7

Further topics

A time-resolved image of the local number density of a cold atomic gas. Black spots correspond to vortices, where the density vanishes. Proceeding left to right, we see the forward time evolution of the cold atomic gas. Understanding the chaotic dynamics of annihilating vortices requires a theory of superfluid turbulence, discussed in §7.4. Quantum systems out of equilibrium will be one topic discussed in this chapter. Image taken from [504] with permission.

7.1 Probe branes

7.1.1 Microscopics and effective bulk action

In this section we discuss a class of holographic models that are obtained from microscopically consistent string theoretic backgrounds in the bulk. They have an extra ingredient relative to the Kaluza-Klein compactifications discussed in §1.8: the probe branes. The original motivation for considering probe branes came from the desire to include dynamical degrees of freedom in the fundamental representation (quarks) in holographic models for QCD [458]. The types of construction considered in §1.8, including for example the string theory background AdS5 × S5, are dual to quantum field theories such as 𝒩 = 4 SYM that only contain adjoint degrees of freedom (gluons). We will see that probe branes are also interesting objects in the context of quantum matter. For now, we digress from condensed matter physics temporarily. The upshot of the following motivational paragraphs will be the action (7.1) and (7.2) below that we shall be adding to the bulk theory. A microscopic motivation is necessary in this case to justify the particular nonlinear form of the action. For a more detailed microscopic description, see [125].

Recall from §1.4 that 𝒩 = 4 SYM in d = 3 with gauge group SU(N) is the effective theory describing the low energy excitations of a coincident stack of N D3-branes, in the regime where gravitational backreaction can be neglected. The massless degrees of freedom come from open strings stretching between pairs of the branes. Such open strings carry two indices (one for each brane they end on) and hence are in the adjoint representation of SU(N). If we want to construct quarks, which transform in (anti-)fundamental representations of the gauge group, we need an additional object on which an open string can end, leaving a single endpoint on the D3 brane stack. In particular, we can place another Dp brane somewhere – wherever the new Dp brane intersects the D3 brane stack, massless “quarks” exist due to strings stretching between D3 and Dp branes.

The crucial simplification of the probe brane limit is to consider many fewer than N of the Dp branes. In general one considers Nf N Dp branes. The subscript f stands for ‘flavor’, as each Dp brane corresponds to a distinct flavor of quark. To avoid clutter we will set Nf = 1. Increasing the string coupling as described in §1.4 causes the N D3 branes to backreact gravitationally on the spacetime and generate the AdS5 × S5 near horizon geometry. However, in the limit in which Nf N, the Dp branes are not themselves heavy enough to backreact and hence they remain present in the backreacted spacetime. That is to say, to add Nf quarks to 𝒩 = 4 SYM, one must add Nf Dp branes into the dual AdS5 × S5 spacetime. We will now describe what this means in practice.

Certain placements of the Dp brane in AdS5 × S5 preserve 𝒩 = 2 supersymmetry, as discussed in [458]. It is helpful to preserve supersymmetry because otherwise the probe branes typically have an instability in which they ‘slide off’ the internal cycles of S5 on which they are placed. Two commonly studied configurations are:

  1. To obtain quarks propagating in all d = 3 spatial dimensions of the QFT, we place a D7 brane on AdS5 × S3. The choice of S3 ⊂ S5 plays a physical role that we will elucidate below.
  2. To obtain quarks propagating on a d = 2 dimensional defect in the d = 3 QFT, we place a D5 brane on AdS4 ⊂ AdS5, wrapping two dimensions along an S2 ⊂ S5. The degrees of freedom on the defect can exchange energy and momentum with the higher dimensional excitations of 𝒩 = 4 SYM [463; 195; 255].

To describe the Dp brane, we must add additional bulk degrees of freedom that propagate on the p + 1 dimensional worldvolume of the brane. There are two types of fields on the worldvolume. Firstly there are scalar fields that describe how the brane is embedded into the spacetime. Secondly there is a U(1) gauge field. As always in holography, this gauged symmetry corresponds to a global U(1) symmetry of the boundary theory. The global symmetry is sometimes referred to as ‘baryon number’, as it counts the density of quarks in the theory. In the following we will show how the dynamics of the U(1) gauge field on the brane leads to novel transport effects that are not captured by the Einstein-Maxwell-dilaton class of theories with action (4.43) that we have considered so far. For brevity, we shall mostly ignore the scalar field dynamics. These can be consistently neglected in considering the Maxwell field dynamics, in the setups we consider, once the embedding of the Dp brane is given.

The new feature of the brane action is that it describes a nonlinear theory for the Maxwell field strength Fαβ on the brane. Here αβ denote worldvolume indices on the Dp brane. D branes can be shown to be described by the Dirac-Born-Infeld (DBI) action [631; 632]:

Here 2πα is the fundamental string tension and gαβ is the induced metric on the brane (from its embedding into the spacetime). Also recall (1.14) and (1.16), and note that the brane tension 7.1) at hand, computations are done using the standard holographic dictionary.

A single probe brane will negligibly backreact on the geometry, hence the name ‘probe’. This is consistent with the fact that the free energy of the D3 branes scales as N2, compared to N for the fundamental quarks dually described by the probe brane. We also note for future reference that other thermodynamic and hydrodynamic quantities for the quark matter will also scale ∼ N. The ability to neglect backreaction at large N makes this class of models very tractable – allowing a wider range of observables to be computed – but also means that the large N limit will miss a lot of physics that is relevant for transport.

