Transport in Conformal Quantum Fluids Thomas Schaefer, North Carolina State University
Why study transport in (nearly perfect) quantum fluids? Transport without quasi-particles? Model system for QGP, strange metals, etc. Fluid dynamics is the universal effective description of non-equilibrium many body systems. Description is most effective in nearly perfect fluids. Fluid-gravity correspondence: Can (strongly coupled) fluids teach us something about quantum gravity?
Hydrodynamics Hydrodynamics (undergraduate version): Newton s law for continuous, deformable media.
Fluids: Gases, liquids, plasmas,... Hydrodynamics (postmodern): Effective theory of nonequilibrium long-wavelength, low-frequency dynamics of any many-body system. non conserved density conserved density τ τ micro τ λ τ τ micro : Dynamics of conserved charges. Water: (ρ, ɛ, π)
Not your grandfathers fluid Consider a many body system with σ 1/k 2 Can be made using Feshbach resonances in dilute atomic gases. Systems remains hydrodynamic despite expansion
Effective theories for fluids (Unitary Fermi Gas, T > T F ) L = ψ ( i 0 + 2 2M ) ψ C 0 2 (ψ ψ) 2 f p t + v x f p = C[f p ] ω < T t (ρv i) + x j Π ij = 0 ω < T s η
Effective theories (Strong coupling) L = λ(iσ D)λ 1 4 Ga µνg a µν +... S = 1 2κ 2 5 d 5 x gr +... SO(d + 2, 2) Schr 2 d AdS d+3 Schr 2 d t (ρv i) + x j Π ij = 0 (ω < T )
Outline I. EFT: Gradient expansion II. EFT: Fluctuations III. Models of fluids: Kinetic theory & QFT IV. Models of fluids: Holography V. Analyzing fluids: How to measure η/s VI. Analyzing fluids: How to measure D s
I. Gradient expansion (simple non-relativistic fluid) Simple fluid: Conservation laws for mass, energy, momentum ρ t + j ρ = 0 ɛ t + j ɛ = 0 π i t + x j Π ij = 0 Ward identity: mass current = momentum density j ρ ρ v = π Constitutive relations: Gradient expansion for currents Energy momentum tensor Π ij = P δ ij + ρv i v j + η ( i v j + j v i 23 ) δ ij k v k + O( 2 )
Conformal fluid dynamics: Symmetries Symmetries of a conformal non-relativistic fluid Galilean boost x = x + vt t = t Scale trafo x = e s x t = e 2s t Conformal trafo x = x/(1 + ct) 1/t = 1/t + c This is known as the Schrödinger algebra (= the symmetries of the free Schrödinger equation) Generators: Mass, momentum, angular momentum M = dx ρ P i = dx j i J ij = dx ɛ ijk x j j k Boost, dilations, special conformal K i = dx x i ρ D = dx x j C = dx x 2 ρ/2
Spurion method: Local symmetries Diffeomorphism invariance δx i = ξ i (x, t) δg ij = L ξ g ij = ξ k k g ij +... Gauge invariance δψ = iα(x, t)ψ δa 0 = α ξ k k A 0 A k ξk δa i = i α ξ k k A i A k i ξ k + mg ik ξk Conformal transformations δt = β(t) δo = βȯ 1 2 βo O More recent work: Newton-Cartan geometry Son, Wingate (2006), Jensen (2014)
Example: Stress tensor Determine transformation properties of fluid dynamic variables δρ = L ξ ρ δs = L ξ s δv = L ξ v + ξ Stress tensor: Ideal fluid dynamics Π 0 ij = P g ij + ρv i v j, P = 2 3 E First order viscous hydrodynamics δ (1) Π ij = ησ ij ζg ij σ ζ = 0 σ ij = ( i v j + j v i + ġ ij 2 ) 3 g ij σ σ = v + ġ 2g Son (2007)
Simple application: Kubo formula Consider background metric g ij (t, x) = δ ij + h ij (t, x). Linear response δπ xy = 1 2 Gxyxy R h xy Harmonic perturbation h xy = h 0 e iωt Kubo relation: G xyxy R = P iηω +... η = lim ω 0 [ 1 ω ImGxyxy R (ω, 0) Gradient expansion: ω P η s η T. ]
II. Beyond gradients: Hydrodynamic fluctuations Hydrodynamic variables fluctuate δv i (x, t)δv j (x, t) = T ρ δ ijδ(x x ) Linearized hydrodynamics propagates fluctuations as shear or sound δvi T δvj T ω,k = 2T ρ (δ ij ˆk νk 2 iˆkj ) ω 2 + (νk 2 ) 2 δv L i δv L j ω,k = 2T ρ ˆk iˆkj shear ωk 2 Γ (ω 2 c 2 sk 2 ) 2 + (ωk 2 Γ) 2 sound v = v T + v L : v T = 0, v L = 0 ν = η/ρ, Γ = 4 3 ν +...
