This research has been co-financed by FP7-REGPOT no , the EU program Thales MIS and by the European Union (European Social

Transcription

1 This research has been co-financed by FP7-REGPOT no , the EU program Thales MIS and by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF), under the grants schemes Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes. 1

2 Quantum Magnets 2015 Workshop 18 September 2015 Ballistic conductivity, thermalization and the AdS/CFT correspondence CCTP University of Crete APC, Paris 7 QCN 2-

3 Bibliography Ongoing work with Jie Ren (Princeton Crete Hebrew), Fransisco Peña-Benitez, (Madrid Crete Perugia) Li Li (Beijing Crete) and many discussions with Xenophon Zotos (Crete) and published work: J. Ren (Crete), E. Kiritsis (Crete) arxiv: [hep-th] F. Peña-Benitez (Crete), E. Kiritsis (Crete) arxiv: [cond-mat.str-el] Based on earlier work E. Kiritsis, B. S. Kim and C. Panagopoulos (Crete) arxiv: [cond-mat.str-el] C. Charmousis, B. Gouteraux (Orsay), B. S. Kim and R. Meyer (Crete). arxiv: [hep-th] 3

4 Introduction Transport and in particular its incarnation via Conductivity is one of the most important observables in condensed matter systems It is relatively easy to define and measure. It tells us a lot about the dynamics of (charge) carriers in a medium. For low enough external sources it is controlled by the retarded current two-point function via the Kubo formula. 4

5 Integrability and transport It is known that for low-dimensional integrable models transport can have exceptional properties. The most striking such property is ballistic transport associated with a δ-function peak at zero frequency. Such a δ-function could be simply explained if momentum is a conserved quantity (or the system is space translation invariant) AND has non-zero charge density AND is boost invariant (or galilean-boost invariant)). The argument is that in a translation-invariant system, a boost is a symmetry and therefore an infinite current can be generated from an infinitesimal electric field. In the presence of integrability, there are further conserved quantities beyond momentum that can produce a ballistic conductivity. 5

6 Spin and strong correlations The Kondo effect has taught as the spins and electrons can interact strongly in the IR. Several materials on the border with magnetism offer a host of exotic phenomena including high-t c superconductivity. They almost always involve strong interactions of the electrons providing a nontrivial challenge for theory. They are suspected to host quantum critical regions that control important parts of their phase diagram. Their normal phases ( strange metals ) still remain mysterious, to a large extend because theory cannot control the associated strong interactions. The presence of fermions makes Monte-Carlo simulations difficult and inhibits the brute-force numerical solution of the theoretical problems. 6

7 A technique for strong coupling I will discuss a technique that applies to a special class of theories (in any dimension) which are at strong coupling, and which can provide analytic or numerical solutions. There are three important ingredients: (a) Large N (b) Strong interactions (c) Toy : the models should be thought of as toy models that can teach us how to control some strong coupling effects (d) There is also an (somewhat) unexpected link: Integrability in D = d + 1 > 2 (or d > 1) 7

8 Large N The large N technique is a well known tool in condensed matter : Take N-copies of a field Ψ Ψ i, i = 1, 2, N assume some SO(N) or SU(N) symmetry and take N. This always simplifies the theory rendering it (typically) solvable. If we complicate matters by introducing a (gauge) interaction that acts on the copies then we obtain SO(N) or SU(N) gauge theories. Such gauge theories are known to describe the strong (N=3) and the weak (N=2) interactions in particle physics. 8

9 Unlike the initial story: (a) The dynamics of such theories is usually strongly coupled and complicated. (b) The large-n limit (N aka t Hooft limit) although semiclassical is/was NOT solvable. In 1974 t Hooft gave some arguments that the large-n theory should be describable as string theory with coupling constant. g s 1 N As N the string theory is weakly coupled. 8-

10 The gauge-theory/string-theory (gravity) correspondence J. Maldacena in 1997 conjectured that: A highly (super)symmetric SU(N) gauge theory in 3+1 dimensions is equivalent (dual) to a 10-dimensional string theory, in curved background. It is a duality of practical importance because at large-n When the gauge theory is strongly coupled The string can be approximated as point particles gravity 9

11 Although it is formally a conjecture, it has been tested in many contexts, well beyond the original context. There is even a constructive intuition that any QFT is dual to some form of string theory and vice versa. Some further intuition: copies (or colors) interact strongly, but colorless boundstates of the copies interact weakly (with strength 1 N ) The bottom line: The strongly-coupled large-n saddle point can be reliably calculated by solving (appropriate) gravitational equations (usually) in one-dimension higher. 9-

12 This extra dimension can be interpreted as the renormalization group scale of the Quantum Field Theory. Near the boundary (the IR of gravity) corresponds to the UV of the QFT. The IR divergences of gravity are in 1-1 correspondence with the UV divergences of the dual QFT. A radially dependent gravity solution corresponds to an RG Flow of the QFT. Ψ e d d x ϕ(x) O(x) Ψ = e S Ψ(ϕ(x)) This is a precise and quantitative formulation of the (holographic) duality. 9-