Hence, we may place the Dp brane on top of the AdS5 (Schwarzschild) geometry, treating gαβ as fixed in (7.2). The induced metric gαβ is immediately found by restricting the AdS5 × S5 metric (1.15) to the probe brane worldvolume:

where k is defined as the dimension of the sphere which the brane wraps, and L cos θ is the radius of the Sk. Here we have allowed the size of the wrapped sphere to be radially dependent, although in several of the examples we consider below we will simply have θ = 0. The effective description of the Maxwell field dynamics on the extended pk + 1 dimensions of the brane is given by integrating the full DBI action over the internal space to obtain

7.3).

Upon Taylor expanding (7.1) in the limit of small Fab, we readily recover the Maxwell action with electromagnetic coupling

7.5) shows that the effective coupling e is also radially dependent, somewhat analogously to the Z(Φ) factor in an EMD model such as (4.43).

7.1.2 Backgrounds

With the induced metric fixed to be (7.3) we must find the background field θ(r) describing the brane embedding, as well as the electrostatic potential At(r) that will determine the dual charge density in the usual way. Evaluated on such configurations, the reduced action (7.4) becomes

We have restricted to the case of ‘space-filling’ branes with pk + 1 = d + 2 for concreteness. The dimensionless prefactor of the 1.16) in this case). The overall normalization of the action will not be important for us.

The bulk excitation described by θ(r) is dual to an operator that gives a mass to the quarks in the dual field theory (see e.g. [488]). For massless quarks we set θ = 0. At low temperatures and zero charge density, massive quarks means that the fundamental degrees of freedom described by the brane are gapped (bound into mesons). Analogously to our discussion in §2.4 above, we should expect the brane not to reach down into the far IR geometry. Indeed, in these cases the Dp brane “caps off” at some critical radius where θ(rcrit) = π/2, and the brane does not extend below this radius [570; 18; 460]. With a nonzero charge density, this capping off does not occur even at low temperatures [488]. Thus there are gapless charged degrees of freedom in this case, even with a mass for the quarks. This is analogous to massive free fermions with a chemical potential larger than the mass. We shall focus on the massless case with θ = 0, and we will comment on the effects of a nonzero mass where relevant.

With θ = 0, the background profile for At is easily seen to be

4.24), while about backreaction of the brane on the background and, potentially more tractably, the possibility that stringy α′ effects will become important [368].

The chemical potential corresponding to (7.7) is given by 7.7) that

= 0. The quantity (7.8) will appear in a couple of places shortly.

The thermodynamics of probe branes is easily computed using standard techniques. Somewhat unconventional powers of temperature arise, see e.g. [459], although these should be understood as corrections to the order N2 thermodynamics of the adjoint sector degrees of freedom. One relevant fact is that probe branes at nonzero charge density have a nonvanishing zero temperature entropy density, see e.g. [461; 455; 459],

7.9) is formally subleading in 1/N. This is immediately reminiscent of the zero temperature entropy density (4.34) of extermal AdS-RN black holes. The way the entropy arises technically in the probe brane case is that the free energy of the quarks is essentially given by the length that the brane extends from the boundary down to the horizon. This length grows linearly with low temperatures as the horizon recedes from the boundary. This result also holds with a quark mass. In the following §7.1.3 we will see another instance in which the low energy DBI dynamics resembles that of a z = ∞ scaling theory, even without backreaction onto the spacetime metric (which has z = 1 in the present case).

7.1.3 Spectral weight at nonzero momentum and ‘zero sound’

As previously, charge and current correlation functions are obtained from time- and space-dependent perturbations of the background described in the previous §7.1.2. Because there is no coupling to metric perturbations in the probe limit, these fluctuations obey equations similar to the those discussed in §3.4.1 for the charge dynamics of a zero density critical point. The novel effect is the nonlinearity of the DBI action that couples the perturbations to the background in a new way. Specifically, equations (3.47a) and (3.47b), with z = 1, are replaced by

3.47a) and (3.47b) upon setting , removing the nonlinearities. Versions of these equations have been considered in several papers including [455; 501; 502; 183; 310; 38].

A key effect of the DBI nonlinearities is seen as follows. At zero temperature f = 1 and the near horizon limit corresponds to r → ∞. In both of the above equations we then see that 7.10b) that is valid at large k, following [38].

A WKB computation of the spectral density at large momenta proceeds as outlined around equation (4.95) above. The computation can be done at any temperature. In both the longitudinal and transverse channels the low energy spectral weight (4.67) following from (7.10) is found to be

Recal that f is a function of r/r+. Rescaling r = r+7.11) then gives [38]

7.8) above. This computation shows that zero temperature, zero frequency spectral weight is present for all momenta k < k. The scale of k is set by the charge density. The nonzero momentum spectral weight is more dramatic than that of extermal RN found in (4.76) and (4.79), which vanished as a power law at low temperatures or frequencies. The spectral weight found in (7.12) is precisely what might have been expected of a 2kF singularity from a Fermi surface theory that has been smeared out by strong interactions.

