3
Quantum critical transport
The optical conductivity of graphene, σ(ω) = σ_{1}(ω) + iσ_{2}(ω), at several gate voltages. In the high frequency limit, all the plots tend to a constant. This is a consequence of the scale invariance at the particle-hole symmetric point. This chapter will consider the consequences of quantum criticality for transport in zero density matter. Image adapted from [521] with permission.
3.1 Condensed matter systems and questions
Let us consider the simplest case of transport of a conserved U(1) “charge” density ρ(x, t) (we will explicitly write out the time coordinate, t, separately from the spatial coordinate x in this section). Its conservation implies that there is a current J_{i}(x, t) (where i = 1…d) such that
We are interested here in the consequences of this conservation law at T > 0 for correlators of ρ and J_{i} in quantum systems without quasiparticle excitations.
We begin with a simple system in d = 1: a narrow wire of electrons which realize a Tomonaga-Luttinger liquid [294]. For simplicity, we ignore the spin of the electron in the present discussion. In field-theoretic terms, such a liquid is a CFT2. In the CFT literature it is conventional to describe its correlators in holomorphic (and anti-holomorphic) variables constructed out of space and Euclidean time, τ. The spacetime coordinate is expressed in terms of a complex number z = x + iτ, and the density and the current combine to yield the holomorphic current 𝒥. A well-known property of all CFT2s with a conserved 𝒥 is the correlator
3.2 Standard approaches and their limitations
We now describe a number of approaches applied to the computation of σ(ω) for CFT3s, and especially to the Wilson-Fisher fixed point of the superfluid-insulator transition described by (2.1). Similar methods can also be applied to other transport coefficients, and to other CFTs. Each of the methods below has limitations, although their combination does yield significant insight.
3.2.1 Quasiparticle-based methods
We emphasized above that interacting CFT3s do not have any quasiparticle excitations. However, there are the exceptions of free CFT3s which do have infinitely long-lived quasiparticles. So we can hope that we could expand away from these free CFTs and obtain a controlled theory of interacting CFT3s. After all, this is essentially the approach of the 𝜖 expansion in dimensionality by Wilson and Fisher for the critical exponents. However, we will see below that the 𝜖 expansion, and the related vector 1/M expansion, have difficulty in describing transport because they do not hold uniformly in ω. Formally, the 𝜖 → 0 and the ω → 0 limits do not commute.
Let us first perform an explicit computation on a free CFT3: a CFT with M two-component, massless Dirac fermions, C, with Lagrangian
The form of the current correlator of the CFT3 is considerably more subtle at T > 0. In principle, this evaluation only requires replacing the frequency integral in (3.17) by a summation over the Matsubara frequencies, which are quantized by odd multiples of πT. However, after performing this integration by standard methods, the resulting expression should be continued carefully to real frequencies. We quote the final result for σ(ω), obtained from the current correlator via (3.10)
We can now ask if the zero frequency delta function is preserved once we move to an interacting CFT. A simple way to realize an interacting CFT3 is to couple the fermions to a dynamical U(1) gauge field, and examine the theory for large M. It is known that the IR physics is then described by a interacting CFT3 [48; 752; 137; 672; 685; 684; 422; 421] and its T = 0 properties can be computed in a systematic 1/M expansion (we emphasize that this is a ‘vector’ 1/M expansion, unlike the matrix large N models considered elsewhere in this book). The same remains true at T > 0 for the conductivity, provided we focus on the ω ≫ T region of (3.18) (we will have more to say about this region in the following subsection). However, now collisions are allowed between the quasiparticles that were not present at M = ∞, and these collisions cannot be treated in a bare 1/M expansion. Rather, we have to examine the effects of repeated collisions, and ask if they lead to charge diffusion and a finite conductivity. This is the precise analog of the question Boltzmann asked for the classical ideal gas, and he introduced the Boltzmann equation to relate the long-time Brownian motion of the molecules to their two-particle collision cross-section. We can apply a quantum generalization of the Boltzmann equation to the CFT3 in the 1/M expansion, where the quasiparticles of the M = ∞ CFT3 undergo repeated collisions at order 1/M by exchanging quanta of the U(1) guage field. The collision cross-section can be computed by Fermi’s golden rule and this then enters the collision term in the quantum Boltzmann equation [167; 672].
One subtlety should be kept in mind while considering the analogy with the classical ideal gas. The free CFT3 has both particle and anti-particle quasiparticles, and these move in opposite directions in the presence of an applied electric field. So the collision term must also consider collisions between particles and anti-particles. Such collisions have the feature that they can degrade the electrical current while conserving total momentum. Consequently, a solution of the Boltzmann equation shows that the zero-frequency conductivity is finite even at T > 0, after particle-anti-particle collisions have been accounted for [167; 673]. As in the usual Drude expression for the conductivity, the zero frequency conductivity is inversely proportional to the collision rate. As the latter is proportional to 1/M, we have σ(0) ∼ M. Similarly, we can consider the frequency dependence of the conductivity at ω ≪ T, and find that the width of the peak in the conductivity extends up to frequencies of order T/M. So the zero frequency delta function in (3.18) has broadened into peak of height M and width T/M, as sketched in Figure 3.1. It required a (formally uncontrolled) resummation of the 1/M expansion using the Boltzmann equation to obtain this result. And it should now also be clear from a glance at Figure 3.1 that the ω → 0 and M → ∞ limits do not commute at T > 0.
Schematic of the real part of the conductivity of a CFT3 of M Dirac fermions coupled to a U(1) gauge field in the large M limit. Similar features apply to other CFT3s in the vector large M limit. The peak at zero frequency is a remnant of the quasiparticles present at M = ∞, and the total area under this peak equals (T ln2)/2 as M → ∞.
A similar analysis can also be applied to the Wilson-Fisher CFT3 described by (2.1), using a model with a global O(M) symmetry. The Boltzmann equation result for the zero frequency conductivity is σ(0) = 0.523M, where the boson superfluid-insulator transition corresponds to the case M = 2 [672; 755].
Our main conclusion here is that the vector 1/M expansion of CFT3s, which expands away from the quasiparticles present at M = ∞, yields fairly convincing evidence that σ(ω) is a non-trivial universal function of ω/T. However, it does not appear to be a reliable way of computing σ(ω) for ω < T and smaller values of M.
3.2.2 Short time expansion
Transport involves the long time limit of the correlators of conserved quantities, but we can nevertheless ask if any useful information can be obtained from short time correlators. For lattice models with a finite Hilbert space at each site, the short time expansion is straightforward: it can be obtained by expanding the time evolution operator e^{−iHt/ℏ} in powers of t. Consequently, the short time expansion of any correlation function only involves integer powers of t.
