70 Chapter 3. Holographic methods at finite temperature
Applying (3.117) and assuming∆+ > ∆−, ∆+ >0we obtain the two-point function of op-eratorsOby an expansion in the radial coordinaterand taking the boundary limit afterwards
hOOi ∝ B
A +contact terms. (3.121)
The poles of the retarded correlator thus correspond to the zeroes of the connection coeffi-cientA. On the other handAis determined by the coefficients of the equation of motion for the field fluctuationφ and therefore A = 0 is a particular choice of boundary condition for that field fluctuationφ. As an example consider∆− = 0, ∆+ = 2and B(r, ω,q) ∝ r3 and rbdy=∞. Then
hOOi ∝ lim
r→rbdy
r3 −2Br−3
A+Br−2 +contact terms. (3.122) Now we are ready to connect our holographic considerations back to the gravity definition of quasinormal modes given above (3.116). Comparing the two approaches we conclude that the condition for having quasinormal modes coming from gravity (3.116) and the boundary condition for the field fluctuation in AdS/CFTA = 0are identical. For this reason the quasi-normal frequencies of black hole excitations are identical to the poles of the retarded two-point correlator of their AdS/CFT-dual operators.
4
Holographic thermo- and hydrodynamics
In this chapter I present my (partly unpublished) own work on introducing a chemical isospin and baryon potential (cf. section 3.2.2) into the thermal N = 4 Super-Yang-Mills theory coupled to fundamental matter as described in section 2.3. We were the first to consider the non-Abelian part of the flavor gauge group in the context of AdS/CFT with a finite charge den-sity [1, 2] and the results are summarized and considerably enhanced especially in sections 4.2 and 4.4.
In the upcoming section 4.1 we will start out with an application of the Kubo formula for heat conductivity derived in the previous chapter 3.2.3. The rest of this chapter considers the D3/D7-brane setup with a background flavor gauge field introduced on the D7-brane as de-scribed in section 2.3, section 2.4 and section 3.2.2. In section 4.2 we first take an analytical approach to get some exact results for massless quarks, while in chapter 5 we will use nu-merical techniques. In order to do so we have to employ a small-frequency/small-momentum approximation coined the hydrodynamic expansion ( cf. equation (3.24), and those following it). These requirements are then relaxed and in section 4.3 the background and its thermody-namics are generalized to non-zero quark masses in a setup where also arbitrary frequencies / momenta of the perturbations (cf. chapter 5) are treatable. The price for this generalization is that we have to use numerical techniques in order to find the (massive) D7-brane embeddings as analyzed in [56]. In this context we will review the thermodynamics at finiteU(1)baryon density [42] or finite baryon chemical potential [52] in section 4.3. Investigating the effects of isospin and the non-Abelian part of the flavor group we will develop the thermodynamics for the non-Abelian partSU(Nf)of the full flavor gauge symmetryU(Nf). We find a significant impact of isospin on the hydrodynamics as well as on thermodynamics.
71
72 Chapter 4. Holographic thermo- and hydrodynamics
4.1 Application of the Kubo formula
The purpose of the calculation ahead is to understand and check the non-equilibrium methods introduced in the previous chapter 3.2.3. This general understanding is needed in the com-ing chapter 6 and all our asides on diffusion or other non-equilibrium phenomena. We will keep the computation as general as possible and only in the very end we apply the result to a conformal field theory in order to check it. The present computation may be seen as a prepa-ration to apply similar calculations to more QCD-like theories in order to find their transport coefficients.
In [109] a relatively general treatment of the problem ahead is given. The problem is simply how to (carefully) carry out the integrals inside a Kubo formula. Hosoya does this for the Kubo formula giving the shear viscosity; whereas we are actually interested in the heat conductivity.
But having this sample calculation at hand let’s follow it and we will see that the steps for our Kubo formula will walk exactly the same path (up to some constant factors).
The viscosity Kubo formula is [109]
η =−1 5limǫ→0
Z0
−∞
dt1eǫt1
t1
Z
−∞
dt′ Z∞
−∞
dk0
2π eik0t′Π(k˜ 0), (4.1) whereΠ(k˜ 0)is a 2-point correlation function only depending onk0 out of the integration vari-ables. For the shear viscosity this correlator is the energy-momentum tensor 2-point function hTijTiji. The ǫ appearing here comes from the non-equilibrium thermodynamics formalism and it parametrizes the (small) deviation from thermal equilibrium. Since we will see that it formally has exactly the same effect as an ordinary QFT regulator, I will call it the thermal regulator. Speaking about the ordinary QFT regulators, as is common habit, in (4.1) the field theory regulator is not explicitly written. We put it back in byk0 → (1−iǫ0)k0 in order to keep track of all the poles appearing.
