Hydrodynamic Modes of Incoherent Black Holes

Transcription

1 Hydrodynamic Modes of Incoherent Black Holes Vaios Ziogas Durham University Based on work in collaboration with A. Donos, J. Gauntlett [arxiv: 1707.xxxxx, 170x.xxxxx] 9th Crete Regional Meeting on String Theory July 2017 Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

2 Table of Contents 1 Motivation 2 Green s Functions Perspective 3 Holography DC Conductivity Diffusive Mode 4 Outlook Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

3 Transport in Inhomogeneous Media Conductivity matrix characterizes linear response of system to external sources. Apply constant, time-dependent electric field E j and thermal gradient ζ j and read off the U(1) and heat currents: ( ) ( J i σ ij T α Q i = ij ) ( ) Ej T ᾱ ij T κ ij Translation invariance divergent DC conductivities ( δ(ω) + i ω when ω 0. For physical, finite conductivities, we need a mechanism for momentum to dissipate. Explicit breaking of translational invariance by introduction of spatially dependent sources. ζ j ) Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

4 Transport in Inhomogeneous Media Strongly broken translational invariance hydrodynamics dominated by energy and charge diffusion. Holography is useful: Hydrodynamic modes in incoherent setups. New ground states (Q-Lattices, Helical lattices,... ), DC Conductivity from black hole horizons, 1... Proposed bounds on diffusion: 2 D v 2 τ Einstein relations D κ/c hold in translational invariant case, 3 need to investigate inhomogeneous case. 1 [A. Donos and J. Gauntlett 15], [E. Banks, A. Donos and J. Gauntlett 15] 2 [Hartnoll 15] 3 [Forster 90] Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

5 Green s Functions Consider a finite temperature QFT with a conserved current J µ, preserving translational invariance. 3 Can Fourier transform the retarded two point functions G AB (t, x; t, x ) iθ(t t ) [ A(t, x), B(t, x ) ] G AB (ω, k) Define the charge susceptibility and the AC conductivity χ(ε k) lim G J t Jt (ω, ε k), σij ω 0 1 (ω) lim ε 0 iω G J i Jj (ω, ε k) We write G J t J t (ω, ε k) = N(ω, ε k) χ(ε k) iω + N(ω, ε k) 3 [Forster 90] Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

6 Green s Functions N(ω, ε k) is an analytic function of ω away from the real axis. The Ward identity µ J µ = 0 and time reversal invariance imply that N = ε 2 k ik j σ ij (ω) χ(0) + O(ε 3 ) Assume that N has no poles at ω = 0 and that DC conductivity is finite σ ij DC = lim ω 0 σij (ω) Then we find a diffusive pole of G J t J t at ω = iε 2 D(k) +..., D(k) = k ik j σ ij DC χ(0) Can also obtain Einstein relations for multiple conserved currents. Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

7 Green s Functions Let s place our theory on a space with lattice symmetry x x + L. We can perform a continuous and a discrete Fourier transform G AB (t, x; t, x ) G AB (ω, k, {n j }) and focus on G AB (ω, k) G AB (ω, k, {0}) = dx dx G AB (ω, x, x )e ik(x x) Then we can repeat the previous analysis. Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

8 Holographic Setup Bulk action: Einstein, but can generalise to include Maxwell and coupling to scalars. 1 Background static black holes: ds 2 = U(r)G dt 2 + F U(r) dr 2 + g ij dx i dx j 0 r 1 [A. Donos and J. Gauntlett 15], [E. Banks, A. Donos and J. Gauntlett 15] Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

9 Holographic Setup Bulk action: Einstein, but can generalise to include Maxwell and coupling to scalars. 1 Background static black holes: ds 2 = U(r)G dt 2 + F U(r) dr 2 + g ij dx i dx j AdS boundary asymptotics Regular horizon g ij = g (0) ij (x) +... U = 4πT r +... g ij (r, x) = r 2 ḡ ij (x) +... U = r [A. Donos and J. Gauntlett 15], [E. Banks, A. Donos and J. Gauntlett 15] Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

