A holographic view on physics out of equilibrium

DCPT-10/21

We review the recent developments in applying holographic methods to understand non-equilibrium physics in strongly coupled field theories. The emphasis will be on elucidating the relation between evolution of quantum field theories perturbed away from equilibrium and the dual picture of dynamics of classical fields in black hole backgrounds. In particular, we discuss the linear response regime, the hydrodynamic regime and finally the non-linear regime of interacting quantum systems. We also describe how the duality might be used to learn some salient aspects of black hole physics in terms of field theory observables.

1 Introduction

Astounding amount of progress and understanding in physics has been achieved by studying special systems in equilibrium, which are by definition ‘non-dynamical’, i.e. independent of time. One important reason for the prevalent focus on time-independent systems is that they are of course far simpler to study than dynamical ones. Indeed, vast majority of exactly tractable systems are of this kind. Known exact solutions are seldom fully generic and typically admit a large degree of symmetry, usually including time-translation invariance. Nevertheless, while the study of such non-dynamical, stationary situations might seem rather looking-under-the-lamppost type strategy, it has proved to be a very successful one. Although due to an incredible complexity of nature around us such exactly-tractable systems are at best only approximations to the real world, they are often remarkably relevant and useful. Dynamically-evolving systems tend to equilibrate, and in absence of external forcing they typically settle down to stationary configurations.333 Implicit in this statement is an assumption that the systems under consideration are sufficiently ergodic. Thus stable stationary solutions can reveal the late-time physics of a generic system. Moreover, many interesting physical processes such as phase transitions, which we observe occurring dynamically, can be well-studied without invoking any explicit time-dependence at all.

Nevertheless, not many would dispute that most interesting physical phenomena do involve non-trivial temporal dynamics. Not only are dynamically evolving systems ubiquitous in nature, but they are the raison d’être for everything we see around us. Hence the need to understand dynamics scarcely needs motivation. However, progress so far has unfortunately been hindered by lack of adequate techniques. Typically, one resorts to perturbative methods, though the regime of their validity places severe limitations on their applicability. Alternately, one might try to “put the system on a computer” and evolve numerically, but the computational cost involved is usually too astronomical to allow for convenient extraction of the physics. In certain cases one may circumvent both of these limitations by mapping the system into a more tractable one. We will see that all these ingredients come into play in the present Review.

So far our somewhat self-evident remarks have been rather abstract and general. We will now specify the particular context we wish to address. We will concentrate on exploring the dynamics of a certain class of quantum field theories, focusing especially on the strong coupling regime. Although we have best understanding of field theories at weak coupling, strongly-coupled systems are ubiquitous in nature, ranging from typical condensed matter systems, to quark-gluon plasmas created in high-energy experiments, and in fact play a role in most areas of physics. Apart from the evident applicability, there is also the rewarding aspect of serendipity related to the present cutting-edge experiments involving these systems.

Which properties of strongly coupled field theories do we wish to understand, and which ones can we hope to understand? Fortuitously there is a substantial overlap between the answers to both questions. In particular, it is both interesting and tractable to extract certain universal features, as we will discuss below. Conversely, we seldom can, nor want, to calculate the detailed microscopic dynamics, due to the sheer level of complexity; it is generally much more instructive to take a coarse-grained view of the system. As a result we will often focus on obtaining a low energy effective description for such strongly interacting systems. From a conventional renormalization group (RG) picture, it is then clear that one expects similar low energy physics for systems within the same universality class. One such low energy theory which we will discuss in some detail is hydrodynamics, or more generally fluid dynamics, which is expected to be a good description as long as the local fluid variables vary slowly compared to the microscopic scale, i.e. at long wavelengths and small frequencies.

Hence in studying dynamics of strongly coupled field theories, we are simultaneously exploring the dynamics of fluids. Our study is then bolstered by the insights which we have already acquired from hydrodynamics. On the other hand, despite decades of theoretical as well as numerical, observational, and experimental scrutiny which fluid dynamics has received, there are still many deep questions which remain to be answered, especially involving dynamical evolution. For example, one of the famous Clay Millennium Prize Problems concerns the global regularity (existence and smoothness) of the Navier-Stokes equations [1]. Intriguingly, the solutions often include turbulence, which, in spite of its practical importance in science and engineering, still remains one of the great unsolved problems in physics.

As already mentioned above, understanding dynamics in strongly coupled field theories, or their effective description in terms of fluids, is an exceedingly hard problem. One of the key strategies has been to focus on field theories which admit a holographic dual, and use this dual description to extract the physical properties of the field theoretic system under consideration. The prototypical case is the AdS/CFT correspondence [2, 3, 4] which relates the four-dimensional Super Yang-Mills (SYM) gauge theory to a IIB string theory (or supergravity) on asymptotically AdS spacetime.444 The AdS/CFT correspondence is comprehensively reviewed in the classic reviews [5, 6]. See also [7] for a nice review with emphasis on ‘applied holography’. As is well known, this is a strong/weak coupling duality; the strongly-coupled field theory can be accessed via the semi-classical gravitational dual, which has obvious computational, as well as conceptual, advantages. If we wish to know how a given strongly-coupled system behaves, we need not devise methods to calculate this in the field theory at all – a rather daunting, if not impossible, task – we only need to translate the system into the dual language and calculate the classical evolution using Einstein’s equations.

While the SYM is fundamentally distinct from the ‘real-world’ systems which we would ultimately like to understand, it can serve as a useful toy model to motivate and study new classes of strongly coupled phenomena. This philosophy has led to the correspondence being applied to QCD via the so called AdS/QCD approach (for reviews see [8, 9, 10, 11]), and more recently to condensed matter systems, often dubbed AdS/CMT, where a variety of physical effects raging from superfluid transitions to non-Fermi liquid behaviour are being actively investigated. An excellent account of these efforts can be found in the reviews [12, 13, 7, 14]. Of course, what makes this enterprise fascinating is the degree to which the computations in the AdS/CFT framework agree with experimentally measured quantities in the real world. One such quantity which received much attention is the shear viscosity of the quark-gluon plasma; based holographic computations, [15] suggest that the dimensionless ratio of viscosity to entropy density () has a universal lower bound, , which subsequently has been subject to intense scrutiny; for the current status in string theory see [16, 17].

Perhaps the most intriguing – and promising – aspect of the gauge/gravity correspondence is highlighted by its holographic nature: the theories which are dual to each other are naturally formulated in different number of dimensions.555 The holographic principle was proposed much before the advent of the AdS/CFT correspondence and was motivated in part by the peculiar non-extensive nature of black hole entropy in [18, 19]. This automatically implies that the effective degrees of freedom on the two sides of the correspondence are related in a highly non-local (and hitherto rather mysterious) fashion. This of course makes the task of extracting physics of one theory from its dual formulation even more formidable than merely doing calculations in the dual; indeed, much of the AdS/CFT-related efforts of the last decade have concentrated on elucidating the dictionary between the two sides. Yet at the same time this very feature, which might initially appear as an unwanted complication, lies at the heart of the enormous potential the duality holds for solving the system. The effects we seek to unravel are often very complex, emergent phenomena which cannot (in practice, or even in principle) be described in terms of the fundamental degrees of freedom. In other words, they require a different description. This by itself does not of course guarantee that the holographic description is the desired one, but the fact that out of such formidably complicated repackaging of the information a new simple and elegant picture arises, is at least promising. It may well transpire that the complex emergent phenomena we seek to understand, at least those which are in some sense fundamental, appear simple when re-written in the dual language.

One classic indication that the above hope is more that just a wishful thinking may be found in the manner in which the radial direction of the bulk emerges from the boundary field theory, sometimes referred to as the scale/radius, or UV/IR, duality [20]. From everyday experience, we are well-aware that physics likes to organize itself by energy (or length) scales; processes occurring at widely-separated scales do not interact very much. In the holographic dual, this hierarchy of scales, mathematically expressed in terms of Wilsonian RG, is neatly packaged in terms of an emergent radial direction of the bulk geometry (this idea is at the heart of the holographic renormalization group developed initially in [21]). But rather than just serving as a mnemonic for the energy scale, this dimension takes on a life of its own: it mixes with the other directions in a fully covariant fashion! Taking the bulk perspective, the field theory’s decoupling of scales is simply a manifestation of bulk locality. So this simple and natural feature of the bulk description has far-reaching implications for the boundary description.

