By Andreas Karch, physics professor at University of Washington
Having recently made my first appearance in the comment section of a physics blog about the general question of "what has AdS/CFT done for us lately?", Luboš offered me the opportunity to write a guest post about the general topic of "AdS/anything", which I'll prefer to call "Applications of Holography". This entry will be loosely based on variety of very similar colloquium level talks I've given about this topic in the last two years. While I'll try to keep this blog post mostly free of technicalities, readers interested in a slightly more detailed version can refer to my Proceedings for my talk at the PANIC 11 or to the slides for my talk at STRINGS 12.
First off, let me define what the tool is we are using. "AdS/CFT" or the "gauge/gravity correspondence" or "holography" are different words used for more or less the same interesting set of conjectures. Holography is the statement that a large class of strongly coupled field theories has an equivalent description in terms of a classical theory of gravity (in one higher dimension). The conjecture comes with an explicit dictionary. For any quantity of interest in these field theories we can formulate a classical gravitational problem whose solution gives the desired answer in that field theory. This way holography provides us with a series of solvable toy models of strong coupling dynamics.
By toy models here I mean theories that are not actually realized in nature, but share some important qualitative features with the real thing - just like the famous spherical cow. Studying toy models can be important when addressing the real question is simply out of reach. In this case toy models can provide us with quantitative guidance and important qualitative insights. Ideally, they will help us develop new ways to think about the problem at hand so we can eventually develop new techniques to actually solve the problem thrown at us by nature. I will describe a sequence of examples where holography has been used this way.
What sort of questions in physics are currently outside theorist's ability to calculate? Clearly strong coupling is one complication; our prime calculational tool we have all been raised with since our undergraduate days is perturbation theory. Perturbation theory relies on the existence of a small parameter one can expand in. When interactions are weak, the leading order answers can be obtained by considering a bunch of non-interacting particles move around, and the fact that they do every now and then talk to each other can be taken into account systematically, order by order, in that small parameter. But if there is no small parameter, this method clearly fails. Of course other techniques exist at strong coupling. Lattice gauge theory has been used very successfully to calculate many properties of the strong interactions as described by QCD. By discretizing space and time, we turn the very formal path integral of quantum mechanics into a product of ordinary integrals, basically one for each spacetime point. This is something a computer can calculate. The technical problem is that these integrals are still too many to handle for a cutting edge computer - the way lattice gauge theory is implemented in real simulations is via Monte Carlo techniques, also often referred to as importance sampling. In order to calculate the integral, we evaluate the integrand at random points and sum those up. As long as the integrand is real and positive, we can get a good approximation to the full answer by making sure we get a sufficient sample of points in the region where the integrand is largest. This method clearly fails if the integrand can take negative values or is complex. Here the final answer often involves cancelations between different points in the integration region and finite samples give no useful information. This, in a nutshell, is the "sign problem". So when is the weight in the path-integral not a real positive quantity? The path integral measure is \(\exp(iS_{\rm action})\), so in general it is complex. For static questions, we can Wick rotate to Euclidean time and the exponential becomes real. This is the realm of lattice gauge theory as it is mostly performed today. Masses, matrix elements, finite temperature thermodynamics are all under control, often with percent level accuracy. But time-dependent phenomena are mostly out of reach. So are even static problems in which we turn on a finite density for fermions, which also leads to a complex weight even in equilibrium. This is especially a problem for simulations of strongly correlated condensed matter systems where of course one wants to have a finite number of charge carriers present. In all these situations, the insights gained from the holographic toy models can be an important first step towards gaining qualitative insights into how a strongly coupled system should behave. Ideally, this will pave the way to formulate new, purely field theoretic tools that will allow one novel ways to attack these old problems.
Before discussing a series of examples of holographic successes, let me spend one paragraph on discussing what sort of strongly coupled theories have holographic duals, as there exist certain misconceptions about this. The first holographic duality gave a solution for \(\NNN=4\) supersymmetric Yang-Mills theory (SYM) with a large number of colors \(N\) and a large 't Hooft coupling \(\lambda = g_{\rm YM}^2 N\). This is a supersymmetric, conformal field theory. Neither of those properties is crucial. Among the zoo of holographically dual pairs, we know lots of examples of confining gauge theories with or without supersymmetry. What all these field theories with holographic dual share are the following two properties: for one, they have to be taken in a limit where the number of local degrees of freedom is large. If they are based on an \(SU(N)\) gauge theory, this is typically achieved by a large \(N\) limit. This ensures that the theory behaves classically in the right variables. Here the right variables are the gravitational dual. The other property is that the field theory (let us talk about a confining example for concreteness) has a large separation of scales: the lightest spin 1 meson has to have a mass much lower than the lightest spin 2 meson. What this ensures is that the dual holographic theory is based on supergravity, as opposed to a classical string theory. If we could study even classical string theory on AdS space, large \(N\) QCD would be solved. But we can't. These days many people calling themselves string theorists essentially study little beyond general relativity - at least as far as applications of holography are concerned. Real QCD has no such separation of scales. The lightest spin 2 and spin 1 mesons come in at \(1275\MeV\) and \(750\MeV\) respectively. QCD may have a holographic dual, but it is not a supergravity theory. I has to be a full fledged string theory. In that sense, field theories with a holographic dual (by which I here mean a classical theory of gravity, not a string theory) are toy models.
