This paper now has almost one thousand citations. What changed, and why is this result now regarded as so interesting? To understand this, we return to what was calculated: a measure of the viscosity of something like the experimental quark-gluon plasma in theories that are, while not identical, related to the actual strong force. The calculation was also at large values of the coupling, in the precise region where it is extremely hard to calculate in the actual strong force. For reasons that are too technical for this book, this problem was also not amenable to the put it on a computer and press enter’ approach. While there are aspects of the strong force at strong coupling that can be solved through cracking the whip and slave-driving a harnessed team of CPUs, this is not one of them.
While brute force could not be used, there had existed analytical approaches to computing this viscosity, which were based on using techniques applicable at weak coupling and extrapolating to strong coupling. Both the AdS/CFT approach and the weakly-coupled analytical approaches gave estimates for the shear viscosity of the quark-gluon plasma. These estimates were then tested by measurements at actual experiments such as the Relativistic Heavy Ion Collider at Brookhaven National Laboratory. The surprise was that the measurement of the viscosity disagreed badly with that computed in the traditional’ approach – but agreed approximately with that computed using AdS/CFT. The viscosity of the quark-gluon plasma was surprisingly lower than expectations – and the only approach that seemed to get it right was that of AdS/CFT.
It is essentially this fact that drew so much attention to this work and so many physicists to this area. Physicists like to calculate. It is what they do. AdS/CFT offered an approximate model of the quark-gluon plasma that gave approximate agreement with various aspects of experimental data. The AdS/CFT model was amenable to calculation, and also to extension through the generation of further accessible variants. It offered a whole new set of insights into how to think about the strong force at finite temperature. As a bonus, it was connected to ideas about quantum gravity, but this was inessential and one could also take a purely pragmatic view. It was relevant to the quark-gluon plasma, and even if all one cared about was the quark-gluon plasma, that was enough.
Certainly it was not a perfect or precise approximation, but it also seemed better than the otherwise available options. There are many things wrong’. The theory the calculations were done in is not the actual strong force. The number of particles in this theory is far greater than in the actual strong force. There are additional types of particles that are not present for the real strong force: for example, scalar particles. The interactions are different. The symmetries are different. It is a model – but a model that has some agreement with the data.
However, this is precisely the modus operandi of physics: start with simplified and calculable models, and then expand through progressively adding refinements that make the models less and less simplified. This is how physics works. There is now a significant community working on using AdS/CFT as an approximate way to understand the strongly coupled quark/gluon plasma at high temperature. They use it both as a computational tool and for conceptual insight. Is it ‘right’? It is useful, and that is what matters.
8.4 APPLICATIONS TO CONDENSED MATTER PHYSICS
The quark-gluon plasma belongs to particle physics, and quantum field theory is certainly a major part of particle physics. Quantum field theory has however spread its affections widely, and it is a subject with many lovers.
In particular, it is also pervasive throughout condensed matter physics. This is the branch of physics that describes phases of matter, both common and uncommon, that consist of lots and lots and lots of atoms. These include old favourites from school such as solids, liquids and gases, but also more exotic examples such as superconductors and topological insulators. These different phases have radically different properties and can behave in bizarre ways. What they have in common is that their meaningful existence requires many atoms to be present.
Confronted with a lone atom, it makes as much sense to regard it as a liquid atom or a solid atom as it does to view it as a rat atom or an elephant atom. By itself, it is just an atom. When many atoms are put together though, the behaviour of the collective exhibits distinctive properties. This is directly analogous to the movements of a crowd at a music festival or the flight of a murmuration of starlings. For this reason, condensed matter physics is also often called many-body physics. It is the science of how macroscopic properties of matter arise from the collective microphysics of large numbers of atoms.13
The number of bodies – atoms or molecules – involved in typical macroscopic systems is not just large. It is enormous. A glass of water contains enough atoms to put ten thousand of them on every square millimetre of the earth’s surface. If we tried to write down the equations for each atom, the sun would be dead and cold before we were one millionth of the way through. The differences between solids and liquids, real as they are, cannot be found by direct assault on the equations of each individual atom.
This is an old problem, with a history that predates the advent of quantum mechanics. The motion of an ideal gas – separate individual atoms interacting with the rigid formality of billiard balls – was understood in Victorian times. This was a classical problem, but in many cases it is now necessary to consider the simultaneous effects of both quantum mechanics and the presence of many bodies. For close-packed atoms in a solid, with an average separation of somewhere around a nanometre, the distances are small enough that quantum mechanics cannot be ignored.