Probe branes show a further feature reminiscent of Fermi surfaces: linearly dispersing collective modes at zero temperature. In an ordinary Fermi liquid, these modes essentially correspond to the “sloshing” of the Fermi surface at fixed density. In contrast, in an ordinary sound mode, the charge density (along with the pressure) will oscillate at finite density. The holographic mode is found by solving the perturbation equations in the longitudinal channel (7.10a) at small momenta and frequencies and zero temperature. This can be done using, for instance, the method described in §3.4.3 above. One finds a low frequency quasinormal mode with the dispersion [455]

7.8) above. These zero temperature sound modes have been called holographic ‘zero sound’ by analogy with the modes that exist in a Fermi surface. It is noteworthy that, as we noted in §5.4.2 for analogous T = 0 sound modes in backreacted EMD models, these modes coexist with low energy spectral weight at nonzero momentum. Perhaps a direct connection can be established in holographic models.

It is instructive to track the quasinormal modes in the complex plane upon increasing the temperature [86; 183]. Unlike the backreacted models discussed in §5.4.2, the probe brane modes do not couple to metric fluctuations and hence are not involved in conventional T > 0 sound modes (as occurs in the backreacted models). Therefore the ‘zero sound’ mode of probe branes cannot cross over to the hydrodynamic nonzero temperature sound mode. Instead what one finds is that the pair of ‘zero sound’ modes has a decreasing propagation speed and increasing attenuation constant. At a critical temperature, the propagation speed vanishes, and the two poles begin moving in opposite directions along the imaginary frequency axis. At high temperatures, the pole that moves towards the real axis becomes the ordinary hydrodynamic charge diffusion mode. The other pole is non-hydrodynamic. However, the presence of a non-hydrodynamic pole on the real axis will have a consequence that we discuss in the following section: a Drude peak in the optical conductivity (despite the fact that the probe brane does not conserve momentum, which can be lost to the bath).

The ‘zero sound’ modes are seen in other probe brane models, including ones with a mass for the fundamental matter [501]. An interesting effect is seen if the background bath of adjoint matter (the geometry in which the probe brane is embedded) has z ≠ 1. There the ‘zero sound’ pole in the Green’s function takes the form [418]

where c is a complex constant. This shows that if z > 2 the mode is overdamped and does not exist as a sharp excitation. This condition is generalized in the presence of a hyperscaling violating background to z > 2(1 − θ/d) [612; 202].

7.1.4 Linear and nonlinear conductivity

Here we compute the electrical conductivity in these probe brane models, following [462]. Before presenting the computation, however, we can think about how we expect the N scaling of various quantities to affect the result. Recall the general hydrodynamic result (5.43) and the N scalings discussed in §7.1.1:

Here we used the fact that the charge dynamics is determined by the probe sector (which is order N) whereas the energy and pressure are dominated by the adjoint bath (which is order N2). Hence, at leading order in the large N limit, we will only be able to compute σQ and we will not see the effects of ‘momentum drag’. This is naturally understood from the fact that in the probe brane approximation, the backreaction of the geometry is neglected. We hence cannot recover the “boost” perturbation in the bulk corresponding to the δ function in (5.43), as we did in §5.6.5. This lack of backreaction on the geometry does come with an interesting “advantage” – we may compute σ beyond linear response without worrying about the heating of the black hole at nonlinear orders. So below, we will compute the conductivity σ ≡ J/E, but with J a non-linear function of the electric field E.

The method we will use to compute the dc conductivity was developed by [462] and is similar in spirit to the method used in §3.4.2 above. The DBI equations of motion imply radially conserved currents, assuming that all bulk fields are spatially homogeneous:

7.2). Here the raised indices mean that the whole matrix g + 2παF is to be inverted, not that the indices are raised with the inverse metric. These quantities are, in fact, equal to the expectation value of the charge current operator in the boundary theory. As above we will focus on the case of massless quarks and with the background geometry being the AdS-Schwarzchild geometry given in (2.37) and (2.42) (in real time, with z = 1 and θ = 0). Upon applying the electric field Ax = −Et + ax(r), and assuming a background gauge field At(r), we obtain the set of equations

We focus on the second equation. As f → 0 at the black hole horizon, the objects in square brackets on both sides are negative. However, as r → 0 near the boundary, both brackets are positive. Since they multiply manifestly positive quantities, these square brackets must both vanish at the same radius r = r. Setting the left hand side to zero implies

In this last equation we have set Jt = ρ for consistency with our earlier notation throughout.

At a quantum critical point, the nonlinear conductivity is subtle because the various limits T → 0, ω → 0 and E → 0 need not commute. These issues are explored for probe brane theories in [464]. If we take the linear response, E → 0, limit of (7.19) at fixed temperature, then clearly rr+, the horizon. The dc conductivity then looks intriguingly like a nonlinear version of the ‘mean-field’ dc conductivity obtained previously in (5.186). In particular the first term in the square root is suggestive of a ‘pair creation’ term while the second term in the square root depends explicitly on the charge density. This interpretation of the first term can be made more rigorous here: if a mass is given to the quarks then the first term picks up an additional factor that depends on the profile θ(r). This factor becomes small if the mass of the quarks is large compared to the charge density [462], consistent with the fact pair production is now costly. However, as we have seen in (7.15) above, the probe brane conductivities do not know about momentum conservation and so the second term cannot be interpreted as a momentum drag or ‘Drude-like’ term, despite appearances.

When the second term in the square root in (7.20) dominates (for instance with a mass for the quarks, as just described, or at temperatures low compared to the charge density) then the conductivity is proportional to the charge density. While this is natural in a limit of dilute charge carriers, the fact that it is true in the opposite limit of large densities is a nontrivial property of the DBI action in the bulk. Generalizing (7.20) to cases where the background metric has a general z [368], and using (2.44) to relate r = r+ to the temperature, one obtains in this large density limit

In particular, z = 2 leads to T-linear resistivity.