Hydro Loops: Breakdown of second order hydro Correlation function in hydrodynamics G xyxy S = {Π xy, Π xy } ω,k ρ 2 0 {v x v y, v x v y } ω,k v T ρ ρ v T vl v L v L ρ ρ ρ vt Match to response function in ω 0 (Kubo) limit ρ G xyxy R = P + δp iω[η + δη] + ω 2 [ητ π + δ(ητ π )] with δp T Λ 3 δη T ρλ η δ(ητ π ) 1 ω T ρ 3/2 η 3/2
Hydro Loops: RG and breakdown of 2nd order hydro Cutoff dependence can be absorbed into bare parameters. Non-analytic terms are cutoff independent. Fluid dynamics is a renormalizable effective theory. ( ) 2 ( ρ P Small η enhances fluctuation corrections: δη T η ρ ) 1/2 Small η leads to large δη: There must be a bound on η/n. Relaxation time diverges: δ(ητ π ) 1 ω ( ρ η ) 3/2 2nd order hydro without fluctuations inconsistent.
Fluctuation induced bound on η/s η/s 1.0 0.8 0.25 η(ω)/s 0.20 fluctuations 0.6 0.15 0.4 0.10 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (η/s) min 0.2 0.05 kinetic theory T/T F ω/t F 0.0 0.5 1.0 1.5 2.0 spectral function non-analytic ω term Schaefer, Chafin (2012), see also Kovtun, Moore, Romatschke (2011)
IIIa. Kinetic theory Microscopic picture: Quasi-particle distribution function f p (x, t) ρ(x, t) = dγ p gmfp (x, t) π i (x, t) = dγ p gpi f p (x, t) Π ij (x, t) = dγ p gpi v j f p (x, t) Boltzmann equation ( t + pi m x i ( g il ġ lj p j + Γ i p j p k ) ) jk m p i f p (t, x, ) = C[f] C[f] = Solve order-by-order in Knudsen number Kn = l mfp /L
Kinetic theory: Knudsen expansion Chapman-Enskog expansion f = f 0 + δf 1 + δf 2 +... Gradient exp. δf n = O( n ) Knudsen exp. δf n = O(Kn n ) First order result Bruun, Smith (2005) δ (1) Π ij = ησ ij η = 15 32 (mt )3/2 π Second order result Chao, Schaefer (2012), Schaefer (2014) [ δ (2) Π ij = η2 Dσ ij + 2 ] P 3 σij ( v) + η2 P [ ] 15 14 σ i k σj k σ i k Ωj k + O(κη i j T ) relaxation time τ π = η/p
Frequency dependence, breakdown of kinetic theory Consider harmonic perturbation h xy e iωt+ikx. Use schematic collision term C[f 0 p + δf p ] = δf p /τ. δf p (ω, k) = 1 2T iωp x v y iω + i v k + τ 1 0 Leads to Lorentzian line shape of transport peak f 0 p h xy. η(ω) = η(0) 1 + ω 2 τ 2 0 Pole at ω = iτ 1 0 (τ 0 = η/(st )) controls range of convergence of gradient expansion. High frequency behavior misses short range correlations for ω > T.
