Lectures on holographic methods for condensed …

1 [hep-th] Lectures on holographic methods forcondensed matter physicsSean A. HartnollJefferson Physical Laboratory, Harvard University,Cambridge, MA 02138, notes are loosely based on Lectures given at the CERN Winter School onSupergravity, Strings and Gauge theories, February 2009 and at the IPM String Schoolin Tehran, April 2009. I have focused on a few concrete topics and also on addressingquestions that have arisen repeatedly. Background condensed matter physics materialis included as motivation and easy reference for the high energy physics discussion of holographic techniques progresses from equilibrium, to transport andto [hep-th] 17 Jan 2010 Contents1 Why holographic methods for condensed matter ? Why condensed matter ? .. Quantum criticality .. : The Wilson-Fisher fixed point .. : Spinons and emergent photons .. to nonconventional superconductors ..102 Applied AdS/CFT Geometries for scale invariant theories.

2 : So what iszin the real world? .. Finite temperature at equilibrium .. Finite chemical potential and magnetic field at equilibrium .. Relevant operators .. Expectation values .. Dissipative dynamics close to equilibrium .. Example: How to compute electrical and thermal conductivities .. Comparison to experiments in graphene ..443 The physics of spectral Relation to two point functions and symmetry properties .. Causality and vacuum stability .. Spectral density and positivity of dissipation .. Quantum critical dynamics with particle-hole symmetry .. Quantum critical transport with a net charge and magnetic field .. A simple treatment of impurities ..574 holographic What is a superconductor? .. Minimal ingredients for a holographic superconductor .. Superconducting phase .. fields ..715 Potential and limitations of the holographic approach7311 Why holographic methods for condensed matter ?

3 Why condensed matter ?Why, on the eve of the LHC, should high energy and gravitational theorists be thinkingabout phenomena that occur at energy scales many orders of magnitude below their usualbandwidth? Three types of answer come to , the AdS/CFT correspondence [1] is a unique approach to strongly coupled fieldtheories in which certain questions become computationally tractable and conceptually moretransparent. In condensed matter physics there are many strongly coupled systems that canbe engineered and studied in detail in laboratories. Some of these systems are of significanttechnological interest. Observations in materials involving strongly correlated electronsare challenging traditional condensed matter paradigms that were based around weaklyinteracting quasiparticles and the theory of symmetry breaking [2]. It seems reasonable tohope, therefore, that the AdS/CFT correspondence may be able to offer insight into someof these nonconventional , condensed matter systems may offer an arena in which many of the fascinat-ing concepts of high energy theory can be experimentally realised.

4 The standard modelLagrangian and its presumptive completion are unique in our universe. There will or willnot be supersymmetry. There will or will not be a conformal sector. And so on. In con-densed matter physics there are many effective Hamiltonians. Furthermore, an increasingnumber of Hamiltonians may be engineered using, for instance, optical lattices [3]. As wellas novel realisations of theoretical ideas, ultimately one might hope to engineer an emergentfield theory with a known AdS dual, thus leading to experimental AdS/CFT (and reversingthe usual relationship between string theory and the standard model).Thirdly, and more philosophically, the AdS/CFT correspondence allows a somewhatrearranged view of nature in which the traditional classification of fields of physics byenergy scale is less important. If a quantum gravity theory can be dual to a theory withmany features in common with quantum critical electrons, the question of which is more fundamental is not a meaningful question.

5 Instead, the emphasis is on concepts that havemeaning on both sides of the duality. This view has practical consequences. For instance,seeking a dual description of superconductivity one realises that there might be loopholesin black hole no-hair theorems and one is led to new types of black hole Lectures will be about the first type of answer. We shall explore the extent towhich the AdS/CFT correspondence can model condensed matter Quantum criticalityAlthough quantum critical systems are certainly not the only condensed matter systems towhich holographic techniques might usefully be applied, they are a promising and naturalplace to start. Quantum critical points have a spacetime scale invariance that providesa strong kinematic connection to the simplest versions of the AdS/CFT , a lack of weakly coupled quasiparticles often makes quantum critical theoriesdifficult to study using traditional methods . Outside of AdS/CFT there are no modelsof strongly coupled quantum criticality in 2+1 dimensions in which analytic results forprocesses such as transport can be critical theories arise at continuous phase transitions at zero temperature.