Formulae similar to (7.20) can be obtained for probe brane models in a background magnetic field [606].

The equations in (7.18) describe a stationary state in the presence of a nonlinear electric field. Perturbation about this state allows the computation of thermal fluctuations of the current operator or ‘current noise’ [705]. For simplicity considering d = 2 and Jt = 0 – i.e. at a zero density quantum critical point – an explicit calculation [705] reveals that this out of equilibrium thermal noise obeys an ‘equilibrium’ fluctuation-dissipation relation at an emergent temperature T, related to r in (7.19) by T = 1/πr. That is r rather than the horizon radius sets the effective temperature, c.f. (2.43).

In models with backreaction, nonlinear reponse necessarily induces a time dependence due to Joule heating. The growth of the black hole horizon in time due to entropy production in a nonlinear electric field can be described exactly in certain simple cases [404].

Finally, let us comment on the optical conductivity σ(ω). This can be obtained from the linearized perturbation equations given in the previous §7.1.3. The conductivity exhibits a Drude-like peak at low frequencies [368]. The peak becomes narrow (of width less than T) at low temperatures and, in fact, at zero temperature the peak becomes a delta function δ(ω) if z < 2. This can be seen from setting k = 0 in the expression (7.14) for the current-current Green’s function. This delta function is not related to momentum conservation, as momentum does not couple to probe brane excitations. Instead, it suggests that probe branes have an additional conserved quantity at T = 0 that is relaxed at any nonzero temperature [177]. Recently, it has been shown that this operator is Ji itself [133]. At low temperatures, the resulting linearized hydrodynamics of the current Jμ is

where τ is the relaxation time in the Drude peak in the conductivity, vs is the zero sound speed, and χJJ is the current susceptibility, as described in §5.6; explicit expressions for all three coefficients may be found in [133]. The resulting equations, analogous to a fluid with weak momentum relaxation, can be used to derive both a Drude peak in the conductivity, and a weakly relaxed zero sound mode at nonzero temperature.

The divergence and Drude peak discussed in the previous paragraph depend upon a nonzero charge density. In fact, at zero charge density, it was shown in [464] that in d = 2 boundary space dimensions the self-duality argument of §3.4.6 extends to the nonlinear DBI action. This implies that the full nonlinear conductivity σ(ω, E, T) is independent of frequency in that case.

7.1.5 Defects and impurities

Probe Dp branes that are localized in the boundary spatial dimensions can be used to model defects or impurities that contain their own localized dynamical degrees of freedom (in the limit where these defects do not backreact on the ambient theory). A classic condensed matter model of such a system is the Kondo model, which couples a single impurity spin to a Fermi liquid. Holographic systems can generalize this class of models to describe a single impurity interacting with an ambient strongly coupled field theory. See [675] for a general discussion of defects coupled to higher dimensional CFTs and the relationship to probe branes.

Some aspects of Kondo physics were captured in a holographic probe brane model in [257]. In particular, an effect analogous to the screening of the impurity spin at low temperatures in the Kondo model was realized by a holographic superconductor-type instability on the defect. The condensation of a charged scalar reduces the electric flux in the IR, leading to the impurity transforming in a lower dimensional representation of SU(N) in the IR. This is the screening.

An earlier probe brane discussion of Kondo physics can be found in [343]. In that construction the screening of the defect degrees of freedom at strong coupling was dually described by the breakdown of the probe limit and the backreaction of the defect branes on to the geometry. Such backreaction is described by a geometric transition [526; 203] in which the brane itself disappears (the screening) and is replaced by various higher-form fluxes that thread new cycles that emerge in the geometry.

An interesting new phenomenon occurs when probe brane and anti-brane defects are placed a finite distance R apart [447]. This is analogous to two “opposite” impurities spatially separated in the field theory description. When TR 1, each brane extends into the horizon. However, when TR 1, the minimal free energy solution corresponds to a single brane which begins and ends on the AdS boundary, only extending a finite distance into the bulk. A preliminary study of the physics of this “dimerization” transition also considered the possibility of a lattice of defects and anti-defects whose low temperature dimerization can lead to random configurations [447; 448], possibly leading to glassy physics.

Rather than treating the probe branes themselves as defects, one can also consider probe brane actions (such as DBI) with inhomogeneous boundary conditions. The discussion turns out to be very similar to that in §5.10.1, and similar phenomenology holds: see e.g. [665; 427].

7.2 Disordered fixed points

It is possible that upon adding marginal or relevant disorder to a quantum field theory, there is an RG flow to a “disordered fixed point” where the disorder strength is either finite or infinite [586]. A disordered fixed point is possible in principle because disorder breaks translation invariance at all length scales. In the case of disordered statistical field theories (in which all dimensions including ‘time’ are disordered), disordered fixed points can be accessed via standard tools that combine the replica methods with perturbative field theory: see [750] and references therein. However in the quantum mechanical case in which only the spatial dimensions are disordered, these methods typically do not work. This is because upon writing down the RG equations one finds [673; 479]

towards long distance scales, taking us outside of the perturbative regime of validity in the IR. This is to be contrasted with situations like the Wilson-Fisher fixed point where the second term in the beta function above has the opposite sign and so can balance the classical term. Even if some disordered quantum theory with the right sign quadratic term in the beta function were found – for instance by effectively reducing the role of the time dimension [111; 186] – perturbative field theory would not be able to access the strongly disordered physics that is necessary to discuss incoherent transport of §5.8, or a possible localization transition. In this section we will outline how holography may give rise to strongly disordered fixed points that can be studied without the replica trick or perturbation theory.

