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arXiv:1202.6008 · Fortschritte der Physik

Holography and condensed matter

Francesco Benini

Published Feb 29, 2012• 12 citations
arXivHTMLPDF

Abstract

Abstract This is a review of recent developments in the application of AdS/CFT methods to some condensed matter problems. In particular we present the holographic description of a local quantum critical state, related to (non)‐Fermi liquids and the strange metal, that appears in large N CFTs with a gravity dual at finite density and zero temperature, and explore its properties by probing it with fermionic operators. Further we discuss possible bosonic and fermionic instabilities, leading to s‐, p‐ or d‐wave holographic superconductors and “electron stars”. Finally we present a realization of local quantum criticality with an impurity problem.

DOI
10.1002/prop.201200041

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Over the past decade, ultracold gases have been studied extensively, both theoretically and experimentally. This is due to the discovery that the relevant measure for the interatomic interactions in such systems can be controlled experimentally by simply applying a magnetic field. Consequently, these gases can be conveniently studied in both the weakly and the strongly coupled regime. In particular, it is possible to tune the magnetic field such that the interaction strength between the atoms diverges. This is called the unitarity limit. Interestingly, at unitarity the system becomes almost scale invariant, which means that the only scales left in the problem are set by the temperature and the density of the gas. The unitary gas then exhibits universal behavior. It therefore constitutes a perfect candidate for the application of the AdS/CFT duality, also known as holography, to a nonrelativistic condensed-matter system. According to this duality, a strongly coupled, scale-invariant system can be described using a dual gravitational model with one additional spatial dimension. In this thesis we apply the AdS/CFT duality to construct holographic models for ultracold gases at unitarity. To this end, we start by studying properties of a well-known holographic model of a condensed-matter system, namely the holographic superconductor. We then show how we can use a very similar model to describe elementary fermions with a nonzero mass in holography. Subsequently, we study these holographic models in the nonrelativistic limit. We find that the resulting models exhibit thermodynamic properties that are qualitatively similar to those of a unitary Fermi gas. Lastly, we take some first steps towards holographically modelling Bose gases at unitarity. In particular, we show that we can describe massive elementary bosons using AdS/CFT.

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Introduction

Jan Zaanen, Yan Liu, Ya-Wen Sun +1 more

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-T c superconductors. First seen around thirty years ago, these have defied any reasonable explanation despite countless attempts resting on the available mathematical techniques.

Cambridge University Press eBooks•Oct 31, 2015•0 citations
Abstract This is a review of recent developments in the application of AdS/CFT methods to some condensed matter problems. In particular we present the holographic description of a local quantum critical state, related to (non)‐Fermi liquids and the strange metal, that appears in large N CFTs with a gravity dual at finite density and zero temperature, and explore its properties by probing it with fermionic operators. Further we discuss possible bosonic and fermionic instabilities, leading to s‐, p‐ or d‐wave holographic superconductors and “electron stars”. Finally we present a realization of local quantum criticality with an impurity problem.
I begin with a review of quantum impurity models in condensed matter physics, in which a localized spin degree of freedom is coupled to an interacting conformal field theory in \\(d\\) = 2 spatial dimensions. Their properties are similar to those of supersymmetric generalizations which can be solved by the AdS/CFT correspondence; the low energy limit of the latter models is described by a \\(AdS_2\\) geometry. Then I turn to Kondo lattice models, which can be described by a mean- field theory obtained by a mapping to a quantum impurity coupled to a self-consistent environment. Such a theory yields a 'fractionalized Fermi liquid' phase of conduction electrons coupled to a critical spin liquid state, and is an attractive mean-field theory of strange metals. The recent holographic description of strange metals with a \\(AdS_2 \\times R^2\\) geometry is argued to be related to such mean-field solutions of Kondo lattice models.
I begin with a review of quantum impurity models in condensed matter physics, in which a localized spin degree of freedom is coupled to an interacting conformal field theory in d = 2 spatial dimensions. Their properties are similar to those of supersymmetric generalizations which can be solved by the AdS/CFT correspondence; the low energy limit of the latter models is described by an AdS 2 geometry. Then I turn to Kondo lattice models, which can be described by a mean-field theory obtained by a mapping to a quantum impurity coupled to a self-consistent environment. Such a theory yields a 'fractionalized Fermi liquid' phase of conduction electrons coupled to a critical spin liquid state, and is an attractive mean-field theory of strange metals. The recent holographic description of strange metals with an AdS 2 × R 2 geometry is argued to be related to such mean-field solutions of Kondo lattice models.
I review two classes of strong coupling problems in condensed matter physics, and describe insights gained by application of the AdS/CFT correspondence. The first class concerns non-zero temperature dynamics and transport in the vicinity of quantum critical points described by relativistic field theories. I describe how relativistic structures arise in models of physical interest, present results for their quantum critical crossover functions and magneto-thermoelectric hydrodynamics. The second class concerns symmetry breaking transitions of two-dimensional systems in the presence of gapless electronic excitations at isolated points or along lines (i.e. Fermi surfaces) in the Brillouin zone. I describe the scaling structure of a recent theory of the Ising-nematic transition in metals, and discuss its possible connection to theories of Fermi surfaces obtained from simple AdS duals.
Over the past decade, ultracold gases have been studied extensively, both theoretically and experimentally. This is due to the discovery that the relevant measure for the interatomic interactions in such systems can be controlled experimentally by simply applying a magnetic field. Consequently, these gases can be conveniently studied in both the weakly and the strongly coupled regime. In particular, it is possible to tune the magnetic field such that the interaction strength between the atoms diverges. This is called the unitarity limit. Interestingly, at unitarity the system becomes almost scale invariant, which means that the only scales left in the problem are set by the temperature and the density of the gas. The unitary gas then exhibits universal behavior. It therefore constitutes a perfect candidate for the application of the AdS/CFT duality, also known as holography, to a nonrelativistic condensed-matter system. According to this duality, a strongly coupled, scale-invariant system can be described using a dual gravitational model with one additional spatial dimension. In this thesis we apply the AdS/CFT duality to construct holographic models for ultracold gases at unitarity. To this end, we start by studying properties of a well-known holographic model of a condensed-matter system, namely the holographic superconductor. We then show how we can use a very similar model to describe elementary fermions with a nonzero mass in holography. Subsequently, we study these holographic models in the nonrelativistic limit. We find that the resulting models exhibit thermodynamic properties that are qualitatively similar to those of a unitary Fermi gas. Lastly, we take some first steps towards holographically modelling Bose gases at unitarity. In particular, we show that we can describe massive elementary bosons using AdS/CFT.
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-T c superconductors. First seen around thirty years ago, these have defied any reasonable explanation despite countless attempts resting on the available mathematical techniques.