20 results
9 - Holographic photoemission and the RN metal: the fermions as probes
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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Summary
The general nature of the holographic strange metals as discussed in the previous chapter took some time to be appreciated. Triggered by the studies dating from 2007 by Herzog, Kovtun, Sachdev and Son of quantum critical transport which we described in chapter 7, the AdS/CMT development as focussed on quantum matter started in 2008 with the discovery that the RN metal is quite susceptible to spontaneous symmetry breaking: this is the holographic superconductor [347, 348]. What thoroughly accelerated the interest in the condensed matter applications of AdS/CFT was the discovery in 2009 of the “MIT–Leiden fermions” [349, 350]. This will be the topic of this chapter. In hindsight, this was a highlight of the crossdisciplinary exchange between condensed matter physics and string theory. Driven by experimental progress during the last twenty years or so, angular-resolved photoemission (ARPES) (and in principle also its “unoccupied-states sibling” inverse photoemission) has acquired a very prominent status as a means to “observe” strongly interacting systems in solids. It has been further fortified by the quite recent development of scanning tunnelling spectroscopy (STS), which can be seen as the real-space partner of ARPES. Together these spectroscopies have produced a barrage of serendipitous surprises during the last 15 years, and now play a prominent role in the flourishing of the experimental study of the strongly correlated electron systems.
Both methods probe the single-fermion two-point function. For string theorists with their traditional focus on high-energy experimentation and cosmology, the importance of the single-fermion propagator as an observational tool is less obvious. Its unique powers come into play in finite-density systems. Compared with the collective “bosonic” current responses which were in the foreground in the previous two chapters, it yields complementary but yet quite different information regarding the vacuum structure of the interacting system. One advantage is of a rather pragmatic nature. In practice one can measure the bosonic responses often only at small momenta, where general “hydrodynamics” principles tend to equalise the outcomes – see the discussion of zero sound and optical conductivity of the RN metal in section 8.3. On the other hand, photoemission (and its siblings) accesses the full kinematical range associated with the electrons in solids: state-ofthe- art spectrometers probe length scales ranging from multiple nanometres to the sub-Ångström scale and energies ranging from sub-kelvin to electron volts.
Holographic Duality in Condensed Matter Physics
- Jan Zaanen, Yan Liu, Ya-Wen Sun, Koenraad Schalm
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A pioneering treatise presenting how the new mathematical techniques of holographic duality unify seemingly unrelated fields of physics. This innovative development morphs quantum field theory, general relativity and the renormalisation group into a single computational framework and this book is the first to bring together a wide range of research in this rapidly developing field. Set within the context of condensed matter physics and using boxes highlighting the specific techniques required, it examines the holographic description of thermal properties of matter, Fermi liquids and superconductors, and hitherto unknown forms of macroscopically entangled quantum matter in terms of general relativity, stars and black holes. Showing that holographic duality can succeed where classic mathematical approaches fail, this text provides a thorough overview of this major breakthrough at the heart of modern physics. The inclusion of extensive introductory material using non-technical language and online Mathematica notebooks ensures the appeal to students and researchers alike.
Frontmatter
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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2 - Condensed matter: the charted territory
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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Summary
This book does not pretend to be a comprehensive treatise of condensed matter physics. This chapter (and the next one) are just intended to lay down a common ground for physicists of various backgrounds relating to the condensed matter aspects of holography. The style will be descriptive, if not sketchy: there will be no boxes with detailed computations and so forth and instead we intend to guide the reader into the condensed matter literature. This chapter is on solid ground, dealing with those aspects of the physics that are at present quite well understood, while being at the same time of relevance to the holographic constructions in the later chapters. In the next chapter we will delve into the unknown where the challenges for holography reside. We do urge also the condensed matter expert to have a look, since the relations with the AdS/CMT results of chapters 6–13 often involve a rather modern understanding of the condensed matter canon, which goes beyond the standard textbooks.