Phrased more prosaically, it is not just to be hoped-for, but rather it is guaranteed, that certain complex phenomena must have a very simple and natural explanation in the dual picture. After all, the two dual theories are just different descriptions of the same physical system: there is no absolute notion of which side is ‘more fundamental’ than the other. Whichever side we use as a starting point, there are certain fundamental properties or principles underlying the theory that we understand in simple terms; yet these will be mapped into the dual framework, wherein they will appear as highly non-trivial. If we were to come upon them from the other side – which due to their inherent importance is perhaps not entirely unlikely – it would appear rather magical that in the dual language they suddenly become simple. Of course, this is not to say that all complex phenomena must admit a simple description in some reformulation, but it is not so outrageous to expect that the fundamentally important ones do.

The preceding philosophical interlude was meant to motivate the use of gauge/gravity duality to understand certain field theories at strong coupling in terms of their gravitational dual, beyond the mere fact that we can usually calculate in classical gravity more easily than in the strongly-coupled field theory. But the gauge/gravity correspondence likewise has profound implications when applied in the other direction. The field theory, albeit strongly coupled, provides a definition of quantum gravity with asymptotically AdS boundary conditions. Hence even in extreme regimes where classical gravity breaks down, and where we don’t yet have the tools to understand the physics within string theory directly, the field theory remains well-defined. This means that once we achieve sufficient understanding of the dictionary between the two sides, we can elucidate the long-standing quantum gravitational puzzles by re-casting them into the field theoretic language.

Let us briefly indicate some of the gravitational questions we ultimately hope to answer, since these have posed the underlying motivation in many of the works mentioned in this Review. A central question of quantum gravity concerns the fundamental nature of spacetime. We have come to realize that spacetime is an emergent concept, but exactly how it emerges remains a mystery. Happily, certain aspects of this emergence can be conveniently explored using the AdS/CFT framework, including ones pertaining to the present theme of out-of-equilibrium dynamics.

To set the stage, let us first recall several well-understood classical highlights of the correspondence. According to the AdS/CFT dictionary, different asymptotically AdS spacetimes manifest themselves by different states in the boundary field theory. For example, the vacuum state in the CFT corresponds to the pure AdS spacetime. Metric perturbations which maintain the AdS asymtopia are related to the stress-energy-momentum tensor expectation value in the CFT. More importantly, putting a black hole in the bulk has the effect of heating up the boundary theory. Specifically, a large666 AdS is a space of constant negative curvature, which introduces a length scale, called the AdS scale , corresponding to the radius of curvature. The black hole size is then measured in terms of this AdS scale; large black holes have horizon radius . Schwarzschild-AdS black hole corresponds to (approximately) thermal state in the gauge theory. This can be easily conceptualized as the late-time configuration a generic state evolves to: in the bulk, the combined effect of gravity and negative curvature tends to make a generic large-energy configuration collapse and form a black hole which then quickly settles down to the Schwarzschild-AdS geometry. On the other hand, in the field theory, a generic large-energy excitation will thermalize. At the level of this coarse entry in the dictionary we see that heating the field theory corresponds to black hole formation in the dual gravity, and the subsequent thermalization corresponds to the black hole settling down to a stationary state.

While appealingly simple, this level of understanding is far too coarse to allow us to extract the more interesting aspects. We need to probe the AdS/CFT dictionary further to uncover what happens in regions where the classical description of the black hole breaks down, such as near the curvature singularity, or in the more general dynamical situations. Ultimately, we would like to answer such questions as: Which CFT configurations admit a dual description in terms of classical spacetime? What types of spacetime singularities are physically allowed? How are the disallowed singularities resolved? How is spacetime causal structure encoded in the dual field theory?

Emergence of time is, if anything, even more mysterious than the emergence of space. Not only do we have more satisfactory toy models of the latter than of the former, but conceptually the problem of time is one of the deepest problems in quantum gravity. The ‘time’ (conjugate to the Hamiltonian) which quantum mechanics uses for evolution is ingrained in the fundamental formulation of the system; it is there from the start rather than being emergent. We can take this evolution parameter to be simply the time of the non-dynamical (and non-emergent) background on which the CFT lives. What is then the relation between this CFT time and the notion of time in the dual bulk spacetime? For bulk spacetimes which are globally static we tend to associate the two; but already for general dynamical spacetimes there is no natural identification even classically, since there is no uniquely-specified foliation of the bulk. Even if we can construct geometrically defined spacelike slices through the bulk (such as volume-maximizing ones), there is a-priori no reason that the bulk events localized on a given slice should be dual to boundary events localized at the corresponding boundary time; to the contrary, we have many indications that the correspondence is much more temporally non-local. Nevertheless, we still expect that time-dependence on the boundary will be manifested by time-dependence in the bulk, and vice-versa. Indeed, focusing on certain characteristic features of a given time-dependence can enable us to elucidate the gauge/gravity dictionary further. Thus, our overarching motif of considering dynamical systems in strongly coupled field theories will naturally translate to studying bulk spacetime dynamics.

We now return to our initial comments regarding time-dependence being difficult to handle. The reader might observe that these general comments applied equally to the bulk side of the correspondence as well as to the boundary side, and might therefore wonder what have we gained by translating one hard problem into another hard problem. The purpose of this Review is to indicate what in fact we have gained by using a holographic dual, and to outline some of the methods that have been used to obtain further understanding of dynamical systems. We will structure our presentation of the various approaches according to the severity of time-dependence they can handle, i.e. how strongly out-of-equilibrium evolutions they apply to.

A useful starting point is to consider a well-understood global equilibrium situation, and try to understand the response of the system under small deviations from this equilibrium. If the amplitude of such deviations is suitably small everywhere, the system may be studied using a linear response theory, which will be the focus of §3 and §4. We will briefly review the basic concepts in linear response theory in §3 focussing on two main aspects: the use of retarded correlation function in equilibrium to extract dynamics in the presence of small fluctuations, and the behaviour of fast probes in an equilibrated medium as modeled by Langevin dynamics. In §4 we will see how this physics can be mapped into the gravitational arena. The linear response regime is in fact simplest to understand from the AdS/CFT correspondence, for the computation of correlation functions is the best understood part of the AdS/CFT dictionary. We will also describe the behaviour of probes and their stochastic dynamics by drawing connection with semi-classical dynamics in black hole backgrounds.

While surprisingly powerful, linear response theory requires small deviations from global equilibrium, leaving more general dynamics inaccessible. To go beyond linear response, we will first focus on long-wavelength IR physics, where the coarse-grained description of an interacting field theory is provided by fluid dynamics. Fluid dynamics can of course describe fluids well out of global equilibrium, as long as the fluid variables, such as local temperature, fluid velocity, etc., make sense. In particular, the spatial and temporal variations of these variables must occur on much longer scales than the microscopic ones, but the amplitudes need not be small. Borrowing notation from kinetic theory, let us denote this microscopic scale , and the typical scale on which the fluid variables vary . Then the regime in which the fluid description is meaningful, or equivalently the regime where the system attains local thermal equilibrium everywhere, is given by the condition ; we will refer to this as the ‘long-wavelength’ regime. How does the fact that we have a fluid description of a given configuration help? As we will see in §5, it conveniently constitutes a large truncation of the relevant degrees of freedom describing the system. Using the recently-formulated fluid/gravity correspondence [22], there is a one-to-one mapping between any such solution to fluid dynamics and a bulk solution to Einstein’s equations describing a large non-uniform and dynamically evolving black hole. Note that material covered in sections §4.14.2 and §5 has previously been reviewed in [23] and [24, 25], respectively.

§4§5§6Dynamical regime
Fig. 1: Rough indication of the regimes of validity of the linear response theory and the fluid/gravity correspondence, in the space of perturbations from global thermodynamic equilibrium, labeled by the amplitude of perturbations and the wave number (or frequency) , relative to the microscopic scale. We have indicated the relevant sections of the paper where the different regimes are discussed from the holographic perspective.

Fig. 1 illustrates the two regimes of validity discussed so far. Linear response theory is valid for small deformations from global equilibrium, while fluid/gravity correspondence can be used in the long-wavelength regime. The main points to note are that these two regimes are a-priori distinct, and that they still leave the more interesting region of large deviations from global equilibrium out of reach.

The general story of understanding physics out of equilibrium then involves consideration of deformations that take us to the non-linear regime in amplitude and frequency. This translates to the full-blown study of gravitational dynamics, including the fascinating physics of black hole formation. In §6 we will review recent progress in understanding various aspects of such analyses. Finally in §7 we will take the opportunity to address interesting questions about gravity: what does our understanding of physics of strongly coupled field theories teach us about gravitational dynamics? Our focus in this brief section will be to describe various attempts in the past decade to extract features of the bulk geometry in terms of field theory observables. We conclude with a discussion in §8.