Let us recall how the notion of relevant versus irrelevant operators is generalized to treat disordered couplings. In field theory this is called the Harris criterion [342]. We can give a holographic derivation of this criterion [14; 535; 373], which essentially amounts to the same power counting argument one would apply directly in the field theory. Consider Einstein gravity coupled to a massive scalar field. Neglecting the backreaction of scalar fields on the geometry, a bulk scalar field corresponding to a disordered boundary source takes the form

and Δ(kr) can be expressed in terms of modified Bessel functions. Its precise form is tangential to our discussion. Even before averaging over disorder realizations, we immediately know that the metric will feel the first perturbative corrections due to this scalar hair at . Of course, it may be the case that the perturbation is large compared to the background, in which case perturbation theory has failed. Let us diagnose whether or not such a perturbative treatment about AdS is consistent in the UV (r → 0). This can be done quite simply [535]. Einstein’s equations read

We have assumed that we may average over disorder realizations to extract the long wavelength behavior of the metric. The final step follows from straightforward scaling arguments – importantly, the power of r multiplying 7.26) is small compared to the left hand side – and hence perturbation theory is sensible – so long as

7.27) is known as the Harris criterion for the relevance of disorder. In the case of more general hyperscaling-violating theories, (7.27) generalizes to [535]

. Here we present a simpler argument for this resummation. For simplicity, we focus on the case d = 1. Consider the following convenient ansatz for the metric (in units where L = κ = 1):

At leading order in perturbation theory, we may treat the right hand side of (7.30) as a small source, using A = 1/r2 as for AdS, which is manifestly homogeneous. Employing scale invariance and the precise form of Δ we find that at second order in perturbation theory:

7.30) is

[371]. Nonetheless, it would be desirable to have a more intrinsic IR understanding of the physics at work. This seems to require the development of more sophisticated techniques for solving the bulk Einstein equations with strong inhomogeneities.

With a view to understanding incoherent transport in strongly disordered gapless systems, the thermal conductivity κ has been computed in these disordered fixed points, as well in new disordered fixed points constructed from relevant disorder [373]. It was found that the thermal conductivity appears to exhibit log-periodic oscillations in temperature. Log-periodicity is symptomatic of discrete scale invariance and associated complex scaling exponents. It is possible, then, that these backgrounds then have instabilities in the general class studied in Chapter 6. A possible endpoint of such putative instabilities are geometries with fragmented horizons, of the kind mentioned in §5.10.2.

7.3 Out of equilibrium I: Quenches

Our focus so far has been on the nature of quantum matter at zero and nonzero temperature and density, and on the consequences of perturbing states of quantum matter a little away from equilibrium, leading to transport coefficients like the electrical conductivity. The final two subsections will briefly consider quantum matter far from equilibrium. This first section will focus on quenches in field theories in d ≥ 2 spatial dimensions, where there are almost no techniques from field theory to apply. As we will see, far from equilibrium dynamics will tend to drive quantum matter to finite temperature states. By studying dynamics in black hole backgrounds, and/or black hole formation, holographic models make it possible to treat such thermal effects from first principles. For this reason, the holographic approaches we derive below are a valuable tool, allowing for precise results beyond ‘dimensional analysis’. One caveat to keep in mind is that dissipation in holographic models is into a strongly interacting large N ‘bath’. This may or may not be the dissipative mechanism of interest in other circumstances.

A simple way to drive a system far from equilibrium is through a quantum quench. The idea is as follows: consider a time dependent Hamiltonian

with f(t) a function which varies over time scales τ which are often quite fast. For simplicity, most research focuses on the case where f(t) either interpolates between f(−∞) = 0 and f(∞) = 1 (quench from one state to another), or where f(±∞) = 0 but f(0) = 1 (pulsed quench). We will see examples of both in this section.

The case where f(t) = sin(ωt)Θ(t) is also interesting, though we will have little to say on it in this review. Such a drive generically does work on a system, possibly until it reaches infinite temperature. See [55] for a holographic study. On the other hand, let us also note the existence of curious (non-driven) states in field theory which do not thermalize (at large N) despite oscillating in time with non-vanishing energy density. This is manifested holographically in fully nonlinear solutions of Einstein’s equations in asymptotically Anti-de Sitter spacetime which oscillate: for example, gravitational analogues of standing waves which are nonlinearly stabilized [210; 116].

7.3.1 Uniform quenches

Let us begin with the case where H0 describes a conformal field theory, and

𝒪 is the zero momentum mode of a relevant operator of dimension 0 < Δ ≤ d + 1, and we consider the limit of small λ. The exact behavior of f(t) is not important. This setup was considered in a series of papers [93; 117; 168; 169], and we summarize the main conclusions. The most interesting observation is that the energy density added to the CFT scales as

ℰ(x) a function of its dimensionless parameter. The limit of interest is

7.35) implies the physics in this limit is universal.