We start with a subject that is regarded as advanced in condensed matter physics, but where the connections with holography are most obvious and unproblematic: the zero-density “bosonic” quantum field theories describing the physics of quantised order-parameter fields (section 2.1). In this long section we will first introduce the repertoire of such theories, emphasising the physical context where they arise. Mainly for synchronisation purposes we will then briefly discuss the popular (in condensed matter) Abelian–Higgs or “(particle–)vortex” duality in 2 + 1 dimensions. We perceive this as an instructive metaphor for the condensed matter physicist to appreciate the weak–strong duality property of the holographic duality. It should also be of interest for the high-energy physicist to learn about the context where such dualities are in the foreground in condensed matter. We then turn to a theme that is at the core of AdS/CMT: the strongly interacting conformal quantum field theories that arise as the effective description of the physics at continuous zero-temperature quantum phase transitions. The quantum critical state realised right at the phase transition is the theatre of the“un-particle physics” which is ruled by the principle of scale (or even conformal) invariance of the quantum dynamics which is also the grand motive underlying much of the holographic description of matter.
10 - Holographic superconductivity
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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Summary
The AdS/CMT pursuit aimed at addressing the physics of finite-density quantum matter started seriously in 2008 with the discovery of holographic superconductivity, as first suggested by Gubser [347] and subsequently implemented in an explicit minimal bottom-up construction by Hartnoll, Herzog and Horowitz [348, 367]. This triggered a large research effort in the string-theory community. The underlying physics of spontaneous symmetry breaking means that this aspect of AdS/CMT is now quite well understood theoretically, much more so than, for instance, the fermion physics of chapters 9 and 11.
Holographic superconductivity is quite an achievement, however, since it is also far more than the straightforward physics of symmetry breaking. From the condensed matter perspective it should be viewed as the first truly mathematical theory for the mechanism of superconductivity that goes beyond the Bardeen–Cooper– Schrieffer (BCS) theory. As we emphasised in chapter 2, the Cooper mechanism, the central wheel of the BCS theory, critically depends on the normal state being a Fermi liquid. We continued by arguing in chapter 3 that the BCS vacuum structure does not need a Fermi-liquid “mother”. Starting from the RVB wave function Ansatz, we illuminated the case in which the BCS vacuum structure should be viewed in full generality as a long-range entangled state formed from a charge-2e Bose condensate living together with a Z2 spin liquid that is responsible for the “spinon” Bogoliubov excitations.
Holography takes this a step further by demonstrating that a generalisation of the Cooper mechanism is at work in the holographic strange metals which were the focus of the previous chapters. As in the Fermi-liquid case, the fermion pair/order parameter channel is singled out as the source of the instability. The phenomenology is very similar, up to the point that one can construct holographic Josephson junctions. One can contemplate s-wave superconductors but also p- and d-wave pairs and so forth. Crucially, a gap opens up at Tc showing a BCS (mean-field) temperature evolution, while at low temperature one might find long-lived Bogoliubov fermions. The differences from BCS are that this gap opens in the incoherent “unparticle” excitations of the strange metal – the sharp Bogoliubov particle poles develop only deep in the superconducting state. Also the rules determining the transition temperature drastically change: a “high” Tc becomes easy to accomplish.
4 - Large-N field theories for holography and condensed matter
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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The profound puzzles posed by quantum critical metals with Planckian dissipation and long-range entanglement, as observed in cuprates and heavy-fermion systems, cry out for a novel point of view. Holography can provide this new perspective. This book will propose that its concrete manifestation in terms of the AdS/CFT correspondence gives qualitatively new insights into these puzzles. The reason is that holography has to be understood above all as a “weak–strong” duality between two different descriptions of the same physics. In this regard it is qualitatively similar to the Kramers–Wannier or Abelian–Higgs duality we reviewed in chapter 2, but it takes the notion to a new level: it relates quantum field theories to a dual description that includes the gravitational force. For an extremely strongly coupled field theory, the weakly coupled theory is now Einstein's theory of general relativity. Vice versa, a strongly interacting gravitational theory has an equivalent description as a weakly coupled quantum field theory.
General relativity inherently contains the notion of a dynamically fluctuating space-time. The remarkable way in which this emerges in holography is by incorporating the renormalisation-group structure of the quantum field theory into the dualisation. As we previewed in the introduction, the renormalisation-group scale becomes part of the geometrical edifice as an additional space dimension in the gravitational theory.
It is still baffling that a quantitative duality relation can exist between two theories in different space-time dimensions. This paradox is resolved, however, by the holographic principle of quantum gravity. This lesson from black-hole physics insists that gravitational systems are less dense in information than conventional quantum field theories in a flat non-dynamical space-time, to the degree that the former can be encoded in a “holographic screen” with one dimension less. The dynamics of this “screen” can be thought of as the dynamics of the dual field theory.