The study of quantum many-body systems has been carried out by many fabulously smart people over many decades. I will not and could not review it in any detail. Much of it is too tangential to this book, and my knowledge of the history is only of the mythological kind produced by textbooks and the names of Nobel Prize winners. I move straight to two important and relevant conclusions: first, quantum mechanics is essential, and second, quantum field theory is essential. The quantum field theories in this case are not the ‘fundamental’ quantum field theories that the Standard Model purports to be: these are practical, emergent theories that arise in the description of the physics of the many many atoms present.14 In the Standard Model, displacements of the electric and magnetic fields from zero turn, through quantum mechanics, into particles of light: photons, the quanta of electromagnetism. In condensed matter physics, the displacements of atoms from their equilibrium positions in an atomic lattice also turn, through quantum mechanics, into particles: phonons.
Whether this physics is fundamental’ or emergent’, it is again a requirement to grok quantum field theory that sits athwart the road to understanding. This fact underlies a long history of intellectual interchange between particle physics and condensed matter physics, starting as the centrality of quantum field theory to both subjects arose in the 1960s.15 One of the key questions of that era was how quantum field theory behaved across different scales. As discussed in chapter 3, the conceptual framework for this was provided by the Nobel Prize winner Kenneth Wilson, who straddled both particle physics and condensed matter physics and solved important problems in both subjects. Wilson was a deep thinker who published rarely. In his own words – and university administrators are here warned of offensive material –
[Cornell] gave me tenure after only two years and with no publication record. In fact, there was one or two papers on the publications list when I was taken for tenure and Francis Low complained that I should have made sure there was none. Just to prove that it was possible.
In terms of realising quantum field theories, condensed matter physics has clear advantages over particle physics. Particle physics is stuck with one universe, and it is not simple to make new ones. There is only one quantum field theory whose behaviour we get to see experimentally – the Standard Model. Now, this is less restrictive than it sounds, because the behaviour of the Standard Model changes depending on the energies of the collisions used to test it and the particles that are collided
. Despite being just a single quantum field theory, it can be both strongly coupled and weakly coupled; it can involve both relativistic and non-relativistic physics; it can involve both fundamental particles and emergent composites made from more fundamental constituents.
Nonetheless, the Standard Model is still just one quantum field theory. In contrast, condensed matter physicists can ‘make’ new quantum field theories by considering different materials, with different impurities, at different temperatures and pressures. Rather than ordering a new universe, they merely need to order materials from the suppliers’ catalogue. While this does not allow for an infinity of choices, it does provide an enormous permutation of options.
Condensed matter physics then involves a plenitude of quantum field theories: theories at weak coupling, theories at strong coupling, theories with fermions, theories with bosons, theories with two dimensions, theories with three dimensions, theories with non-relativistic particles and theories with (effectively) relativistic particles. Just as for particle physics, some of these are easier to understand than others. Theories with weakly coupled interactions are preferable to theories with strong interactions. It is both simpler to understand the physics, and quicker to perform the calculations.
Weakly coupled quantum field theories underlie some of the biggest successes of condensed matter physics. For example, most examples of superconductivity can be described in this language. Superconductivity is the phenomenon by which a material loses all resistance to the passage of electrical current. In normal’ superconductivity, a material with few impurities is gradually cooled down. The temperature is reduced towards absolute zero. Once it gets below a certain critical temperature, it undergoes a sudden transition whereby its electrical resistivity vanishes. Current can flow freely without dissipating any heat.
Within a superconductor, electrical current can flow continuously for thousands of years without dissipating energy. Wow – what an amazing world, that has such features in it! Why does this happen?
The cleanest route to the physics of superconductivity is through quantum field theory. A quantum field theory can be written down that describes the behaviour of electrons and atoms in a material with the ability to superconduct. By studying the interactions between electrons and their resulting tendency to pair up, it is possible to see why and when superconductivity occurs. The field theory also allows a quantitative demonstration of the unusual properties of superconductors such as the absence of any internal magnetism and the vanishing of electrical resistance.
Normal superconductors were discovered in 1911 by the wonderfully named Dutch physicist Heike Kamerlingh Onnes. The transition temperature below which superconductivity occurs varies, but is not higher than around thirty degrees above absolute zero. For a long time, this also remained the experimental situation. Following the theoretical explanation of superconductivity in the 1960s, the study of superconductivity appeared to be entering a slow and gentle decline.
However, this all changed utterly in 1986. Out of the blue and inside the reagents cupboard, new superconductors were suddenly discovered with transition temperatures up to three times higher than those previously known – the so-called ‘high temperature superconductors’. This electrified the subject, and the strength of the shock is illustrated by the fact that the Nobel Prize for this discovery was awarded only one year later, in 1987.
It is still not known what is the underlying theoretical cause of high temperature conductivity. This problem remains one of the biggest open problems in condensed matter physics. However, whatever the answer is, it is not a simple weakly coupled field theory. Perhaps, the answer requires a quantum field theory that is intrinsically strongly coupled.
