Black holes, Stokes flows and DC transport at strong coupling Talk - PowerPoint PPT Presentation

“Black holes, Stokes flows and DC transport at strong coupling” Talk at the “Oxford Holography Group” Aristomenis Donos Durham University March 2015 Based on work with J. Gauntlett : 1506.01360 J. Gauntlett and E. Banks: 1507.00234 J. Gauntlett, T. Griffin and L. Melgar: 1511.00713

Outline 1 Motivation/Setup 2 Holography 3 Summary / Outlook

The Result Things a black hole horizon knows about: Temperature A Entropy s = 4 G s Shear Viscosity (sometimes) η = 4 π [Son, Starinets, Policastro] � σ � T α We will add DC conductivities T ¯ α ¯ κ

Charge transport in real materials Drude peak σ Incoherent metal Mott insulator ω eV Materials with charged d.o.f. can be Coherent metals with a well defined Drude peak Insulators Incoherent conductors of electricity Interactions expected to become important in the incoherent phase → Possible description in AdS/CFT?

The Cuprates The Cuprates are real life example of : Incoherent transport Anomalous scaling of conductivity and Hall angle with T [Blake, AD] σ B =0 DC ∝ T − 1 , θ H ∝ T − 2

Electrons as a soup Recent evidence for high viscosity in strongly interacting electrons. [1508.00836] , [1509.04165] , [1509.05691] Hydrodynamics accurate in the high T , momentum (quasi-) conserving regime [Hartnoll, Kovtun, Muller, Sachdev] Incoherent transport is away from this limit

Electrons as a soup Macroscopic effects of viscosity [1509.05691] ✯ 0 ✯ 0 ✟ ∇ i v i = 0 v j ∇ j v i + 2 η ∇ j ∇ ( j v i ) − ∇ i p = − g, ✟ + ✟✟✟ ✟✟ ∂ t v i v z = ( g/ 4 η ) ( R 2 − ρ 2 ) ⇒ σ DC ≈ R 2 /η btw This is a Stoke’s flow

Drude Model Put the lattice back! Lattice scattering (Drude physics) Average momentum obeys τ � p � ⇒ σ = nq 2 p � = qE − 1 τ � ˙ 1 − ıωτ ⇒ σ DC ≈ τ ≈ l m m Microscopically σ = G JJ ( ω ) / ( ıω )

Viscosity vs Lattice Scattering Im w Don’t need quasi-particles to have Drude physics. Coherent metals arise when mo- mentum relaxation is slow with Re w dominant pole on imaginary axis. [Hartnoll, Hofman] 1.0 0.5 0.8 0.4 0.6 0.3 Re [ σ ] Im [ σ ] 0.4 0.2 0.2 0.1 0.0 0.0 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 ω ω

Fourier/Ohm law We have electric currents J i and a thermal current Q i = − T it − µ J i Transport coefficients are packaged in Ohm/Fourier law � J � � � � � σ αT E = Q αT ¯ κT ¯ − ( ∇ T ) /T With ∇ T a temperature gradient

Setup In D = 4 Einstein-Maxwell with AdS asymptotics: L EM = R − 1 4 F µν F µν + 12 4 = − U ( r ) dt 2 + U ( r ) − 1 dr 2 + r 2 � ds 2 dx 2 1 + dx 2 � 2 A = a ( r ) dt Background black hole has temperature T , energy E , pressure P , entropy s and charge q .

Setup Introduce periodic lattice (deformation) on the boundary Focus on simple black hole topologies More general statements [AD, Gauntlett, Griffin, Melgar]

Setup Deform by chemical potential µ 0 and magnetic field B Hold at finite temperature T Introduce periodic sources that can relax momentum: Local chemical potential ∇ µ Local temperature ∇ T Magnetic impurities Local stress + rotation Probe with external electric field ∇ δµ = E and thermal gradient −∇ δT/T = ζ to extract conductivities

RG/Holographic picture , HSV, ... ? I Charge dominated RG flows, translations restored in IR → Coherent transport II Lattice dominated RG flows, translations broken in IR → Incoherent transport [AD, Hartnoll] [AD, Gauntlett]

Conductivity from Q-lattices [AD, Gauntlett] 3.0 T � Μ� 0.100 60 T � Μ� 0.100 T � Μ� 0.0502 50 2.5 T � Μ� 0.0503 T � Μ� 0.00625 40 T � Μ� 0.0154 T � Μ� 0.00118 Re � Σ � Re � Σ � 2.0 T � Μ� 0.00671 30 1.5 20 10 1.0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Ω � Μ Ω � Μ 0.70 0.100 0.65 0.050 0.60 0.55 0.020 Ρ Ρ 0.50 0.010 0.45 0.40 0.005 0.35 0.001 0.002 0.005 0.010 0.020 0.050 0.100 0.00 0.02 0.04 0.06 0.08 0.10 T � Μ T � Μ Can model Metal - Insulator transitions Similar story for inhomogeneous lattices [Rangamani, Rozali, Smyth]

Currents At Equilibrium For homogeneous systems we have T 0 , µ 0 , � B 0 , ... First consider hydrodynamic limit Weakly break translations µ 0 → µ 0 + δµ ( x ) , B 0 → � � B 0 + δ � B ( x ) , T → T + δµ ( x ) In hydrodynamic limit, magnetization becomes local δ � M = ∂ µ � M 0 δµ ( x ) + ∂ T � M 0 δT ( x ) + · · · ⇒ Presence of local magnetization currents J = � � ∇ × δ � M Similar for heat currents