The situation is more subtle in theories without an ultraviolet cutoff because the naive expansion of the time evolution operator leads to ultraviolet divergencies. For CFTs these ultraviolet divergencies can be controlled by a renormalization procedure, and this is expressed in terms of a technology known as the operator product expansion (OPE). Detailed reviews of the OPE are available elsewhere [248; 613], and here we only use a few basic results to examine its consequences for the two-point correlator of the current operator at short times. After a Fourier transform from time to energy, the short time expansion translates into a statement for the large frequency behavior of the conductivity of CFT3s, at frequencies much larger than T [465]
This formula remains true outside of the quantum critical fan in many circumstances. Furthermore, the ratio c_{1}/c_{2} is universal and depends only on d and ν.
The high frequency expansion (3.19) does not directly place any constraints on the conductivity at low frequencies, ω ≪ T. However, in combination with the sum rule in (3.13), we can make an interesting observation. The 1/M expansion for the Wilson-Fisher CFT, and the numerical results presented below, show that the constant b_{1} > 0. Consequently we have Re[σ(ω)] < 𝒦 at large enough ω. The sum rule in (3.13) now implies that there must be a region at smaller ω where we have Re[σ(ω)] > 𝒦. If we make the assumption that Re[σ(ω)] −𝒦 crosses zero only once (this assumption is seen to fail in some of the holographic examples considered below), we can conclude that Re[σ(ω)] should have the form in Figure 3.2, with the two shaded regions having equal area. The qualitative form of Re[σ(ω)] has similarities to the quasiparticle result in Figure 3.1, although we have not made any use of quasiparticles in the present argument.
Schematic of the real part of the conductivity of a CFT3 as constrained by the OPE in (3.19) and the sum rule in (3.13). The areas of the shaded regions are equal to each other.
3.2.3 Quantum Monte Carlo
Quantum Monte Carlo studies perform a statistical sampling of the spacetime configurations of the imaginary time path integral of quantum systems. So they are only able to return information on correlation functions defined at imaginary Matsubara frequencies. However, even in such Euclidean studies there is the potential for a ‘sign problem’, with non-positive weights, because of the presence of complex Berry phase terms [673]. For the CFT3s being studied here such sign problems are usually absent; however, the problems with a non-zero chemical potential to be considered later in our review invariably do have a sign problem.
For the Wilson-Fisher CFT3 defined by (2.1), there is an especially elegant model for efficient Monte Carlo studies. This is the lattice regularization provided by the Villain model. This model is defined in terms of integer valued variables, j_{i, μ}, on the links of a cubic lattice. Here i labels the sites of a cubic lattice, and μ = ±x,±y,±z represents the six links emerging from each site. Each link has an orientation, so that 2.1). These currents must be conserved, and so we have the zero divergence condition
where Δ_{μ} is the discrete lattice derivative; this is the analog of Kirchoff’s law in circuit theory. Finally, the action of the Villain model is simply [745]
There have been extensive numerical studies of the Villain model, and sophisticated finite-size scaling methods have yielded detailed information on σ(iω_{n}) at the Matsubara frequencies, ω_{n}, which are integer multiples of 2πT [736; 756; 136; 292; 291; 465]. Note especially that Monte Carlo does not yield any information at ω_{n} = 0 (this is consequence of the structure of the Kubo formula for the conductivity, which is only defined at non-zero ω, and reaches ω = 0 by a limiting process). So the smallest possible frequency at which we can determine the conductivity is 2πTi. The results at these, and higher Matsubara frequencies are all found [465] to be in excellent agreement with OPE expansion in (3.19). So the conclusion is that the quantum Monte Carlo results are essentially entirely in the large ω regime, where the results of the OPE also apply. The simulations can therefore help determine the values of the numerical constants b_{1, 2} in (3.19). While this agreement is encouraging, the somewhat disappointing conclusion is that quantum Monte Carlo does not yield any more direct information on the low frequency behavior than is available from the OPE.
3.3 Holographic spectral functions
In the next few sections we will cover holographic computations of the retarded Green’s functions for operators in quantum critical systems at nonzero temperature. In general we will be interested in multiple coupled operators and so we write, in frequency space,
A sum over B is implied. Here δ⟨𝒪_{A}⟩ is the change in the expectation value of the operator 𝒪_{A} due to the (time and space dependent) change in the sources δh_{B}. From the holographic dictionary discussed in §1.5, we know that expectation values and sources are given by the near boundary behavior of the bulk fields {ϕ_{A}} dual to the operators {𝒪_{A}}. Therefore
To compute the Green’s function (3.24) in a given background, one first perturbs the background fields
Linearizing the bulk equations of motion leads to coupled, linear, ordinary differential equations for the perturbations δϕ_{A}(r). Once these linearized equations are solved, the asymptotic behavior of fields can be read off from the solutions and the Green’s function obtained from (3.24). A crucial part of solving the equations for the perturbations is to impose the correct boundary conditions at the black hole horizon. One way to understand these boundary conditions is to recall that the retarded Green’s function is obtained from the Euclidean (imaginary time) Green’s function by analytic continuation from the upper half frequency plane (i.e. ω_{n} > 0). An important virtue of the holographic approach is that this analytic continuation can be performed in a very efficient way, as we now describe.
3.3.1 Infalling boundary conditions at the horizon
Starting with the Euclidean black hole spacetime (2.37), we described how to zoom in on the spacetime near the horizon in the manipulations following (2.38). To see how fields behave near the horizon, we can consider the massive wave equation in the black hole background (2.47) at r ∼ r_{+}. The equation becomes
For ω_{n} > 0, it is clear that the regular solution is the one that decays as r → r_{+} (that is, the positive sign in the above equation). Once this condition is imposed on all fields, the solution is determined up to an overall constant, which will drop out upon taking the ratios in (3.24) to compute the Green’s function.
The real time equations of motion are the same as the Euclidean equations, with the substitution iω_{n} → ω. In particular, the regular boundary condition at the horizon analytically continues to the “infalling” boundary condition
corresponding to modes that carry energy towards r = r_{+} as t → ∞. That is, they are modes that fall towards the black hole (rather than emerge out of the black hole). This is indeed the required behavior of physical excitations near an event horizon, and can also be derived directly from the real time, Lorentzian, equations of motion. Infalling modes are required for regularity in the Kruskal coordinates that extend the black hole spacetime across the future horizon (see e.g. [350]). The derivation via the Euclidean modes makes clear, however, that infalling boundary conditions correspond to computing the retarded Green’s function.
The infalling boundary condition (3.28) allows the retarded Green’s function to be computed directly at real frequencies. This is a major advantage relative to e.g. Monte Carlo methods that work in Euclidean time. The infalling boundary condition is also responsible for the appearance of dissipation at leading order in the classical large N limit, at any frequencies. This is also a key virtue relative to e.g. vector large N expansions, in which ω → 0 and N → ∞ do not commute at nonzero temperatures, as we explained above. Infalling modes have an energy flux into the horizon that is lost to an exterior observer. The black hole horizon has geometrized the irreversibility of entropy production. Physically, energy crossing the horizon has dissipated into the ‘order N^{2}’ degrees of freedom of the deconfined gauge theory.