In this chapter we will first provide a brief account of the conceptual and historical background of the holographic principle and in particular its manifestation within string theory (section 4.1). This is where the origin of the AdS/CFT correspondence lies. Fortunately one does not need all this material to understand holographic duality practically. In the remainder of this chapter we approach holography from a constructive angle instead, as was first put together in the excellent review [5].
1 - Introduction
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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A tour guide of holographic matter
A quake is rumbling through the core of physics. Suddenly apparently unrelated areas appear to have a common ground, showing an eerie capacity to fertilise each other. In physics such occasions are invariably propelled by novel mathematical machinery and the present case is no exception. This new mathematical contraption is “holographic duality” (or “anti-de Sitter/conformal field theory correspondence”), which was originally discovered in string theory in the 1990s. Until recently its use was limited to the historic scope of string theory – particle physics and quantum gravity. At a breathtaking pace it has since rolled out over many of the subject areas of modern fundamental physics, even yielding new insights into old subjects such as the nineteenth-century theory of hydrodynamics.
Several books of this kind could be written, and are being written, highlighting how anti-de Sitter/conformal field theory (AdS/CFT) impacts on various fields in physics. This book will focus on a prominent area where the developments have been particularly stunning. This is the application to equilibrium condensed matter physics. This started in 2007, and in a matter of a few years condensed matter theory was rewritten in a different mathematical language. This language is the one that one would perhaps least expect: general relativity. On its own a rewriting of condensed matter might not sound like a great advance, no matter how unconventional the language. However, the correspondence makes it possible to explore regimes of quantum many-body physics that are completely inaccessible with conventional techniques. In particular we refer to non-Fermi-liquid states of matter formed in finite-density systems of strongly interacting fermion systems. The holographic mathematics here becomes particularly expressive, suggesting that a general principle of a new kind is at work. It appears that this principle relates to the physics of compressible quantum matter: the notion that the nature of this state of matter is governed by a macroscopic quantum entanglement involving allof its constituents. This discovery is not just remarkable on its own. “Holographic strange matter” also has tantalisingly suggestive resemblances to the mysterious phenomena observed experimentally in strongly interacting electron systems that have been realised in special materials such as the high-Tc superconductors. First seen around thirty years ago, these have defied any reasonable explanation despite countless attempts resting on the available mathematical techniques.
12 - Breaking translational invariance
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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Symmetry comes first in physics and we have seen its prominent role in holography. When one wants to compare the results and insights gained in holography up to this point with the physics of electron systems in solids, there is one crucial form of symmetry that does not match. The holographic systems described so far all live in the Galilean continuum while electron systems live in crystals. In crystals, the background lattice formed by the ions breaks the Galilean invariance of space.
This chapter is dedicated to the effects of translational symmetry breaking on holographic matter. At the time of writing, this issue is still being intensely pursued as a research subject. Although rapid progress is being made, it is far from completely settled. The reasons are of a “technical” nature. The breaking of translations in the boundary implies that this symmetry is also broken in the spatial directions in the bulk. Finding solutions of Einstein's equations when translational symmetry is broken is a very challenging exercise – in the absence of this symmetry the non-linear nature of general relativity comes out in force. However, this hard work pays off for the boundary physics: it adds quite a bit of condensed matter realism to the holographic computations, especially when it comes to transport properties.
As we will review in section 12.2, the optical conductivity of the holographic strange-metal/superconductor system in the presence of a lattice potential looks surprisingly similar to the experimental conductivities measured in cuprates, in fact much more so than any other theoretical result that has appeared in the 25-year history of the subject. In section 12.5 two equally credible holographic mechanisms that offer potential explanations for the famous linear-in-temperature resistivity of the cuprate strange metals as discussed in section 3.6.2 will be introduced. Although less is known to date regarding the effects of a lattice on the fermion spectral functions, some first results shed a quite surprising light on the effects of Umklapp on the strange-metal “un-particles” (section 12.3). The discovery of a new form of “algebraic” insulating state will be revealed in section 12.4. Under the influence of a unidirectional periodic potential, these un-particles form states that are in one direction insulating, while they stay metallic in the other directions.
14 - Outlook: holography and quantum matter
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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We are nearing the end of this journey through the landscape of holographic matter. It is a colourful place where there is much to be seen. Since the first direct application to condensed matter in 2007 [24], the pace of exploration of this landscape has been remarkable. The rich sceneries we have described in this book have all been discovered in the last seven years. But what does it all mean? In truth, this is still a mystery at present.
