually
parametrized by the shear viscosity η and the bulk viscosity ζ. The dissipative part of the
stress tensor becomes Πij= −2η(uij−1
In a transversely isotropic fluid, there are five independent components of the tensor
ηijkl. For concreteness we choose the symmetry axis to be along the x-axis. The non-zero
components are given by
3δijul
l) − ζul
lδijwhich is the well-known result.
ηxxxx= ζx− 2λ,ηyyyy= ηzzzz= ζy−λ
ηyyzz= ζy−λ
ηxyxy= ηxzxz= ηxy.
2+ ηyz,
ηxxyy= ηxxzz= λ,
2− ηyz,
ηyzyz= ηyz,
(D.3)
The non-zero off-diagonal components of the stress tensor are given by
Πxy= −2ηxyuxy,
Πyz= −2ηyzuyz,
Πxz= −2ηxyuxz,
(D.4)
So far, we only considered the contribution to the stress tensor due to the dissipation
via viscosity and found the terms in the constitutive equation which contain the velocity of
the normal fluid uµ. In general, also terms depending on the derivative of Nambu-Goldstone
boson fields vµ= ∂µϕ, on the superfluid velocity and on the velocity of the director may
contribute to the dissipative part of the stress tensor. Here the director is given by the
vector pointing in the preferred direction. However these terms do not contribute to the
off-diagonal components of the energy-momentum tensor for the following reasons: (1)
a shear viscosity due to the superfluid velocity leads to a non-positive divergence of the
entropy current [49,50] and (2) no rank two tensor can be formed out of degrees of freedom
of the director if the gradients of the director vanish [43]. In our case the second argument
– 34 –
Page 36
is fulfilled since the condensate is homogeneous and the fluctuations depend only on time.
These degrees of freedom will generate additional transport coefficients, but they do not
change the shear viscosities. Thus we can write Kubo formulae which determine the shear
viscosities in terms of the stress energy correlation functions.
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