2 The AdS/CFT dictionary

Before we delve into the details of physics out of equilibrium, we pause to recall some salient facts about the AdS/CFT correspondence. As already mentioned, this is well explained in the classic reviews [5, 6], so accordingly we will be brief and simply collect facts that are necessary for the current discussion.

The essence of the AdS/CFT correspondence is that strongly coupled field theory dynamics is recorded in terms of string theory (or classical gravity if the field theory admits an appropriate planar limit) with appropriate asymptotically Anti-de Sitter boundary conditions. Spacetimes which are asymptotically AdS can be thought of as deformations of pure AdS by normalizable modes in the supergravity description. The framework is also sufficiently rich to allow for deformations of the field theory, for instance by turning on relevant operators.777 This type of deformation can be used to study field theories on curved backgrounds as described in detail in [26] (see also [27, 28, 29] for recent investigations). However, from the bulk perspective, these are large deformations, involving asymptotically “locally AdS” geometries. The basic dictionary relates the field theory observables, which are gauge invariant operators (local or non-local), to their counter-parts in the string (or gravitational) description. For instance, local gauge invariant single-trace operators formed out of the fundamental fields of the gauge theory such as map to single-particle states in the bulk spacetime, while non-local operators such as Wilson loops map to string or D-brane world-sheets.

In this language, pure AdS spacetime characterizes the UV fixed point of a quantum field theory. The fixed points of interest are field theories with at least superconformal symmetry.888 While non-supersymmetric AdS compactifications also arise in string theory, they are seldom stable. Some notable exceptions which satisfy perturbative stability conditions, i.e. spectrum statisfying the Breitenlohner-Freedman bound [30], were described in [31]; however to the best of our knowledge non-perturbative stability of such vacua have not been established (for instance note that non-supersymmetric quotients of AdS do suffer from non-perturbative instabilities [32]). The central charge of this CFT is given in terms of the geometry of the AdS spacetime. In the familiar examples arising from string theory one typically encounters spacetimes of the form AdS supported by various fluxes, where is generically a compact manifold (which is -dimensional in string theory or analogously -dimensional for M-theory solutions).

The field theories are characterized by a dimensionless coupling constant(s) and the gauge group. We will use the ’t Hooft coupling parameter , while the information regarding the rank of the gauge group is given by the central charge of the CFT. The latter is in turn given in terms of the volume of . Schematically,

(2.1)

Denoting the AdS curvature scale by and taking into account the basic relation between the 10-dimensional Planck and string length scales [33]

(2.2)

we obtain

(2.3)

For instance, in the celebrated duality between SYM and Type IIB string theory on AdS one finds

(2.4)

As described above, deformations away from the UV fixed point correspond to deformations from pure AdS spacetime. Normalizable modes in the bulk spacetime would be related to deformations that are engineered by giving vacuum expectation values to the dual field theory operator. Non-normalizable modes in the bulk are non-fluctuating sources that can be used to deform the field theory Lagrangian.999 In certain situations one encounters a choice of boundary conditions to impose on bulk fields. A case in point is that of massive scalar fields with mass lying close to the Breitenlohner-Freedman bound [30, 34]. We assume that a particular choice has been made among the various possibilities and refer to normalizable and non-normalizable with respect to this choice of boundary conditions. Finally, in our considerations we will not allow irrelevant deformations of the CFT, as these would correspond to destroying the AdS asymptotics.

2.1 Regimes of interest

The AdS/CFT correspondence is a profound correspondence between two quantum theories; but for general values of the parameters these theories are complicated and beyond computational control. The set of limits we focus on to gain control and insight is the following.

  • or equivalently : quantum corrections are suppressed and the gravitational theory becomes classical. This also has the added merit of suppressing string interactions since in the planar limit.

  • : stringy () corrections are suppressed, so the bulk theory is simply Einstein gravity interacting with other fields, while the boundary theory describes dynamics of local single trace operators. At this level, the field theory dynamics still looks complicated and highly nonlocal: the single trace operator expectation value is related to the asymptotic fall-off of the corresponding bulk field. Thus although the bulk fields evolve according to the 2-derivative field equations, the dynamics on the boundary contains infinite number of (spatial and temporal) derivatives.

Hence in the large-, large- regime we are essentially down to studying classical gravitational dynamics in order to elucidate aspects of the field theory at strong coupling. This is of course a drastic simplification of the problem, but nevertheless it warrants further simplification to gain tractability. Classical gravitational dynamics in asymptotically AdS spacetime involves an infinite number of fields arising from Kaluza-Klein (KK) modes on the compact space . Fortunately, further truncation can often be achieved by the magic of consistent truncation [35]. The basic philosophy behind consistent truncations is to find an appropriate ansatz for Type II or M-theory fields, which can acturally be reproduced by examining the dynamics of a truncated set of fields in dimensions. The most familiar example of consistent truncation are the gauged supergravity theories which keep only the lightest modes under the KK reduction. More complicated examples including massive fields have been constructed recently [36, 37] and play an important role in the applied AdS/CFT correspondence.

One hassle with consistent truncations is that the lower-dimensional theory, and hence the dynamics of the dual field theory, are model dependent, i.e., dependent on the choice of the internal manifold (and in general on the fluxes turned on). Although this prevents one from making universal statements, it has the opposite advantage of incorporating richer dynamics. As we describe later, such extensions are in fact necessary in order to study the behaviour of field theories in grand canonical ensembles with prescribed chemical potentials. Nevertheless, it is useful to ask whether one can further distill the essential features of the correspondence to a minimal classical gravitational Lagrangian, and use this to identify universal features that are shared by a wide variety of QFTs. This is in fact possible and is achieved in the simplest imaginable manner: by studying the dynamics of pure gravity in AdS spacetimes. Let us pause to review that argument before proceeding.

Given any solution of the form AdS one can in fact argue that there is an universal sub-sector which simply comprises pure gravitational dynamics in AdS, as can be seen by Kaluza Klein reducing on the compact space . Clearly any non-trivial solution obtained in dimensions uplifts to a solution of the full string theory equations of motion. As a result, we always have a consistent truncation of the complicated lower dimensional Lagrangian to just Einstein gravity with negative cosmological constant, where the only dynamical mode is the graviton. Since in the field theory dual this graviton mode corresponds to the stress tensor expectation value, this truncation provides a decoupled sector with universal dynamics for the stress tensor. In other words, the stress tensor obeys the same equations of motion in each of such infinite class of strongly coupled field theories. From the field theory perspective, the different theories are simply characterized by their differing central charge. Hence apart from an overall normalization we will usually be probing dynamics across a wide class of theories.

The complete dynamics of the stress tensor, which in particular allows one to compute all the -point functions of in the prescribed state, is still a complicated and non-local system from the field theory perspective. After all, this dynamics requires one to be able to solve for the dynamics of the non-linear Einstein-Hilbert Lagrangian (with negative cosmological constant) in dimensions. Therefore to simplify things further, in §5 we will take one further limit: we will focus on configurations wherein the stress tensor varies sufficiently slowly compared to the local equilibration length scale (the long-wavelength limit). Such configurations will then be locally thermalized, allowing for a much simpler description in terms of fluid dynamics. In other words, the equations for the stress tensor in this limit reduce to generalized Navier-Stokes equations [22].

Once one has an understanding of the dynamics of the field theory stress tensor, or equivalently the bulk dynamics of pure gravitational degrees of freedom, one can enlarge the system to include other fields. For instance, a natural extension of the canonical ensemble in field theories with conserved charges is to include chemical potentials for the said charges and examine the behaviour of the grand canonical ensemble. Since global symmetries in the field theory map to gauge fields in the bulk spacetime, one naturally ends up studying the behaviour of Einstein-Maxell or Einstein-Yang-Mills type theories (again with negative cosmological constant of course). In this context one can furthermore consider the behaviour of composite gauge invariant operators of the field theory carrying various quantum numbers in these ensembles. The gravitational problem then generalizes appropriately to the physics of the dual bulk fields. This general scheme has recently been applied to study the phase structure of field theories at finite temperature and density (i.e., non-zero chemical potential) and has unearthed many interesting features which share qualitative similarities with the dynamics of superconducting phase transitions, non-Fermi liquid behaviour etc., which has been reviewed in [12, 13, 7, 14].

Our strategy in the following will be to consider the simplest setting of just gravitational physics in the bulk, since as argued above this sector is universal across a wide variety of field theories. To this end, our bulk analysis will involve the dynamics of Einstein gravity with a negative cosmological constant, i.e.,

(2.5)

Einstein’s equations are given by101010 We use upper case Latin indices to denote bulk directions, while lower case Greek indices will refer to field theory (or “boundary”) directions. Finally, we will use lower case Latin indices to denote the spatial directions in the boundary.