IIIb. Quantum Field Theory = The diagrammatic content of the Boltzmann equation is known: Kubo formula with Maki-Thompson + Azlamov-Larkin + Self-energy Can be used to extrapolate Boltzmann result to T T F viscosity η [ h n] 200 100 50 20 10 5 2 1 0.5 classical 2.77 T 3/2 viscosity η(ω=0) 0.2 0.1 0.2 0.5 1 2 5 10 20 T [T F ] η(ω) [ h n] 100 10 1 0.1 0.01 0.001 0.1 1 10 100 1000 ω [E F / h] T=10 T= 5 T= 2 T= 1 T=0.5 T=0.151 Enss, Zwerger (2011), see also Levin (2014)
Operator product expansion (OPE) Short time behavior: OPE η(ω) = n O n ω ( n d)/2 O n (λ 2 t, λx) = λ n O n (t, x) Leading operator: Contact density (Tan) O C = C 2 0ψψψ ψ = ΦΦ C = 4 η(ω) O C / ω. Asymptotic behavior + analyticity gives sum rule [ 1 dw η(ω) O ] C π 15π = E mω 3 Randeria, Taylor (2010), Enss, Zwerger (2011), Hoffman (2013)
IV. Holography DLCQ idea: Light cone compactification of relativistic theory in d+2 p µ p µ = 2p + p p 2 = 0 p = p2 p + = 2n + 1 2p + L Galilean invariant theory in d+1 dimensions. String theory embedding: Null Melvin Twist AdS d+3 NMT Schr 2 d Iso(AdS d+3 ) = SO(d + 2, 2) Schr(d) Son (2008), Balasubramanian et al. (2008) Other ideas: Horava-Lifshitz (Karch, 2013)
Schrödinger Metric Fluctuations δg y x = e iωu χ(ω, r) satisfy wave equation (u = (r + /r) 2 ) χ (ω, u) 1 + u2 f(u)u χ (ω, u) + u f(u) 2 w2 χ(ω, u) = 0 Retarded correlation function G R (ω) = βr3 + v f(u)χ (ω, u) 4πG 5 uχ(ω, u). u 0 Viscosity from Kubo relation η s = 1 4π Adams et al. (2008), Herzog et al. (2008)
Spectral function η(ω)/s 0.14 0.12 0.10 0.08 0.06 0.04 0.02 η(0)/s = 1/(4π) η(ω ) ω 1/3 0.0 0.5 1.0 1.5 2.0 ω/(πt ) Kubo relation (incl. τ π ): G R (ω) = P iηω + τ π ηω 2 + κ R k 2 τ π T = log(2) 2π AdS 5 : τ π T = 2 log(2) 2π Range of validity of fluid dynamics: ω < T Sch 2 : Cannot be matched to relaxation type hydro? Schaefer (2014), BRSSS (2008)
Quasi-normal modes 1.0 0.5 0.5 1.0 2 Re w 4 2 2 4 1 Re w 4 2 6 3 8 Im w 4 Im w Sch 2 2 AdS 5 QNM s are stable, Im λ < 0. Pole at ω it limits convergence of fluid dynamics. Modes overdamped in Sch 2 2. Schaefer (2014), Starinets (2002), Heller (2012)
V. Experiments: Elliptic flow σ σ µ Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy σ σ O Hara et al. (2002)
Determination of η(n, T ) Measurement of A R (t, E 0 ) determines η(n, T ). But: gas n(x) fluid The whole cloud is not a fluid. Can we ignore this issue? v(x) transition regime heat current viscous stress δπ( x) No. Hubble flow & low density viscosity η T 3/2 lead to paradoxical fluid dynamics. Q = σ δπ =
Possible Solutions Combine hydrodynamics & Boltzmann equation. Not straightforward. Hydrodynamics + non-hydro degrees of freedom (E a ; a = x, y, z) E a t + j a ɛ = P a 2τ E t + j ɛ = 0 P a = P a P E = a E a τ small: Fast relaxation to Navier-Stokes with τ = η/p τ large: Additional conservation laws. Ballistic expansion.
