AdS/condensed matter

Written by PhD Candidate Daniel K. O'Keeffe and A.W. Peet

Like many human beings younger than a certain age, we grew up watching (and/or re-watching) Star Trek: The Next Generation. One of the many technological wonders shown off in the show was the holodeck; a room whose walls were made out of some kind of advanced holographic projection system which could construct hyper-realistic three dimensional images of just about anything, from landscapes to people. The whole system was interactive and was frequently depicted being used to play out intricate games where entire fantasy worlds could be constructed and manipulated.

Such a technology is, as of today, nothing more than science fiction, but it serves to demonstrate the basic idea behind holography, an important concept of modern theoretical physics. The essence of a hologram is that its two-dimensional surface somehow encodes information about a three dimensional world. If you have ever been a science museum, you have probably seen a hologram on display; when laser light is shone on a surface with a certain pattern stuck to it, a three dimensional image can be produced in front of the plate. (Even at a low-tech level, we have some intrinsic idea of this notion as well: consider a reflection of a crowded subway car in one of the windows of the train. You are really just looking at a reflection, but somehow you are able to understand that this two dimensional projection you are seeing contains tons of information about the subway car you are in, the position of the other passengers, their distance apart. Even dynamical information such how they are moving about is, in principle, accessible through the reflection.)

Holograms are really cool and involve well-understood aspects of light and its interaction with solids. But the idea of holography in physics may run even deeper than that. The specific concept we will delve into here is the modern notion that a gravitational theory on some background geometry, referred to as the bulk, can be physically equivalent to a quantum field theory (QFT) without gravity living on the boundary. QFTs are of course ubiquitous in modern physics; they are used to describe e.g. the standard model of particle physics and condensed matter systems. The gravity theory and the QFT are said to be dual to each other in the sense that there is a one-to-one mapping between physical quantities in one theory and physical quantities in the other, both kinematical and dynamical.

Let us examine how a gravitational theory and a non-gravitational theory in one lower dimension could possibly be physically equivalent. The key is to think about the entropy. In a regular $d$-dimensional QFT, the maximum entropy in a region scales like the volume of that region $E_{QFT} \sim L^{d}$, where $L$ is a characteristic length scale. In a theory with gravity, the maximum amount of entropy in a region $V$ is equal to the entropy, $S_{BH}$, of the biggest black hole that can fit inside the region. To see why this is true, consider a setup in which you have some mass, less than the mass required to form a black hole, and an entropy greater than that of a black hole. Start adding mass to the system; this causes both the total mass and the entropy to increase. If you keep doing this, eventually you will form a black hole, at which point the entropy of the system will need to magically decrease down to $S_{BH}$, violating the second law of thermodynamics.

The black hole entropy is given by the famous Bekenstein-Hawking formula $$ S_{BH} = {\frac{A_{horizon}}{4 G_{N}}} \,. $$ $A_{horizon}$ is the area of the black hole event horizon, which is a special surface in spacetime. Events below the horizon cannot affect observers above it. $G_{N}$ is the Newton constant in $d+1$ dimensions. (Here, and in the following, we will use units in which $\hbar= c = k_{B} = 1$.) Notice that the black hole entropy is proportional to an area, rather than a volume, which gives us the first hint that a holographic equivalence might be possible. Because an entropy that scales like an area in a $d+1$ dimensional theory and another entropy that scales like volume in a $d$ dimensional theory could in fact be the same thing wearing different disguises.

So let us ask a more technical question:if you do have a holographic system, how exactly is information about a spacetime region encoded in the boundary system, and vice versa? To make this sort of mysterious idea concrete, we need to have model systems where analytical (and numerical) calculations are possible. This was the major advance that Juan Maldacena made in 1997 with his discovery of the AdS/CFT Correspondence using the toolbox of string theory. AdS/CFT relates a gravity theory (actually a string theory, to which ordinary gravity is a low-energy approximation) on a special spacetime geometry called Anti de-Sitter space (AdS) to a field theory with a high degree of symmetry called conformal symmetry.