13 sym and Integrability The prototype duality: AdS/CFT correspondence. The QFT is N=4 SU(N) sym theory in 3+1 dimensions. It is exactly scale invariant at the quantum level (CFT). It has maximal supersymmetry (16 conserved spinor (super)charges). It has a single complex parameter (λ gy 2 MN, θ). Its global symmetry is O(1, 5) O(6). It is a non-trivial interacting quantum field theory. 10

14 The dual string theory has ten dimensions and the strings move in a curved very symmetric space AdS 5 S 5 The symmetry of the dual sym theory is realized geometrically. The gravitational solution with the metric of AdS 5 S 5 represents the vacuum saddle point of the dual QFT. Correlators can be computed by putting sources at the boundary, and computing the response, by solving the gravitational equations. We now know that to leading order in the N expansion this QFT is integrable. It is suspected that this remains so, order by order in the 1/N expansion. The integrability works in an unusual way. 10-

15 The spectrum of (anomalous) dimensions is given by an infinite - dimensional system of TBAs that each looks like the one for integrable spin chains. Gromov+Kazakov+Viera, 2009 It can be expanded at small λ and reproduces sym Perturbation theory up to four loops. It can be expanded at large λ and reproduces string theory to two loops. It can be solved numerically at all λ. Recently the calculation of correlators has been formulated as an integrable system in an expansion near the collinear limit. Basso+Caetano+Cordova+Sever+Viera, 2014 It seems that in the next few years the theory will be solved. So far we do not know if the theory has an infinite number of conventional conserved charges. There is a similar example in 2+1 Dimensions: Chern-Simons Theory coupled to matter. N=8 supersymmetric This seems harder to solve than then 3+1 dimensional theory. 10-

16 Finite temperature The finite temperature (T > 0) canonical ensemble is represented by another solution in the dual string theory: the AntideSitter-Schwartzschild black hole. This explains why black holes in gravity follow thermodynamics laws. The phase transition between the AdS phase and the black hole phase is a standard first order phase transition. Hawking+Page It corresponds to the confinement-deconfinement transition in the gauge theory. Dissipation happens naturally because of the black-hole horizon. 11

17 11-

18 Large numbers CFTs in 3,4 and 6 dimensions. In 1+1 dimension we know infinite classes of CFTs. Many of them are solvable. We know there exist more, but they are not solvable so far. The full set of 1+1 CFTs is unknown. In 2+1 dimensions, until recently we knew 1 non-trivial CFT: The Wilson- Fisher fixed point. Now we know infinite classes of them, most in the large N limit. In 3+1 dimensions the story is similar. In 5+1 dimensions there are no weakly coupled (interacting) CFTs. Now we know infinite classes of 5+1 dimensional CFTs with no weak coupling limit. 12

19 Scaling General quantum critical theories are characterized by exponents: [ ] ds 2 = r 2θ dr 2 d r 2 + d x2 r 2 dt2 r 2z Invariance under r λ r, x i λ x i, t λ z t, ds λ θ d ds Dynamical or Lifshitz exponent, z, (z=1 is Anti de Sitter space) the hyperscaling-violation exponent θ, (S T d θ z ) Charmousis+Gouteraux+Kiritsis+Kim+Meyer, 2010, Gouteraux+Kiritsis, 2011 There are two more (anomalous) exponents associated with charge and transport. conduction exponent ζ, and a cohesion exponent χ. E. Kiritsis+B. Gouteraux,

20 Dynamics far from equilibrium The duality is very well suited to study far from equilibrium dynamics and thermalization in the gravitational language. We consider the theory in its vacuum state and then perturb it by a time dependent coupling constant L QF T + dt d 3 x f 0 (t) O(x) t T tt = f 0 O The approach to equilibration is controlled by the expectation values T tt (t), O (t). We expect that if the system thermalizes then O (t ) T r[ρ thermal O] 14

21 Thermalization at strong coupling To calculate the observables at strong coupling we will assume the holographic (AdS/CFT) correspondence (aka gauge/gravity duality). Thermalization corresponds to black hole formation in the bulk spacetime. 15

22 Holographic YM The Holographic model we will use has a behavior similar to YM in 4d Gursoy+Kiritsis+Nitti, Gubser+Nellore The gravitational action contains a metric and a scalar. S IHQCD = M 3 d 5 x [ g R 4 3 ( ϕ)2 V (ϕ) They are the fields dual to the most important YM operators ϕ T r[f 2 ] coupling constant g µν T µν ] The theory like QCD has confinement, a mass gap and a discrete spectrum of excitations (bound states of glue=glueballs). We will take for computational simplicity the operator dual to ϕ to have UV dimension 3. This is not expected to affect qualitatively the results. 16