Currents At Equilibrium

Currents At Equilibrium In the non-hydrodynamic limit k = ∂ t is a symmetry L k ∗ J = 0 ⇒ i k ( d ∗ J ) + d ( i k ∗ J ) = 0 d ( i k ∗ J ) = 0 Assuming R t × M D − 1 topology i k ∗ J = d ∗ D M + ω with ω harmonic. Currents relax � i k ∗ J = 0 ⇒ ω = 0 ⇒ J i = ∂ j ( √ g D − 1 M ij ) C D − 2 Similarly for the heat current Q i = T iµ k µ − k µ A µ J i = ∂ j ( √ g D − 1 M ij T )

DC conductivities from BH horizons Bulk theory is Einstein-Maxwell Consider E/M charged, static black branes ds 2 = − UG ( dt + χ ) 2 + F U dr 2 + ds 2 (Σ d ) A = a t ( dt + χ ) + a i dx i ds 2 (Σ d ) = g ij ( r, x ) dx i dx j Asymptotically, r → ∞ U → r 2 , F → 1 a t ( r, x ) → µ ( x ) , a i ( r, x ) → a i ( x ) G → ¯ g ij ( r, x ) → r 2 ¯ G ( x ) , g ij ( x ) , χ i ( r, x ) → ¯ χ i ( x ) Local µ , B , T , mag impurities, surface forces

DC conductivities from BH horizons For the perturbation write δ ( ds 2 ) = δg µν ( r, x ) dx µ dx ν − 2 tGUζ i dtdx i , δA = δa µ ( r, x ) dx µ − tE i dx i + ta t ζ i dx i E ( x i ) and ζ ( x i ) are closed forms ζ is boundary temperature gradient E is boundary electric field Count functions: g µν → 1 2 ( d + 2) ( d + 3) − ( d + 2) functions A µ → ( d + 2) − 1 functions

Radial Hamiltonian Imagine radial foliation by hypersurfaces e.g. normal to ∂ r Radial evolution Hamiltonian is sum of constraints � N H + N µ H µ + D G + b . t . H ∂ r = At infinity they yield Ward identities ∇ µ � T µν � = F µν � J ν � , ∇ µ � J µ � = 0 , � T µµ � = anom Meaningful but not closed system without hydro

DC conductivities from BH horizons Projection of metric h µν and gauge field b µ on r = ε surface Conjugate momentum densities π µν and π µ with respect to ∂ r “Evolution” equations h µν = δH ∂ r π µν = − δH ∂ r ˙ δπ µν , ˙ δh µν b µ = δH ∂ r π µ = − δH ∂ r ˙ δπ µ , ˙ δb µ

DC conductivities from BH horizons And constraints H ν = D µ t µν − 1 2 f νρ j ρ = 0 G = D µ j µ = 0 With t µν = ( − h ) − 1 / 2 π µν and j µ = ( − h ) − 1 / 2 π µ . Continuity equations on the surface

DC conductivities from BH horizons Examine constraints close to the horizon Impose infalling conditions Define v i ≡ − δg (0) w ≡ δa (0) it , , t p ≡ − 4 πT δg (0) G (0) − δg (0) it g ij rt (0) ∇ j ln G (0)

DC conductivities from BH horizons Constraints on the horizon give H t ⇒ ∇ i v i = 0 ∇ 2 w + ∇ i ( F (0) ik v k ) + v i ∇ i a (0) = −∇ i E i G ⇒ t H j ⇒ 2 ∇ i ∇ ( i v j ) + a (0) t ∇ j w − ∇ j p ji v i + F (0) t v i + F i (0) k v k ) + 4 πT dχ (0) ji ( ∇ i w + a (0) = − 4 πT ζ j − a (0) t E j − F (0) ji E i Solve for a Stokes flow on the curved black hole horizon Closed system of equations in d dimensions Nowhere made hydro assumptions! Related (?) work [Damour][Thorne, Price][Eling, Oz][Bredberg, Keeler, Lysov, Strominger]

DC conductivities from BH horizons Electric Current Define J i = √− gF ir At r → ∞ gives field theory current densities J i ∞ Anywhere in the bulk � √− gF ji � + √− g F ij ζ j ∂ r J i = ∂ j ∂ i J i = J i ζ i

DC conductivities from BH horizons Heat Current Let k = ∂ t and define G µν = − 2 ∇ [ µ k ν ] − k [ µ F ν ] σ A σ − 1 2 ( φ − θ ) F µν and Q i = √− gG ir At r → ∞ gives field theory heat current densities Q i δT it δJ i � � � � ∞ = − − µ Anywhere in the bulk � √− gG ji � + 2 √− gG ij ζ j + √− gZF ij E j ∂ r Q i = ∂ j ∂ i Q i = 2 Q i ζ j + J i E i

DC Conductivities from BH horizons For the background ( E i = ζ i = 0 ) we have = ∂ j M ( B ) ij J ( B ) i = ∂ j M ( B ) ij , Q ( B ) i T ∞ ∞ with the magnetizations � ∞ � ∞ dr √− g F ij , dr √− g G ij M ij M ij ( x ) = − T ( x ) = − r + r + satisfying ∂ i J ( B ) i ∂ i Q ( B ) i = 0 , = 0 ∞ ∞ and giving and no fluxes!

DC Conductivities from BH horizons Back to perturbations we write... (0) + ∂ j M ij − M ( B ) ij ζ j J i ∞ = J i (0) + ∂ j M ij T − M ( B ) ij E j − 2 M ( B ) ij Q i ∞ = Q i ζ j T The “transport components” of the currents are then [Cooper, Halperin, Ruzin] J i ∞ = J i Q i ∞ = Q i (0) , (0) Important point is ∂ i J i ∂ i Q i ∞ = 0 , ∞ = 0 ⇒ Meaningful to examine fluxes through d − 1 cycles!