(2.6)

If the bulk AdS spacetime geometry is some negatively curved -dimensional Lorentzian manifold, , with conformal boundary , then the field theory lives on a spacetime of dimension in the same conformal class as . Choosing an appropriate conformal frame, one may identify and and speak of the field theory as living on the AdS boundary. From the standpoint of the bulk theory, the choice of metric on fixes a boundary condition that the bulk solution must satisfy.111111 In standard AdS/CFT parlance, this amounts to fixing the non-normalizable mode of the bulk graviton to obtain the desired metric on . The correspondence is simplest to state for conformal field theories in dimensions where the trace anomaly vanishes, but with appropriate care, the correspondence also holds in the presence of a trace anomaly, and it can accommodate non-conformal deformations.

In general, one could have multiple bulk spacetimes whose boundary is the spacetime on which our field theory lives. In such cases, the AdS/CFT prescription of [4, 38] requires that one view all such possibilities as saddle points for the string theory (or gravity) path integral and one is instructed to sum over all such possibilities. Of course, the saddles might exchange dominance as one changes the boundary manifold; this can be viewed as a phase transition of the field theory as one dials an external parameter (in this case the geometry of the non-dynamical spacetime which it lives on). We will shortly encounter an example of such phase transitions for field theories on compact spatial volume.

2.2 Field theories in the canonical ensemble & thermal equilibrium

Finite temperature physics in the field theory can be realized by coupling the system to a heat bath, or more precisely by looking at the thermal density matrix

(2.7)

where is the field theory Hamiltonian. The dual spacetime should have a natural thermal interpretation. It is a well-known fact going back to the seminal works of Bekenstein and Hawking [39, 40] that black hole spacetimes with non-degenerate event horizons naturally exhibit features associated with thermal physics; thereby one is led to expect that black hole spacetimes to play a role in describing the dual of a finite temperature field theory, which is further supported by the intuition mentioned in §1 that endpoints of generic evolutions should match in the dual descriptions. It is, however, logically possible that one also has to consider ‘thermal geometries’ (such as thermal AdS) which just have the Euclidean time circle periodically identified.

To understand this issue better it is useful to think of the thermal density matrix by working in Euclidean time. On the field theory side one can achieve finite temperature by putting the theory on where the Euclidean time circle has period and is the spatial manifold. For compact , such as for , one does indeed have two candidate bulk spacetimes satisfying the boundary conditions [38]. The first is the so-called thermal AdS spacetime which is AdS with periodically identified Euclidean time coordinate,

(2.8)

The other saddle point is the static, spherically symmetric Schwarzschild-AdS spacetime,

(2.9)

for which the Hawking temperature is related to the size of the horizon , as

(2.10)

These two geometries, (2.8) and (2.9), exchange dominance at [38]: at low temperature, the thermal ensemble is dual to the thermal AdS spacetime and has a free energy of , while at high temperature, the correct dual is the Schwarzschild-AdS which has a free energy of . This phase transition is referred to as the Hawking-Page transition [41] and is best thought of as a confinement-deconfinement transition (since the jump in the free energy is large at large central charge as required for the planar limit). There is furthermore strong evidence that the transition persists to weak coupling where it has been identified as a Hagedorn transition [42].

For most of our discussion, however, we are going to be interested in the dynamics of field theories on non-compact spacetimes; our focus will typically be on Minkowski spacetime . In this case there is no phase transition: the flat space limit is essentially the same as the high temperature limit for conformal field theories. As a result, the relevant geometry dual to the thermal density matrix (2.7) in the field theory is the planar Schwarzschild-AdS black hole (where WLOG121212 Note that in addition to the choice of AdS length scale, the presence of full invariance of the solution combined with the underlying isometry of AdS allows us to rescale the coordinates and set the horizon to be located at . we have fixed and ):

(2.11)

The causal structure of this solution is easily determined: the spacetime has a spacelike curvature singularity at , cloaked by a regular event horizon at , and a timelike boundary at . This simple solution is of course static and translationally invariant in the boundary directions parameterized by . One can in fact generate a -parameter family of solutions by boosting in with normalized -velocity and scaling . This generates stationary black holes whose horizon size is given (after scaling) by and the boost velocity enters into the Killing generator of the horizon via

(2.12)

It will turn out that a more convenient form of the metric is one which is manifestly regular on the horizon as well as being boundary-covariant. We can obtain such a form by starting from (2.11), then change to ingoing Eddington coordinates to avoid the coordinate singularity on the horizon: where , and finally ‘covariantize’ by boosting: , , where is the spatial projector, . This leads to the form of the metric we will use later:

(2.13)

The event horizon is now at , which in turn is related to the temperature via the large limit of (2.10),

(2.14)

Once the bulk black hole solution is determined, it is straightforward to use the holographic prescription of [43, 44] to compute the boundary stress tensor. To perform the computation we regulate the asymptotically AdS spacetime at some cut-off hypersurface and consider the induced metric on this surface, which (up to a scale factor involving ) is our boundary metric . The holographic stress tensor is given in terms of the extrinsic curvature and metric data of this cut-off hypersurface. Denoting the unit outward normal to the surface by we have

(2.15)

For example, for asymptotically AdS spacetimes, the prescription of [44] gives

(2.16)

where is the extrinsic curvature of the boundary. Implementing this procedure for the metric (2.13) we learn that the AdS/CFT correspondence maps this bulk solution to an ideal fluid characterized by temperature and fluid velocity . In particular, the induced stress tensor on the boundary is

(2.17)

where is an overall normalization that is proportional to the central charge of the field theory. Note that this stress tensor is traceless, , as indeed is expected for a CFT.

2.3 Equilibrium in grand canonical ensembles

A natural extension of the above framework is to consider systems at finite density by generalizing (2.7) to

(2.18)

where are chemical potentials for the conserved charges . The choice of chemical potentials is of course restricted by the global symmetries of the field theory. If we insist on restricting attention to the universal dynamics of the stress tensor alone, then the only possible generalization is to consider chemical potentials for rotations . For a field theory on one in general has independent rotations and each of these can be given a non-zero angular velocity .

However, if we are willing to add matter fields in the bulk, i.e. extend (2.5) to include additional (non-gravitational) degrees of freedom, then we can generalize the discussion to include more interesting chemical potentials. First of all, we note that in order to incorporate chemical potentials, the field theory must admit conserved currents. Conserved currents in the field theory map to gauge symmetries in the bulk spacetime.131313 This is in fact reminiscent of the theorem “No global symmetries in string theory”, which follows from the observation that conserved currents in spacetime lead to world-sheet vertex operators of weight and respectively, which in turn imply massless particle states in spacetime. For every conserved field theory current one therefore has a bulk gauge field . The bulk gauge field obeys some equations of motion in the AdS spacetime and its non-normalizable mode in the near-boundary expansion corresponds to the boundary chemical potential .

Typically, large- field theories with holographic duals have conserved charges; for instance SYM has a global symmetry and one can consider the grand canonical ensemble with (s) corresponding to Cartan subgroup of this non-abelian global symmetry group. In fact, for a wide variety of superconformal field theories in one has three conserved charges which geometrically can be related to the fact that these field theories arise from compactifications of Type IIB supergravity on toric Sasaki-Einstein manifolds. A simple example to keep in mind is the Einstein-Maxwell Lagrangian (with perhaps Chern-Simons terms in odd bulk dimensions) which can be used to study the grand canonical ensemble with charge chemical potentials, which can be obtained via a consistent truncation of gauged supergravity theories.

Given an appropriate definition of the field theory grand canonical ensemble, the equilibrium solution corresponding to this ensemble can be found by looking for static black holes carrying the appropriate charges. For Einstein-Maxwell theory these would be just the Reissner-Nordstrom-AdS (RNAdS) black holes, and the thermodynamic properties of these black holes correspond to the equilibrium thermodynamics of the field theory.

In the presence of matter, one could have interesting phase transitions over and above the analog of the Hawking-Page transition discussed in §2.2. For pure Maxwell field interacting with gravity, these include charge redistribution as discussed originally in [45, 46], or the more recent examples involving Chern-Simons dynamics [47]. Inclusion of charged scalars (and sometimes neutral scalars) can also lead to interesting phase transitions as originally pointed out by [48], which plays an important role in the physics of holographic superconductors [49, 50].

3 Linear response theory

Having reviewed the basic framework of the AdS/CFT correspondence we now turn to describing the essential features of linear response theory. This will set the stage for our discussion of deviations away from equilibrium in §4.