23 Quench dynamics The quench profile is: f 0 (t) = f 0 δf 0 e t2 2τ 2 Two basic parameters: δf 0 and τ. For numerical simplicity we start with the theory in a thermal state with a small black hole in the bulk (5d) space. ie. The smallest the initial black hole, the closest we are to the initial (confining) ground state of the theory. 17

24 Thermalization in QFT = Formation of an apparent horizon in gravity dual. There are three possible characteristic times involved. 17-

25 We find the following The characteristic time associated with the intermediate non-linear regime is negligible compared to τ and T RD. This seems to be a generic occurrence in holography/gravity and a clean explanation is still lacking. Therefore T thermalization 1 Γ For adiabatic perturbations, τ 1 the system does NOT oscillate but goes continuously to the final-state black hole. 17-

26 17-

27 The ring-down phase 18

28 The temperature dependence of the decay width Γ for the lowest lying scalar quasi-normal mode in several states of our theory. The blue circles are large black branes whose temperature is an integer multiple of T c. The orange squares correspond to the minimum temperature black brane (top) and the smallest black hole we perturb in our study (bottom). The ratio Γ/πT approaches (the dashed line) at high temperatures, which coincides with the expected value for perturbations of AdS 5 Schwarzschild by a dimension 3 scalar operator 18-

29 Scaling 19

30 Fast Quenches Buchel+Lehner+Myers+Niekerk, Das+Galante+Myers 20

31 Finite Density Solutions At finite charge density the saddle-point solutions associated with many holographic strongly coupled theories are translationally invariant. The simplest example is given by the finite density solution associated with a large-n holographic CFT: it is the Charged (Reissner-Nördstrom) black hole. Near the boundary (UV) the solution is close to AdS space (UV Conformal invariance). However the charge density breaks the scaling symmetry of the QFT. In the IR the solution has an emergent scaling symmetry, that scales time only. The effective speed of light has gone to zero in the IR. The system has non-zero entropy density at T = 0. 21

32 Conductivity Conductivity can be calculated in the linear regime from the correlators of the currents σ ij (ω, k) = 1 d p xdt e iωt i k x J i (t, x)j j (0, 0) iω or more generally as the (non-linear) response to an external electric field J i = σ ij E j Translational invariance and finite density imply a pole at zero frequency for the conductivity. ( σ(ω) K δ(ω) + i ) + ω K is the Drude weight (T-dependent) 4ρ 2 K = ε(t ) + p(t ) and vanishes as ρ 0. The formula is valid for gapless theories. This shares some features to what is seen in integrable systems, but its emergence seems different. 22

33 General features of holographic conductivities When translational invariance is broken the δ-function is resolved to a Drude peak. ( K δ(ω) + i ) Kτ ω 1 iωτ +, 1 τ = scattering rate The DC conductivity has the schematic form σ DC σ pair + σ drag Karch+O Bannon, Blake+Tong, Donos+Gauntlett The first term σ pair is non-zero even when ρ = 0. Karch+O Bannon Because of this it was interpreted as a conductivity due to pair production (problematic in many respects) What contributes to it does not contribute to thermal transport. 23

34 Donos+Gauntlett It is finite, even when there is no momentum dissipation (does not contribute to the Drude weight). The rest σ drag τ is due to momentum-dissipating interactions and is proportional to the charge density. If more than one mechanisms of momentum dissipation are at work, σ drag = I σ I drag, τ = I τ I (inverse Mathiessen law) Hints from Donos+Gouteraux+Kiritsis The above are suggestive but the true story appears to be more complicated and not fully understood. Gouteraux+Richardson 23-

35 Conductivity Spectra Many holographic dual theories can be surveyed by using effective holographic theories. Charmousis+Gouteraux+Kiritsis+Kim+Meyer Various types of AC conductivity spectra have been found that show the differences of transport properties at strong coupling The insulators above may or may not have a gap, and a continuum above it. 24

36 This looks like a charge supersolid (but is not) 24-

37 IR irrelevant momentum dissipation can be introduced in the theory above in order to remove the ballistic transport. The result depends on the original theory and the IR dimension of the dissipating interaction. Kiritsis+Ren In one class of cases, the δ-function disappears, while the spectrum remains discrete. This seems like a new mechanism for an insulator at strong coupling. In another class of cases the δ-function persists at T=0 despite the momentum dissipating interaction. Here we expect that at T > 0 it will disappear. In a third class of cases that involve a superconducting instability the δ-function now is there due to the presence of the superfluid. However the spectrum is discrete and this ressembles more a supersolid. 24-

