This book is about a particular interface between condensed matter physics, gravitational physics and string and quantum field theory. The defining feature of this interface is that it is made possible by holographic duality, to be introduced briefly in the first chapter. This interface has been called ‘AdS/CMT’ – a pun on the name of the best understood version of holography, the AdS/CFT correspondence. Let us be clear from the outset, however, that neither AdS nor CFT are essential features of the framework that we will develop.
Holographic condensed matter physics encompasses a body of work with a wide range of motivations and objectives. Here are three different informal and zeroth order descriptions of the remit of this book:
- For the condensed matter physicist: AdS/CMT is the study of condensed matter systems without quasiparticles. In particular, AdS/CMT provides a class of models without an underlying quasiparticle description in which controlled computations can nonetheless be performed.
- For the relativist: AdS/CMT is the study of event horizons in spacetimes whose asymptopia acts as a confining box. Asymptotically AdS spacetime is the simplest such box. Of particular interest are charged horizons, as well as various flavors of novel ‘hairy’ black holes.
- For the string or field theorist: AdS/CMT is the study of certain properties of large N field theories and their corresponding holographically dual spacetimes. Of particular interest are dissipative properties and the phases of the theory as a function of temperature and chemical potentials.
Of course our reader should feel no need to place herself in a particular category! The point is that under the umbrella of ‘AdS/CMT’ one will find, for example, discussions of experimental features of non-quasiparticle transport, ways around no-hair theorems for black holes, and discussions of objects from string theory such as D-branes. We aim to address all these perspectives in the following. While we have tried to make our discussion intelligible to readers from different backgrounds, for basic introductory material we will refer the reader to the literature. We also note two recent books [32; 771] which cover related topics on applications of holography to condensed matter physics.
The book is broken down into the following chapters. In Chapter 1 we introduce the essential workings of the holographic correspondence. The presentation is grounded in a historical context that explains the fundamental challenge – physics without quasiparticles – that motivates the holographic approach to quantum matter, as well as the key ingredients that make holography tick. We outline Maldacena’s original string-theoretic argument for holographic duality in §1.4, but otherwise our discussion throughout the book does not require any familiarity with string theory (although there is some ‘stringy’ context to §1.8 and §7.1.1). This chapter shows how to relate observables on the two sides of the duality and explains how the emergent, gravitating, spatial dimension in the bulk is related to the energy scale of the boundary renormalization group flow.
Chapters 2 and 3 are concerned with strongly interacting quantum matter at a fixed commensurate density. Such systems can often be described by conformal field theories, or by more general scale-invariant theories, and provide the simplest instances of matter without quasiparticles. The holographic discussion is motivated by reviewing the varieties of quantum critical theories that arise in commensurate quantum matter (in Chapter 2) and the challenges in understanding their dynamics by conventional means (in Chapter 3). Chapter 2 describes the geometrization of scaling symmetry by holography, as well as the holographic manifestation of basic energy scales that break the symmetry: temperature and mass gaps. Chapter 3 develops the holographic description of dissipative dynamics, grounded in the irreversibility of black hole mechanics. The frequency-dependent conductivity σ(ω) is calculated in various models and the importance of quasinormal modes is stressed. These two chapters are closely connected and also develop basic tools and concepts used in later chapters.
Chapter 4 is concerned with compressible quantum matter, with a variable density. Such states are obtained by ‘doping’ the commensurate matter of Chapters 2 and 3 with a chemical potential. The variable charge density leads to a wealth of new and challenging phenomena. We start by reviewing the best-understood compressible phase, Fermi liquids, and go on to discuss the theoretical challenges of ‘non-Fermi liquids’. The holographic description of compressible phases leads directly to the study of charged horizons and especially extremal, zero temperature horizons. A key concept is the near-horizon geometry, which exhibits emergent scaling symmetry. We show how the different holographic backgrounds can be characterized by the momentum-space structure of low energy charged excitations. This is one aspect of an important question for this chapter: the extent to which notions such as a Fermi surface remain (or not) as organizing principles in non-quasiparticle systems. These results will feed into the discussion of transport in the following Chapter 5.
Chapter 5 explores transport in compressible quantum matter. This is substantially more subtle than the zero density transport of Chapter 3, yet accounts for much of the anomalous behavior of unconventional metals. The complication is that in a compressible regime, transport is intimately tied up with the fate of momentum relaxation. In order to properly understand the holographic results, we give an extensive and self-contained description of hydrodynamics and of the memory matrix formalism. When momentum relaxation is slow, these frameworks can be combined with the results on holographic spectral weight from Chapter 4 to very directly obtain results for thermoelectric transport, without solving any additional bulk equations. The chapter goes on to consider strong momentum relaxation through a combination of analytical and numerical methods. The chapter ends with an overview of the Sachdev-Ye-Kitaev model for AdS2 physics, a microscopic model with remarkable similarities to the previously discussed holographic results.
Chapter 6 develops the holographic description of symmetry broken phases. These ordered phases emerge from non-quasiparticle quantum critical regimes. We discuss the spontaneous breaking of both internal and spacetime symmetries, as well as the response of the ordered phases to bosonic and fermionic probes. This chapter is mostly independent of Chapter 5, and instead builds upon concepts from Chapters 2, 3 and 4.
Chapter 7 considers several topics that extend the material of previous chapters. The first 7.1 discusses ‘probe branes’. This is a string-theory-motivated way to introduce certain nonlinearities into the bulk action for the Maxwell field, with some remarkable consequences. The discussion here is mostly a further development of the themes from Chapters 3 and 4. The second §7.2 discusses the holographic realization of strongly disordered fixed points and their connection with the transport themes of Chapter 5. The final sections discuss aspects of far-from-equilibrium dynamics through holography, going beyond the linear response regime of previous chapters. Holography maps non-equilibrium quantum dynamics of strongly interacting systems to the classical nonlinear dynamics of gravity.
Finally, in Chapter 8 we end with some brief comments on the relevance of the topics covered in this book to experimental challenges in specific quantum materials. In previous chapters we have refrained from experimental discussion, framing the holographic results as a response to clearly defined theoretical limitations of conventional methods. We argue that through their role as solvable model systems without quasiparticles, holographic theories have led to different experimental perspectives on unconventional quantum matter than those suggested by a weakly interacting intuition.
We are greatly indebted to our collaborators and colleagues over many years, who have built the holographic edifice described in this review. We are grateful to Joe Polchinski for initially asking us to write a review on holography and condensed matter. We would like to thank Richard Davison, Blaise Goutéraux, Andreas Karch, Jorge Santos and Jan Zaanen for discussions relevant to the writing of this book. We also thank Chi-Fang Chen, Oliver DeWolfe, Nicodemo Magnoli, Philip Philips and Koenraad Schalm for helpful feedback on a draft. This research was supported by the NSF under Grant DMR-1360789 and the MURI grant W911NF-14-1-0003 from ARO. AL also acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4302. SAH is partially supported by a DOE Early Career Award. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. SS also acknowledges support from Cenovus Energy at Perimeter Institute.
Sean A. Hartnoll, Andrew Lucas and Subir Sachdev
Stanford, CA and Cambridge, MA, September 2017