One aspect is crystal clear. A whole new side of Einstein gravity has been discovered, by focussing on what the correspondence has to tell us about matter. Inspired by common-sense condensed matter questions on the boundary, the holographists had to look for unusual solutions of the Einstein equations in the bulk. In this search, they discovered whole new categories of unexpected gravitational universes. The most prominent of these is the “hairy black hole” describing holographic superconductivity: it flagrantly violates the no-hair theorem. Much more followed, up to the very recent realisation that translational-symmetry breaking in the boundary goes hand in hand with massive gravity in the bulk. These discoveries may only be the tip of the iceberg. They nearly all concern equilibrium matter dual to stationary solutions of the gravity in the bulk. Currently, specifically in the context of AdS holography, there has been astonishing progress in numerical time-dependent solutions in gravity. This also brings the non-equilibrium physics in the boundary into reach. The spectacular results for the duals of classical and superfluid turbulence in section 10.2.1 which are deduced from black holes with fractal horizons demonstrate the unexpected richness that exploration in this direction may reveal. AdS/CMT has, at the very least, given us tremendous new insights into the physics of black holes. Clearly, deeply buried in the equations of general relativity, there is still an enormous potential wealth for holography to uncover.
The UHOs: the unidentified holographic objects
In spite of its motivational role, the mystery is the condensed matter side of AdS/CMT.
Contents
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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11 - Holographic Fermi liquids: the stable Fermi liquid and the electron star as holographic dual
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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Summary
In chapter 9 we introduced the holographic description of single-fermion propagators in the finite-density Reissner–Nordström metals. We discussed in particular how sharp Fermi surfaces can arise due to the approximate confinement of fermionic excitations in the potential well created by the geometrical domain wall. The interaction of these fermionic quasi-bound states with the strongly coupled AdS2 IR gives these a finite lifetime. Eventually it decays into the quantum critical horizon, and this imbues the state with its non-Fermi-liquid properties. For particular parameter choices these fermionic responses can closely resemble the “marginal-Fermi-liquid” spectral functions suggested by photoemission experiments in the cuprate strange metals. However, these computations involved an approximation in the form of the probe limit, which assumed that the bulk fermions do not influence the bulk physics.
But the reader is now familiar with holographic superconductivity, a context where the limitations of the probe limit become very explicit: the violation of the BF bound by the fluctuations of a scalar field in the bulk signals an instability of the vacuum, and one has to recompute the response after the backreaction of the scalar field has been fully accounted for. Ignoring this amounts to computing the physics of an unstable, false vacuum. We emphasised in chapter 9 that a similar issue arises in the parameter regime of the fermion scaling dimension and charge where the holographic Fermi surface is formed. Invariably the fermion propagators show in this regime a log-oscillatory behaviour at small momenta, which is caused by a BF-bound violation of the bulk fermions. The physics in the bulk behind this log-oscillatory behaviour is Schwinger pair production in the background of the charged black hole. Although the physics is less straightforward than for the bosonic fields, it does indicate that the system is unstable – the Fermi surfaces of chapter 9 are properties of a false vacuum. To stabilise the bulk it has to be that the fermionic states in the bulk get occupied. In the true ground state the bulk fermions have to have a macroscopic effect themselves, and the only way this can be accomplished is by forming finite-density fermionic matter. The bulk fermions are non-interacting to leading order in 1/N, but they are still subject to Fermi–Dirac statistics. Thus a finite-density Fermi gas has to form in the bulk with a charge and energy density that will modify the gauge fields and the geometry.
7 - Holographic hydrodynamics
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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3 - Condensed matter: the challenges
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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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.
Preface
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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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.
13 - AdS/CMT from the top down
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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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.
5 - The AdS/CFT correspondence as computational device: the dictionary
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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References
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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8 - Finite density: the Reissner–Nordström black hole and strange metals
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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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.
Index
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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6 - Finite-temperature magic: black holes and holographic thermodynamics
- Jan Zaanen, Universiteit Leiden, Yan Liu, Universidad Autónoma de Madrid, Ya-Wen Sun, Universidad Autónoma de Madrid, Koenraad Schalm, Universiteit Leiden
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- Book:
- Holographic Duality in Condensed Matter Physics
- Published online:
- 05 November 2015
- Print publication:
- 05 November 2015, pp 176-221
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- Chapter
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Summary
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].