Infalling boundary conditions were first connected to retarded Green’s function in [699]. We derived the infalling boundary condition from the Euclidean Green’s function. An alternative derivation of this boundary condition starts from the fact that the fully extended Penrose diagram of (Lorentzian) black holes has two boundaries, and that these geometrically realize the thermofield double or Schwinger-Keldysh approach to real time thermal physics [397; 696].
3.3.2 Example: spectral weight of a large dimension operator
We will now illustrate the above with a concrete example. We will spell out here in some detail steps that we will go over more quickly in later cases. To start with we will put k = 0 and obtain the dependence of the Green’s function on frequency. The scalar wave equation (2.47) in the Lorentzian signature black hole background becomes
To develop an intuition for an equation like the above, it is often useful to put the equation into a Schrödinger form. To do this we set ϕ = r^{d/2}ψ, so that (3.30) becomes
The boundary at r = 0 is not an asymptotic region of the Schrödinger equation. There will be normalizable and non-normalizable behaviors of ϕ as r → 0. To compute the Green’s function we will need to fix the coefficient of the non-normalizable mode and solve for the coefficient of the normalizable mode.
A key physical quantity is the imaginary part of the retarded Green’s function (also called the spectral weight) of the operator 𝒪 dual to ϕ. The imaginary part of the retarded Green’s function is a direct measure of the entropy generated when the system is subjected to the sources δh_{B} of (3.23). Specifically, with an assumption of time reversal invariance (see e.g. [350]), the time-averaged rate of work w done on the system per unit volume by a spatially homogeneous source driven at frequency ω is given by
We now illustrate how this spectral weight is given by the amplitude with which the field can tunnel from the boundary r = 0 through to the horizon. Thus, we will exhibit the direct connection between dissipation in the dual QFT and rate of absorption by the horizon.
The analysis is simplest in the limit of large mL ≫ 1. Recall that this corresponds to an operator 𝒪 with large scaling dimension. In this limit (3.30) can be solved in a WKB approximation. In the large mass limit, the potential (3.32) typically decreases monotonically from the boundary towards the horizon. There is therefore a unique turning point r_{o} at which V(r_{o}) = ω^{2}. The infalling boundary condition together with standard WKB matching formulae give the solution
In this limit V = (mL)^{2}f/r^{2z}. Expanding the above solution near the r → 0 boundary, the retarded Green’s function is obtained from the general formula (3.24). In particular, the imaginary part is given by
From the above expression the expected limits of high and low frequency are immediately obtained. Note that for ωr^{z}_{+} ≪ 1 then r_{o} ∼ r_{+}, whereas for ωr^{z}_{+} ≫ 1, r_{o} ∼ ω^{−1/z}. Thus
On general grounds χ′′ must be an odd function of ω. This can be seen in a more careful WKB computation that includes the order one prefactor. Relatedly, to obtain (3.36) we used the fact that the scaling dimension (2.26) is given by Δ = mL in the limit mL ≫ 1. We also used the expression (2.44) for the temperature. The full result (3.35) gives an explicit and relatively simple crossover between the two limits in (3.36).
More generally, away from the WKB limit, (3.30) can easily be solved numerically. Later we will give an example of the type of Mathematica code one uses to solve such equations. We should note, however, that for general values of Δ (e.g. not integer, etc.) the boundary conditions often need to be treated to a very high accuracy to get stable results, as discussed in e.g [192]. Also, one should be aware that when the fast and slow falloffs differ by an integer, one can find logarithmic terms in the near boundary expansion. These must also be treated with care and, of course, correspond to the short distance logarithmic running of couplings in the dual QFT that one expects in these cases.
3.3.3 Infalling boundary conditions at zero temperature
In (3.28) above we obtained the infalling boundary condition for finite temperature horizons. Zero temperature geometries that are not gapped will also have horizons in the far interior (or, more generally, as we noted in our discussion of Lifshitz geometries in §2.2 above, mild null singularities). To compute the retarded Green’s function from such spacetimes we will need to know how to impose infalling boundary conditions in these cases. This is done as before, by analytic continuation of the Euclidean mode that is regular in the upper half complex frequency plane. It is not always completely straightforward to find the leading behavior of solutions to the wave equation near a zero temperature horizon, especially when there are many coupled equations. The following three examples illustrate common possibilities. In each case we give the leading solution to ∇^{2}ϕ = m^{2}ϕ near the horizon.
1. Poincaré horizon. Near horizon metric (as r → ∞):
2. Extremal AdS_{2} charged horizon (see Chapter 4). Near horizon metric (as r → ∞):
3. Lifshitz ‘horizon’. Near horizon metric (as r → ∞):
Finally, occasionally one wants to find the Green’s function directly at ω = 0 (for instance, in our discussion of thermal screening around (2.49) above). The equations at ω = 0 typically exhibit two possible behaviors near the horizon, one divergent and the other regular. At finite temperature the divergence will be logarithmic. Clearly one should use the regular solution to compute the correlator.
3.4 Quantum critical charge dynamics
So far we have discussed scalar operators 𝒪 as illustrative examples. However, a very important set of operators are instead currents J^{μ} associated to conserved charges. The retarded Green’s functions of current operators capture the physics of charge transport in the quantum critical theory. We explained around (1.40) above that a conserved U(1) current J^{μ} in the boundary QFT is dually described by a Maxwell field A_{a} in the bulk. Therefore, to compute the correlators of J^{μ}, we need an action that determines the bulk dynamics of A_{a}.
3.4.1 Conductivity from the dynamics of a bulk Maxwell field
As we are considering zero density quantum critical matter in this section, we restrict to situations where the Maxwell field is not turned on in the background (we will relax this assumption in Chapter 4 below). Therefore, in order to obtain linear response functions in a zero density theory, it is sufficient to consider a quadratic action for the Maxwell field about a fixed background. The simplest such action is the Maxwell action (with F = dA, as usual)
so we will start with this. We noted in §1.8 that Einstein-Maxwell theory can be obtained as a consistent truncation of explicit bulk theories with known QFT duals.
Before obtaining the Maxwell equations we should pick a gauge. A very useful choice for many purposes in holography is the ‘radial gauge’, in which we put A_{r} = 0. Among other things, the bulk Maxwell field in this gauge has the same nonzero components as the boundary current operator J^{μ}. The Maxwell equations about a black hole geometry of the form (2.37) can now be written out explicitly. Firstly, write the Maxwell field as
where without loss of generality we have taken the momentum to be in the x direction. The equations of motion, ∇_{a}F^{ab} = 0, become coupled ordinary differential equations for the a_{μ}(r). By considering the discrete symmetry y → −y of the background, where y is a boundary spatial dimension orthogonal to x, we immediately see that the perturbation will decouple into longitudinal (a_{t} and a_{x}) and transverse (a_{y}) modes. It is then useful to introduce the following ‘gauge invariant’ variables [500], that are invariant under a residual gauge symmetry A_{μ} → A_{μ} + ∂_{μ}[λe^{−iωt+ikx}],
leading to the following two decoupled second order differential equations
When k = 0 the transverse and longitudinal equations are the same, as we should expect. These equations with z = 1 have been studied in several important papers [638; 386; 500; 391], sometimes using different notation. The gauge-invariant variables avoid the need to solve an additional first order equation that constrains the original variables a_{x} and a_{t}.