Anisotropic Hydrodynamics: Comparison with Boltzmann Aspect ratio A R (t) = ( r 2 / r2 z ) 1/2 (T/T F = 0.79, 1.11, 1.54) Dots: Two-body Boltzmann equation with full collision kernel Lines: Anisotropic hydro with η fixed by Chapman-Enskog High temperature (dilute) limit: Perfect agreement! Boltzmann, Pantel et al (2015); AVH1 hydro code, M. Bluhm & T.S. (2015)
Elliptic flow: High T limit Quantum viscosity η = η 0 (mt ) 3/2 2 T /T F = 0.79, 1.11, 1.54 Cao et al., Science (2010) Bluhm et al., PRL (2016) fit: η 0 = 0.282 ± 0.02 theory: η 0 = 15 32 π = 0.269
Anisotropic fluid dynamics analysis 1.50 1.45 AR(t * ) 1.40 1.35 1.30 1.25 0.6 0.8 1.0 1.2 1.4 1.6 1.8 E/(NE F ) A R = σx/σy as function of total energy. Data: Joseph et al (2016). E/(NE F ) 0.6 is the superfluid transition. Grey, Blue, Green: LO, NLO, NNLO fit. η = η 0 (mt ) 3/2 { 1 + η 2 nλ 3 + η 3 (nλ 3 ) 2 +... }
Reconstruct η/n and η/s 8 1.0 6 0.8 η/n 4 η/s 0.6 0.4 2 0.2 0 0.5 1.0 1.5 2.0 T/T F 0.2 0.4 0.6 0.8 1.0 T/T F Left: η/n (Red band) Right: η/s (Red band) Tc 0.17T F. Joseph et al. (Black points). Enss et al. (Dashed line). Kinetic theory at low and high T (blue dashed) η(t T c ) = (0.265 ± 0.02)(mT ) 3/2 η 0 (th) = 15 32 π = 0.269 η/n Tc = 0.41 ± 0.15 η/s Tc = 0.56 ± 0.20
VI. Spin Diffusion 2 100 8 6 4 md s /ħ 2 10 8 6 4 2 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 T/T F Simple scaling at high T ( T ) 3/2 P exp( Γt) Γ s ω2 z Γ D S = 6.3 m T F Sommer et al. (2011) Theory predicts Ds = 1.1( /m)(t /T F ) 3/2, Bruun (2010), Sommer et al. (2011)
Spin Diffusion at high temperature Kinetic theory (T > T f ) D 1/n 1.5 1.0 Get large spin current j s D M in dilute corona. Predict spin drag Γ s 1.8E F (0) Γ red ( TF T ) 1/2 with Γ red > 200. Experiment 11.3 z 0.5 0.0-0.5-1.0-1.5 0.0 0.5 1.0 1.5 ρ
Spin Hydrodynamics 4 0.225 0.200 0.175 2 0.150 d z 2 4 6 8 10 12 d z 0.125-2 0.100 0.075-4 t 2 4 6 8 10 t Crossover from ballistic to diffusive behavior for D const Decay of spin dipole mode for diffusion constant D 1/n D = β, β = (1000, 5, 2, 1, 0.05) D = β T (mt ) 3/2 /n, β T = (0.2, 0.1, 0.05, 0.02) Find Γ red 11.0, in agreement with experiment
Time to pour yourself a good fluid Questions to ponder: Fluid dynamics as an E(F)T? Unfold temperature, density dependence of η/s, D s and κ. Best way to do hydro+? LBE, stoch hydro. Ahydro, Quasi-particles or quasi-normal modes?
Appendix I: Beyond conformal symmetry
Bulk viscosity and conformal symmetry breaking Conformal symmetry breaking (thermodynamics) 1 2E 3P = O C 12πmaP 1 6π nλ3 λ a How does this translate into ζ 0? Momentum dependent m (p). Bulk viscosity ζ = 1 24 2π λ 3 ( zλ a Im Σ(k) zt Re Σ(k) zt λ a T ɛ k Erf T ɛ k F D ) 2 ζ ( 1 2E 3P ( ɛk T ) ( ɛk ) 2 η T T )
Shear viscosity and conformal symmetry breaking Consider shear viscosity at a ( ) λ 2 η = η 0 {1 + O a 2 + O ( ) zλ a } +... Medium effects at O(zλ/a): Self energy, in-medium scattering Π(P, q) = η/η 1 0.3 0.2 0.1 t = 1 t = 2 t = 3 0-0.4-0.2 0 0.2 0.4 1/(k F a) Minimum shear viscosity achieved on BEC side Bluhm, Schaefer (2014)