Consider a generic quantum system characterized by a unitary Hamiltonian acting on a Hilbert space in equilibrium. The system could be in a pure state or more generally in a density matrix ; the only requirement is that the system be in a stationary state. We now wish to perturb the system away from equilibrium and analyze the dynamics. A general deformation can be thought of as a change in the evolution operator and one is left with having to examine the dynamics with respect to this new Hamiltonian. Things however simplify if we can focus on deviations which are small in amplitude – this is the regime of linear response theory, which we now briefly review. For a beautiful account see the original derivation by Kubo [51, 52] and the review [53].

In the linear response regime, one imagines the system being perturbed by a weak external force. To wit, one can write where is the external force (with time dependence explicitly indicated) and is the canonical conjugate operator to the force. In the linear response regime the amplitude of the force is constrained to be small, , while we allow arbitrary temporal dependence, which in particular could involve high frequency modes being excited; see Fig. 1. We essentially wish to do time dependent perturbation theory (being careful of causality in relativistic theories) for such deviations away from equilibrium.

3.1 Response functions for deviations from equilibrium

To monitor the departure from the stationary state, we can pick some observable, say the expectation value of a local operator in the theory.141414 One could also consider non-local operators and more exotic observables such as entanglement entropy; we will discuss the latter in §7. For concreteness, we will assume that the system was in a density matrix before we perturbed it. Stationarity of this state demands that . Turning on the perturbation will cause a deformation to the density matrix. Suppose that we encounter a new density matrix , which now has to satisfy Hamiltonian evolution with respect to , i.e.,

(3.1)

For small amplitude perturbations, we can assume that and solve for formally in terms of a time ordered integral expression:

(3.2)

which is derived by solving the linearized version of the evolution equation (3.1).

Once we have the change in the density matrix, it is easy to compute the change in any measurement occurring due to the perturbation. We can estimate the response of the system by examining the expectation value of some observable , which can be obtained directly as:

(3.3)

It is conventional to define the response function (sometimes called the after-effect function) as the change in the operator expectation value for a delta function perturbation; i.e., for . From (3.3) one recovers

(3.4)

where we have tried to make clear in the notation the idea that one measures the response of to a perturbation caused by the deformation due to a force conjugate to . By taking a Fourier transform of the response function, one arrives at the admittance:

(3.5)

It should be clear from the above discussion that this formalism is sufficiently general to accommodate arbitrary linear changes in a given dynamical system. Non-linear corrections can be explored in perturbation theory, generalizing familiar ideas from time dependent perturbation theory in quantum mechanics.

3.2 Retarded correlators & Kubo formulae

The perturbations discussed so far are explicit perturbations on the system caused by some external force . One can as well envisage a perturbation driven purely by thermal fluctuations, which are not a-priori related in any obvious way to external forces acting on the system. However, thermal fluctuations can be measured by looking at the response of the extensive thermodynamic variables to variations in the local energy or charge densities. These in fact can also be treated in linear response and lead to the famous Kubo formulae for the transport coefficients. We now give a brief overview of these concepts valid in any field theory; in §4 we will demonstrate how these formulae can be applied in the AdS/CFT context.

Let us return to the response function given in (3.4) and extract some essential features that we will use in later analysis. While that discussion was sufficiently general, it is worth translating this into more familiar language. Physically we wish to monitor the behaviour of the expectation value of some operator when the system is subject to some perturbation. We can imagine this occurring via a direct coupling of the operator to some sources , whose effect is to change the action via:

(3.6)

The response of the system due to this change can be obtained by rewriting the result (3.4) slightly. Since the system will respond only after the perturbation, causality demands that we simply convolve the retarded Green’s function of the operator to the sources that causes it to deviate from stationarity. In particular, we have

(3.7)

where the retarded correlation function is defined as usual via

(3.8)

Therefore given the retarded correlation functions, one can immediately infer, in the linear response regime, the manner in which the system under consideration reacts to the external disturbance caused by the sources. One will often be interested in the behaviour of the Green’s functions in momentum space. To this end we define the Fourier transform of the retarded correlators via

(3.9)

where we have assumed translational invariance and defined the -vector .

A class of observables that are interesting to examine are those corresponding to conserved currents in the theory, viz., the energy-momentum tensor or generic conserved global charges . If we consider systems in thermal equilibrium, where deviations from thermality are engineered by density or charge fluctuations, then one is naturally led to studying the retarded Green’s functions of these conserved currents:

(3.10)

where we have assumed that the density matrix in question is appropriate to the ensemble under consideration. The correlators indicated on the RHS in (3.10) are therefore the thermal correlators, with perhaps fixed chemical potentials.

Interesting physical quantities characterizing the system can be extracted by examining the momentum dependence of the retarded correlation functions of the conserved currents. The momentum space correlators have non-trivial analytic behaviour, with poles in the complex plane. One can read off the dispersion relation for the associated modes, by solving for . Since thermodynamic systems typically incorporate dissipative effects, these dispersion relations typically have imaginary pieces which capture the rate at which the system relaxes back to equilibrium. Stability of the quantum system demands that the perturbations damp out exponentially in time. This translates to the poles of the retarded Green’s functions lying entirely in the lower half plane of the complex frequency space. We will see shortly that these poles of the retarded Green’s functions are in fact associated with the quasinormal modes of black hole geometries (which describe how a black hole settles back to its quiescent equilibrium state) via the AdS/CFT correspondence.

To be specific, let us consider a four dimensional conformal field theory and examine the stress tensor retarded correlator. It is useful to take into account the stress tensor conservation equation which implies a Ward identity .151515 We are here ignoring contact terms which can appear in the Ward identities. In momentum space these will give rise to analytic pieces and will therefore be irrelevant to our discussion of of poles. It is in fact convenient to pick a direction in momentum space, say , and describe modes as being longitudinal or transverse to this choice of momentum. One can then show that the transverse components of the stress tensors have the behaviour

(3.11)

while the longitudinal modes behave as

The correlators are thus completely described by three scalar functions of momenta, for , which exhibit the aforementioned poles.

Out of the set of poles of the retarded Green’s function, of special interest are those that capture the late time behaviour. These necessarily involve small imaginary parts in (since the modes with large imaginary parts damp out quickly and therefore have short half-lives). These special set of poles are the hydrodynamic poles; they capture, and in fact provide a basis for, a complete description of the interacting quantum system via linearized hydrodynamics.161616 Note that hydrodynamics is a good approximation to any physical system close to equilibrium, for fluctuations that are sufficiently long in wavelength. We discuss this in detail in §5. In the low-frequency regime, i.e., for , it turns out that the function defined in (3.11) is non-singular, while the other correlation functions exhibit poles. In fact, at low frequencies only the pieces of the correlation functions that correspond to conserved quantities can be singular, for it is these modes which due to the conservation law take a long time to relax back to equilibrium.

The hydrodynamic poles of the system are characterized by having dispersion relations which satisfy the constraint as . Fluctuations transverse to the direction of momentum flow, captured by , give rise to dispersion relation of the form

(3.13)

which is characteristic of a diffusive mode, with diffusion constant . For energy-momentum transport this is the shear mode, with the diffusion constant being related to the shear viscosity of the system via , where and are the equilibrium energy density and pressure, respectively. The longitudinal component, given by , has poles at locations

(3.14)

which describe sound propagation in the medium with velocity and attenuation . Note that the sound mode is the only propagating mode in the hydrodynamic limit.

While one can examine the analytic structure of the retarded correlators in momentum space and extract interesting transport properties, it is useful to obtain direct formulae for them. These are the famous Kubo formulae. For instance to find the shear viscosity of the system, one simply takes an appropriate zero frequency limit of the retarded correlator, i.e.,

(3.15)

Similar expressions can be written down for the charge conductivity, etc..

3.3 Brownian motion of probes and Langevin dynamics

Our discussion thus far has been anti-chronological in a historical sense, for we have used general notions of quantum field theories and statistical mechanics to arrive at the response of the system to external perturbations. Historically, these ideas were first explored in kinetic theory, where it was realized that one could systematically account for the deviations away from purely thermal behaviour. We will refrain from repeating these ideas here in the context of departures from equilibrium of the system as a whole, but instead use these concepts to describe the physics of a single probe particle in a thermal medium. We have in mind a projectile that moves through some plasma medium. The medium itself will be taken to be in a thermal ensemble and we will be interested in the manner in which the probe particle loses energy to the medium. Moreover, it is also well known that even if the particle attains equilibrium with the plasma, it will continue to be buffeted by thermal fluctuations from the medium and undergo random motion. This is the famous stochastic Brownian motion, which is best described in the limit of a heavy probe particle in a thermal medium.