6 Azero temperature phase transition is a nonanalyticity in the ground state of an (infinite)system as a function of some parameter such as pressure of applied magnetic field. Thequantum critical point may or may not be the zero temperature limit of a finite temperaturephase transition. Note in particular that the Coleman-Mermin-Wagner-Hohenberg theorem[4] prevents spontaneous breaking of a continuous symmetry in 2+1 dimensions at finitetemperature, but allows a zero temperature phase transition. In such cases the quantumphase transition becomes a crossover at finite , as the continuous quantum critical point is approached, the energy of fluctu-ations about the ground state (the mass gap ) vanishes and the coherence length (or othercharacteristic lengthscale) diverges with specific scaling properties. In a generic nonrela-tivistic theory, these two scalings (energy and distance) need not be inversely related, as wewill discuss in detail below. The quantum critical theory itself is scale critical points can dominate regions of the phase diagram away from the pointat which the energy gap vanishes.

7 For instance, in regions where the deformation away fromcriticality due to an energy scale is less important than the deformation due to a finitetemperatureT, < T, then the system should be described by the finite temperaturequantum critical theory. This observation leads to the counterintuitive fact that the imprintof the zero temperature critical point grows as temperature is increased. This phenomenonis illustrated in figure get a feel for quantum critical physics and its relevance, we now discuss several1An example of a solvable 1+1 dimensional model with a quantum critical point is the Ising model in atransverse magnetic field, see chapter 4 of [5].2In certain 2+1 dimensional systems the quantum critical point can connect onto a Berezinksy-Kosterlitz-Thouless transition at finite temperature. Also, strictly infiniteNevades the theorem as fluctuations 1: Typical temperature and coupling phase diagram near a quantum critical two low temperature phases are separated by a region described by a scale-invarianttheory at finite temperature.

8 The solid line denotes a possible Kosterlitz-Thouless transi-tion. Figure taken from reference [6].examples of systems that display quantum criticality. These will include both lattice modelsand experimental setups. Our discussion will be little more than an overview the readeris encouraged to follow up the references for details. We shall focus on 2+1 dimensions,as we often will throughout these Lectures . In several cases we will explicitly write downan action for the quantum critical theory. Typically the critical theory is strongly coupledand so the action is not directly useful for the analytic computation of many quantitiesof interest. Even in a largeNor (for instance)d= 4 expansion, which effectivelymake the fixed point perturbatively accessible, time dependent processes, such as chargetransport, are not easy to compute. This will be one important motivation for turningto the AdS/CFT correspondence. The correspondence will give model theories that sharefeature of the quantum critical theories of physical interest, but which are amenable toanalytic computations while remaining strongly Example: The Wilson-Fisher fixed pointLet be anNdimensional vector.

9 The theory described by the actionS[ ] = d3x(( )2+r 2+u( 2)2),(1)becomes quantum critical asr rc(in mean field theoryrc= 0 but the value getsrenormalised) and is known as the Wilson-Fisher fixed point. At finiteNthe relevant4couplinguflows to large values and the critical theory is strongly coupled. The derivativein (1) is the Lorentzian 3-derivative ( signature ( ,+,+)) and we have set a velocityv= 1. This will generally not be the speed of now briefly summarise two lattice models in which the theory (1) describes thevicinity of a quantum critical point, as reviewed in [6, 5].The first model is an insulating quantum magnet. Consider spin half degrees of freedomSiliving on a square lattice with the actionHAF= ij JijSi Sj,(2)where ij denotes nearest neighbour interactions and we will consider antiferromagnets, >0. Now choose the couplingsJijto take one of two values,JorJ/gas shown infigure 2. The parametergtakes values in the range [1, ).]

10 Figure 2: At largeg, the dashed couplings are weaker (J/g) than the solid ones (J). Thisfavours pairing into spin singlet dimers as shown. Figure taken from reference [6].The ground state of the model (2) is very different in the two limitsg 1 andg .Atg= 1 all couplings between spins are equal, this is the isotropic antiferromagneticHeisenberg model, and the ground state has N eel order characterised by Si = ( 1)i ,(3)where ( 1)ialternates in value between adjacent lattice sites. We can na vely picture thisstate as the classical ground state in which neighbouring spins are anti-alligned. Here is a three component vector. The low energy excitations about this ordered state are spin5waves described by the action (1) withN= 3 and 2fixed to a finite value. Spin rotationsymmetry is broken in this the limit of largeg, in contrast, the ground state is given by decoupled dimers. Thatis, each pair of neighbouring spins with a couplingJ(rather thanJ/g) between them formsa spin singlet.