38 Outlook The holographic duality between quantum field theories and gravity/string theories, provides a novel tool to address equilibrium and non-equilibrium phenomena at strong coupling. Because of this it may be useful beyond its original area of inception. At finite density, it provides a rich landscape of possibilities for conductivity, some of them rather exotic, and most of which are not well understood. It realizes novel types of insulators and supersolid-like behavior. It has provided several interesting arenas to describe thermalization and out-of-equilibrium phenomena like quenches. It has also provided a simple geometrical way to calculate entanglement entropy using gravity, that established links with the quantum information field. Ryu+Takayanagi Questions related to the integrability of d > 2 CFTs are still to be fully addressed. 25

39 . THANK YOU 26

40 This research has been co-financed by FP7-REGPOT no , the EU program Thales MIS and by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF), under the grants schemes Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes. 27

41 Effective Schrödinger potentials Schrödinger potential for the different regimes of interest. RETURN 28

42 Overview of the solutions The gravitational action: and S = M 3 d 5 x g [ R 1 2 ( ϕ)2 + V (ϕ) Z 1(ϕ) 4 F 2 1 Z 2(ϕ) 4 F 2 2 with F i µν µa i ν νa i µ V (ϕ) IR V 0 e δϕ, Z 1 (ϕ) IR Z 10 e γ 1ϕ, Z 2 (ϕ) IR Z 20 e γ 2ϕ. The metric ansatz has helical symmetry and we parametrize the scaling solutions as : [ 2 ( ω λ ) ] r 2ω 3 2, A 1 r ζ z ds 2 = r 2θ 3 where dt2 z 2z + L2 dr 2 + ω 1 r r 2z 2 ω 1 = dx 1, ω 2 = cos(kx 1 )dx 2 +sin(kx 1 )dx 3, ω 3 = sin(kx 1 )dx 2 cos(kx 1 )dx 3 ] 29

43 Critical lines vs critical points in Cuprates Kim+Kiritsis+Panagopoulos Anomalous Criticality in the Electrical Resistivity of La 2 x Sr x CuO 4. R.A. Copper et. al

44 Breaking Translation Invariance Charge lattices lead to non-linear (gravitational) PDEs that are in general very difficult to solve. Horowitz+Santos+Tong (2012), Lin+Liu+Wu+Xiang (2013), Donos+Kiritsis (2013) Some other string mechanisms of translation invariance breaking (momentum dissipation) are simpler to analyze: An effective phenomenological treatment of momentum dissipation using massive gravity. Vegh (2013), Davison (2013), Blake+Tong (2013), Davison+Schalm+Zannen (2013) Interactions of charge with a bulk sector that carries most of the energy (DBI probe approximation) Karch+O Bannon (2007), Charmousis+Gouteraux+Kiritsis+Kim+Meyer (2010) Interactions with string axions that model a kind of homogeneous disorder. Andrade+Withers (2013), Gouteraux (2013), Donos+Gauntlett (2013) The use of random field disorder in holography. Hartnoll+Herzog (2008), Davison+Schalm+Zaanen (2013), Lucas+Sachdev+Schalm (2014) Study of saddle points with helical symmetry. Kachru et al. (2012), Donos+Gauntlett (2012), Donos+Hartnoll (2012) 31

45 DC conductivity The general structure of the holographic DC conductivity at strong coupling: σ DC = (σ pp DC )2 + (σ drag DC )2, σ DC = σ pp DC + σdrag DC σ pp is the pair-creation contribution: it exists also when Q = 0. σ drag is the drag contribution. It originates from the force and dissipation a fractionalized charge (a string) feels as it is moving in the strongly coupled medium. Karch+O Bannon, (2007) 32

46 Ground-states with helical symmetry Work with A. Donos and B. Gouteraux, to appear The gravitational action: S = M 3 d 5 x g [ R 1 2 ( ϕ)2 + V (ϕ) Z 1(ϕ) 4 F 2 1 Z 2(ϕ) 4 F 2 2 ] and with F i µν µ A i ν ν A i µ V (ϕ) IR V 0 e δϕ, Z 1 (ϕ) IR Z 10 e γ 1ϕ, Z 2 (ϕ) IR Z 20 e γ 2ϕ. The metric ansatz has helical symmetry and we parametrize the scaling solutions as : [ 2 ( ω λ ) ] r 2ω 3 2, A 1 r ζ z ds 2 = r 2θ 3 where dt2 z 2z + L2 dr 2 + ω 1 r r 2z 2 ω 1 = dx 1, ω 2 = cos(kx 1 )dx 2 +sin(kx 1 )dx 3, ω 3 = sin(kx 1 )dx 2 cos(kx 1 )dx 3 33

47 Anisotropic saddle points (a) IR marginal current (ζ = θ 2 2z 2 ) S a T 2z 2 +2 θ z vanishes at T 0 For the DC conductivity σdc a T ζ 2 z σ DC along the helical axis vanishes always at T = 0 (power insulators) For the IR limit of AC conductivity we find σac a (T = 0) ω θ 2z 2 4 z This is the same power as the DC conductivity and σac a (T = 0) 0 as ω 0. 34