Near the boundary at r → 0, the asymptotic behavior of a_{μ} is given by:
Note that when z ≠ 1, as expected, the asymptotic falloffs are different for the timelike and spacelike components of a_{μ}. So long as d > z, it is straightforward to add boundary counterterms to make the bulk action finite. We can then, using the general formalism developed previously, obtain the expectation values for the charge and current densities:
A quantity of particular interest is the frequency dependent conductivity
As we have noted above, the dissipative (real) part of the conductivity will control entropy generation due to Joule heating when a current is driven through the system. We shall compute the optical conductivity (3.50) shortly, but first consider the physics of certain low energy limits.
3.4.2 The dc conductivity
The dc conductivity
The dc conductivity can be obtained by directly applying a uniform electric field, rather than taking the k, ω → 0 limit of (3.47b). Consider the bulk Maxwell potential
The bulk Maxwell equations can be written as 3.52) one immediately has that
We have thereby identified a radially conserved quantity. Furthermore, near the boundary
From (3.53) and (3.54) it follows that we can obtain the field theory current response ⟨J_{x}⟩ if we are able to evaluate at any radius. It turns out that we can compute it at the horizon. It is a standard trick that physics near black hole horizons is often elucidated by going to infalling coordinates; so we replace t in favor of the ‘infalling Eddington-Finkelstein coordinate’ v by
Using the relation (2.44) between r_{+} and T, we obtain
The same argument given above, applied to certain perturbations of the bulk metric rather than a bulk Maxwell field, gives a direct proof of a famous result for the shear viscosity η over entropy density s in a large class of theories with classical gravity duals [431]:
Here η plays an analogous role to the dc conductivity σ in the argument above. The shear viscosity was first computed for holographic theories in [637] and the ratio above emphasized in [498; 495].
3.4.3 Diffusive limit
The longitudinal channel includes fluctuations of the charge density. Because the total charge is conserved, this channel is expected to include a collective diffusive mode [450]. This fact is why the longitudinal equation (3.47a) is more complicated than the transverse equation (3.47b). It is instructive to see how the diffusive mode can be explicitly isolated from (3.47a). We will adapt the argument in [712].
Diffusion is a process that will occur at late times and long wavelengths if we apply a source to the system to set up a nontrivial profile for the charge density, turn off the source, and then let the system evolve. Therefore diffusion should appear as a mode in the system that (i) satisfies infalling boundary conditions at the horizon and (ii) has no source at the asymptotic boundary. We will see in §3.5 below that these are the conditions that define a so-called quasinormal mode, which correspond precisely to the poles of retarded Green’s functions in the complex frequency plane. More immediately, we must solve the longitudinal equation (3.47a) with these boundary conditions, and in the limit of small frequency and wavevector.
We cannot simply take the limit ω → 0 of the longitudinal equation (3.47a). This is because taking ω → 0 in this equation does not commute with the near horizon limit r → r_{+}, at which f → 0. As we take the low frequency limit, we need to ensure that the infalling boundary condition as r → r_{+} is correctly imposed. This can be achieved by writing
To isolate the diffusive regime, consider ω ∼ k^{2}. This corresponds to taking 3.63). The full solution to the order we are working can now be written
This is the anticipated diffusive mode. Some comments on this result:
1. Beyond finding the diffusive mode (3.65), one can also find the full diffusive part of the longitudinal channel retarded Green’s functions, giving a density Green’s function of the form (3.6), as was originally done in [638; 386].
2. The diffusion constant in (3.65), unlike the dc conductivity (3.58), is not given purely in terms of horizon data. To find the diffusive mode we had to explicitly solve the Maxwell equations everywhere. This is an example of the phenomenon of semi-holography mentioned in our discussion of Wilsonian holographic renormalization in §1.6.4. Because there is only one diffusive mode compared to the many (order ‘N^{2}’) gapless modes of the IR theory, the diffusive mode is ‘not powerful’ enough to backreact on the dynamics of the IR fixed point theory. To some extent this is an artifact of the large N limit of holography. The consequence is that the diffusive mode is not fully described by the dynamics of the event horizon, but is rather a quasinormal mode in its own right, with support throughout the spacetime. A holographic Wilsonian discussion of the diffusive mode can be found in [431; 266; 602].
3. When d < z, the diffusivity is UV divergent and (3.65) no longer holds [611]. This gives an extreme illustration of the previous comment: in this case the diffusivity is dominated by non-universal short distance physics, even while the dc conductivity (3.58) is captured by the universal low energy dynamics of the horizon.
4. A basic property of diffusive processes is the Einstein relation σ = χD, where χ is the charge susceptibility (compressibility). The susceptibility is obtained from the static, homogeneous two point function of a_{t}. The easiest way to do this is to show that the following a_{t} perturbation solves the linearized Maxwell equations (with ω = 0):
where μ is a small chemical potential. The susceptibility is given by ∂_{μ}ρ(μ, T) as μ → 0. Combining (3.66) and (3.49a), and using ⟨J^{t}⟩ = ρ we find
3.4.4 σ(ω) part I: Critical phases
We have emphasized in §3.1 and in (3.33) above that the real part of the frequency-dependent conductivity at zero momentum (k = 0)
Consider first a bulk Maxwell field in a scaling geometry with exponent z as discussed in §3.4.1 above. Putting k = 0 in the equations of motion (3.47a) or (3.47b) for the Maxwell potential leads leads to
We must solve this equation subject to infalling boundary conditions at the horizon. Given the solution, equations (3.48b) and (3.49b) for the near-boundary behavior of the field imply that the conductivity (3.50) will be given by
The overall normalization is unimportant. It follows immediately from the formula for the conductivity (3.70), using only the fact that at the boundary f(0) = 1, that
In particular, there is no dependence on ω/T! In the language of §3.1, this means that the, a priori distinct, constants characterizing the diffusive (ω → 0) and zero temperature (ω → ∞) limits are equal in this case: 𝒦 = 𝒞_{χ}𝒞_{D}, first noted in [391]. We have obtained this result for all z in d = 2. We will see in §3.4.6 that the lack of ω dependence is due to the electromagnetic duality enjoyed by the equations of motion of the bulk 3+1 dimensional Maxwell field, which translates into a self-duality of the boundary QFT under particle-vortex duality. Meanwhile, however, this means that in order to obtain more generic results for the frequency-dependent conductivity, we will need to depart from pure Maxwell theory in the bulk.