So what are these toy models good for? Let me give a long list of examples where insights gained from holography have actually have had some impact on how people think about certain physical problems. In this post I will mostly focus on questions motivated by heavy-ion collisions and only briefly touch upon some more recent "AdS/CMT" results later on.
QCD, the theory of the strong interactions, obviously also governs heavy-ion collisions. QCD becomes weakly coupled at very high energies. But around the temperature where nucleons melt into quarks and gluons, QCD still seems to be rather strongly coupled. The equilibrium properties of this strongly coupled soup of quarks and gluons (at least at zero baryon number) can be mapped out with lattice QCD. But a heavy-ion collisions is anything but "static". Two ions (gold/gold at RHIC, lead/lead at the LHC) are smashed into each other to create an exploding fireball of molten quark-gluon liquid. To understand what is going on in the non-equilibrium environment of the rapidly expanding fireball is exactly one of the arenas where holographic toy models can be of some use. Insights that have been gained include the following:
Viscosity: The shear viscosity of a fluid determines the force two fluid layers moving past each other exert on each other. More precisely, it gives the force per unit area per unit velocity gradient. As such, it should be measured (in SI units) in Pa s. A more convenient unit is the centipoise (cp), which is 1/1000 Pa s. In these units our favorite fluid, water, has a viscosity of 1 cp, air about 0.02 cp and honey (depending on your favorite brand) anywhere between 2000 cp and 10,000 cp. One (in)famous experiment measuring viscosity is the pitch drop experiment from the University of Queensland. It's claim to fame is that it is the world's longest continuously running laboratory experiment. It was started in 1927 and since then has seen 8 drops fall. Despite a video camera being installed after drop 7, no one has seen a drop fall at this machine - the camera malfunctioned at drop 8 (it's slightly younger sister in Ireland recently took a video of a drop falling. Check it out on youtube). One can estimate the viscosity of pitch from this somewhat bizarre experiment at 230 billion cp - a seemingly ridiculously large number. So it might come somewhat as a surprise when, after the viscosity of the quark-gluon plasma had been inferred from the experiments at the RHIC heavy-ion collider experiment at Brookhaven National lab came in at about 100,000 billion cp (3 orders of magnitude larger than that of pitch), the press release accompanying it announced the discovery of an "extremely low viscosity" that makes this plasma the "most nearly perfect fluid". What this was referring to was earlier holographic calculations by Policastro, Son and Starinets and Kovtun, Son and Starinets finding that the viscosity to entropy ratio takes a universal value (1/4 pi) in many holographic models and is larger than that in all fluids known to mankind. So this ratio (and not the viscosity itself) turned out to be the best quantity to measure how well a fluid flows (and to be really small in the quark gluon plasma, but gigantic in pitch). Not only did holography pin point the best observable to use, it also gave a ballpark figure. At this stage, it is almost impossible to go to a talk about heavy-ion physics without a plot that includes the "KSS" bound as a benchmark value. It clearly has influenced strongly how we think about viscosity. The one thing it however is not is, well, a bound. As the ratio becomes large at weak coupling and takes a universal in strongly coupled holographic models KSS thought it may well be a universal bound. We know this not to be true today. And the reason we do is once more holography. \(\NNN=2\) SYM based on an \(Sp(N)\) gauge theory with a hypermultiplet in the anti-symmetric tensor representation as well as 8 fundamental hypermultiplets has a holographic dual not of the type considered in KSS and violates the "bound", albeit only by a little. So the universal character of the KSS value still stands.