48 (b) IR Irrelevant current (z = 3 2 z 2). For the DC conductivity S b T (2z 2 +2 θ) z, σdc b T ζ 2 z It is sometimes diverging (metal) and sometimes vanishing (insulator). σ DC comes always for the pair creation term. Therefore in the metallic case we expect no Drude peak (incoherent metals). For the AC conductivity at zero T: σ a AC (T = 0) ωmin{n 1,n 2 } n 1 = ζ + 3z 2 3z 2, n 2 = θ2 + 96z 2 76θz z 2 2 z

49 Whenever the system is an insulator, n 1 dominates and the the AC and DC exponents are equal. When n 2 dominates, the system is metallic with a decaying low-frequency AC tail. A δ(ω) may bridge this behavior. When the system is metallic and n 1 dominates, it can have: 1. Diverging AC frequency power tail and match the DC scaling 2. Diverging AC frequency power tail without matching the DC scaling 3. Decaying AC frequency power tail. Whenever σ AC ω 0 then σ DC T 0 or σ DC T 2 34-

50 The effective temperature The effective temperature on the brane is not t for non-trivial E: t eff = t J 2 A(t) 2 A(t) 2t 2 + t 5 ( t 2 A(t) + 2t ) 2 2t 3 ( t 5 A(t) + J 2 ) Dependence of the world-volume temperature T eff with the parameters J and t, the effective temperature is always bigger than the background temperature T. 35

51 The AC conductivity in a holographic strange metal Kim+Kiritsis+Panagopoulos The bulk is the AdS Schwarzschild black hole. The charge is described by a standard DBI action coupled to the metric S DBI = N The ansatz for the ground state is d 5 x det(g + F ) A = (Ey + h + (u))dx + + (b 2 Ey + h (u))dx + (b 2 Ex + h y (u))dy, It is a stationary solution in lightcone coordinates with a nontrivial charge density and a light-cone electric field F +y = E The DBI equations can be solved exactly and the conductivity computed à la Karch-O Bannon. 36

52 The DC conductivity The parameters are E, J +, T. cuprates. E is similar to the doping parameter of They can be combined in one scaling variable t and parameter J. t = πlt 2 E, J2 = and the DC conductivity becomes J 2 + (2N ) 2 2(E) 3, σ DC = σ 0 σ 2 DR + σ2 QC, σ 2 DR = J2 t 2 A(t) We will also define the ratio, A(t) = t t 4, σ 2 QC = t3 A(t). q σ2 DR σ 2 QC that determines in which regime we are. 37

53 The conductivity is as follows: ρ = (σ 0 J) 1 t q 1, t 1 linear 2σ 1 0 J t2 q 1, t 1 quadratic σ 1 0 t 3/2 q 1, t 1 regime I 2σ 1 0 t 1/2 q 1, t 1 regime II Drude regime (DR) pair production regime (QC) (1) 37-

54 Center: Location of the four regimes in the space of parameters (J, t) log-log scale. The black line q = 1, separates the DR respect to the QC regime. The magenta line represents the region with q = 10 2 and the red one q =

55 La 2 x Sr x CuO 4 Cooper et al. Science (2009) Kim+Kiritsis+Panagopoulos 37-

56 The AC conductivity From the AC conductivity equations one can compute the asymptotics The large ω behavior ( b 2 ln T ) 1 σ(ω) = i ( 2π 3 The generalized relaxation time defined as ) 7/3 2 Γ (1/3) Γ (7/3) ω 1 3 e iπ/6 σ(ω) σ DC ( 1 + iτω + O(ω 2 ) ), can be computed analytically but has a complicated formula. There is another intermediate asymptotic scaling regime, in which σ ω 1 2. It was found numerically, and it coincides with a feature in the effective Schrödinger potential 38

57 The generalized relaxation time τ as a function of the scaling temperature variable t.the left plot shows the behaviour of the dimensionless quantity Tτ (T is the temperature) deep in the Drude regime (DR). The right plot shows the behavior in the Quantum Critical regime (QCR) at zero density. Dots show numerical data and the continuous line is the analytic formula obtained in the text using perturbation theory. 38-

58 Conductivity for three different temperatures in the QC (up) and in the DR (down) regimes. The left figures show the real part of the conductivity while the right figures the imaginary part of the conductivity. Continues lines represent the numerical data, dashed lines correspond to a fit to the Drude peak formula. 38-

59 The colored continuous lines show the Drude fitting using the analytic computation of the generalized relaxation time. Straight black and dashed gray lines show the UV and the intermediate power law behavior of the AC conductivity. 38-

60 Absolute value (left) and argument (right) part of the conductivity. The blue continuous line shows the Drude fit using the analytic computation of the generalized relaxation time. Straight black and dashed gray lines show the UV and the intermediate power law behavior of the AC conductivity. 38-