Let us therefore consider the Langevin equation, which is the simplest model describing a non-relativistic Brownian particle of mass in one spatial dimension:

(3.16)

Here is the (non-relativistic) momentum of the Brownian particle at position and time , and . The two terms on the right hand side of (3.16) represent friction and random force, respectively, and is a constant called the friction coefficient. One can think of the particle as losing energy to the medium due to friction, and concurrently getting a random kick from the thermal bath, modeled by the random force. We assume the latter to be simply white noise with

(3.17)

where is a constant. Note that the separation of the force into frictional and random parts on the right hand side of (3.16) is merely a phenomenological simplification – microscopically, the two forces have the same origin, namely collision with the fluid constituents.

Assuming equipartition of energy at temperature , one can derive the following time evolution for the square of the displacement [54]:

(3.18)

where the diffusion constant is related to the friction coefficient by the Sutherland-Einstein relation:

(3.19)

We can see that in the ballistic regime, , the particle moves inertially () with the velocity determined by equipartition, . On the other hand, in the diffusive regime, , the particle undergoes a random walk (). The transition is not instantaneous because the Brownian particle must collide with a certain number of fluid particles to get substantially diverted from the direction of its initial velocity. The crossover time between the two regimes is the relaxation time

(3.20)

which characterizes the time scale for the Brownian particle to forget its initial velocity and thermalize. One can also derive the important relation between the friction coefficient and the size of the random force :

(3.21)

which is the simplest example of the fluctuation-dissipation theorem and arises precisely because the frictional and random forces have the same origin.

In spatial dimensions, the momentum and force in (3.16) are generalized to -component vectors, and (3.17) is then naturally generalized to

(3.22)

where . In the diffusive regime, the displacement squared scales as

(3.23)

On the other hand, the Sutherland-Einstein relation (3.19) and the fluctuation-dissipation relation (3.21) are independent of .

Let us now return to the case with one spatial dimension (). The Langevin equation (3.16), (3.17) captures certain essential features of physics, but nevertheless is too simple to describe realistic systems, since it assumes that the friction is instantaneous and that there is no correlation between random forces at different times (3.17). If the Brownian particle is not infinitely more massive than the fluid particles, these assumptions are no longer valid; friction will depend on the past history of the particle, and random forces at different times will not be fully independent. We can incorporate these effects by generalizing the simplest Langevin equation (3.16) to the so-called generalized Langevin equation [53, 55],

(3.24)

The friction term now depends on the past trajectory via the memory kernel , and the random force is taken to satisfy

(3.25)

where is some function. We have in addition introduced an external force that can be applied to the system.

Let us briefly indicate how, faced with such a system with only under our control, we would extract the physical information, namely and , as well as the characteristic timescales involved in collision and relaxation. The latter will offer more direct insight into the nature of the medium under consideration. To analyze the physical content of the generalized Langevin equation, it is convenient to first Fourier transform (3.24), obtaining

(3.26)

where are Fourier transforms, e.g.,

(3.27)

while is the Fourier–Laplace transform:

(3.28)

If we take the statistical average of (3.26), the random force vanishes because of the first equation in (3.25), and we obtain

(3.29)

where is called the admittance. The strategy, then, is to first determine the admittance , and thereby , by measuring the response to an external force. For example, if the external force is

(3.30)

then is simply

(3.31)

For a quantity , we define the power spectrum by

(3.32)

Note that is independent of in a stationary system. The knowledge of the power spectrum is equivalent to that of 2-point function, because of the Wiener–Khintchine theorem:

(3.33)

Now consider the case without an external force, i.e., . In this case, from (3.26),

(3.34)

Therefore, the power spectrum of and that for are related by

(3.35)

Hence, combining (3.31) and (3.35), one can determine both and appearing in the Langevin equation (3.24) and (3.25) separately. However, these two quantities are not independent but are related to each other by the fluctuation-dissipation theorem, generalizing the relation (3.21), cf., [53].

For the generalized Langevin equation, the analog of the relaxation time (3.20) is given by

(3.36)

If is sharply peaked around , we can ignore the retarded effect of the friction term in (3.24) and write

(3.37)

The generalized Langevin equation (3.24) then reduces to the simple Langevin equation (3.16), so that corresponds to the thermalization time for the Brownian particle.

Another physically relevant time scale, the microscopic (or collision duration) time , is defined to be the width of the random force correlator function . Specifically, let us define

(3.38)

If , the right hand side of this precisely gives . This characterizes the time scale over which the random force is correlated, and thus can be thought of as the time elapsed in a single process of scattering. In many cases, it is natural to expect that

(3.39)

since, after all, we indicated that it takes a heavy probe many collisions to thermalize. Typical examples for which (3.39) holds include settings where the particle is scattered occasionally by dilute scatterers as described by kinetic theory, and settings where a heavy particle is hit frequently by much smaller particles [53]. However, as we will discuss in §4.3.2, for the Brownian motion dual to AdS black holes, the field theories in question are strongly coupled CFTs and in fact (3.39) does not necessarily hold. There is also a third natural time scale given by the typical time elapsed between two collisions. In the kinetic theory, this mean free path time is typically between the single-collision and relaxation time scales, ; but again, this hierarchy is not expected to hold beyond perturbation theory.

The basic message to take away from this discussion is that the linear response regime is accessible once one understands the dynamics in equilibrium. The response functions are simply given in terms of Green’s functions evaluated in the stationary configuration.

4 Linear response from AdS/CFT: Probes of thermal plasma

We now proceed to put together the toolkits we presented in the preceding two sections. In §3 we have described the basic methods employed in non-equilibrium statistical mechanics to understand the physics of systems out of equilibrium, viz., linear response theory. One of the fundamental tenets in this approach is that for small-amplitude deviations, one can compute relevant observables by computing appropriate correlation functions in the equilibrium ensemble. Since the time-evolution part of the problem has been effectively dealt with in this manner, one is left with a much easier task in general.

However, the computation of equilibrium correlation functions is not as trivial as it sounds, especially in circumstances where the underlying quantum system is intrinsically strongly coupled. One therefore requires some further insight to deal with such situations. Fortunately, for a class of field theories which have holographic duals, the gauge/gravity correspondence comes to rescue, as indicated in §2. In fact, one the earliest developed technologies within the correspondence was the recipe to compute correlation functions of gauge invariant local operators in field theories using their dual gravity picture. In the present context this means that the gauge/gravity correspondence provides an efficient way to compute the correlation functions relevant for the linear response theory directly in terms of classical computations in an asymptotically AdS spacetime.

We will begin by a brief review of the techniques employed to compute correlation functions in the AdS/CFT correspondence, followed by a discussion of lessons learnt by examining the linear response regime. Finally, we will discuss how one can monitor the behaviour of probe motion (both ballistic and stochastic) in a strongly coupled plasma medium.

4.1 Computing correlation functions in AdS/CFT

Let us consider a local gauge-invariant single trace operator with conformal dimension in the boundary CFT. To compute correlation functions of this operator, one would deform the CFT action by adding a term and obtain the generating function of the correlators as a functional of the sources . The requirement that the deformation term be dimensionless implies that has scaling dimension .

In the AdS/CFT correspondence, a given boundary operator maps to a bulk field whose spin is determined by the Lorentz transformation property of the operator in question. The bulk field , with being the radial coordinate in AdS, has mass . For scalar operators one has the relation [4]171717 This is not required to be true for where is the Brietenlohner-Freedman mass, providing the lower bound on the mass of a scalar field in AdS. In the said range, one can equally well associate bulk field of mass with a boundary operator of dimension as discussed in [34], which corresponds to an alternative quantization of fields in AdS [30].

(4.1)

Similarly, for a -form operator on the boundary, the relation between the conformal dimension of the operator and the mass of the bulk field is given as:

(4.2)

Given this map between fields and operators we can go ahead and use the AdS/CFT correspondence to compute the generating function of correlation functions,

(4.3)

where we have schematically indicated the path integral over the quantum fields of the CFT.