Higher order transport: Hydrodynamics is only an approximation valid for long wavelength fluctuations around equilibrium. For very short wavelength, viscous relativistic hydrodynamics becomes acausal. The same happens for example already with the diffusion equation. One time, two space derivatives in the equation imply a non-relativistic dispersion relation where frequency goes as wavenumber squared, so the speed of a wave (frequency over wavenumber) goes to infinity at large wavenumber. This is not a problem conceptually, as hydrodynamics is only supposed to be valid at small wavenumber (that is, at large wavelength). But when modeling hydro on a computer, one needs to tell the computer what to do even at short distances. The acausalities of leading order hydro translate into numerical instabilities. A simple way to fix those is to include higher order terms. Ideally the values of the higher order terms have no impact on the simulation - they are just introduced to stabilize the numerics. If second order terms in a derivative expansion turned out to give important contributions, one obviously should not stop there and the whole long wavelength approximation underlying hydrodynamics would be called into question. To confirm this is not happening, one often varies the higher order transport terms by a factor of 2 up and down to make sure that the simulation gives the same answers. Of course, varying them over orders of magnitude will eventually have to have effects. So all one needs for a stable running of the code is a rough estimate of what the higher order transport terms should be - an ideal question for holography. In some state of the art numerical simulations of hydrodynamics for heavy-ion collisions (which are needed to, say, extract the viscosity from actual data) holographic values for higher order transport coefficients are routinely used as a default.
Anomalous Hydro: Hydrodynamics is non-equilibrium physics (albeit near equilibrium). So understanding the transport coefficients involves real-time correlation functions - the sort of quantities that are challenging on the lattice. It has recently been understood that there is however a small subset of transport coefficients that, in contrast, can be precisely determined even in QCD as they are related to anomalies. These transport coefficients are typically referred to as the chiral magnetic and the chiral vortical effect (CME and CVE). In the presence of a left/right asymmetry in the plasma (which should happen every now and then due to fluctuations) a background magnetic field or a non-trivial eddy current in the plasma will drive an electric current (and hence charge separation). The strength of this effect is uniquely fixed by a simple triangle Feynman diagram and can be calculated in any quantum field theory - be it strongly coupled or not. Whether these phenomena have been or should be seen in heavy-ion collisions is a topic of active debate. The biggest unknown is that we do not understand how large a charge asymmetry one should expect from event-to-event fluctuations. But theoretically the field theory derivations of CME and CVE are extremely clean and insightful - we have learned something about transport in strongly coupled theories that is universally applicable. More recently, these two effects have even been claimed to be potentially relevant in condensed matter systems as well. The so called Weyl metals are conjectured materials that are governed by chiral fermions and hence anomalies. Consequently, they would exhibit the anomaly driven transport phenomena as well. These days one can tell this whole story without any reference to holography. Purely field theoretic arguments (postulating existence of a entropy current or, alternatively, postulating existence of an equilibrium configuration in non-trivial but static backgrounds) can give a complete derivation of both CME and CVE. But the fact that especially the CVE is intrinsically tied to an anomaly was first uncovered using holography. The CVE was found to be uniquely fixed in terms of a bulk Chern-Simons term - the holographic manifestation of anomalies. This came as a complete surprise, but it motivated researchers to look for a connection, which eventually was established. To me, this is really one of the prime examples of how holography should work. We learn something new and surprising about strongly coupled systems using the holographic toy models, which then motivates us to learn something new about the real world using more traditional techniques.
Energy Loss: one of the most important probes of what's going on inside the quark-gluon plasma are energetic probes. Sometimes, in the collision of two nuclei that created the quark-gluon plasma there will be isolated collisions of partons (quarks or gluons inside the incoming nuclei) that create a pair of particles with energy much higher than the typical thermal particle in the fireball. How these jets travel through the plasma and lose energy to it can provide a great probe of the plasma properties. One good quantity to calculate theoretically is the stopping distance as a function of energy. In all weakly coupled plasmas (including standard electro-magnetic plasmas) the stopping distance goes as the square root of the energy times a coefficient, the jet quenching parameter, that depends on the details of the theory. Early efforts in understanding jet energy loss at strong coupling have gone into trying to understand this jet quenching parameter. But what has become clear from the holographic toy models is that the change to strong coupling has much more severe consequences - not only the prefactor changes, but in fact the power law itself changes. Instead of E to the 1/2, we can get a completely different exponent like 1/3 or 1/4. How well this compares with data is something that isn't clear yet. The important point here however is that holography helped us to even understand what the diagnostic of strong coupling should be - a change in power law. Whether the jet interacts weakly or strongly with the plasma doesn't just depend on the plasma, it also depends on the energy of the jet. Asymptotic Freedom of QCD guarantees that at sufficiently high energies the coupling will become weak. The irony of this is that as far as seeing strong coupling in jet physics is concerned, the LHC is a worse experiment than it's predecessor. The LHC generates much higher jet energies and perturbatively inspired models seem to be doing a decent job of explaining the data - even though the final verdict on whether we've really seen the crossover into a weakly coupled regime is certainly still out. At RHIC energies there seem to be some clear indications however that the power laws have to be changed.