61 Lessons learned When the conductivity is dominated by the drag mechanism (momentum dissipation) there is a clear Drude peak. When the conductivity is dominated by the pair-production mechanism there is no Drude peak. There is an associated scaling tail in the conductivity (here σ ω 1 3). This always survives beyond the Drude peak as it falls off slower than ω 1. It is controlled by the pair-production contribution and is there even if Q = 0. It has the features seen in experiment by Van der Marel et al.: a) Constant phase matching the power falloff. b) Temperature independence. c) It implies that the QC point is hyperscaling-violating. These properties seem to hold more generally and beyond the system under study. 38-

62 General Scaling Charmousis+Gouteraux+Kiritsis+Kim+Meyer Gouteraux, Kiritsis+Pena-Benitez Consider EMD solutions with general scaling exponents z, θ, ζ. U(1)) and gapless charge excitations). (unbroken The conductivity is obtained by solving for the fluctuations of the gauge field, δa i = a i (r)e iωt. ( ) [ ] 1 grr gtt r Z a grr i + ω 2 Q2 g rr Z g tt g rr g tt Zgxx 2 a i = 0 By a field and coordinate redefinitions it can be mapped into a Schrödinger problem ψ + V eff ψ = ω 2 ψ, V eff (x) = V 1 (x) + Q 2 V 2 (x) V 1 (x) 1 x 2, V 2 1 x 2a, x a 1 It is the coefficient of 1/x 2 that controls the scaling power of the AC conductivity. Goldstein+Kachru+Prakash+Trivedi (2010) 39

63 The Charge density is supporting the IR geometry. Q 2 is fixed in terms of z, θ. Both terms in the potential contribute at the same order. σ ω m, Arg(σ) π m 2 m = 2(z 1) + d θ z In his case m is always positive. The case where the IR geometry is AdS 2 can be obtained for z giving m = 2. For hyperscaling violating semilocal geometries, we must take, θ, z with z θ = η fixed. m = 2 + η > 0 39-

64 Contour plots to illustrate the region in the parameter space where the exponent m takes negative values for the single charged model. Left: Conductivity in the charged case for d = 2. Right: Conductivity in the charged regime for d = 3. The allowed values for the parameters are bounded by the gray mesh. The negative values for m are outside the permitted region. 39-

65 The Charge density is a probe in the IR geometry. The charge term is subleading or absent. σ ω m, Arg(σ) π m 2 m = z + ζ 2 z 1, It allows for negative values in m but always m 1 (unitarity bound) For an AdS 2 IR geometry the exponent can be obtained by an z giving m = 0. For hyperscaling violating semilocal geometries we must take θ, z with z θ = η fixed and obtain m = d 2 d η > 0 Finally, for the gauge field conformal case we obtain m = 0 when d =

66 Contour plots to illustrate the region in the parameter space where the exponent m takes negative values for the probe charge density case. The plots above correspond to d = 2. Left: m for κ = 5. Right: m for κ = 10. The allowed values for the parameters are bounded inside the gray mesh. Green dashed lines are the contour levels where m = 2/3 and blue dashed lines m = 1/3. 39-

67 Contour plots to illustrate the region in the parameter space where the exponent m takes negative values for the probe charge density case. The plots above correspond to d = 3 Left: m for κ = 5. Right: m for κ = 10. The allowed values for the parameters are bounded inside the gray mesh. Green dashed lines are the contour levels where m = 2/3 and blue dashed lines m = 1/3. 39-

68 The numerical data 40

69 Holographic QCD Holographic models were developed that describe with rather good accuracy the strong coupling physics of 3+1 YM theory. Gursoy+Kiritsis+Nitti, Gubser+Nellore The gravitational action contains a metric and a scalar. S IHQCD = M 3 d 5 x [ g R 4 3 ( ϕ)2 V (ϕ) They are the fields dual to the most important YM operators ϕ T r[f 2 ] g µν T µν ] The potential is in one-to-one correspondence with the YM β-function. The theory like QCD has confinement, a mass gap and a discrete spectrum of excitations (glueballs). 41

70 Panero 41-

71 Therefore the picture of the thermalization process is as follows: We will take for computational simplicity the operator dual to ϕ to have UV dimension 3. This is not expected to affect qualitatively the results. 41-

72 Generic Scaling Geometries: A classification of all QC points in holography Charmousis+Gouteraux+Kim+Kiritsis+Meyer (2010), Gouteraux+Kiritsis (2011), Huisje+Sachdev+Swingle, (2011), Iizuka+Kachru+Kundu+Narayan+Sircar+trivedi+Wang (2012) Assume translation and rotational invariance in space and time. The QC points generically break hyperscaling invariance and are characterized by several exponents. Two appear in the metric (z, θ). z dynamical exponent θ hyperscaling violation exponent ds 2 = r 2θ d dt 2 r 2z + dr2 + dx dx dx2 d r 2 42