The statement of the AdS/CFT correspondence asserts that this generating functional is given by the partition function of the string theory with the fields prescribed to take on the boundary values at the boundary of AdS. In particular, in the limit when classical gravity is a good approximation, the string partition function simply reduces to the on-shell action of gravity evaluated on the solution to the field equation. The on-shell action is usually divergent since we are turning on mode that is non-normalizable to act as a source. To ensure that we capture the correct physics, we can regulate the AdS spacetime at and demand that we satisfy the boundary condition there, i.e., demand as . Thus one arrives at the relation derived in [3, 4]:

(4.4)

or in the limit where classical gravity in the bulk is a good approximation

(4.5)

The general scheme we have outlined above works well for computing Euclidean correlation functions in asymptotically AdS spacetimes, but there are certain subtleties to keep in mind while computing retarded correlation functions. A clear prescription was initially given in [56] for which a nice supporting argument based on Schwinger-Keldysh contours was provided in [57]. More recently, these arguments have been revisited in [58] and a compact expression for computing two-point functions was provided in [59, 60]. Formal studies of these correlation functions from a holographic renormalization scheme and general contour prescriptions for higher-point functions were discussed in [61, 62]; recently three-point functions at finite temperature were computed in [63].

Since we will be primarily interested in addressing issues in linear response theory, let us record here the prescription derived in [59], relating the retarded Green’s function to a simple ratio involving the field and its conjugate momentum. In particular, for massive sscalar fields in asymptotically AdS spacetimes,

(4.6)

Here is the canonical momentum conjugate to the field (which itself is dual to the operator under consideration) under radial evolution in AdS. Furthermore, is the on-shell solution to the appropriate wave equation subject to the boundary conditions that it be regular in the interior181818 In the case of spacetimes with horizons, this amounts to demanding that the field be infalling at the horizon, as we explain in §4.2. and approaching some chosen boundary value at the boundary of the AdS spacetime. The constraints (i.e. switching off the source) and the limit are necessary to obtain the boundary observable as one anticipates from (4.5). We should note that this formula has been written down after taking into account the intricacies of the holographic renormalization and hence one is instructed to extract the finite part of the bulk calculation. For details on these techniques we refer the reader to [64, 65].

4.2 Retarded correlators and black hole quasinormal modes

Given the utility of the AdS/CFT correspondence in computing correlation functions, let us now return to the issue of linear response around a given equilibrium configuration. As we have seen in §2.2 and §2.3, physics of thermal equilibrium is captured by stationary black hole spacetimes in the dual geometric description. Therefore, linear response behaviour in the field theory translates directly to the behaviour of linearized fluctuations of bulk fields on AdS black hole backgrounds.

One of the first steps in this direction was taken in [66], who pointed out the connection between AdS black hole quasinormal modes and the rate at which disturbances away from equilibrium re-equilibrate. Since this connection underpins much of the linear response theory we are about to describe, we will pause to recall the basics of the quasinormal mode spectrum in black hole spacetimes.

Physically, quasinormal modes correspond to the late-time “ringing” of the black hole geometry. In particular, perturbations of the black hole undergo damped oscillations, whose frequencies and damping times are entirely fixed by the geometry and the nature of the propagating field, i.e., the modes are determined by the linearized wave operator and are independent of the initial perturbation. In fact, it is well understood that black hole spacetimes, owing to the presence of an event horizon into which the fields can dissipate, act as open systems; the corresponding spectrum of fluctuating modes is complex. The reason for this behaviour is intuitively easy to understand. In classical general relativity, the event horizon acts as a one-way membrane; fields fall into the black hole but do not emerge out. Mathematically, this translates to an infalling boundary condition on fields at the horizon in this black hole background. These same fields are also required to be normalizable near the AdS boundary, for one wishes to retain the AdS asymptotics (and therefore in the field theory side retain the UV fixed point CFT unperturbed by relevant or irrelevant operators). Quasinormal modes for a classical field (suppressing Lorentz indices) are defined as eigenfunctions of the linearized fluctuation operator which acts on in the black hole background, satisfying these boundary conditions i.e, ingoing at the horizon and normalizable at infinity.191919 An excellent summary of known spectra of quasinormal modes for various fields in diverse black hole backgrounds, together with their implication for both astrophysical black holes and in the AdS/CFT context can be found in [67]. See also [68] for an earlier discussion of black hole quasinormal modes.

As initially described in [66], the quasinormal modes of AdS black holes capture the rate at which the field theory, when perturbed away from thermal equilibrium, returns back to the quiescent equilibrium state. In this context, one usually concentrates on the lowest set of modes, as these dominate the long-time behaviour. Nevertheless, it is possible to give a clear interpretation to the entire quasinormal mode spectrum. As was pointed out in [69] in the context of 1+1 dimensional boundary CFTs and asymptotically AdS BTZ black holes, the entire quasinormal mode spectrum maps to the poles of the retarded Green’s functions of operators in the canonical ensemble. This was extended to higher dimensions in the seminal work of [56] and further elaborated upon in [70].

While the relation between quasinormal modes and poles of retarded Green’s functions holds in general for any operator in the dual field theory, it takes on interesting hues for the case when the dual operator corresponds to a conserved current. As discussed in §3.2, the analytic structure in the retarded Green’s functions of the stress tensor which corresponds to the hydrodynamic modes of the system, has complex dispersion relations characterized by the long-wavelength behaviour as . This was explicitly verified by the computation of gravitational quasinormal modes in planar AdS black hole backgrounds, which have translationally invariant horizons and allow for arbitrarily long-wavelength modes. On the contrary, global AdS black holes have horizons of spherical topology and correspond to field theories living in finite volume on . One then encounters IR effects coming from the finiteness of spatial volume which precludes the existence of quasinormal modes with vanishing frequencies. In order to see the hydrodynamic behaviour one has to scale the curvature to zero as well, which reduces the problem to the planar case; one can systematically account for the curvature corrections as we indicate in §5.

The fact that black hole quasinormal mode spectrum admits modes with hydrodynamic dispersion relation leads one to suspect that one can use the gravity analysis to compute properties of the fluid description. Indeed as we have sketched in §3.2, one can use the behaviour of the retarded Green’s functions at zero momentum to learn about transport coefficients like viscosity. This was first carried out in the AdS/CFT context in [71, 72]. The analysis was ground-breaking in that it not only verified the general intuition that one can relate the classical dynamics in a black hole background to the physics of strongly coupled plasmas, but it also paved the way for what is perhaps the most famous conjecture in the subject, viz., the bound on the ratio of shear viscosity to entropy density, , [15]. For a large class of two-derivative theories of gravity one finds by direct computation that this bound is in fact saturated, which prompted [15]. Understanding its implications and its raison d’être has been the focus of a large body of literature, which we cannot do justice to here and point the reader to the excellent review article [23] for developments till a couple of years ago.

This rather small value of shear viscosity obtained in the holographic computations has been instrumental in forging connections with ongoing experimental efforts to understand the state of matter, the quark-gluon plasma (QGP), produced in heavy-ion collisions and RHIC and soon at LHC. Fits to data from the STAR detector at RHIC suggests that the QGP behaves close to the deconfinement transition in QCD as a nearly-ideal fluid with very low viscosity (see [73] for a discussion of near perfect fluidity in physical systems). This has prompted a concentrated effort in the literature and spurred the growth of the AdS/QCD enterprise; we refer the reader to the reviews [9, 11] for these developments.

Recently, this bound has been shown to be violated in higher-derivative theories of gravity: there are example toy-models such as Gauss-Bonnet gravity [74, 75] and other higher-derivative theories [76, 77, 78, 79, 80, 81] and also some string theory inspired constructions of large- superconformal theories [82, 17]. The general consensus at the stage of writing this review seems to be that the bound, whilst robust in the two-derivative approximation (which corresponds to the strong coupling, large- theory), could in general be violated by finite- (string interactions) and also perhaps by (large finite coupling) effects.

The quasinormal mode analysis can also be used to go to higher orders in the hydrodynamic expansion. After all, for a planar black hole one has a spectrum of poles of the retarded Green’s function which can be used to extract an exact non-linear dispersion relation beyond the leading long-wavelength approximation. Such techniques were first employed in the analysis of [83] who computed certain second-oder transport coefficients. Similar analyses were also carried out in [84, 85]. In an nice calculation [86] described the behaviour of low lying quasinormal modes (the modes closest to the real axis) and displayed how it exchanges dominance with the next quasinormal mode at some finite value of momentum. This in particular indicates the regime of validity of the linear hydrodynamic approximation; for the higher quasinormal modes, while still giving poles of the retarded Green’s functions, are not part of the effective hydrodynamic theory.

In summary, there is a direct relation between the physics of black hole quasinormal modes and the retarded Green’s functions of local gauge invariant operators. For a given operator on the boundary, one identifies the corresponding bulk field and computes its quasinormal mode spectrum to infer the location of the poles of the retarded Green’s function. From the retarded Green’s functions of conserved currents one learns that the long-wavelength behaviour of the interacting CFTs which fall within the purview of the AdS/CFT correspondence is described by linearized hydrodynamics. Thus black hole quasinormal modes provide a powerful computational technique to learn about the dynamics of strongly coupled gauge theories and their relaxation back to equilibrium. They also confirm the intuition that the thermal behaviour of strongly coupled field theories can be captured in effective field theory by viewing the system as a plasma medium.