In the remainder of this subsection, we will restrict ourselves to the important case of CFT3s. That is, we put d = 2 and z = 1. The discussion in §3.2.2 showed that the existence of a relevant deformation of the quantum critical theory plays an important role in the structure of σ(ω), because it determines the leading correction to the constant ω/T → ∞ limit. Such an operator will always be present if the critical theory is obtained by tuning to a quantum critical point. However, a quantum critical phase, by definition, does not admit relevant perturbations. In such cases one may continue to focus on the universal sector described in the bulk by the metric and Maxwell field. It is an intriguing fact that the simplest holographic theories lead naturally to critical phases rather than critical points.
An especially simple deformation of Maxwell theory that does not introduce any additional bulk fields is
A word of caution is necessary before proceeding to compute in the theory (3.73). Generically, the bulk derivative expansion will only be controlled if the coupling constant γ is parametrically small. Considering a finite nonzero γ while neglecting other higher derivative terms requires fine tuning that may not be possible in a fully consistent bulk theory. Remarkably, it can be shown, both from the bulk and also from general QFT arguments, that consistency requires [401; 399; 594; 145; 400]
Satisfying the bound (3.74) by no means guarantees that the bulk theory is a classical limit of a well defined theory of quantum gravity [123]. However, it does mean that no pathology will arise at the level of computing the retarded Green’s function for the current operator in linear response theory about the backgrounds we are considering. Therefore, we can go ahead and use the theory (3.73) as tool to generate Green’s functions that are consistent with all necessary CFT axioms, that satisfy both of the sum rules (3.13) and (3.15), and for which we know the parameter b_{2} appearing in the large frequency expansion (3.19).
The equations of motion for the Maxwell potential A following from (3.73) are easily derived. The background geometry will be the AdS-Schwarzschild spacetime (i.e. the metric (2.37) with emblackening factor (2.42), with z = 1, θ = 0 and d = 2). As above, the perturbation takes the form A = a_{x}(r)e^{−iωt}dx. The Weyl tensor term in the action (3.73) changes the previous equation of motion (3.69) to [759]
The only effect of the coupling γ is to introduce the second term in brackets. To obtain this equation of motion, one should use a Mathematica package that enables simple computation of curvature tensors (such as the Weyl tensor). Examples are the RGTC package or the diffgeo package, both easily found online.
Near the boundary, the spacetime approaches pure AdS_{4}, which has vanishing Weyl curvature tensor, and hence the new term in the action (3.73) does not alter the near boundary expansions of the fields. Therefore, from a solution of the equation of motion (3.76), with infalling boundary conditions at the horizon, the conductivity is again given by (3.70). The following Mathematica code solves equation (3.76) numerically.
% emblackening factor of background metric
fs[r_] = 1 - rˆ3;
% equation of motion
eq[ω_,γ_] = Ax’’[r] + ωˆ2/f[r]ˆ2 Ax[r] + f’[r]/f[r] Ax’[r]
+ (12rˆ2 γ Ax’[r])/(1+4rˆ3 γ) /.f → fs;
% series expansion of A_{x} near the horizon. Infalling boundary conditions
Axnh[r_,ω_,γ_] = (1-r)ˆ(-iω/3) (1+((3+8γ(6-iω)-2iω)ω)
/(3(1+4γ)(3i+2ω))(1-r));
% series expansion of A_{x}′
Axnhp[r_,ω_,γ_] = D[Axnh[r,ω,γ],r];
% small number for setting boundary conditions just off the horizon
𝜖 = 0.00001;
% numerical solution
Asol[ω_,γ_]:= NDSolve[{eq[ω,γ]==0, Ax[1-𝜖]==Axnh[1-𝜖,ω,γ], Ax’[1-𝜖]
== Axnhp[1-𝜖,ω,γ]},Ax,{r,0,1-𝜖}][[1]];
% conductivity
σsol[ω_,γ_]:= 1/i/ω Ax’[0]/Ax[0] /.Asol[ω,γ];
% data points for σ(ω)
tabb[γ_]:= Table[{ω, Re[σsol[ω, γ]]},{ω,0.001,4,0.05}];
% make table with different values of γ
tabs = Table[tabb[γ],{γ,-1/12,1/12,1/12/3}];
% make plot
ListPlot[tabs, Axes →False, Frame →True, PlotRange →{0, 1.5}, Joined →True, FrameLabel →{3ω/(4πT),eˆ2 Re σ},
RotateLabel →False, BaseStyle →{FontSize →12}]
The output is plots of the conductivity σ(ω) for different values of γ. These plots are shown in Figure 3.3. In performing the numerics, it is best to rescale the coordinates and frequencies by setting r = r_{+}3.74) on γ. Therefore, within this simple class of strongly interacting theories, the magnitude of the ‘Damle-Sachdev’ [167] low frequency peak is bounded.
Frequency-dependent conductivity computed from the bulk theory (3.73). From bottom to top, γ is increased from −1/12 to +1/12.
3.4.5 σ(ω) part II: Critical points and holographic analytic continuation
It was emphasized in [465] that in order to describe quantum critical points rather than critical phases, it is essential to include the scalar relevant operator 𝒪 that drives the quantum phase transition. A minimal holographic framework that captures the necessary physics was developed in [595]. The action is
The two couplings α_{1} and α_{2} in the theory (3.77) implement two important physical effects. The first, α_{1}, ensures that 𝒪 acquires a nonzero expectation value at T > 0. This is of course generically expected for a scalar operator, but will not happen unless there is a bulk coupling that sources ϕ. Because C_{abcd} = 0 in the T = 0 pure AdS_{4} background, this term does not source the scalar field until temperature is turned on. While the Weyl tensor thus is a very natural way to implement a finite temperature expectation value, the C_{abcd}C^{abcd} term is higher order in derivatives and therefore should be treated with caution in a full theory. The second, α_{2}, term ensures that there is a nonvanishing C_{JJ𝒪} OPE coefficient. Both α_{1} and α_{2} must be nonzero for the anticipated term to appear in the large frequency expansion of the conductivity – i.e. for the coefficient b_{1} in (3.19) to be nonzero. Plots of the resulting frequency-dependent conductivities can be found in [595].
An exciting use of the theory (3.77) is as a ‘machine’ to analytically continue Euclidean Monte Carlo data [756; 465; 595]. The conductivity of realistic quantum critical theories such as the O(2) Wilson-Fisher fixed point can be computed reliably along the imaginary frequency axis, using Monte Carlo techniques. However, analytic continuation of this data to real frequencies poses a serious challenge, as errors are greatly amplified. One way to understand this fact is that, as we will see in §3.5 shortly, the analytic structure of the conductivity in the complex frequency plane in strongly interacting CFTs at finite temperature is very rich [759]. In particular there are an infinite number of ‘quasinormal poles’ extending down into the lower half frequency plane.