The universality of this result follows from the fact that this quench is “fast”. Let us begin with the case Δ < d + 1. Holographically, the time scale of black hole formation (or any other gravitational backreaction) associated with the quench is set by λ−1/(d+1−Δ) ≡ τλ. From (7.36) τλ τ. Therefore, the quench dynamics is characterized by the fast dynamics of a scalar field in AdS, and the resulting slow dynamics of black hole formation (but with properties like 𝜖 already fixed on the fast time scale). In order to fix 𝜖 ∼ λ2, we employ a “hydrodynamic” Ward identity (c.f. (5.79))

CFT = 0 as it is relevant. Once the λ scaling of 𝜖 is fixed, the τ scaling follows from dimensional analysis. (7.35) is valid for both free theories and strongly interacting theories [168]. Of course, holography provides strongly-coupled models where (x) can actually be computed.

In the case of Δ = d + 1 (a marginal operator), [93] was able to say much more about the resulting geometry. Now λ itself is a dimensionless parameter, and at leading order in the perturbative expansion in λ, they found that the metric could be well approximated by

3.55). The function appearing above is an interpolating function between 0 at v = −∞, and 1 at v = ∞. The coefficient M is given by (4πT*/d)d+1, where T* is the final temperature of the geometry.

When the force acts only over a short time, the function appearing in the metric (7.38) is effectively

This metric is called the AdS-Vaidya metric. It is analogous to the original Vaidya metrics describing the geometry associated with infalling null dust [737], and is an exact solution to Einstein’s equations sourced by infalling pressureless dust. The field theory interpretation of (7.39) is clear, albeit quite surprising: beyond time t = 0 (in the boundary), the theory appears to thermalize instantaneously! The reason is that we have added a very small amount of energy quickly. If one adds an energy density 𝜖 ∼ τd−1 (by setting λ = 1), then thermalization does not occur instantaneously. This was observed numerically in [141], albeit in a slightly different model. More recently, [47; 273] have studied the dynamics of pure states in CFT2s of very high energy: while they look thermal at N = ∞, non-perturbative corrections in 1/N restore unitarity (no thermalization). It will be interesting to understand this physics more carefully from the bulk perspective, where such effects are non-perturbative in quantum gravity.

Another example of a spatially homogeneous quench is as follows. Consider a holographic superfluid at an initial temperature T < Tc with a source for the superfluid order parameter which is pulsed analogously to above [89]. The pulse injects energy into the superfluid and the late time dynamics of the order parameter are described by

Figure 7.1. When 0 < λ < λ1, both γ and ω are positive, and 𝒪SF(∞) > 0; when λ1 < λ < λ2, ω = 0 but B > 0; when λ > λ2, ω = 0 and B = 0. Hence, for λ < λ2, the final state is a superfluid, while for λ > λ2 it is a normal fluid.

Figure 7.1
A sketch of the dynamics of the order parameter in a holographic superfluid quench. Top row: 𝒪 as a function of t. Bottom row: location of the lowest lying QNMs, and their motion upon increasing λ. Columns left to right: 0 < λ < λ1; λ1 < λ < λ2; λ > λ2.

As we have seen repeatedly in this review, this late time dynamics is naturally understood by studying the lowest lying quasi-normal modes (QNMs) of the bulk geometry. In the superfluid phase, the existence of a Goldstone mode is manifested as a mode with γ = ω = 0, and so the relaxation in (7.40) is governed by the QNM with next smallest imaginary part. The “motion” of these QNMs with increasing λ is shown in Figure 7.1. This reveals the holographic origin of the dynamical transition at λ = λ1. When the purely damped (ω = 0) mode collides with the Goldstone mode, these modes separate off the ω = 0 axis and γ is increasing with increasing λ. This corresponds to a quench which has injected enough energy so that the final state is in the normal phase. These holographic dynamics reproduce the expectations of weakly interacting field theories [75], but it is instructive to see how the QNMs of “superfluid” black holes recover this dynamics even at strong coupling.

7.3.2 Spatial quenches

Let us now consider a slightly more complicated set-up: suppose we take two copies of a conformal field theory in d spatial dimensions, one (L) defined for x < 0 and the other (R) defined for x > 0. Now, suppose that we prepare L in a thermal state at temperature TL, and R in a thermal state at TR. Without loss of generality,1 we suppose that TL > TR. At time t = 0 we allow these theories to couple. The set-up is depicted in Figure 7.2.

Figure 7.2
A quench connecting two heat baths. Left: two CFTs placed next to each other at different temperature. Right: a NESS forms as energy flows from the left to right bath.

What is often found in such quenches that the theory reaches a non-equilibrium steady state (NESS), where the local density matrix near x = 0 is given by exp[−∑λiHi], for all conserved quantities Hi [655]. For the case of a conformal field theory, the answer is known exactly [87]. For |x| < t, the local density matrix takes the remarkably simple form

ρNESS only contains two conserved quantities, despite the existence of an infinite number. There is a very simple explanation: CFT2s consist of decoupled left and right moving theories, and so at the speed of light the left movers at TR from the right bath and right movers at TL from the left bath “mix” near the interface. This argument is rigorous and can be shown using CFT technology [87]; see [88] for a holographic derivation.

It was pointed out using hydrodynamic arguments in [88; 131] that the NESS could exist in any dimension. Let us consider a CFT for simplicity, but the argument generalizes naturally, and focus on the case where TLTR to simplify the mathematics. In this limit the system is almost in equilibrium and we can linearize the equations of hydrodynamics. These equations lead to elementary sound waves and we conclude that a NESS forms – with density matrix analogous to (7.41) – for Figure 7.2. The nonlinear calculation is more subtle but a NESS can be shown to form within hydrodynamics [545; 707].

