^{1}

^{3}.

It is well known that a Rindler observer measures a nontrivial energy flux, resulting in a thermal description in an otherwise Minkowski vacuum. For systems consisting of large number of degrees of freedom, it is natural to isolate a small subset of them and engineer a steady state configuration in which these degrees of freedom act as Rindler observers. In Holography, this idea has been explored in various contexts, specifically in exploring the strongly coupled dynamics of a fundamental matter sector, in the background of adjoint matters. In this article, we briefly review some features of this physics, ranging from the basic description of such configurations in terms of strings and branes, to observable effects of this effective thermal description.

Thermodynamics is ubiquitous. Typically, for a collection of large number of degrees of freedom, be it strongly interacting or weakly interacting, a thermodynamic description generally holds across various energy scales and irrespective of whether it is classical or quantum mechanical. The underlying assumption here is a notion of at least a

While thermodynamics has a remarkable reach of validity, equilibrium is still an approximate description of Nature, at best. Most natural events are dynamical in character. Of these, a particular class of phenomena can be easily factored out, that of systems at steady state. While steady state systems are not strictly in thermodynamic equilibrium, they can be described in terms of stationary macroscopic variables. For such systems, there is a nonvanishing expectation value of a flow, such as an energy flow or a current flow, which does not evolve with time. Typically, such states can be reached

In this article, we will consider a similar situation. The prototype will consist of a

In the framework of quantum field theory (QFT), a similar construction was considered by Feynman-Vernon in [

Consider the basic idea behind the thermofield double. Consider a quantum mechanical system, with a Hamiltonian

Given such a pure density matrix, let us compute the reduced density matrix while integrating out one copy of the system. Thus we obtain

A similar picture holds true in the Holographic framework, which will be the primary premise in our subsequent discussions. In [

The Penrose diagram corresponding to an eternal black hole in AdS. The left and the right boundaries are where the dual CFT is defined, which are schematically denoted by

Let us now crystalize our discussion towards our specific goal. All of the above discussions are assumed to go through the framework of QFT. In general, of course, for weakly coupled QFT systems, explicit perturbative calculations are sometimes feasible, although those are certainly not simple for processes involving real-time dynamics. Furthermore, the existence of such a small coupling is far from guaranteed in Nature,

An interesting avenue is to explore the Gauge-Gravity duality, or the Holographic principle, or the AdS/CFT correspondence [

During the past couple of decades, a wide range of research has been carried out in this framework, in the context of quantum chromodynamics (QCD) as well as several condensed matter-type systems. Popularly, these efforts are sometimes dubbed as AdS/QCD or AdS/CMT literature. By no stretch of imagination, we will attempt to be extensive: for some recent reviews, see,

The Holographic framework, which can also be viewed as the most rigorous definition of a theory of quantum gravity, is a promising avenue to explore complicated gauge theory dynamics, qualitatively. There already exists a large literature analyzing various dynamical features of thermalization, quench dynamics for strongly coupled systems. We will not attempt to enlist the references here, however, presenting one example of new results, such as scaling laws in quantum quench processes [

Even though remarkable progress has been made in understanding dynamical issues, they remain rather involved and, in general, far more complex than equilibrium physics. Thermal equilibrium is particularly simple since it can be macroscopically described in terms of a small number of intensive and extensive variables. Intriguingly, for steady state configurations, which are neither in precise thermal equilibrium nor fullly dynamical, an

As mentioned earlier, we have one

The adjoint matter sector in

Within this framework, many interesting physics have been uncovered within the probe fundamental sector, specially the thermal physics; see,

A schematic arrangement of the steady state. The

The nonlinear dynamics of the brane, along with the U

Based on these ideas, a lot of interesting physics has been explored over the years. Here we will merely present a few broad categories and some representative references, which we will not explicitly discuss in this review. In [

This article is divided into the following parts: In the next section, we begin with a brief description of how nonlinearity can result in a black hole like causal structure, and how this is currently being investigated to understand features of QCD. In Section

The standard folklore concept of temperature is certainly in describing equilibrium thermal systems, at least in a local sense. In physics, there are various ways to define the temperature of a system: In thermodynamics, temperature is defined as an intensive variable that encodes the change of entropy with respect to the internal energy of the system. In kinetic theory, the definition of temperature can be given in terms of the equipartition theorem for every microscopic degree of freedom. In linear response theory, temperature can also be defined in terms of the fluctuation-dissipation relation.