The conductivity of holographic models, in contrast, can be directly computed with both Euclidean signature (imposing regularity at the tip of the Euclidean ‘cigar’) and Lorentzian signature (with infalling boundary conditions at the horizon). Solving the holographic model thereby provides a method to perform the analytic continuation, which is furthermore guaranteed to satisfy sum rules and all other required formal properties of the conductivity. A general discussion of sum rules in holography can be found in [332]. The model (3.77) allows an excellent fit to the imaginary frequency conductivity of the O(2) Wilson-Fisher fixed point. The fit fixes the two parameters α_{1} and α_{2} in the action. The dimension Δ of the relevant operator at the Wilson-Fisher fixed point is already known and is therefore not a free parameter. Once the parameters in the action are determined, the real frequency conductivity σ(ω) is easily calculated numerically using the same type of Mathematica code as described in the previous subsection.
Further discussion of the use of sum rules and asymptotic expansions to constrain the frequency-dependent conductivity of realistic quantum critical points can be found in [758].
3.4.6 Particle-vortex duality and Maxwell duality
All CFT3s with a global U(1) symmetry have a ‘particle-vortex dual’ or ‘S-dual’ CFT3, see e.g. [767]. The name ‘particle-vortex’ duality can be illustrated with the following simple example. Consider a free compact boson θ ∼ θ + 2π, with Lagrangian
Here w_{n} ∈ ℤ denotes the winding number of a vortex located at (x_{n}, y_{n}), and 3.78), understood to be inside a path integral. We begin by adding a Lagrange multiplier J^{μ}:
with F = dA and a normal vector parallel to the worldline of the nth ‘vortex’. Evidently, the compact boson is the same as a free gauge theory interacting with charged particles. The vortices have become the charged particles, which leads to the name of the duality.
Above, we see that (3.78) and (3.81) are not the same. In some cases, such as non-compact ℂP^{1} models [587] or supersymmetric QED3 [454], the theory is mapped back to iteslf by the particle-vortex transformation. These theories can be called particle-vortex self-dual. We will see that the bulk Einstein-Maxwell theory (3.43) is also in this class, at least for the purposes of computing current-current correlators. We firstly describe some special features of transport in particle-vortex self-dual theories.
In any system with current conservation and rotational invariance, the correlation functions of the current operators may be expressed as [391]
with p^{μ} = (ω, k). No Lorentz invariance is assumed. K^{L} and K^{T} determine the longitudinal and transverse parts of the correlator. The second term in the above equation is defined to vanish if μ = t or ν = t. In [391] it was argued that particle-vortex self-duality implies
at all values of ω, k and temperature T. 𝒦 is the constant ω → ∞ value of the conductivity, as defined in §3.1 above, and must be determined for the specific CFT. The key step in the argument of [391] is to note that particle-vortex duality is a type of Legendre transformation in which source and response are exchanged. This occurred in the simple example above when we introduced 2J^{μ} = 𝜖^{μνρ}∂_{ν}A_{ρ}. When sources and responses are exchanged, the Green’s functions relating them are then inverted. Self-duality then essentially requires a Green’s function to be equal to its inverse, and this is what is expressed in (3.83).
Setting k → 0 in the above discussion, spatial isotropy demands that K_{L}(ω) = K_{T}(ω) = σ(ω). Hence, (3.83) in fact implies that for self-dual theories:
This is not a symmetry of the full quantum mechanical Maxwell theory partition function, as electromagnetic duality inverts the Maxwell coupling. However, all we are interested in here are the equations of motion. To see how (3.85) acts on the function σ(ω) we can write
We have used isotropy in the second-to-last equality. We have written 3.85) means that G satisfies the same equations of motion as F. Therefore, we must have
To paraphrase the above argument: from the bulk point of view, the conductivity is a magnetic field divided by an electric field. Electromagnetic duality means we have to get the same answer when we exchange electric and magnetic fields. This fixes the conductivity to be constant.
The argument of the previous paragraph can be extended to include the spatial momentum k dependence and recover (3.83) from bulk electromagnetic duality (3.85), allowing for the duality map to exchange the longitudinal and transverse modes satisfying (3.47a) and (3.47b) above [391]. Furthermore, an explicit mapping between electromagnetic duality in the bulk and particle-vortex duality in the boundary theory can be shown at the level of the path integrals for each theory. See for instance [767; 567; 391].
The inversion of conductivities (3.86) under the duality map (3.85) is useful even when the theory is not self-dual. We noted above that for small deformations γ, the theory (3.73) is mapped back to itself with γ → −γ under (3.85). We can see the corresponding inversion of the conductivity clearly in Figure 3.3.
3.5 Quasinormal modes replace quasiparticles
3.5.1 Physics and computation of quasinormal modes
We have emphasized around (2.4) above that zero temperature quantum critical correlation functions are typically characterized by branch cut singularities. These branch cuts smear out the spectral weight that in weakly interacting theories is carried by dispersing quasiparticle poles, at ω = 𝜖(k), on or very close to the real frequency axis. Holographic methods of course reproduce this fact, as in equation (1.38) above. In contrast, at T > 0, all strongly interacting holographic computations have found that the retarded Green’s function is characterized purely by poles in the lower half complex frequency plane (causality requires the retarded Green’s function to be analytic in the upper half plane):
Quasinormal modes do not indicate the re-emergence of quasiparticles at T > 0. Instead, the imaginary and real parts of ω_{⋆} are typically comparable, so that each modes does not represent a stable excitation. Long-lived quasinormal modes, close to the real axis, can generically only arise in special circumstances (as we shall see): as Goldstone bosons, Fermi surfaces and hydrodynamic modes. Occasionally, poles that are somewhat close to the real axis can produce features in spectral functions. More typically, the spectral density will be seen to receive contributions from a large number of quasinormal modes. Nonetheless, quasinormal modes have a rich and constrained mathematical structure and, in the bulk, capture essential features of gravitational physics.^{2} As T → 0, the poles coalesce to form the branch cuts of (2.4). Some quasinormal modes can survive as poles in the T = 0 limit, as we shall see in our later descriptions of nonzero density states of holographic matter.
Perturbative computations at weak coupling lead to branch cuts in T > 0 correlation functions. It is an open question whether these are artifacts of perturbation theory – that the cuts break up nonpertubatively into finely spaced poles at arbitrarily weak nonzero coupling – or whether the cuts become poles at some finite intermediate coupling or perhaps only in the infinite coupling limit [365; 659; 312]. In any case, in this section we elaborate on the description of strongly interacting T > 0 retarded Green’s functions given holographically in the form (3.88).
The quasinormal frequencies can be computed, in principle, from the formula for the holographic retarded Green’s function in (3.24) above. Specifically, poles of the retarded Green’s function occur at frequencies ω_{⋆} such that the corresponding perturbation δϕ(r) of the bulk solution [96; 699]
- Satisfies infalling boundary conditions (3.28) at the horizon.