A conformal field theory (CFT) is a quantum field theory that has extra symmetries. The set of transformations encoding these symmetries include the usual Poincare symmetries (translations, rotations and Lorentz boosts) as well as scaling transformations (dilations) and a set of special conformal transformations. Invariance under scaling transformations means that the theory is the same when spacetime is rescaled as $x^{\mu} \rightarrow \lambda x^{\mu}$ for some constant $\lambda$. The superscript $\mu$ is a label for the different space and time dimensions. The CFT lives in $d$ diemensions and the gravity dual lives in $d+1$ dimensions, so $\mu$ runs from $0\ldots d$ labelling all dimensions of the field theory. If the field theory has this enhanced set of symmetries, and we want it to be dual to a theory with gravity, then the gravity dual had better also possess the same enhanced set of symmetries. This is where the AdS part comes in.

AdS is a solution to Einstein's equations of general relativity with a negative cosmological constant. General Relativity describes the dynamics of spacetime and matter within it geometrically. This geometry is encoded in a quantity called the metric tensor $g_{\mu \nu}$. It can be thought of as a matrix of rank $D \times D$, where $D$ is the number of dimensions of spacetime ($D=d+1$). Physically, the metric encodes how lengths are measured in a geometry which is not necessarily flat; this is reflected by the line element $$ ds^{2} = \sum_{\mu, \nu} g_{\mu \nu} dx^{\mu} dx^{\nu} \, . $$ For example, a flat Euclidean space in $D=3$ dimensions has a metric with is just a $3 \times 3$ diagonal unit matrix: $ds^{2} = dx^{2} + dy^{2} +dz^{2}$, which is just encoding the length of a straight line trajectory in three dimensions. The structure of the line element becomes richer if you also include time as a dimension, as is required in special relativity. In this case, $ds^{2} = -c^{2} dt^{2} + dx^{2} + dy^{2} + dz^{2}$ which is called Minkowski space (the speed of light, $c$, has been restored here for clarity). There are three different types of behaviour possible with this Minkowski metric: If $ds^{2} \lt 0$, the trajectory is said to be timelike. These are trajectories followed by massive particles, which travel with velocity less than the speed of light in vacuum. Events separated by a timelike path are causally related. If $ds^{2} > 0$, the trajectory is said to be spacelike. Events separated by a spacelike path are not causally related as that would require faster than light propagation. Finally, if $ds^{2} = 0$, the trajectory is said to be null. These are relevant to massless particles (e.g. photons) which travel exactly at the speed of light.

General relativity encodes the structure of much more complex geometries in the metric. The one we are interested in is AdS. In $D=d+1$ dimensions, the metric is $$ ds^{2} = \frac{r^{2}}{L^{2}} \left ( -dt^{2} + dx_{1}^{2} + \cdots dx_{d-1}^{2} \right ) + \frac{L^{2}}{r^{2}} dr^{2} \, . $$ $L$ is a constant called the AdS radius of curvature (it sets the length scale in the geometry). The coordinate $r$ is called the radial coordinate, and it is the extra dimension that the gravity theory has that dual field theory does not. You can see that the first term on the right hand side looks just like the Minkowski space metric, expect with $r^{2}/L^{2}$ in front of it. This extra factor is called a conformal factor. Now, we have said that the dual field theory should live on a space that is related to the boundary of this spacetime geometry. AdS has a boundary at $r \rightarrow \infty$, where, up to the conformal factor, the geometry is Minkowski space, where the dual field theory lives. The peculiar thing about AdS is that massless particles can actually propagate from anywhere in the interior and reach the boundary in finite time (affine parameter), and come back again. This is different from what one usually expects in flat spacetime, where the boundary is just infinitely far away and unreachable. This property of AdS gives us some hint that the boundary can know what is happening in the rest of the spacetime and vice versa.