Outside the realm of equilibrium thermal description, the notion of a temperature can sometimes be generalized. This is a vast and evolving topic in itself and we will refer the interested to reader to,

Intriguingly though, strong coupling seems to suggest a much simpler scenario. Most of our review will concern with the standard strongly coupled systems (and toy models) within the framework of gauge-gravity duality, where a fluctuation-dissipation based temperature is already explored in [

As described in [

Now, QCD or any such non-Abelian gauge theory is inherently nonlinear and produces a nontrivial medium for itself. In this case, the effective action can be fixed to be (see the discussion in [

Based on the hints above, the basic conjecture was made in [

The resulting essential consequences are as follows: color confinement and vacuum pair production leads to the event horizon. The only information that can escape this horizon is a color-neutral thermal description with an effective temperature. The resulting hadronization is essentially a result of such successive tunnelling processes. Also, there is no clear notion of “thermalization” through standard kinetic theoretic collision processes. A thermal description emerges as a consequence of the strongly coupled dynamics and the pair production. We will end our brief review here, since this is still an active field of research and is an evolving story.

In this article, we will primarily review the progress made and important results obtained in the framework of holography, the AdS/CFT correspondence [

To do so, we briefly review the standard understanding of how this correspondence (or, duality) emerges. Closed string theories describe a consistent theory, and in the low energy limit, this consistent description can be truncated to various supergravity theories, in general. On the other hand, open string theories naturally arise as boundary conditions which contain the information of the string end point. The canonical construction begins with a stack of

For a stack of

Alternatively, a stack of D

A schematic presentation of a generic D-brane construction. A certain

The construction above can now be generalized by introducing additional degrees of freedom. In the dual gauge theory, this corresponds to introducing additional matter sector transforming in different representations of the SU

A schematic presentation of a generic D-brane construction. A certain

The prototype model in our subsequent discussion is based on the dynamics of this additional set of branes, which capture the dynamics of open long strings. There is a “bath” with a large number of degrees of freedom which is provided by the stack of

Before moving further, let us note the following: The fundamental matter sector, essentially, is described by long, open strings. It is not unreasonable to assume that there is a sensible limit in which the probe D-brane becomes irrelevant and the essential physics can be captured by the dynamics of explicit strings. Indeed, for specific models, this limit can be made precise: For example, if we introduce a mass of the

We begin with reviewing a general description of a string worldsheet which is embedded in an AdS-background, following closely the treatment of [

The dynamics of the string is governed by the Nambu-Goto action:

Given the above action in (

The left diagram corresponds to

The string with a single end point at the UV, which is what is described above, can be written in a particularly recognizable form [

We can describe the string profile in terms of the embedding space; however, in this case it is straightforward to adopt a Poincaré patch description of the profile. This can simply be done by using,

Here we will discuss the particular case, in which an infinitely massive quark (

The causal structure induced from the embedding in (

Patching a retarded solution and an advanced solution has been discussed extensively in [

We have shown here various string profiles in

A simple but instructive example is to consider the string end point moving in a circle of constant radius. Evidently, the end point does undergo an acceleration. Let us review this based on [

In the global AdS patch, the boundary theory is kept at a finite volume and likewise develops a mass gap. Thus, a rotating string will not develop an event horizon on the worldsheet for any value of the frequency, unlike the Poincaré-slice result. In this case, for sufficiently large angular frequency

A thermal effective description ensures that the string end point will undergo a stochastic motion, because of the thermal fluctuations. This results in a Brownian motion for the string end point. In [

Given a thermal system, how fast a small perturbation relaxes to the thermal value sets the thermalization time scale for such small excitations. This is closely related to the scrambling time for the system, which determines the speed at which a quantum system spreads a localized information. In [

The notion of ergodicity and thermalization are intimately related. In a semi-classical description, the physics of thermalization is further related to a particular notion of growth of

Given a classical system, a phase space description is provided in terms of canonical coordinates and momenta variables:

A schematic representation of the chaos diagnostic function(s). Here, qualitatively,

It is not simple to calculate higher point OTOCs, in general and there are currently a handful of tractable examples. Nevertheless, in [

Given the black hole like causal structure on the string worldsheet, it is therefore expected that such a saturation will hold on the worldsheet horizon as well. Indeed, it was explicitly shown in [

The soft sector physics, described by (

The four-point effective vertex contains two interaction vertices with the soft sector, along with a propagator of the same soft sector. The dashed horizontal line is the soft sector propagator, and the solid lines are external hard modes.