- Is normalizable at the asymptotic boundary, so that δϕ^{(0)} = 0 in (2.24).
Satisfying the two conditions above typically leads to an infinite discrete set of complex frequencies ω_{⋆}. In particular, the infalling boundary conditions may be imposed for real or complex frequencies.
A technical challenge that arises in numerical studies is the following: in the lower half complex frequency plane, the infalling mode grows towards the horizon while the unphysical ‘outfalling’ mode goes to zero (this fact is seen immediately from (3.28)). This means that for frequencies with large imaginary parts, imposing infalling boundary conditions requires setting to zero a small correction to a large term, which can require high precision. This problem is especially severe for infalling boundary conditions at extremal horizons (as in e.g. (3.40)), where one must set to zero an exponentially small correction to an exponentially large term. To circumvent this problem, various numerical methods have been devised to directly obtain the quasinormal modes in asymptotically AdS spacetimes. These include the use of truncated high order power series expansions [403] and a method using continued fractions [711]. For extremal horizons, a more general method due to Leaver is necessary [511], as described in [193].
Analytic computations of quasinormal modes are possible using WKB methods, suitably adapted to deal with complex frequencies. These methods were pioneered in [585]. While formally capturing modes at large frequencies, the WKB formulae often give excellent approximations for ω_{⋆} down to the lowest or second-lowest frequency. Comprehensive results for planar AdS black holes using this techniques can be found in e.g. [599; 269]. For example, for a scalar operator in a CFT3 with large scaling dimension Δ ≫ 1, cf. §3.3.2 above, one has [269]
These are results for zero momentum, k = 0. One corollary of this result is that there are infinitely many quasinormal frequencies, extending down at an angle in the complex frequency plane. The poles exist in pairs related by reflection about the imaginary frequency axis. This follows from time reversal invariance, according to which G^{R}(−ω^{*}) = G^{R}(ω)^{*}.
The quasinormal modes of scalar operators in quantum critical theories with z > 1 have been studied in [724] as function of scaling dimension Δ of the operator and dimension d of space. The modes have momentum k = 0. When z < d, the quasinormal modes are qualitatively similar to those in CFTs (i.e. with z = 1). That is, they extend down in the complex frequency plane similarly to (3.89). However, for z ≥ d all quasinormal frequencies are overdamped, with purely imaginary frequencies. This is illustrated in Figure 3.4. The density of states in these quantum critical theories, on dimensional grounds, satisfies ρ(ω) ∼ ω^{(d−z)/z} [724]. Therefore, d < z implies a large number of low energy excitations. Decay into these excitations may possibly be the cause of the overdamped nature of the modes. Precisely at z = d, the quasinormal frequencies may be found analytically [724]. They have a similar integrable structure to those of CFT2s [96; 699].
Quasinormal poles as a function of z. Hollow dots denote quasinormal frequencies at z = 1. The arrows show the motion of the poles as z is increased. For z ≥ d the modes are all overdamped, lying along the negative imaginary axis. All modes have momentum k = 0. [Figure adapted with permission from [724]]
An example where a low-lying quasinormal mode imprints a feature on the conductivity along the real frequency axis is the theory (3.73), Maxwell theory modified by a coupling to the Weyl tensor, that we studied above. Figure 3.5 plots |σ(ω)| in the lower half complex frequency plane. This plot is obtained using the same Mathematica code that produced Figure 3.3 above. A higher order series expansion has been used to set the initial conditions close to the horizon and a higher WorkingPrecision has been used in numerically solving the differential equation. In the figure we see that γ = 1/12 leads to a pole on the negative imaginary axis whereas γ = −1/12 leads to a zero on the negative imaginary axis. These are responsible for the peak and dip seen in Figure 3.3, respectively. The poles and zeros are exchanged in the two plots of Figure 3.5. This is consistent with the fact that electromagnetic duality inverts the conductivity (3.86), and also, for γ ≪ 1, sends γ → −γ. We thus learn that the zeros of the conductivity in the complex frequency plane are also important, as they become the poles of the particle-vortex dual theory. A similar zero-pole structure to that of Figure 3.5 is seen in other cases where particle-vortex duality acts in nontrivial way, such as in a background magnetic field [353]. We will discuss magnetic fields and particle-vortex duality further in §5.7 below.
|σ(ω)| in the lower half plane, with γ = 1/12 (left) and γ = −1/12 (right). The dominant features are poles and zeros, which are shaded darker than the background. The plot has been clipped at e^{2}|σ| = 2, and so poles are associated with the solid/sharply shaded regions.
Beyond peaks or dips caused by specific poles or zeros, one can use the lowest few poles and zeros to construct a ‘truncated conductivity’ that accurately captures the low frequency behavior of the conductivity along the real frequency axis [759].
In the case of γ = 0, pure Maxwell theory in the bulk, there are no quasinormal modes. Recall that the conductivity (3.72) is featureless in this case. It is instructive to see how the quasinormal modes plotted in the previous Figure 3.5 annihilate in a ‘zipper-like’ fashion as γ → 0. Figure 3.6 shows that, as γ is lowered, pairs of zero and poles move towards the imaginary frequency axis. Once the pair reaches the axis, one moves up and the other moves down the axis. The pole or zero that moves up then annihilates, as γ → 0, with a zero or pole that is higher up the axis. The poles and zero annihilate precisely at the Matsubara frequencies , with n = 1, 2, 3, … [759].
Motion of poles and zeros as γ is decreased. Crosses are poles and circles are zeros. Pais of poles and zeros move to the imaginary frequency axis, and then move up and down the axis to annihilate with a zero or pole. In these plots w = ω/(4πT). Figure taken with permission from [759]
A very important class of quasinormal poles are those associated with hydrodynamic modes. Hydrodynamic modes have the property that ω_{⋆}(k) → 0 as k → 0. These modes are forced to exist on general grounds due to conservation laws (we will discuss hydrodynamics in Chapter 5) and can usually be found analytically. In fact, relativistic hydrodynamics can be derived in some generality from gravity using the ‘fluid/gravity’ correspondence [92]. We have already computed a hydrodynamic quasinormal mode in §3.4.3 above. In (3.65) we found the diffusion mode
In later parts of this review we shall come across other circumstances where ω_{⋆} → 0: Goldstone modes, zero temperature sound modes and Fermi surface modes.
3.5.2 1/N corrections from quasinormal modes
We have said that quasinormal poles define the ‘on-shell’ excitations of the strongly interacting, dissipative finite temperature system. Virtual production of these excitations in the bulk should be expected to contribute to observables at subleading order in the large N expansion (which, we recall, is the semi-classical expansion in the bulk). Occasionally such a ‘one-loop’ effect in the bulk can give the leading contribution to interesting observables. In the boundary theory, these effects either correspond to non-equilibrium thermodynamic fluctuations or else to physics that is suppressed because it involves a number of modes that remains finite in the large N limit (e.g. there might only be one Goldstone boson).