4.3 Probes in the plasma: Dissipation and stochastic motion

Thus far, we have discussed the behaviour of retarded correlation functions of local, gauge invariant operators using the AdS/CFT correspondence. These analyses allow us to picture the interacting, thermal field theory as a plasma medium. As discussed in §3.3, it is useful to ask how do probe particles introduced into such plasmas behave? This question is not only interesting from a theoretical viewpoint, but also from a pragmatic standpoint. For instance, in the case of the QGP, one is interested in knowing how much energy is lost by a quark produced in the deep interior of the plasma as it traverses outward. There, one does not have reliable computational methods owing to the strongly coupled nature of the plasma, but as we shall see, in the AdS/CFT framework one can again distill this question to a simple classical computation. In this section we will explore the various attempts in the literature aimed at addressing this question, starting with the ballistic motion of quarks in the plasma and then turning to a discussion of the stochastic Brownian motion of stationary probes.

4.3.1 Energy loss and radiation of moving projectiles

To understand the energy loss of probe particles in plasma medium, we introduce an external probe in the form of an external quark or meson (for non-abelian plasmas) into the medium. Such probes are holographically modeled by an open string in the bulk geometry; here the geometry of interest is an asymptotically AdS black hole, which as we have described above provides the thermal medium. Let us understand the set-up in more detail. One of the open string end-points is pinned on the boundary of the AdS spacetime. Since this end-point carries the usual Chan-Paton index, it corresponds to the external quark we have introduced into the system. Heuristically, the external quark has a flux tube attached to it; in the holographic description one can view this flux tube as the open string world-sheet which extends into the bulk spacetime. Monitoring the motion of the external quark through the plasma thus amounts to studying the classical dynamics of string world-sheet in a black hole background. For mesonic probes we consider open strings with both end-points stuck on the boundary of AdS. One could also consider other probes such as monopoles or baryons (which are heavy in the large- limit); these would correspond to D-branes living in the bulk.

The first steps to understand the energy loss for probes in plasmas holographically were carried out in the seminal papers [87, 88, 89, 90, 91, 92, 93, 94], by considering the dynamics of probe strings as described above. A brief summary of these accounts can be found for instance in [9]. The general philosophy in these discussions was to use the probe dynamics to extract the rates of energy loss and transverse momentum broadening in the medium, which bear direct relevance to the physical problem of motion of quarks and mesons in the quark-gluon plasma.

The simplest computation of the energy loss was originally considered in [87, 89]. The idea was to examine a probe in the boundary moving with a constant velocity under the influence of an external force, applied so as to compensate for the frictional force acting on the probe and to maintain a steady state. From this steady state speed one can then recover the friction constant . On the bulk side, the problem reduces to a classical solution of Nambu-Goto action for the string, with one end-point being pulled with velocity along the AdS boundary. The constancy of velocity at the boundary is again maintained by an external force, in this case a constant electric field that drags the string end-point. By examining the classical solution of the Nambu-Goto action with these boundary conditions, one finds that while the quark moves forward with constant velocity, the bulk string world-sheet trails behind, see Fig. 2. Since there is no natural place for the string world-sheet to end in the bulk spacetime, it simply dips into the horizon.202020 This is however not captured by the constant- snapshot of the dragged string represented by Fig. 2 where only the static region outside the horizon is visible, so the string looks like it ‘freezes’ onto the horizon, analogously to the “frozen star” picture of a collapsing black hole. In particular, this implies that the classical world-sheet of the string itself has an induced horizon (which for non-zero velocity will always be outside the spacetime event horizon), a fact that will be important when we address the stochastic motion of the boundary end-point in §4.3.2.

Fig. 2: The trailing string solution in Schwarzschild-AdS spacetime. The curves are drawn for differing values of the quark velocity (specifically from right to left, , and ) on the boundary.

To see the construction in more detail, consider the planar Schwarzschild-AdS black hole (2.11); we are looking for a classical solution to the Nambu-Goto action

(4.7)

with being the world-sheet coordinates . The computation is easily carried out in static gauge and . In order to track the motion of a quark on the boundary, it is convenient to use the ansatz

(4.8)

resulting in the Lagrangian density:

(4.9)

with given by (2.11). The solution is obtained straightforwardly by noting that the conjugate momentum to , is conserved and constant, which allows one to solve via quadratures,212121 The physically correct sign of the square root corresponds to the bulk piece of the string world-sheet trailing the quark on the boundary.

(4.10)

In fact, requiring that the induced metric be timelike and non-degenerate fixes completely. Noting that for and implies that the numerator of (4.10) can vanish somewhere in the bulk of the spacetime. At this point, it must be that the denominator also vanishes. This then constraints to take on the value:

(4.11)

Given the classical string configuration obtained by solving (4.10), one can compute the rate of energy loss by looking at the momentum flow along the string world-sheet. One simply has to compute the net flux of this world-sheet momentum down the string. This results in [87, 89, 95]

(4.12)

where we have translated the result in terms of the temperature which is related to via (2.14) and reinstated the AdS length scale. In the non-relativistic limit, , this means that the friction constant is

(4.13)

If we use the Sutherland–Einstein relation (3.21),222222 Note that, as explained around (3.22), the relation (3.21) does not depend on . we obtain the diffusion constant

(4.14)

The calculation described above breaks down at very large velocities: physically, the force required to keep the quark moving sufficiently fast becomes so large that the quark anti-quark production is unsuppressed [94]. In [87] the more involved analysis of actually trying to see a moving quark slow down explicitly was also undertaken. This involves solving the time dependent equations arising from the Nambu-Goto action (4.7) with the initial conditions of non-zero velocity.

In this context, there is an interesting puzzle: a-priori one would have expected that any particle moving through a plasma will lose energy to the medium and slow down. While we have seen that this occurs for quarks, described by the trailing string configuration (4.10), there is no mechanism for energy loss in the case of a meson moving through a plasma. A meson, modeled as a quark anti-quark pair corresponds to a configuration of an open string with both ends stuck to the boundary; in such situations one finds ‘no-drag’ solutions where the string simply dangles down into the bulk independently of the direction of . More pertinently, for small separations between the quark and anti-quark (such that the meson remains in its bound state), the string stays above the horizon and as a result does not lose energy to the black hole. This feature was observed in different contexts in the analysis of [96, 97, 98]. Likewise it was also noticed in [99] that baryonic particle also propagate without being subject to drag (these authors also derived the drag force on k-quarks and gluons). These probes which seemingly do not suffer from dissipation are to our knowledge rather poorly understood.232323 In fact, the gravitational dual is analogously puzzling. A dragged string above a static black hole is equivalent to a static string above a boosted black hole, since only the relative velocity matters. For the latter configuration, the boost of the black hole produces an ergoregion above the horizon, and it is easy to see that the string cannot dip into this ergoregion. However, one might naively expect the frame-dragging effect to extend past the ergosurface and affect the string, bending it in the direction of the boost. Yet, as manifest from the solutions, no such effect takes place! Analysis of the velocity dependence of the screening length in the plasma for mesons and baryons was explored in [97, 98, 93, 100]. Another mechanism for quark energy loss in the medium based on Cherenkov radiation was proposed recently in [101].242424 We should also remark that there have been attempts to study linearly accelerating particles and their associated radiation in the AdS/CFT context; see [102, 103, 104, 105, 106, 107, 108].

Note that deceleration of the quark provides yet another mechanism for its energy loss, induced by radiation due to the quark’s deceleration. The interesting question of interplay between the two distinct energy loss mechanisms – namely medium-induced (i.e. drag) and acceleration-induced – has been explored in [102, 109], suggesting that these two effects may interfere destructively. More specifically, the authors consider a quark undergoing a circular motion at constant angular velocity. One advantage of such setup is that it can be treated as approximately stationary configuration while nevertheless incorporating acceleration (and in fact the classical world-sheet calculation for circular acceleration remains valid well into the acceleration-dominated regime), in contrast to the above-mentioned cases of linear motion. Interestingly, [109] find that depending on the angular velocity and radius of the quark’s circular trajectory, the energy loss is dominated by either by drag force acting as though the quark were moving in a straight line, or by the radiation due to the circular motion as if in absence of any plasma, whichever effect is larger, with continuous crossover between these regimes occurring at