η=−1
5limǫ,ǫ0→0
Z0
−∞
dt1eǫt1 Z∞
−∞
dk0
2π Π(k˜ 0(1−iǫ0))
t1
Z
−∞
dt′eik0(1−iǫ0)t′
| {z }
h 1
ik0(1−iǫ0)eik0(1−iǫ0 )t′it1
−∞
. (4.2)
Of thet′-integral only the upper limit (t1) remains because for the lower bound (−∞) we get limt′→−∞eip0(1−iǫ0)t′ = limt′→−∞ eik0t′
oscillating|{z}
ei(−i)k0ǫ0)t′
| {z }
→0
. (4.3)
So from this integral we are left with 1
ik0(1−iǫ0)eik0(1−iǫ0)t1. (4.4) Note that the use of the regulator ǫ0 together with the integral gives us a new pole for the k0-integration atk0 = 0. We will see that subsequent integration of this overt1 together with
4.1. Application of the Kubo formula 73
the thermal regulator ǫ will give us yet a different pole structure in the complex k0-plane.
Explicitly carrying out the same procedure as before with this new expression we are left with η =−1
5limǫ,ǫ0→0 Z∞
−∞
dk0 2π
Π(k˜ 0(1−iǫ0)) ik0(1−iǫ0)
Z0
−∞
dt1eǫt1+ik0(1−iǫ0)t1
| {z }
ik0(1−iǫ1 0)−iǫ[1−e(ǫ+ǫ0k0)(−∞)e−ik0)(−∞)]
. (4.5)
This leaves us with thek0-integration and an integrand having two poles1: 1
5(2π)(1−2iǫ0) Z∞
−∞
dk0 Π(k˜ 0(1−iǫ0)) k0
|{z}
≡A
(k0−i ǫ 1−iǫ0
)
| {z }
≡B
. (4.6)
To integrate a function like this the Cauchy-Riemann formula Z
closed contour
f(z)dz
(z−z0)2 = (2πi)∂zf(z)|z=z0 , (4.7) is usually of great help. But to apply it we first need to turn the integrand with two different poles into one with two poles at the same position to match the form of the integrand the Cauchy-Riemann formula. This can be done by introducing Feynman parametersa, bmaking use of the formula
1 AB =
Z1
0
da db δ(a+b−1) 1
(aA+bB)2 , (4.8)
which can be verified by carrying out the integrals on the right hand side. Plugging inA =k0 andB =k0−i1−ǫiǫ
0 we get 1 AB =
Z1
0
dadbδ(a+b−1) (a+b)2
1 (k0−i1−iǫǫ
0
b
a+b)2 , (4.9)
which displays the sought-after second order pole at k0 = i1−ǫiǫ
0
b
a+b. Use of the Cauchy formula and integration over the Feynman parameterbyields
η= i
5limǫ0→0limǫ→0
1 1−2iǫ0
Z1
0
da ∂k0Π˜
k0=i1−iǫǫ
0(1−a) . (4.10) Now first taking theǫ0 ordinary field theory limit gives
i
5limǫ→0
Z1
0
da ∂k0Π˜
k0=iǫ(1−a) , (4.11)
1We have to assume that the functionΠ(k˜ 0(1−iǫ0))introduces no additional poles.
74 Chapter 4. Holographic thermo- and hydrodynamics
and afterwards the thermal regulator limitǫ→0produces η= i
5 ∂k0Π˜
k0=0 . (4.12)
This formula is true for any correlatorΠ˜ which introduces no new poles ink0and which does not depend on any of the time-variables (t′, t1). The Kubo formula for thermal conductivity will only have a different numerical factor and it will contain the current correlatorhJiaJibi(~k) instead of the energy-momentum correlatorhTijTiji(~k). But both are only functions ofk0 as required. And from the Fourier-transformation of (2.31) essentially given by
hJiaJibi=−δab lim
ǫd→0C(ǫd)~k2+ǫd, (4.13) with the dimensional regularization parameterǫdand the coefficientC(ǫd)we can see, that the conformal flavor current correlator contains no poles ink0.
Simply applying formula (4.11) to the conformal flavor current correlator (4.13) (see also [112]) we get the transport coefficient
η= limǫd→0
i
M ∂k0{δabC˜4−ǫd~k2+ǫd}
k0=0
| {z }
C˜4−ǫd(1+ǫd/2)~kǫd2k0|k0=0
= 0, (4.14)
whereM stands for the factors different from the viscosity case. The vanishing of this trans-port coefficient can be traced to the thermal regulator by plugging it in before taking any of the limitsǫd, ǫ0, ǫ →0. Carrying out all integrations and derivatives before taking these three limits, the coefficient vanishes exactly when taking the ’thermal’ limitǫ→0.
Our interpretation of this fact is that the conformal symmetry realized in the correlator does not allow any scale in the theory. In particular conformal symmetry does not allow introduction of an energy scale like the temperature. Plugging in the conformal correlator essentially amounts to setting the temperature T = 0 in the non-equilibrium theory from which the Kubo formula is derived.