There are also other systems in condensed matter physics that involve strong coupling: for example those involving strongly correlated electrons or heavy fermions. In a heavy fermion material, electrons interact so strongly with the magnetic fields of the atoms that they behave as if they were one hundred times heavier than they actually are. The magnetic fields of the material restrict the movement of electrons, increasing the ‘effective’ mass of the electron. Heavy fermion systems can also lead to superconductivity – but again this is a form of superconductivity that is very different from the standard’ type.
It is relatively easy to ‘make’ strongly coupled quantum field theories in condensed matter physics. As we have seen, one of the biggest attractions of the AdS/CFT correspondence is the way it provides a different approach to thinking about strongly coupled quantum field theories, and a different approach to calculating with them. One of the main directions of AdS/CFT research over the last seven years has been to think about applications to condensed matter physics, starting with a 2008 paper by Sean Hartnoll, Christopher Herzog and Gary Horowitz, from a mixture of Santa Barbara and Princeton, explaining how to describe superconductors within the AdS/CFT correspondence. I had been aware of Sean as an undergraduate and graduate student at Cambridge, where he was two years ahead of me, and his untidy long brown hair concealed the smartest brain I was aware of.
Since then, close to a thousand papers have been written on and around this topic, and it represents one of the most active areas of research within this field. There are both reasons both for caution and for optimism.
What are the reasons for caution? Most obviously, the field theories for which calculations can be done using AdS/CFT are not the field theories that actually apply in condensed matter physics. The AdS/CFT theories are normally extensions of the theories that arise in the Standard Model, only with more particles and more symmetry. They are relativistic, and different in many ways from the quantum field theories that arise in condensed matter physics. While something similar also held for the case of the quark-gluon plasma, in this case the differences are starker. For applications to condensed matter physics, it is possible that the only aspect of the AdS/CFT theories in common with the real condensed matter theories is the presence of strong coupling.
At first sight, this appears a total disaster for this research program – calculations can only be done for theories very loosely connected to those generating the actual experimental data. In the end, it may indeed turn out to be a fatal problem.
However, there are also reasons to be optimistic. For any problem, good intuition is extremely important. It is far easier to intuit how a system will behave if you can view it the right way and from the right perspective: once you realise an object has wheels, it is unsurprising when it starts to roll. Most intuition about quantum field theory has been built up through studying examples of weakly coupled theories. While this is fine for studying weakly coupled theories, it may not offer any assistance for strongly coupled examples – and may even be positively misleading.
However, precisely what the AdS/CFT correspondence does offer are calculational methods that apply for large numbers of strongly coupled quantum field theories. These examples allow a development of intuition for the behaviour of such theories, both in terms of the results for the field theory and also by looking at it as a gravitational theory. Even if the calculations are done with the wrong’ theories, it is a reasonable hope that the wrong theories can still provide insights into the characteristic behaviour of strongly coupled field theories, and that this characteristic behaviour will also carry over to the real theories that describe real materials. If you have always lived on the plains of the Midwest and want to go climbing in the Himalayas, a detailed map and guide of the Alps would still be useful. None of the peaks are exactly the same, but it would show you something of what mountains are like.
The use of AdS/CFT can also offer a guide as to which properties of a quantum field theory are generic, and which only occur within a weakly coupled theory. There are some results which are universal for all quantum field theories, and some which are only a feature of weak coupling. The existence of computationally tractable examples of strongly coupled theories is then enormously helpful, as calculation can be used to show which features
survive the transition from strong to weak coupling. Here the benefits of duality enter crucially – it relates theories in which calculations are easy to theories in which calculations are hard. It is only the presence of a dual weakly coupled gravitational theory that allows the relatively rapid performance of calculations in the strongly coupled field theory.
Another danger that does exist comes from an inability to make definite predictions. The disconnect between the theories where one can calculate and the actual real theories describing any given material can lead to a loosening of standards. For this research program, it is both an advantage and disadvantage of condensed matter physics that there are enormous numbers of materials out there. An advantage is that there are lots of targets to study. The downside is that if a calculation in the dual theory agrees with some material, it can be taken as a success – but if it disagrees then the mismatch can be explained away as a result of necessary approximations. However there is no systematic way of analysing how accurate the AdS/CFT models are expected to be, and no way of producing an error bar on the theory calculations compared to real experiments on real materials. In this respect, some of the claims made within this area have involved a degree of hyperbole.
The ideal dream within this line of research would be to produce a genuine dual description of a real material. One could then predict the behaviour of a laboratory material using gravitational calculations. While dreams are often unattainable, a more practical motivation is to understand the many different types of behaviour that strongly coupled field theory can lead to. String theory is attractive here as it provides new approaches to think about strongly coupled systems.
Why String Theory? Page 20