The usual Euclidean determinant formulae for the one-loop correction to black hole backgrounds obscures the physical role of quasinormal modes. The determinant correction to the bulk partition function (1.20) takes the schematic form
The formula (3.92) is useful for calculating the correction to thermodynamics quantities. One may also be interested in non-analytic corrections to Green’s functions. Here quasinormal modes also play an essential role. We will sketch how this works, more details can be found in [124; 345; 46; 264]. A typical one-loop correction ΔG to a Green’s function will be given by the convolution of two bulk propagators. The finite temperature calculation is set up by starting with periodic Euclidean time. Fourier transforming in the boundary directions, but working in position space for the radial direction, the bulk Euclidean propagators take the form G(iω_{n}, k, r_{1}, r_{2}). Thus, schematically,
The second step is to re-express the correction in terms of retarded Green’s functions. This is achieved using standard representations of the sum over Matsubara frequencies as a contour integral. See the references above. For bosonic fields, this leads to, again schematically,
We analytically continued the external frequency iω_{n} → ω and for simplicity set the external k = 0. We are interested typically in small ω ≪ T. The objective now is to identify the most IR singular part of these integrals. This looks challenging, in particular because of the integrals over the radial direction. However, if there is a quasinormal frequency ω_{⋆}(k) that goes to zero, say as the momentum goes to zero, then one can isolate the singular contribution of this mode to the Green’s function [46]. Schematically,
To illustrate how zooming in on the quasinormal mode (3.95) picks out the singular physics of interest, consider the case of long-time tails that appear in the electrical conductivity. Here one is looking for a one-loop correction to the propagator of the bulk Maxwell field (photon). The one-loop diagram is one in which a photon and a graviton run in the loop (there are no photon self-interactions in Einstein-Maxwell theory). The retarded Green’s functions of these bulk field each have a diffusive quasinormal mode, corresponding to diffusion of charge and momentum. Zooming in on these modes, and assuming that the relevant frequencies will have Ω ≪ T, the integrals in (3.94) become, without keeping track of the overall numerical prefactor,
The above outline is essentially enough to isolate the singular effect in many cases. To obtain the correct numerical prefactor, or to convince oneself that there are no other singular contributions, substantially more work is typically necessary [124; 345; 46; 264].
Exercises
3.1. High frequency conductivity. In this exercise, you will derive the high frequency expansion (3.20) holographically, following [546]. Consider the EMD action (4.43) with Z = 1 + αΦ + ⋯, and dilaton Φ dual to a relevant operator of dimension Δ. You may assume that the background is of the form (4.44), and that it is asymptotically AdS, but take p = 0 for simplicity.
a) Find the equation of motion for a perturbation δA_{x}(r)e^{−iωt} about the background. Do a change of variable to R ≡ ωr. Compute δA_{x}(R) exactly in the ω → ∞ limit, and conclude that σ(ω → ∞) = 1/e^{2}.
b) In the R variable, it is efficient to compute small corrections in ω^{−1} to δA_{x}(R). Let 𝒪 be the scalar operator dual to Φ, and g the field theory source that couples linearly to 𝒪. If either g or ⟨𝒪⟩ is non-vanishing, Φ(r) is non-vanishing. Show that to leading order in ω^{−1}, the corrections to σ(ω) due to nonzero g and ⟨𝒪⟩ is captured by including the 𝒪(α) corrections to the equation of motion of part a), but neglecting the IR deformation of the geometry away from AdS. Further explain why δA_{x} is, at this order in ω^{−1}, independent of the precise form of Φ(r) in the entire bulk geometry.
c) Compute σ(ω) to first order in α, and thus derive (3.20), along with explicit expressions for c_{1} and c_{2}.
d) For what values of Δ could the sum rule (3.13) hold when ⟨𝒪⟩ ≠ 0? What about when g ≠ 0? Hint: σ(ω) is analytic in the upper half of the complex plane. When can you evaluate (3.13) by integrating over an infinitely large semicircle in the upper half plane?
3.2. Damle-Sachdev peak from a quasinormal mode. In this exercise you will show that as the coupling to the Weyl curvature γ in (3.73) becomes large, then a quasinormal mode moves close to real axis, resulting in a sharp ‘Damle-Sachdev’ peak in σ(ω). Such peaks are usually characteristic of weakly interacting CFTs. Indeed, the large γ limit is outside of the bound (3.74) for the validity of the strongly interacting holographic theory. Nonetheless, this example serves to illustrate an analytic method of computing quasinormal modes close to the real axis.
a) The equation of motion for fluctuations a_{x} has been given in (3.76) in the text. The objective is to look for a quasinormal mode to this equation at a small frequency ω. To this end, you can follow the same strategy that was pursued in §3.4.3 in the text to find the diffusive mode. Look for a solution to (3.76) of the form
I.e. plug this ansatz into the equation of motion, expand in small ω and solve the resulting differential equation for b(r). Why did you have to factor out the term f(r)^{−iω/(4πT)}?
b) Impose regularity on b(r) near the horizon. Furthermore require that the full solution (3.97) have no source at the boundary r = 0. In this way obtain the γ ≫ 1 quasinormal frequency
Here 2 · 2^{2/3} · 3^{1/2} ≈ 5.4989. Convince yourself that the low frequency expansion you have just used is self-consistent for this mode. What is the consequence of this quasinormal mode for σ(ω)?
3.3. Quasinormal modes of a scalar. Consider a massive scalar dual to an operator of dimension Δ in a Lifshitz geometry with z = d and θ = 0, at nonzero temperature T.
a) Follow the notation of §2.2.1. Change radial coordinate to R = (r/r_{+})^{2z}. Show that the two linearly independent solutions to the scalar equation of motion at k = 0, ω ≠ 0 are
b) Show that the quasinormal modes are [724]
3.4. Free energy of damped harmonic oscillator from quasinormal modes. In this exercise you will verify the formula (3.92) in the simplest possible setting of a damped harmonic oscillator, which has two quasinormal modes, following [193]. The retarded Green’s function takes the form
through a sum over Matsubara frequencies ω_{n} = 2πnT. Specifically,
Obtain the free energy from this formula. There are various ways to do the sum, for instance you can differentiate with respect to m^{2} and then integrate after doing the sum.
b) Write the retarded Green’s function (3.101) as a sum of two poles at ω = ω_{i}, with i = 1, 2. Thereby obtain the free energy from (3.92) as
Check that this agrees with the result you obtained from the Euclidean Green’s function in the previous part, up to non-universal terms.
Notes
^{1} For vectors and antisymmetric tensors, covariant derivatives can be written as ∇_{a}X^{a⋯} = (−g)^{−1/2}∂_{a}((−g)^{1/2}X^{a⋯}).
^{2} A large part of the first gravitational wave signal detected by the LIGO collaboration is the ‘ringdown’ of a quasinormal mode [4]. See the image at the start of Chapter 1.