What about the symmetries we were so focused on previously? It turns out that the AdS metric does actually have the conformal symmetries that we were interested in, realized as isometries of the spacetime. For example, if we rescale the space and time coordinates as $x^{\mu} \rightarrow \lambda x^{\mu}$ and as long as $r \rightarrow \lambda^{-1} r$, the geometry is unchanged; so $\rm{AdS} \rightarrow \rm{AdS}$ under rescaling, i.e. AdS contains the rescaling symmetry. As we move inward the radial direction, $r$, the spacetime looks like a series of copies of Minkowski space with a decreasing size as $r \rightarrow 0$.

This picture leaves us with a few immediate questions. First of all, how do we interpret this extra dimension in AdS, which we called $r$? If the gravitational theory is supposed to represent all of the physics of the field theory and vice versa, then what does this extra dimension mean in the field theory? Also, how does this picture resolve the question we posed earlier: how can the two theories possibly contain the same number of degrees of freedom?. We will answer the second question first.

First of all, on the gravity side, we need to figure out what is the size of the largest black hole that can fit inside of AdS. By using the Bekenstein Hawking formula, the entropy of this black hole will be proportional to the area of its event horizon. The answer is contained in the metric above. The spacetime stretches in the radial direction from $r=0$ to $r=\infty$. The largest black hole we can fit inside this space, then, should stretch across this same radial distance with its event horizon right up at the AdS boundary. The area of the horizon, then, is just along the the boundary directions $x_{1}, x_{2}, \cdots, x_{d-1}$. The area of this surface is then infinity, so the entropy, and thus the number of degrees of freedom is infinity. This seems problematic, does it mean we have done something wrong? Well no, actually, we could have anticipated this result from field theory intuition. In fact, this observation about the diverging number of degrees of freedom will motivate the solution to our first question about interpreting the extra dimension $r$ in the gravity theory.

QFTs are actually full of divergences. When calculating physical observables in a QFT, what you will often find as an answer is infinity. The problem is we have naively assumed that the QFT description we are using is valid at all energy scales. QFTs should be thought of as having different theory parameters at different energy scales. The high energy, or ultraviolet (UV), theory and its parameters flow with decreasing energy to a low energy, or infrared (IR) theory. By properly regulating the theory, we can remove the problematic divergences and get finite results. This process is called renormalization.

Comparing this with the observation we made about the changing size of AdS, we suspect that the extra radial dimension in the gravity theory actually encodes the energy scale of the dual field theory. The boundary of the spacetime geometry corresponds to the UV of the field theory, while the interior corresponds to its IR. This suspicion has been confirmed in a wide variety of model systems, and can be verified in several different ways. Therefore, the divergence we found in bulk gravity theory due to the diverging size of the boundary is really encoding the fact that the dual field theory has UV divergences. We have already mentioned that there is a procedure (renormalization) for regulating these divergences in field theories and the same is true in the gravity theory. The procedure is called holographic renormalization and it provides us with a geometric way of interpreting field theory renormalization.

The AdS/CFT duality we have been describing has its origins in string theory. A more precise statement of the duality is that type IIB superstring theory (which lives in ten dimensions) on $AdS_{5} \times S^{5}$ is dual to $\mathcal{N} = 4$ supersymmetric Yang-Mills theory in four dimensions. $AdS_{5}$ is five dimensional anti de-Sitter space and $S^{5}$ is a five dimensional sphere. The dual field theory possesses both the conformal symmetry we have already discussed as well as supersymmetry (it has $\mathcal{N} = 4$ supersymmetries). Understanding superstring theory on this kind of geometry is a daunting task. In order to be able to say something useful, we need to a handle on the physics being described on the gravity (string theory) side of the duality. It turns out that there is a certain parameter limit in which things simplify. In this particular limit, the string theory is effectively well described by regular Einstein gravity in the AdS background. The duality maps this particular parameter limit to where the field theory is very strongly interacting.