73 There is invariance under: x i λ x i, t λ z t, r λ r, ds λ θ d ds The entropy scales as S T d θ z which gives an interpretation to the hyperscaling violation exponent. There is a third exponent, associated with the charge density, the conduction exponent ζ: Gouteraux+Kiritsis, Gouteraux A t = Q r ζ z If non-zero it also violates hyperscaling. 42-

74 Insulators 1) Band gap insulators, where the conduction band is empty, and there is therefore a gap that prevents current transport. 2) Anderson localization that is effective usually in two dimensions and where strong disorder inhibits conduction. 3) Mott localization, where strong onsite interactions localize electrons. This has been argued to be at work in insulating anti-ferromagnets. 4) A new mechanism at strong coupling: Momentum-dissipating interactions become relevant (strong) in the IR, and they inhibit conduction Donos+Hartnoll This mechanism was generalized to Einstein-Maxwell-Dilaton (EMD) theories with many saddle points found corresponding to insulators, bad metals and conventional metals. Donos+Gouteraux+Kiritsis, Donos+Gauntlett,Gouteraux 43

75 Supersolids A supersolid is a generalization of a superfluid state. It is characterized by a spontaneously broken U(1) symmetry which guarantees a superfluid component (a zero frequency δ-function). It has (spontaneously) broken translational invariance. and is therefore a solid. Therefore the appropriate two-point function has a discretely localized spectral density. If probed at a generic non-zero frequency it is non-responsive. probed at zero frequency it behaves as a superfluid. If it is They have been theoretically anticipated and studied, especially in the last 2-3 years. Legget, Fisher+Nelson, Anderson, Nicolis+Penco+Rosen There are proposed realizations with cold atoms. Keilmann+Cirac+Roscilde There have been claims for presence in solid Helium 4 as well as a recent refutation. RETURN Kim+Chan 44

76 Insulators at finite density There are two ways that a system can be insulating at T=0: Charged excitations are gapped. In that case the conductivity is non-zero only above the gap. There is no gap but the limit ω 0 gives a vanishing conductivity. both cases the operator/mechanism that breaks translation invariance, is relevant in the IR. In Gapped holographic systems are known at zero charge density, and they are in use as models of YM. Witten, 98, Gursoy+Kiritsis+Nitti, 07, Nishioka+Ryu+Takatanagi, 10 At finite density, many saddle-points with discrete spectrum were found, in the classification of QC points in EMD Theories. Charmousis+Gouteraux+Kiritsis+Kim+Meyer, 10, McGreevy+Balasubramanian, 10 45

77 We parametrize the effective holographic action as S = d 4 x g [ R 1 2 ( ϕ)2 V (ϕ) Z(ϕ) 4 F 2 ], V e δϕ, Z e γϕ The generic IR geometry is hyperscaling violating with (z, θ, ζ) being functions of (γ, δ). The IR charge density is also fixed as a function of (γ, δ). Such extremal geometries have naked singularities. One should impose the Gubser constraints that put restrictions on (z, θ, ζ). Such singularities are expected to be resolvable by effects neglected (KK states, stringy corrections, etc.) As we shall see there may be more constraints when we start considering correlation functions. 45-

78 Holographic calculation of the conductivity The conductivity is obtained by solving for the fluctuations of the gauge field, δa i = a i (r)e iωt. ( ) [ ] 1 grr gtt r Z a grr i + ω 2 Q2 g rr Z g tt g rr g tt Zgxx 2 a i = 0 By a field and coordinate redefinitions it can be mapped into a Schrödinger problem ψ + V eff ψ = ω 2 ψ V eff = V 1 + Q 2 V 2 46

79 46-

80 The parameter space The white part is not allowed by the Gubser criterion. The regions A, B, and C are the parameter space allowed by the Gubser criterion. In region A (yellow), the extremal limit is at T 0, and the current-current correlator is gapless. In region B (red), the extremal limit is at T, and the current-current correlator is gapped. In region C (green), the extremal limit is at T, and the current-current correlator is gapless. Region D (enclosed by blue boundaries) is holographically unreliable. 46-

81 The finite temperature picture The gapped systems above are at finite density, and have therefore a zero-frequency δ-function in the AC conductivity: They are perfect conductors. Note also that we are at T = 0. they resemble real metals at T = 0. In this respect as far as the δ-function is concerned Up to T = T min this is the only saddle-point for the system. For T > T min there are also two black holes that are competing at the same temperature. At T = T c > T min there is a first order phase transition to the large black hole phase that is a gapless plasma phase. This is very similar to the confinement-deconfinement phase transition in gauge theories. 47

82 In gapped saddle points, the intuition that the interior geometry is controlling the IR properties of the theory is incorrect. The far-interior (naked singularity) is controlling the UV asymptotics of the spectrum. 47-