10 DC Conductivity from Black Hole Horizon Within linear response, the thermal conductivity relates the heat current flux to a constant thermal gradient source Q i = T κ ij ζj Holographic calculation of thermal DC conductivity: apply time-independent source δg tj ζ j on the boundary. Find (static) black hole quasinormal modes by solving equations of motion for bulk perturbations δg µν (r, x) with infalling boundary conditions at the horizon. Define a subset of bulk modes by the near horizon expansions δg ti = v i (x) +..., δg tr = p(x)/(4πt ) +... Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

11 DC Conductivity from Black Hole Horizon Local current on the horizon given by Q i (0) = 4πT g (0) v i In general the local horizon current differs from the local boundary current, but their fluxes are equal Q(0) i = Q i On horizon, obtain linearised (charged, forced) Navier-Stokes: H i v i = 0, 2 i (i v j) + j p = 4πT ζ j We interpret as v i as velocity and p as pressure of auxiliary horizon fluid. Realisation of membrane paradigm, but no hydrodynamic expansion. Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

12 Diffusive Mode We first note that there is a zero mode solution corresponding to perturbing the temperature T by a constant amount δt δg RT µν = g µν T δt Does not introduce boundary sources and is regular at horizon. To construct a bulk diffusive mode, we need to consider a time-dependent sourceless perturbation. We take [ ] δg µν = e iωt e iεk i x i δgµν RT + εδg µν {1} +... The functions δg {α} µν (x) are periodic on the lattice Bloch decomposition. Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

13 Diffusive Mode In particular ( ) v i = e iεk i x i ε v (1) i + ε 2 v (2) i +... ) p = e iεk i x (4πδT i + ε p (1) +..., δg (0) ij = e iεk g (0) i x i ij T δt +..., ω = ε ω (1) + ε 2 ω (2) +... For time-dependent perturbations with infalling conditions, we find a non-closed system of constraints on the horizon. At order O(ε) we get ω (1) = 0 and i v (1)i = 0, 2 j (j v (1) i) + i p (1) = ik i 4π δt Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

14 Diffusive Mode In particular ( ) v i = e iεk i x i ε v (1) i + ε 2 v (2) i +... ) p = e iεk i x (4πδT i + ε p (1) +..., δg (0) ij = e iεk g (0) i x i ij T δt +..., ω = ε ω (1) + ε 2 ω (2) +... For time-dependent perturbations with infalling conditions, we find a non-closed system of constraints on the horizon. At order O(ε) we get ω (1) = 0 and i v (1)i = 0, 2 j (j v (1) i) + i p (1) = ik i 4π δt But this is the DC problem with source ζ i ik i δt /T! So we know that 4πT g (0) v (1)i = i κ ij k j δt H Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

15 Diffusive Mode Finally, at order O(ε 2 ) we have i v (2)i + ik i v (1)i iω(2) 2 (g (0) ) (0) g ij ij T δt = 0 Note that we can write the thermodynamic susceptibility as a horizon integral from c T δs δt, s = 4π g (0) H After integrating we obtain the dispersion relation satisfying the Einstein relation. ω = iε 2 κij k i k j c +... Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

16 Outlook Comments: Everything generalises to finite charge density case, resulting in the generalised Einstein relations Outlook: Thermodynamic instability dynamical instability Analogous construction of diffusive modes within hydrodynamics on curved manifolds Hydrodynamic modes in models with spontaneous symmetry breaking? Bounds on diffusion from chaos? Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

17 Thank you for your attention! Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15

18 References A. Donos, J. Gauntlett Navier-Stokes Equations on Black Hole Horizons and DC Thermoelectric Conductivity Phys. Rev. D 92, (2015), arxiv: [hep-th] E. Banks, A. Donos, J. Gauntlett Thermoelectric DC conductivities and Stokes flows on black hole horizons JHEP (2015) 2015:103, arxiv: [hep-th] S. Hartnoll Theory of universal incoherent metallic transport Nature Phys. 11 (2015) 54, arxiv: [cond-mat.str-el] D. Forster Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions Advanced book classics (1990) Vaios Ziogas (Durham University) Diffusion on Incoherent Black Holes Kolymbari / 15