This is immediately a very interesting limit. Strongly coupled field theories are not well understood. The standard textbook approach to understanding QFTs is to work at weak coupling where we can understand the behaviour of the theory perturbatively. Such an approach does not work at strong coupling and it can be difficult to say much about QFTs in this limit except via numerical simulations. The gravity dual is described by general relativity, which we understand quite well. In this way, we can use the duality to understand strongly coupled field theories by studying related quantities in Einstein gravity.

As we pointed out, AdS has a peculiar boundary. In particular, when studying the behaviour of fields in AdS, it is not enough to just specify some initial conditions and work out the dynamics, we must also specify conditions at the boundary. The gauge/gravity conjecture equates the generating functional of the QFT to that of gravity theory. The boundary behaviour of fields in the bulk spacetime are to be understood as sources for corresponding objects in the field theory. The generating functional encodes the behaviour of the content of the theories, so if we can understand one side of the duality, we can construct a dictionary to translate to the other side. Armed with this, we have a set of tools to study the behaviour of the strongly coupled QFT.

The correspondence we have described is one particularly well known example. It is not unique, there are many more examples of these kinds of dualities. From a practical perspective, we are interested in gaining insight into a wider class of strongly coupled field theories, particularly those that arise in condensed matter physics. This leads us to adopt a phenomenological approach to the problem. If we start with a gravity model that contains some field content, can we engineer a system that should in principle have a field theory dual that captures properties observed in condensed matter systems? We do not necessarily know exactly how the gravity model we write down descends from string theory, nor do we necessarily know the details of the proposed dual field theory. What we do know is that using the holographic dictionary, we can describe a number of condensed matter phenomena and compare our results to known models (when available). Starting from this bottom up perspective may lead to insight about what kind of ingredients we need to understand field theories relevant to condensed matter physics and point to a stringy origin. It also highlights the ability of a gravity theory to capture non-trivial physics associated with seemingly unrelated systems.

The holographic approach to condensed matter systems is referred to as AdS/CMT. What kind of systems are amenable to this kind of analysis? We will strive here to give an overview of a few interesting applications.

At very low temperatures, condensed matter systems can undergo a quantum phase transition. Phases transitions between states of matter are usually driven by thermal fluctuations; by heating up ice, it melts into water, the phase transition occurred because of an input of thermal energy. At low temperatures, the same mechanism does not have a major effect, instead phase transitions are driven by quantum fluctuations. There is a critical point for the parameter values at which the transition occurs. At this point, the dynamics of the system is strongly coupled (in 2 spatial dimensions) and is known to display conformal invariance. CFT models of the dynamics near this critical point exist already within the condensed matter literature. Something interesting happens when a small temperature is turned on. It turns out that the dynamics of the system is still controlled by the the CFT, it is still strongly coupled and the known models at zero temperature no longer apply.

A holographic approach has been applied to the problem and has shed light on some transport properties, such as frequency-dependent conductivity. Further studies will hopefully point towards a better understanding of what to expect from these kinds of systems.

A stand out feature of the application we described above is the presence of conformal symmetry. This requirement is quite constraining, as many condensed matter systems do not generally possess this special kind of symmetry. If we are going to make progress, then we should look for ways to break various symmetries within holographic models.

Condensed matter systems to do not typically possess relativistic symmetry. Usually, the velocity scales we are working with are low compared to the speed of light, so relativistic invariance can be relaxed. Any proposed holographic dual to such a system should also break relativistic symmetry. There two major directions that have been followed in addressing this problem. One solution is a spacetime geometry with Schroedinger symmetry, which is like a nonrelativistic version of conformal symmetry. Such geometries have been applied to modelling cold atom system. The other approach considers a class of spacetimes called Lifshitz in which space and time scale differently, explicitly breaking relativistic symmetry by construction. In this case, space and time scale as $t \rightarrow \lambda^{z} t \, , \,\, x_{i} \rightarrow \lambda x_{i}$, unlike the usual scaling we discussed in the conformal case. $z$ is called a dynamical critical exponent. This scaling can be further reduced by including a hyperscaling violation parameter $\theta$ to the geometry. Hyperscaling is the property that the free energy of a system should scale as its dimension, $d$. This makes intuitive sense, as we add dimensions we can fit more stuff into the system. When hyperscaling is violated, this scaling relation is broken to $d-\theta$. In other words, the system behaved as if it lived in an effective dimension $d_{eff} = d-\theta $ < $ d$. Such a system may seem exotic and we might wonder exactly what kind of physics can be modelled in this way.