We will end this brief section here, leaving untouched a remarkable amount of recent research in related topics. Chaotic properties of strongly coupled CFTs with or without a holographic dual are a highly evolving field of research. Moreover, the notion of chaos in quantum mechanical systems is a rich field in itself and only some explicit calculations using a particular semi-classical prescription have been made. Thus, we will leave a detailed discussion for future and, for now, shift our attention to the effective thermal description of a D-brane.

Let us consider a stack of

The matter content of

In terms of the dual gauge theory side (in this part, we closely follow the discussions in [

The Lagrangian of the worldvolume theory can be written in the

Degrees of freedom in the dual field theory.

Fields | Components | Spin | | | | |
---|---|---|---|---|---|---|

| | | | | | |

| | | | | | |

| ||||||

| | | | | | |

| | | | | | |

| | | | | | |

| ||||||

| | | | | | |

| | | | | |

The effective action for the probe D7-brane is given by the Dirac-Born-Infeld (DBI) Lagrangian with an Wess-Zumino term. (Here we are using the Lorentzian signature.)

We intend to design a system in which energy is constantly pumped into the probe sector. This can be achieved by exciting the gauge field. (We absorb the factor of

In the presence of

Geometrically, therefore, there is a close parallel to the purely thermal physics. When there is a black hole present in the background, the probe brane can either fall inside the black hole, or stay above it. This depends on the asymptotic separation between the D

In the vanishing electric field limit, the D-brane (denoted by the curves above) can stay above the black hole (right-most) and fall into the black hole (left-most). These two phases are separated by a critical embedding, corresponding to the middle one.

With a nonvanishing electric field, the new radial location

The D

To obtain the fluctuation action, we can expand (

With these, the effective scalar and vector fluctuation actions are given by

The fermionic fluctuations of the D

It is clear that the fluctuations in the D-brane,

With

An effective temperature can be defined from the metric in (

Let us go back to (

A qualitative picture of the Kruskal extension of the open string metric. We have drawn the Penrose diagram in a “square” form, which is not strictly correct. We will elaborate more on the Penrose diagram in later sections. This diagram is taken from [

It is possible to further construct explicit examples of such probe brane configurations, in a background geometry. Since such constructions can be varied in richness, we will not attempt to provide a classification; rather we will present another explicit and instructive example. In this case, we are motivated by the analytical control the model provides us with. We begin with the standard

Fundamental sector introduced in the background of

Brane | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

| ||||||||||

( | – | – | – | – | X | X | X | X | X | X |

| ||||||||||

( | – | – | – | X | – | – | – | X | X | X |

The geometry is given by

To study a nontrivial dynamics of the D

Further note that this thermal imprint continues to hold beyond standard

Continuing on this theme, we can explicitly construct other examples where the adjoint matter sector is not described by a CFT, unlike the

In this section, we will discuss the essential physics, without referring to any explicit example. The basic idea is as follows: Given a low energy closed string data, in string frame,

The essential dynamics is given by the following action, in the notation of [

Around a classical saddle of (

In [

In general, we can arrange the electric and the magnetic field to be (i) parallel or (ii) perpendicular. Assume that we have an

The open string metric can also contain an “ergoplane” and therefore similar related black hole physics. This is explicitly demonstrated with the D

Let us take a simple example, with the background if given by an

In addition to the explicit D-brane constructions that have been discussed before, here we briefly mention some examples in which explicit analytical calculations can be performed, to arrive at the same conclusion. These are the so-called Lifshitz background of the following form:

Around this classical saddle, we consider fluctuations which yield

With the following ansatz:

Our entire discussion is based on the probe limit. Since this is a crucial ingredient in our construction, let us briefly review what this entails. At the level of the equations of motion, the Einstein tensor of a given solution must be parametrically large compared to the stress-tensor of the probe sector. Consider the decoupling limit of

Consider

The basic statement of gauge-gravity duality is an equivalence between the bulk gravitational path integral and the dual field theory path integral:

The quadratic fluctuation part can be schematically represented as

Let us focus on the D-brane sector. In this sector, let us rewrite the generic form of the scalar and vector fluctuations:

Before discussing the Euclidean path integral (

In the Euclidean patch, the path integral yields a thermodynamic description in terms of a partition function. It is now expected that the probe sector thermodynamics will be simply determined in terms of the effective temperature

For the case in consideration, a proposed thermodynamic free energy was discussed in [

Furthermore, the presence of the OSM event horizon and the open string data are

A detailed analysis of the causal structure of such OSM geometry was discussed in [

Penrose diagram corresponding to an OSM embedded in a purely

Let us discuss some aspects of “energy conditions” for the OSM geometry. A simple way to define such conditions is to demand that the OSM geometry can be obtained by solving Einstein equations, with a suitable matter sector and translating the energy conditions on the corresponding matter sector. Thus, imagine

Defined this way, an

In view of (

In a special example, discussed in [

A simple solution of the resulting equations of motion can be obtained for

In this review, we have discussed various examples in which the nonlinear dynamics can give rise to an effective causal structure. When this causal structure is similar to that of a black hole, many similarities to classical as well as quantum properties of black holes become manifest. This bears substantial resemblance to other nonlinear systems that describe gravity-like phenomena, known as

However, our attempts have been to briefly review the essential ideas, some explicit and simple examples on which these ideas are rather manifest. We have certainly not made any attempt to review the vast literature on the physics of probes traveling through a thermal medium in a strongly coupled gauge theory. The basic ingredients used to study this are open strings in a supergravity background. In the probe limit, the corresponding string profile can develop a worldsheet event horizon, which plays a crucial role in determining energy loss, drag force, stochastic Brownian motion of the probe. For a more extensive review on these, we will refer the reader to [

Staying in the probe limit, while there are similarities with a black hole, there are unsettled issues as well. One of the main issues is about the physical meaning of the area of the event horizon in all of the cases we have discussed above. For a black hole, this corresponds to the thermal entropy; moreover, one can prove area increasing theorems (with certain energy conditions for the matter field) in coherence with the second law of thermodynamics. This, however, is in stark contrast with what we have reviewed here. First, we can identify the Euclidean path integral as the corresponding thermal free energy and, therefore, derive the thermal entropy from this free energy. It is rather easy to see that this does not equal the area of the event horizon on the string worldsheet, or of the open string metric geometry. Secondly, as we have explicitly discussed

One intriguing proposal of [

Complexity is a well-defined notion in quantum systems, which measures the minimum number of unitary operations required to reach a certain state starting from a base-state. Based on the eternal black holes in AdS, recently it has been suggested in [

Finally, we end by briefly commenting on one of the most interesting questions in black hole physics: the information paradox. Since the causal structure in the probe sector is closely similar to the causal structure of a black hole, in the eternal patch (

On a string worldsheet, a naíve analysis following [

Finally, we end this review with the mention of the Lieb-Robinson bound, which provides an emergent upper bound on how fast information can propagate in a generic nonrelativistic quantum mechanical system, with local interactions. While this limiting velocity is system specific, it rather intriguingly suggests a

The author declares that there are no conflicts of interest.

We thank Sumit Das, Jacques Distler, Roberto Emparan, Willy Fischler, Vadim Kaplunovsky, Sandipan Kundu, Hong Liu, Gautam Mandal, David Mateos, Andy O’Bannon, Bala Sathiapalan, Julian Sonner, and Javier Tarrío for numerous discussions and conversations on related topics. We specially thank Tameem Albash, Avik Banerjee, Veselin Filev, Clifford Johnson, Sandipan Kundu, and Rohan Poojary on various collaborations that helped form the bulk of this review article. We also thank Mariano Chernicoff, Alberto Guijosa, and Juan Pedraza for giving the permission to use several figures from their papers. This work is supported by the Department of Atomic Energy, Govt. of India.