If this book ever makes it to a second edition, this is most likely to be the chapter that will have to be most thoroughly rewritten. Is there a need in condensed matter physics for a theory that goes beyond the paradigm which we sketched in rough outline in the previous chapter? If so, would the lessons of AdS/CMT which are found in the later chapters be of any relevance for this purpose?

At present the fog of war is still obscuring the battlefield. This war started some thirty years ago with the discovery of high-Tc superconductivity. Before this event, there was a sense that insofar as metals and superconductors are involved the fundamentals could be understood in terms of the “fifties paradigm” of the previous chapter. In the frenzy that followed the high-Tc discovery, experiments showed that strange things were happening. The reaction of the mainstream was to try to tamper with the established paradigm, to accommodate the anomalies. However, Philip W. Anderson, who was very influential back then, took the lead in insisting that new physics is at work in the copper oxide electron systems [86]. This in turn had a great impact on the research agenda. During the subsequent thirty years the field diversified to other materials, while the repertoire of experimental methods employed to study the electrons in solids greatly expanded. Literally millions of papers were written on the subject. But some of the most basic questions formulated in the late 1980s are still awaiting a definitive answer. It is just impossible to do justice to this large and confusing literature in the present context (see e.g. [87]). We will therefore present here a small selection of subjects, which is intended to form a minimal background for the holographist to communicate with the condensed matter community.

Back in the late 1980s the great puzzle was why the superconducting transition temperature could be as high as 150 K, given that the conventional phonon mechanism runs out of steam at 40 K or so. It was also realised early on that the electron systems in cuprates are characterised by unusually strong inter-electron repulsions. An aspect that is well understood in these systems is the microscopic physics.

Not so long ago, two large and quite old fields in physics, string theory and condensed matter physics, were more or less at the opposite ends of the physics building. During the 40 or so years of its history, string theory has developed into a high art of “mathematical machine building”, propelled forwards by the internal powers of mathematics as inspired by physics. Yet, it has suffered greatly for the shortcoming that its theoretical answers are always beyond the reach of experimental machinery. Modern condensed matter physics is in the opposite corner. It has been propelled forwards by continuously improving experiments, which have delivered one serendipitous discovery after another during the last few decades. However, its interpretational framework rests by and large on equations developed 40 years or so ago. There has been an increasing sense that it is these that fall short in trying to explain the strongly interacting quantum many-body systems as realised by electrons in high-Tc superconductors and other unconventional materials.

All this changed dramatically in 2007 when physicists started to feed condensed matter questions to the most powerful mathematical machine of string theory: the holographic duality in the title of the book, also known as the “anti-de Sitter/ conformal field theory” (AdS/CFT) correspondence. This book introduces the explosion of answers that has followed since then.

The first (Jan) and last (Koenraad) of this book's authors are from such opposite corners. As soon as the seminal work of Herzog, Kovtun, Sachdev and Son in 2007 showed that these two subjects have dealings with each other, Jan and Koenraad recognised the potential and met up, almost literally half-way up the stairs. As has been characteristic for the development at large, it took us remarkably little effort to get on speaking terms, despite our superficially very different backgrounds. Shrouded by differences in language, string theory and condensed matter had already been on a collision course for a while, meeting each other on the common ground of quantum criticality/conformal field theory.

In the previous chapters, we focussed mainly on the phenomenological bottom-up models. Here the gravitational bulk theory in four or five dimensions is phenomenologically put together in a similar way to Ginzburg–Landau theory. The actual Lagrangian of the boundary theory thereby remains completely unknown. In addition, it is not clear either whether the bulk theory is a well-defined and selfconsistent quantum gravity theory. The advantage of these bottom-up models is that the gravity theory is relatively simple and one is free to add new ingredients in order to realise different behaviours in the boundary theory. However, to make sure that the phenomena found are self-consistent and/or to understand the dynamics in terms of the dual field theory more fully, it is necessary to find an explicit system in string theory where both the field theory and the exact dual gravity theory are known. Instead of the bottom-up approach, this calls for a top-down approach that starts directly from string/M-theory. The canonical example is the seminal construction by Maldacena, and the generalisations which were subsequently discovered share the property that the action of the dual field theory can be directly identified, including its weakly coupled limit. Since string theory is thought to be a fully consistent quantum theory, this guarantees that any phenomenon described by a top-down theory is physical.