At early times, the dynamics is complicated and possibly non-universal near the interface. Holographic descriptions [31] allow us to access both the early and late time dynamics in a unified approach. One immediate consideration is that if a NESS exists, it is thermal (c.f. (7.41)) in a holographic model: this follows from black hole uniqueness theorems [88]. It is further proposed that knowledge of the heat current in the NESS allows one to determine fluctuations of the heat current, giving the first solvable model of “current noise” in higher dimensions [88].

7.3.3 Kibble-Zurek mechanism and beyond

An even more complicated quench set-up is as follows. Consider a Hamiltonian H with a thermal phase transition at temperature Tc. As an explicit example, we will keep in mind a normal-to-superfluid transition in d = 3. The effective action for the order parameter φ dynamics is an O(2) model:

A priori, a superfluid condensate forms, with |φ|2 = 𝜖/λ, and nothing interesting happens. But what is the phase of the complex number φ? Some thermal fluctuation will lead to the local orientation of φ, but these thermal fluctuations will kick arg(φ) to different values at different points. We will now argue that this dynamics leads to the formation of vortex lines (or more general topological defects in more general models).

The canonical description of the resulting dynamics is due to Kibble [473] and Zurek [780], and so is called the Kibble-Zurek mechanism (KZM). In a theory of dynamic critical exponent z and correlation length exponent ν, the time scale associated with the dynamics is

≪|t|, then we expect that the dynamics this far from the quench is adiabatic – the system is in equilibrium (or as close to it as possible). But when τ(𝜖) t, then the system cannot reach equilibrium during the quench. Hence, we define a freeze-out time when τ(𝜖(tf)) ∼ tf:

On this short time scale the system is far from equilibrium. In particular, in the superfluid phase the early time dynamics (0 < t tf) will be in response to the thermal fluctuations imprinted on the system just above Tc, which are of a characteristic size

where dtop is the dimension of topological defects; for a superfluid, dtop = d − 2.

Recent holographic studies have shown that this argument holds for slow thermal quenches [704; 139]. The advantage of a holographic approach is that it incorporates both quantum and thermal processes, including dissipation, from first principles. To recover this effect in holography is slightly subtle – thermal fluctuations are suppressed by a factor of 1/N2 which vanishes in the classical gravity limit. So [704; 139] simply added (small) random external forcing as a boundary condition to mimic these fluctuations – the resulting dynamics will lead to defect formation across the phase transition. Holographic models recover the KZM in its regime of validity when τQ is long. As [139] also emphasized, holographic methods are well suited to model defect formation for rapid quenches, beyond the regime of validity of KZM. The KZM argument fails once the condensate amplitude |φ(teq)|2 𝜖(teq)2β for a critical exponent β; in the example (7.43) above, β = 1/2. The amplitude of fluctuations, however, may be parametrically suppressed – as in holography, where they are suppressed by 1/N2 corrections. Hence the defects are frozen not at tf, but at teq, which is generally much longer (since it takes longer for the exponentially growing |φ(teq)|2 to reach 𝜖2β. It is then teq that controls the density of defects, and this can lead to (parametrically, in the case of holography) small prefactors in front of (7.48).

When teq τQ tf, the system may always be treated in linear response during the quench as the condensate is far from well-formed. Hence, the final density of defects is governed only by the most unstable modes of the condensate at the final quench temperature. This dynamics is also associated with a characteristic length scale , which is independent of τQ. Thus, the density of vortices is expected to scale as (using β = ν = 1/2, z = 2 for the holographic model of [139])

This result was recovered in holographic numerical simulations, although it should hold more generally.

7.4 Out of equilibrium II: Turbulence

One of the most fascinating open problems in all of physics is the nonlinear and chaotic dynamics of turbulent fluids, a problem which has important practical and theoretical applications for classical fluids like water and air [171]. It has been appreciated more recently that strongly interacting quantum fluids will share many of the same dynamical phenomena. Indeed, given the fluid-gravity correspondence previously discussed, it is natural to expect that dynamical black holes can generically behave turbulently. This has indeed been shown numerically [10; 311] for 3+1 dimensional asymptotically-AdS black holes. The numerical methods necessary to study such dynamical problems are reviewed in [142].

As many of the phenomena associated with such turbulent black holes are familiar from classical turbulence, let us focus on interesting geometrical properties of a “turbulent horizon”.2 A purely gravitational calculation [10] shows that the horizon of such a black hole looks “fractal” over the inertial range (the length scales over which turbulent phenomena appear self-similar). This is reminiscent of the fact that turbulent fluids are more effective at dissipating energy than non-turbulent fluids. Indeed, the rapidly growing horizon is a signature of the fast growth of the entropy in the dual theory.

Although holography has not taught us anything about classical turbulence, it has proven to be more useful in the study of superfluid turbulence. Superfluid turbulence is often called “quantum” turbulence, though this is a lousy name. Although vortices are quantized in a superfluid, most turbulent phenomena are entirely classical in nature and insensitive to vortex quantization: if we place M 1 vortices with circulation Ω next to one another, they will behave much like a classical vortex of circulation MΩ, and create a superfluid velocity field just like a large classical vortex. The important difference between superfluid and classical turbulence actually arises at nonzero temperature. This is perhaps surprising, since most quantum phenomena are more pronounced at zero temperature. At nonzero temperature, the normal fluid and superfluid can exchange energy and momentum, complicating the description of superfluid turbulence.

