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Conductivity in Lifshitz Backgrounds

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Conductivity in Lifshitz Backgrounds Conductivity in Lifshitz Backgrounds David Tong Based on work with Sean Hartnoll, Joe Polchinski and Eva Silverstein ETH, June 2010 Plan of the Talk Motivation Anomalous Properties of Strange Metals Introduction to AdS/CFT Should condensed matter theorists care about black holes? Conductivity from Lifshitz Geometry Strange Metals Unconventional superconductors at optimal doping e.g, cuprates, heavy fermions Suggestions that behaviour is governed by quantum critical point. DC Conductivity Ohm’s law: E~ = ρ~j Strange metals: ρ T ∼ Mackenzie, 1997 Compare to usual scaling 2 electron scattering: ρ T ∼ 5 phonon scattering: ρ T ∼ phonon scattering above Debye temperature: ρ T ∼ Hall Conductivity Now a 2x2 matrix B ~ ~ y j = σE x 2 Strange Metals: σxx/σxy T ∼ Drude Model prediction: σxx/σxy ρ ∼ |B=0 Mackenzie, 1997 AC Conductivity AC Conductivity: ~j(ω) = σ(ω)E~ (ω) Strange metals: as ω → ∞ σ(ω) (i/ω)ν → ν 0.65 ≈ Drude Model prediction Van der Marel et al., 2003 σ(ω ) (i/ω) → Challenge Reproduce this anomalous behaviour in conductivity from a (presumably strongly coupled) field theory. Introduction to Gauge/Gravity Duality Strongly coupled field theories Gravity in higher dimensional spaces Exact examples known 5 N=4 Super Yang-Mills Type IIB string theory on AdS5 xS Strategy: Extract general properties of the correspondence Ask: What can strongly coupled matter do? Gauge/Gravity Duality Field theory lives on the boundary of the space Gravity lives in higher dimensional space Or, if you’re slightly better at drawing.. GR=RG …. or [G,R]=0 r = 0 UV IR r → ∞ Gravitational equations in bulk RG equations on boundary [G,R]=0 r = 0 [G,R]=0 r = 0 GKPW Formula boundary source operator The QFT partition function SQFT+A0 φ Z[A0]QFT = Dφ e− · The source comes alive in theR extra dimension S? [A] Z[A0]QFT e− bulk A A0 ≈ → ¯ ¯ on-shell boundary action conditions Strategy Build up ingredients we want by writing an effective theory in the bulk. e.g. Finite Temperature = Black Hole Conserved Current = Maxwell field Pros: Very simple to compute transport properties in a strongly interacting theory Cons: Can’t build microscopic theories to order But even worse…don’t microscopic degrees of freedom The Set-Up Charged particles moving through a strongly coupled soup Probe Brane Gravitational Background Our boundary theory will live in d=2+1 dimensions We want a current with massive charge carriers We will take the strongly coupled soup to obey Lifshitz Scaling Lifshitz Scaling Non-Relativistic Conformal Invariance z: dynamical critical exponent ~x λ~x t λzt → → ~x = (x, y) : d=2 spatial dimensions A weakly coupled example with z=2: S = dtddx φ˙2 ( 2φ)2 − ∇ R h i Lifshitz Geometry Kachru, Liu, Mulligan ‘08 2 2 2 2 ds2 = L2 dt + dr + dx +dy − r2z r2 r2 ³ ´ boundary r r = 0 → ∞ Hot Lifshitz Geometry 2 2 2 ds2 = L2 f(r) dt2 + dr + dx +dy − r2z f(r)r2 r2 ³ z 1 ´z f(r+) = 0 T f 0(r+)/r − 1/r ∼ + ∼ + boundary horizon r = 0 r = r+ z+2 When needed, we take f(r) = 1 (r/r+) − Adding Probe Branes Brane wraps contractible θ cycle in internal space (r) profile looks like cigar When needed, we take this space to be a sphere boundary horizon r = 0 r = r+ r = r0 Adding Probe Branes 3 SDBI = τ dtd σ det [gab + ∂aθ∂bθ + Fab] − − R p Dual to mass of U(1) gauge field, charge carriers dual to current, J boundary horizon r = 0 r = r+ r = r0 2 In this talk, we’ll drop powers of τ , L , α0 Charge Carriers z Charge carrier = string: m = Egap 1/r0 ∼ 3 Note: m T ( E g a p 1 e V , T 10 10 K ) ∼ → À ∼ string boundary horizon r = 0 r = r+ r = r0 Charge Carriers chemical potential charge density t 2 z Finite charge density: A0 μ J r − + . → − Brane now stretches to horizon. (r) profile looks like tube. Electric field boundary horizon r = 0 r = r+ r = r0 DC Conductivity Karch, O’Bannon ‘07 t 2 z A0 μ J r − + . → − x z Ax Et + J r + . → electric field current Compute full non-linear conductivity J x = σ(E, T ) E Without doing any work….! DC Conductivity 4 t 2 σ(E, T ) = const. + r?(J ) p due to pair creation due to background charge 2 2z+2 where f(r?) = E r ? 2/z t z/2 1 T when E g a p , ( J ) T À ρ = σ J t ∼ Comments: Finite DC conductivity only because we’re ignoring backreaction Linear for z=2 Hall Conductivity O’Bannon ‘07 Similar technique: Find temperature dependence σxx T 2/Z σxy ∼ This is like the Drude result: no sign of anomalous behaviour. Caveat: The term from pair creation actually scales as 4/z σxx T t σxy ∼ J B Good for z=2, but unclear why this term would dominate AC Conductivity Ex(ω) x z Ax(ω) = iω + J (ω)r + . Boundary conditions: E x(ω) at r=0 Ingoing boundary conditions at black hole horizon Compute x Current J (ω) Numerical results for all ω Analytic results for T ω Egap ¿ ¿ AC Conductivity t z/2 1 (J ) ω− z < 2 t 1 σ(ω) J (ω log ω)− z = 2 ∼ ⎧ t 2/z ⎨ J ω− z > 2 ⎩ Note: z=3 gives 2/3 scaling Linearity in Jt is now dynamical Comment on the z=2 Crossover Dimensional Analysis: [x] = 1 [t] = z − − Operators are relevant if [ ] d + z O ≤ dtddx O Examples: Rt t 2 Charge density: [ J ] = d ( J ) relevant for z>d 2 Probe Inertia: d t x˙ irrelevant for z>2 R For z>2, all inertia due to stuff that particle drags around with it Summary Charge carriers moving in strongly coupled Lifshitz backgrounds yields: 2/z DC conductivity: ρ T ∼ 2/z Hall conductivity: σxx/σxy T ∼ 2/z AC conductivity σ ( ω ) 1 / ω for z>2 ∼ Any use? .
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