The disadvantage of the top-down approach is that it is technically much more involved. There are far more fields in the gravity theory, often including whole infinite Kaluza–Klein towers that represent the additional dimensions of string theory. In practice one therefore resorts to a consistent truncation of this full top-down theory. This reduces the number of fields, but in such a way that the solution is still guaranteed to be a solution of the full theory. There is an important caveat, deserving special emphasis: a stable solution in a consistent truncation may turn out to be unstable in the full theory where all the truncated field fluctuations are reinstated. Although it will be ignored henceforth, one should be aware of this potential source of trouble.

The material systems studied in condensed matter laboratories are formed from a finite density of conserved entities such as the number of electrons or (cold) atoms. This is a priori quite distinct from the vacuum states which have been discussed in the previous two chapters. These purely scaleless critical states can be mimicked in the laboratory, but this involves meticulous fine tuning to the critical point. The famous example is the cold-atom superfluid Bose Mott-insulator system [319]. As we discussed in chapter 2, according to the established wisdom of condensed matter physics the zero-temperature states of matter which are understood, are generically stable or “cohesive” states. These typically break symmetry spontaneously, with a vacuum that is a short-range entangled product state. In addition, the Fermi liquid and the incompressible topologically ordered states are at present understood as stable states that are “enriched” by the long-range entanglement in their ground states. Finally, by fine tuning of parameters one can encounter special unstable states associated with continuous quantum phase transitions, but these are understood within the limitations of “bosonic” field theory, which relies on the statistical physics (in Euclidean space-time) paradigm.

The AdS/CMT pursuit was kickstarted in the period 2008–2009, inspired by this finite-density perspective underlying condensed matter physics. The first shot was aimed at spontaneous symmetry breaking: the holographic superconductor.We shall discuss this at length in chapter 10. We will find out there that much of the physics of this iconic zero-temperature state of matter is impeccably reproduced by holography, with the results acting thereby as a powerful confidence builder. However, in the development that followed it became increasingly clear that holography insists on the existence of an entirely new class of “non-cohesive” finite-density states at zero temperature. These emergent quantum critical phases are best called “strange metals” [159, 320]. This quantum criticality is not tied to the conformal invariance and supersymmetry of the zero-density CFT inherent in the AdS/CFT name.

Already shortly after the discovery of the AdS/CFT correspondence it became clear that holography can deal remarkably easily with the finite-temperature physics of the boundary system. The key is that one can account for the thermal physics of the strongly interacting critical state in the boundary by adding a black hole in the deep interior in the AdS bulk. As we will explain in detail in the first section, the temperature of the boundary system turns out to be identical to the Hawking temperature of the black hole living in an AdS space-time. This involves an intriguing and non-trivial twist of the “classic” consideration explaining Hawking radiation. In Hawking's computation one is dealing with quantised fields living in the classical black-hole space-time, whereas in the AdS bulk everything is strictly classical and zero temperature. Instead, via a remarkably elegant construction it is easy to understand that the black-hole bulk geometry “projects” onto the boundary system a finite temperature that is coincident with the Hawking temperature one would find in a bulk with quantised fields.

Having identified the dictionary rule that finite temperature is encoded by the bulk black-hole geometry, it turns out that these black holes also encode in an impeccable way for all the thermodynamics principles governing thermal equilibrium physics. This direct map of the “rules of black holes” to the thermodynamics of a real physical system with microscopic degrees of freedom is why the AdS/CFT correspondence manifests the holographic principle explained in the preceding chapters. The most poignant aspect hereof, as we will also discuss in the first section, is the identification of the Bekenstein–Hawking black-hole entropy with the entropy of microscopic configurations of the boundary field theory.

However, holographic thermodynamics is a lot more powerful than these classic black-hole thermodynamics notions. In later chapters we will show that AdS black holes are very different from their more familiar featureless all-engorging flat-space cousins. AdS black holes are actually able to describe rather rich, reallife phase diagrams of the matter in the boundary. In section 6.2 we will highlight a historically important example of such a phase diagram: the “Hawking–Page” confinement–deconfinement phase transition in a finite volume as discovered early on by Witten [211].