83 Adding Momentum dissipation We use axions (goldstone-bosons for broken translational invariance) as a source of momentum dissipation. S = d 4 x g R 1 2 ( ϕ)2 V (ϕ) Z(ϕ) 4 F 2 Y (ϕ) 2 2 i=1 ( ψ i ) 2, V (ϕ) e δϕ, Z(ϕ) e γϕ, Y (ϕ) e λϕ. ψ 1 = kx, ψ 2 = k y We also choose λ in the region where the axions are irrelevant in the IR. This means that leading IR solution is unaffected by the axions. 48

84 48-

85 Momentum dissipation could remove the zero-frequency δ-function but: (a) We are in the T=0 geometry as long as T < T c (b) The axions/translation symmetry breaking are irrelevant in the deep IR. A detailed analysis of the Drude weight is needed. The fluctuation equations for the conductivity involve not only δa i but also the axions. There is a direct formula for the DC conductivity in the presence of a regular (non-extremal) horizon. σ DC = H(r = r h ) Z h + q 2 k 2 (g xx ) h Y h Z 1 + q2 k 2Z 2 There is also a direct formula for the Drude weight: Π(ω) = fhλ 1 q ( ) Z1 k 2fZ 2 λ 2,, r Π = O(ω 2 ) λ 1 = Z 1 H ( a x q k 2 Z 2 Z 1 b x ) Z 2, λ 2 = Z 2 H (qa x + b x ). Gouteraux, Donos+Gauntlett 48-

86 It can be shown that Π(0) is the Drude weight. We can then establish that: (a) when σ DC at extremality then there is a zero-frequency δ-function, and this happens in the gapless geometries. (b) when σ DC 0 at extremality, then there is no zero-frequency δ- functions and this happens for the gapped geometries. We can also show in general that: The Gubser criterion, and irrelevance of axions in the IR = the conductivity of small near-extremal black-holes is dominated by the momentum dissipation term. It is crucial for all of the above, to exclude holographically unreliable cases. We have therefore found holographic systems with a charged spectrum that is gapped and discrete. The stress-energy correlators are also gapped. These are insulators that share properties of both band-gap insulators and Mott insulators. We believe this is a novel mechanism of insulating behavior. 48-

87 Parameter space for the conducting and insulating phases. 48-

88 An interesting borderline case The two-charge black hole (in 4d) is an analytic solution, obtained from the general STU four-charge BH. Cvetic+Du+Hoxha+Liu+Lu+Lu+Martinez-Acosta+Pope (1999) It has effective EMD functions: (γ = 1, δ = 1) (z = 1, θ ). V (ϕ) = 2(cosh ϕ + 2), Z(ϕ) = e ϕ The T = 0 AC conductivity can be calculated analytically ( ) Q i 4ω 2 Q 2 σ(ω) = i 2ω ω ω+iϵ There is a non-zero Drude weight and a mass gap, with continuous spectrum above. 49

89 The real part of the AC conductivity calculated from the 2-charge black hole in AdS 4 at extremality. There is a δ-function at ω = 0. The solution has well defined IR boundary conditions above the gap only. 49-

90 Detailed plan of the presentation Title page 0 minutes Bibliography 1 minutes Introduction 2 minutes Integrability and transport 4 minutes Spin and strong correlations 6 minutes A technique for strong coupling 7 minutes Large-N 9 minutes The gauge theory/string theory correspondence 13 minutes sym and integrability 17 minutes Finite temperature 20 minutes Large numbers of CFTs 22 minutes Scaling 24 minutes Dynamics far from equilibrium 26 minutes Thermalization at strong coupling 27 minutes 50

91 Holographic Yang-Mills 28 minutes Quench dynamics 33 minutes The Ring down phase 35 minutes Scaling 36 minutes Fast Quenches 37 minutes Finite density solutions 38 minutes Conductivity 40 minutes General features of holographic conductivities 43 minutes The AC conductivity Spectra 47 minutes Outlook 48 minutes 50-

92 Effective Schröndiger Potentiasl 50 minutes Overview of the solutions 52 minutes Critical lines vs critical points Breaking translation invariance 54 minutes 55 minutes DC conductivity 56 minutes Ground-states with helical symmetry 57 minutes Anisotropic saddle-points 61 minutes The effective temperature 63 minutes The AC conductivity in a holographic strange metal 65 minutes The DC conductivity 69 minutes The AC conductivity 78 minutes General Scaling 84 minutes The numerical data 86 minutes Holographic QCD 92 minutes Generic Scaling Geometries: A classification of all QC points in holography 98 minutes 50-

93 Insulators 100 minutes Supersolids 102 minutes Insulators at finite density 104 minutes Holographic calculation of the conductivity 106 minutes The finite temperature picture 108 minutes Adding Momentum dissipation 116 minutes An interesting borderline case 118 minutes 50-