3.3.1 Method
Transport coefficients are computed from Kubo formulae which are based on linear response theory. Imagine a small external perturbation Jj which couples to some operatorOi of a theory with HamiltonianH0. In presence of the perturbation the Hamiltonian is modified to
H(t) =H0(t) +Hext(t) = H0(t) + Z
d(d−1)xOi(x)Jj(x). (3.25) Naturally the expectation value of this operator changes
hOi(x)iJ =hO(x)i0+i Z
dt0Θ(t−t0)h[Hext(t),Oi(x)]i (3.26)
=hOi(x)i0−i Z
ddΘ(t−t0)h[Oi(x),Oj(x0)]iJj(x0), (3.27) where in the last step the expansion of the time evolution operator to linear order in Hext was used. Defining the retarded Green’s function GRij(x, x0) = iΘ(t − t0)h[Oi(x),Oj(x0)]i, we can now write down the Kubo formula in momentum space δhOi(k)i=−GRij(k)Jj(k). (3.28) For the locality in momentum space to valid, we need translational invariance of the original theory H0. Note that the retarded Green’s function vanishes before the perturbation is switched on at t0 < t and is computed with the unperturbed Hamiltonian H0. In words, the Kubo formula gives a linear relation between the response δhO(k)i and the small external perturbation J. The correlator −iGR is identified with the transport coefficient. A canonical example is Ohm’s law. The small external sourceAi induces an electric currentδhJi(k)i. AsAi is proportional to the electric field Ei =−∂tAi ∼iωAi the Kubo formula (3.28) gives
δhJi(k)i=− 1
iωGRij(k)Ej(k)≡σij(k)Ej(k), (3.29) where we assumed that in absence of the external electric field there is no electric current J.
Coming back to the holographic duality, the question we have to address is: How can an external perturbation as discussed above be implemented in the a truly microscopic field theory and what is its gravity dual? The answer is that we do not need to actually implement it in this way. The reason is the fluctuation
3.3. Holographic transport coefficients 39
dissipation theorem [76] which states that the response ofGRof a system to a small external perturbation is given by the retarded Green’s function of a spontaneous microscopic fluctuation around the equilibrium state. This in turn is a standard exercise within quantum field theory. The GKPW formula (2.25) is thus essential for the holographic dual of this procedure. For the causal correlator it states
GRij(x, x0) = −ihTOi(x)Oj(x0)i=−i δ2ZQFT[J]
δJj(x0)δJi(x)
=i δ2Sgravityos [φ]
δJj(x0)δJi(x)
φ(r→∞)=J .
(3.30)
T denotes the time ordering operation. Thus to compute a retarded correlator, and hence transport coefficient, of a given quantum field theory with holographic dual in the language of gravity we first need to perturb the fieldφ, whose boundary value represents J. For the leading order of the response δhOi the gravity action has to be expanded to second order in those perturbations δφ, which results in linear equations of motion for δφ.
As an example let us look again at the electrical conductivity of the finite den-sity model introduced in 3.2. Note that every conceptual step in the following discussion applies to any other transport coefficient as well. In the case of the conductivity the Maxwell field A takes the role of φ and we start by2
Ax →Ax+ax. (3.31)
where ax satisfies the following equation of motion in momentum space r2f a0x0
+ω2
f ax = 2κ2L4(A0t)2ax. (3.32) This equation is valid only for a perturbation constant in space. Close to the black hole horizon rh the equations demand ax to behave as
ax∼e−iωt
aR(r−rh)−iω/4πT +aA(r−rh)+iω/4πT
h(r), (3.33) where T is the equilibrium temperature. Defining ˜r= log(r−rh)/4πT this can be
2The correct way to implement the perturbations is to introduce fluctuations for all the fields of the gravity theory and all of their respective components of perturbations. In the present case, with the background as given in 3.2, however, the equation of motion ofax can be shown to be independent of any other fluctuation.
rewritten as
ax ∼aRe−iω(t+˜r)+aAe−iω(t−˜r). (3.34) In this notation it is clear that aR parametrises a wave that moves towards the black brane horizon and aA one that moves away form it. The latter represents an acausal process because classically nothing can leave the black hole. A finite aA imposes a boundary condition at the past horizon. In contrast, the former describes a causal process and a finite aR imposes a boundary condition at the future even horizon. Choosing aA= 0 is the so-called ingoing boundary condition that is gives rise to the retarded Green’s function of the boundary theory via (3.30).
