AdS/CFT Correspondence with Applications to Condensed Matter

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1 AdS/CFT Correspondence with Applications to Condensed Matter Robert Graham SFB/TR 12: Symmetries and Universality in Mesoscopic Systems 1

2 I. Quantum Field Theory and Gauge Theory II. Conformal Field Theory III. Short Introduction to Supersymmetry IV. General Relativity V. Some String Theory Introduction VI. A hand-waving derivation of AdS/CFT VII. Holographic superconductors VIII. Fermionic gas at unitarity IX. Holografic Viewpoint on Brownian Motion 2

3 Brownian motion in AdS/CFT M. Shigemori, Univ. Amsterdam, de Boer et al. [hep-th/ ] Long-wavelength physics of an interacting field theory on the boundary of an AdS D+1 is described by hydrodynamical Navier-Stokes equation is holographically dual to long-wavelength dynamics, of horizon of AdS black hole 3

4 Let s consider a less coarse-grained description now } hydrodynamics Brownian motion thermodynamics particle immersed in fluid at temperature T. 4

5 Recall Langevin dynamics : in 1 d ṗ = t dt γ (t t ) p (t )+R(t)+K(t) p = mẋ (non-relativistic) γ(t) = memory kernel of delayed friction R(t) = random force (assumed with Gaussian statistics) < R(t) > = 0 < R(t)R(t ) > = κ(t t ) K(t) = external (non-random) force γ(t) and κ(t) related by Kubo s fluctuation-dissipation theorem (Re(γ[ω]) = κ(ω)/2mt)). 5

6 time-scales: relaxation rate γ 0 = dtγ(t) 0 time t rel = 1 γ 0 collision: duration t coll = 0 dt κ(t) κ(0) mean time between collisions t mfp 6

7 In typical weakly interacting cases of kinetic theory t rel t mfp t coll 7

8 Brownian motion in AdS/CFT: Brownian motion in D space-time dimensions needs AdS D+1 for its dual description with planar Schwarzschild-AdS black hole ds D+1 = r2 [ ] h(r) dt 2 L 2 +d x 2 L 2 D 1 + r 2 h(r) dr2 h(r) = 1 ( rh r T = β 1 = D 4πL 2 r H ) D L = AdS radius t, x D 1 = boundary coordinates r = r = r H boundary of AdS horizon of black hole 8

9 AdS-Schwarzschild black brane Dual of Brownian particle = string hanging from boundary to horizon of black brane. Hawking radiation of AdS black brane excites modes of string thermally. = string endpoint x D 1 at r = executes Brownian motion. 9

10 Use probe approximation for the string (i.e. neglect its back-reaction on geometry). For small string coupling g s : thermal effect of gravitons in bulk on string negligible compared to direct excitation of string fluctuations at horizon. 10

11 Use of Nambu-Goto action for the string S NG = 1 2πα dτdσ h(τσ) expanded for small transverse excitations of string, x(t, r) S NG = const.+s 2 +S and use the quadratic part S 2 S 2 = 1 4πα dtdr ( ( t x) 2 h(r) r4 h(r)( r x) 2 L 4 ) Mode expansion x(t,r) = 0 dω ( f ω (r)e iωt a ω +h.c. ) [a ω,a ω ] = 2πδ(ω ω ) 11

12 Mode equation [ ω 2 + h(r) L 4 ( r r 4 ) ] h(r) r f ω (r) = 0 exact solution possible for D = 2; for D > 2 asymptotic solution at least in low frequency limit boundary condition: at AdS boundary r = : introduce UV - cutoff r = r c with Neumann boundary condition r x = 0 as if there were a (D-1)-brane at r = r c Note: only due to the cutoff the string has finite length and mass and can therefore execute its Brownian motion. 12

13 Excitation near horizon x(t,r r H ) 0 [ dω a ω (e ) ] iω[t r ] +e iθ ω e iω(t+r ) +h.c. 2ω = outgoing + infalling with relative phase-shift r = tortoise coordinate: ( ) L 2 dr = r 2 dr h(r) 13

14 Excitation at boundary r c x(t) = 0 dω [ a ω f ω (r c ) e iωt +h.c. ] = Learn about excitations on stretched horizon from Fouriertransform of x(t) at the boundary, and = about outgoing mode correlators near horizon < a ω 1 a ω2 > - or vice-versa, from boundary correlators of field-theory < x(t 1 ) x(t 2 ) > 14

15 Semiclassical use of bulk data to analyse Brownian motion Hawking < a ω a ω > = 2πδ(ω ω ) e βω 1 In D=2 mode equation solvable in closed form s 2 (t) <: [x(t) x(0)] 2 :> = 2α β 2 π 2 L 2 For r c r H, with m = r c /2πα 0 dω ω 1+( βω 2π )2 sin 2 ωt 2 1+ω 2 L 4 e βω 1 s 2 (t) T m t2 (t t rel ) ballistic s 2 (t) α t πl 2 T (t t rel ) diffusive = 2Dt t relax = α m L 2 T 2 15

16 Results for Langevin equation in D=2 with 1 γ[ω] iω = α β 2 m 2πL 2 1 iν/ρ c 1 iνρ c κ(ω) = 4πL2 α β 3 1+ν 2 1+ρ 2 c ν 2 β ω e β ω 1 ν = βω 2π κ(ω) = γ[ω] =, ρ c = r c r H + satisfy fluctuation-dissipation theorem. 0 dt κ(t) e iωt dt γ(t) e iωt 16

17 Results for time-scales: from γ(t), κ(t) : ( t Hooft coupling λ = L 4 /α 2 ) t rel m λ T 2 t coll 1 T t mfp 1 T logλ For λ 1 : the usual case in kinetic theory. For strong coupling λ 1 t rel t mfp t coll t coll t mfp particle collides with many particles in the thermal fluid simultaneously. 17