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  with permission.
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 . 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 .
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 . 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:
- 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.
- 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 p − k + 1 dimensions of the brane is given by integrating the full DBI action over the internal space to obtain
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 p − k + 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. ). 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 . 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
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
7.1.4 Linear and nonlinear conductivity
Here we compute the electrical conductivity in these probe brane models, following . 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  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:
7.1.5 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  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 . 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 . 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 . 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 . 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  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]
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  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
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.
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 . For the case of a conformal field theory, the answer is known exactly . 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 ; see  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 TL ≈ TR 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  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 . 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 .
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  and Zurek , 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
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 . 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 .
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  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. ), 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 , the dynamics of superfluid vortices was studied in a probe limit. In addition to seeing some signatures of turbulence including Kolmogorov scaling laws,  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 . It is not fully understood today whether this behavior is truly turbulent, or simply a signature of ‘overdamped’ classical turbulence, as argued in .
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.  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
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.