The major difficulty with studying superfluid turbulence theoretically is the treatment of dissipation at nonzero temperature. The standard approach is to use a damped Gross-Pitaevskii equation (see e.g. [654]), but this approach is formally unjustified. As we have seen repeatedly throughout this review, holographic methods are naturally adapted to study dissipative dynamics at nonzero temperature. In [9], the dynamics of superfluid vortices was studied in a probe limit. In addition to seeing some signatures of turbulence including Kolmogorov scaling laws, [9] was also able to give a “microscopic” description of vortex pair annihilation, a process in which a vortex of winding number + 1 collides with a vortex of winding number −1, releasing a nonlinear burst of sound which ultimately dissipates away. By keeping track of the energy flux across the black hole horizon, they were able to show that dissipative processes during superfluid turbulence are essentially localized near vortex cores, over the “healing length” defining the vortex core size. This corresponds to the length scale over which superfluid hydrodynamics breaks down. The simple holographic picture is that the vortices punch flux tubes from the boundary down through the bulk superconductor and into the horizon. This allows a localized channel through which energy can dissipate into the horizon. Finally, in classical turbulence in two spatial dimensions, vortices of like sign tend to clump together. This signature was not strongly seen in holography, while other signatures of turbulence were observed, suggesting that the holographic computation may probe a new kind of finite temperature turbulence [9]. It is not fully understood today whether this behavior is truly turbulent, or simply a signature of ‘overdamped’ classical turbulence, as argued in [95].

The localized nature of dissipation in a manifestly nonzero temperature, dissipative theory of superfluid turbulence suggested the emergence of a simple effective description of superfluid turbulence, independent of holography. [140] emphasized this description, which is remarkably simple, as well as its practical consequences. Let Xn(t) denote the spatial position of vortex n at time t; one can argue on very general principles that the dynamics of a dilute mixture of vortices is essentially given by

𝜖ijj log |XnXm| the superfluid velocity flowing through the core of vortex n and Wn = ±1 the winding number of each vortex (higher winding number vortices are unstable and will rapidly break apart – see e.g. [9]), ρs the superfluid density, Ω the quantum of circulation, and η a dissipative coefficient whose determination requires a microscopic calculation, such as holography. The dynamics is described by a single dimensionless parameter

. This provides a simple explanation for the new behavior observed in [9]. Furthermore, the vortex pair annihilation process, which plays a non-trivial role in holographic vortex dynamics, is described by

with N(t) the number of vortices (of both Wn) at time t. A different exponent (5/3) is proposed for classically turbulent flows as [140].

Equation (7.52) is a remarkably simple prediction with ramifications for holographic and non-holographic theories, as well as experiments. Two-dimensional cold atomic gases have allowed us to study superfluid turbulence in two dimensions experimentally [600; 504]. In fact, the effective description advocated in [140] – inspired by the holographic description of vortex annihilation – may help lead to the first experimental detection of a turbulent superfluid flow in two spatial dimensions. While it is easy to experimentally measure the vortex annihilation rate, no other experimentally accessible probe of turbulence is known. It is likely that this effective description will also help to resolve the question of whether the holographic vortex dynamics of [9] is turbulent, albeit with substantial numerical effort.

Current holographic simulations of vortex annihilation “experiments” [238] may not be at low enough T/Tc to be relevant for cold atomic gases. A holographic computation of , including low temperatures, is an interesting open problem.

Exercises

7.1.  Zero sound from matching. In this exercise you will derive the zero sound quasinormal mode (7.13). You will use a different method to the sound mode exercise in Chapter 5. Following [455] the mode can be found by applying the matching method described around Figure (4.6). Namely, when (near horizon) have a parametrically overlapping regime of validity.

a) Solve the equation of motion (7.10a) for longitudinal fluctuations in the near horizon regime . Impose infalling boundary conditions on this solution, which you should have obtained in terms of elementary functions.

b) Now solve the equation of motion (7.10a) in the far regime 1. Keep ω/k fixed so that rk 1 also. This amounts to taking a small ω, k limit of the equation. What terms can be dropped in this limit? Solve the equation in this limit in terms of hypergeometric functions. This solution should have two constants of integration.

c) Expand the hypergeometric solution at large r and thereby match onto your infalling solution expanded at small r. In this way you can fix the ratio of the constants of integration. Then, with these constants fixed, expand the hypergeometric solution out to the boundary at small r. Impose the absence of a source and therby obtain the quasinormal mode dispersion (7.13). This exact same method works with z > 1 to give the dispersion in (7.14).

7.2.  Growing black hole. Consider the AdS4-Einstein-Maxwell theory, holographically dual to a charge-neutral CFT on flat space. Apply a constant electric field Fxt = E. Show that the nonlinear bulk equations, with suitable boundary conditions are exactly satisfied by an AdS-Vaidya geometry:

and describe how r+(v) is related to E. Here v is an Eddington-Finkelstein time coordinate. What is the physical interpretation of these results? [404]

7.3.  Charged disordered fixed point. Consider the AdS4-Einstein-Maxwell theory. Choose the standard Minkowski metric as the boundary conditions on the bulk metric, but choose the boundary conditions on the gauge field to be

Show that . Compute the constant c. [284]

Notes

1 So long as the theory is not chiral in d = 1.

2 We neglect subtleties with defining the black hole horizon in a dynamical spacetime – such issues are less relevant in the fluid-gravity limit.