The equation of motion (3.32) is a second order differential equation. To uniquely fix the solution, two boundary conditions have to be imposed. The first is the ingoing boundary condition. The second is given by the boundary value (non-normalisable mode) abx of ax which is dual to the small external perturbation J in the field theory. Equation (3.32) can be solved numerically only. The resulting on-shell action is of the form
Sgravityos = Z
dω abx(−ω)F(ω, r)abx(ω)
∞ rh
, (3.35)
where F is a differential operator in the radial coordinate r. It was shown in [77]
via a Schwinger-Keldysh approach that the retarded Green’s function is given by GR(ω) = −2 lim
r→∞F(ω, r). (3.36)
It is important to stress that the result in (3.36) is highly nontrivial. We just com-puted a real time correlator in a thermal field theory which is naturally formulated in terms of the Wick rotated time coordinate. Real time computations thus require to deal with complex Euclidean time. To date AdS/CFT is the only framework that allows to study real time properties of strongly correlated systems at all. This is one of the main reasons for the importance of AdS/CFT on a practical level.
Equation (3.36) can be rephrased into a more practical prescription of how to compute the transport coefficient of interest
Sgravityos = Z
dωcabx(−ω)asx(ω)
∞ rh
⇒ GR(ω) =−2casx(ω)
abx(ω), (3.37) where asx is the normalisable mode or subleading term of the boundary expansion
3.3. Holographic transport coefficients 41
of ax and cis some constant. The systematic generalisation to the case of several coupled fluctuations and hence a whole set of coupled operators and sources on the field theory side is explained in detail in [78].
3.3.2 Electrical conductivity in holography
From a gravity theory perspective the electrical conductivity is given by σ(ω) =− 1
iωGRJ J(ω) = 2c asx(ω)
iω abx(ω). (3.38) The Kramers-Kronig relations relate the imaginary and real part of the conduc-tivity3. In particular, a pole in the imaginary part Imσ ∼ ω−1 implies a δ-peak in the real part Reσ ∼ δ(ω) giving rise to an infinitely high DC conductivity.
There are two scenarios in which this happens. The first is the absence of momen-tum dissipation. In this case, once there is an electrical current, understood as a stream of moving charged particles, the current will persist, because there is no mechanism to stop it. This results in perfect, i.e. infinite, DC conductivity. The phenomenological Drude model provides a more quantitative explanation. The average velocity v at which a charged particle moving through a material satisfies
mdv
dt =−mv
τ +e E , (3.39)
in terms of the particle’s mass m and its charge e. The average time τ between collisions parametrises the momentum dissipation. With the current density given by J =env, the electrical conductivity (3.29) behaves as
σ(ω) =ν τ
1−iωτ with ν = e2n
m . (3.40)
In chapter 5 we will find exactly this behaviour in a holographic context. As dissipation is switched off by τ → ∞, the conductivity acquires a poleσ ∼ −1/iω.
Holographically the origin of the pole is the effective mass term of the Maxwell field fluctuation (3.32) generated by the background fieldAt: In the zero frequency
3The Kramers-Kronig relations are given by the following: Reσ(ω) = PπR
dω0Imω0σ(ω−ω0) and Imσ(ω) =−PπR
dω0Reω0σ(ω−ω0), where P is the principal value.
limit the subleading term, on dimensional grounds, behaves as asx ∼r2f a0x
r→∞ ∼2κ2L4r4(A0t)2abx
r→∞, (3.41)
and thus asx ∼ O(ω0) which by equation (3.29) leads to a pole in the imaginary part of the conductivity proportional to the effective mass term4.
Infinite DC conductivity is also a characteristic of superconductors which we in-troduce in the next section.
In appendix A we present a different approach to compute the DC conductivity with holographic methods. It is based on the so-called Einstein relation which allows to express σDC entirely in terms of quantities characterising the thermody-namic equilibrium.