A particular phase of matter, called compressible, is a prime candidate. In traditional condensed matter theory, only a handful of models of compressible matter are known. Experimental evidence suggests that there are other types of compressible matter called strange metals. A ubiquitous example of such phases are high temperature superconductors. These kinds of systems display non-Fermi liquid behaviour, meaning that their behaviour deviates from that of a Fermi-liquid, which is a standard description of most metallic states. The physics of these states of matter is strongly coupled, so a holographic approach is well suited. Several studies have been able to model several types of strange metal behaviour, but a full classification of the possible compressible phases described by holography is an open problem.

There are many other symmetries that we might like to beak as well. One important example is translational invariance. The holographic models we've discussed so far have all preserved translational invariance along the boundary directions. In other words, from the perspective of the dual field theory, every direction along the boundary looks the same; there is no preferred direction. As a consequence, momentum along the boundary directions is conserved. Imagine then a system in which we compute the (DC) conductivity, the answer will be infinity. The reason is simply that since there is no way to dissipate momentum, there is nothing to resist the motion of charges, hence the conductivity is infinity. This is problematic if we want to model a realistic condensed matter system which has finite conductivity. This is not so much a problem as a clue, generically, condensed matter systems do not preserve translation invariance so neither should a holographic model.

There have been several approaches to this problem and it is still an active area of research. Attempts to tackle this problem have included explicitly constructing spacetimes which reduce some of the translational symmetry of the boundary directions. It turns out there are entire families of such geometries and that they may be classified by an old general relativity construction called the Bianchi classification. Some of these solutions have found applications in modelling holographic metal-insulator transitions. Another approach has been to construct holographic lattices, where a periodic solution is setup along the boundary by a source. The conductivities calculated in such models are strikingly similar to that measured in cuprate superconductors.

Another interesting approach is to construct holographic models of disorder. Most realistic condensed matter systems have some degree of disorder which acts to break translational invariance. Building holographic systems which incorporate disorder is technically challenging and a few different approaches have been taken, for example. The hope is that studying such systems will give us clues into the nature of many body localization, which is not well understood from the perspective of condensed matter physics.

There is also evidence that properties of glassy systems can be accessed via holography. Glassy dynamics are notoriously difficult to study within condensed matter physics and the nature of the glass transition remains somewhat mysterious. At a glass transition viscosity tends to diverge as the system undergoes a critical slowing down. The dynamics of the system undergoes a radical change from a liquid to a vitrified state, but this is not a phase transition in the traditional sense. There is no noticeable change in the spatial ordering of the system compared to its original liquid phase. Rather, the system finds itself trapped in a metastable state and is never able to reach thermodynamic equilibrium. Understanding this phenomenon in a dual gravity theory may help shed light on the complex non-equilibrium dynamics that occurs in such a transition and the ingredients necessary for it to occur.

There are many more applications that we haven't had the chance to describe to you in this article. For example, applications to fluid dynamics and turbulence, entanglement entropy, and even QCD. Holography opens up novel avenues for studying old problems in condensed matter physics and will hopefully spur new questions to address. The ultimate fantasy would be a prediction about a previously unknown phase of matter; something that could in principle be tested in a lab. Studies of holography also motivate pushing on developing new techniques in classical gravity, both analytical and numerical. Overall, holographic duality provides a link between seemingly unrelated branches of physics, drawing in field theory and condensed matter physics, gravitational physics, and even quantum information theory, to name a few. Understanding how all of this fits together and exactly what it can teach us about nature is a central problem of modern theoretical physics.

Further reading

Students interested in going further with this topic may like to try Horatiu Nastase's relatively self-contained elementary review of AdS/CFT holography.