Far from equilibrium physics: "Strongly coupled far from equilibrium" for a long time has been synonymous for "impossible to calculate, ever". This has certainly changed with the advent of holography. Of course non-equilibrium physics is hard, even when one has a classical gravity dual. But it's possible. One is basically studying the time evolution of a dynamical bulk space-time where injection of energy leads to the formation of a black hole. This is a tricky numerical GR problem. But ever since my local colleagues, Paul Chesler and Larry Yaffe, have been able to demonstrate that this class of problems is substantially easier than traditional numerical GR problems like black-hole mergers, there has been a small industry of people studying non-equilibrium dynamics this way. The verdict is still out on how much we will learn from this. In heavy-ion collisions, perturbative estimates on how soon hydrodynamics should be an approximate description have been way off - the system behaves hydrodynamically after a much shorter time (less than 1 fermi/c) than anticipated. Can this happen in a strongly coupled plasma? Yes, it can. Holography tells us that. But the more interesting questions are what to do with the wealth of numerical data we know have available on strongly coupled non-equilibrium physics thanks to holography. What are the right questions to ask? How to organize the evolution? I would expect many interesting papers in this area in the next two years. The LHC also gave us a recent experimental surprise in this direction: apparently in some fraction of proton on lead collisions, where participants collide head on and produce lots of particles, the system seems to behave like a fluid just like in lead/lead collisions. Intuition from perturbative studies certainly would have told us that such a small system should not behave like a hydrodynamic system - interactions don't have enough time to make the particles talk to each other before they fly apart. How large does a strongly coupled system have to be before it can behave as a fluid? This sort of qualitative question is definitely within reach of holographic methods. Something else that will happen is that people will develop an understanding of how the points discussed earlier, in particular jet energy loss, will be affected by the strongly time dependent environment in a heavy-ion collision.
Let us move on to the set of holographic calculations that have been done with condensed matter applications in mind. In most condensed matter systems, we start with a very simple quantum mechanical Hamiltonian. A fixed number of non-relativistic particles interacting via Coulomb interactions. But it is this Coulombic electron/electron repulsion that makes condensed matter physics hard. It is in size comparable to the interaction of the electrons with the lattice of ions, so it can not be treated as a small perturbation. Every physics major should have seen the solution to Schrodinger's equation for the Hydrogen atom. This is one of the most important results in basic quantum mechanics. But already helium is too hard. If we ignore the electron/electron repulsion, it's just hydrogen all over and we simply "fill up" the energy levels. But it is the electron/electron interaction that ruins this.
Given that two electrons is already too difficult, one may wonder how one should ever hope to say something meaningful about \(10^23\) electrons in a solid. The basic principle of how people made headway on this question is what we in particle physicists call effective field theory. The idea is that the low energy excitations that govern the transport properties of the solid could be weakly coupled even if the underlying degrees of freedom are not. To pin down the properties of the material, one needs to guess the right low-energy degrees of freedom and then write down the most general action for them consistent with the symmetries of the underlying Hamiltonian (and state). So what could the low energy degrees of freedom be? One natural guess is that one obtains weakly interacting fermion "quasiparticles" with the charge of an electron, but with a finite lifetime and a different mass. These fermionic quasiparticles form a Fermi surface at finite density and weakly scatter off one another. This simple guess forms the basis of Landau's incredible successful Fermi liquid theory. One can even show that in this case the interactions become weaker and weaker as one goes to lower energies - the Fermi liquid is an attractive fixed point. If weakly coupled fermions are one answer, surely weakly coupled bosons could also happen. Sure enough, that is Landau's theory of phase transition, where the relevant low energy degree of freedoms are weakly coupled bosons, fine-tuned to low mass by the vincinity to the phase transition point. But we know experimentally that this can not be all. Materials exist, most prominently high Tc superconductors, that fit neither of the two paradigms. So what other possible low energy phases could be realized in a strongly coupled theory? This is once more an ideal playground for holography as one can address this question already at the level of toy models. Already several novel phases have been constructed, most prominently the "holographic non-Fermi liquids" which can be understood as a Fermi surface coupled to a conformal field theory. But there are many more exciting developments. If there is interest, I'd be happy to describe them in another post (or let Lubos pick another guest blogger to do so).
Let me summarize with my prediction where the field is going. As I hopefully made clear, holography is a powerful tool, but it also has its limitations. Everything is calculable about field theories with holographic dual. This is very unusual in physics and provides a great tool for developing concepts and testing basic ideas. But being all about toy models, it is insufficient to give quantitative descriptions of nature. So ideally we will need to take the lessons learned from the holographic toy models back to the real problem, developing new approaches that incorporate the lessons learned in holography. I think that holography will be a crucial tool that will be with us for quite some time to come, but it is one out of many. I always tell my students that they shouldn't expect to be as lucky as me and count on building a career mostly built on using holography. They need to understand "holography and ...". I anticipate that more and more researchers will appreciate the power of holography and will incorporate it as one of the available tools in their repertoire - but it is clearly not a cure-all that will solve all open problems in theoretical physics.