(Non) Conformal Quantum Fluids. Thomas Schaefer, North Carolina State University
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1 (Non) Conformal Quantum Fluids Thomas Schaefer, North Carolina State University
2 Why study (nearly perfect) quantum fluids? Hard computational problem: Have to determine real time correlation functions. Achieve quantum supremacy? 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?
3 Effective theories for fluids (Unitary Fermi Gas, T > T F ) L = ψ ( i M ) 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
4 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)
5 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
6 I. Gradient expansion (simple non-relativistic fluid) Simple fluid: Conservation laws for mass, energy, momentum ρ t + j ρ = 0 ǫ t + j ǫ = 0 π i t + Π ij = 0 x j Ward identity: mass current = momentum density j ρ ρ v = π Constitutive relations: Gradient expansion for currents Energy momentum tensor Π ij = Pδ ij +ρv i v j +η ( i v j + j v i 23 ) δ ij k v k +O( 2 )
7 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 = dxρ P i = dxj i J ij = dxǫ ijk x j j k Boost, dilations, special conformal K i = dxx i ρ D = dxx j C = dxx 2 ρ/2
8 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)
9 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 = Pg 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 2 ) 3 g ij σ σ = v + ġ 2g Son (2007)
10 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. ]
11 Second order conformal hydrodynamics Second order gradient corrections to stress tensor [ δ (2) Π ij = ητ π Dσ ij + 2 ] 3 σij ( v) +λ 1 σ i k σj k +λ 2 σ i k Ωj k +λ 3 Ω i k Ωj k +O( 2 T) D = 0 + v A ij = 1 2 ( A ij + A ji 2 ) 3 g ij Ak k Ω ij = ( i v j j v i ) New transport coefficients τ π,λ i,γ i Can be written as a relaxation equation for π ij δπ ij π ij = ησ ij τ π [ Dπ ij ( v)πij ] +... Chao, Schaefer (2011)
12 Second order fluid dynamics: Causality Speed of diffusive wave in Navier-Stokes theory v D = ω k = 2η ρ k May encounter v D c s Not a fundamental problem (should impose k < Λ), but a nuisance in simulations. Second order fluid dynamics, relaxation type iω = νk2 1 iωτ π ( resummed hydro ) Limiting speed v D η/(ρτ π ) Find v D c s for τ π = η/p.
13 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 δv T i δv T j ω,k = 2T ρ (δ ij ˆk iˆkj ) δv L i δv L j ω,k = 2T ρ ˆk iˆkj νk 2 ω 2 +(νk 2 ) 2 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 +...
14 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 TΛ 3 δη TρΛ η δ(ητ π ) 1 ω Tρ 3/2 η 3/2
15 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.
16 Fluctuation induced bound on η/s η/s η(ω)/s 0.20 fluctuations kinetic theory (η/s) min 0.2 T/T F ω/t F spectral function non-analytic ω term Schaefer, Chafin (2012), see also Kovtun, Moore, Romatschke (2011)
17 III. 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 jk p j p k m ) ) p i f p (t,x,) = C[f] C[f] = Solve order-by-order in Knudsen number Kn = l mfp /L
18 Kinetic theory: Knudsen expansion Chapman-Enskog expansion f = f 0 +δf 1 +δf 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 = π (mt)3/2 Second order result Chao, Schaefer (2012), Schaefer (2014) [ δ (2) Π ij = η2 Dσ ij + 2 ] P 3 σij ( v) + η2 P [ ] σ i k σj k σ i k Ωj k relaxation time τ π = η/p +O(κη i j T)
19 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.
20 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). ImΣ(k) zt Bulk viscosity ζ = π λ 3 ( zλ a ReΣ(k) zt λ a T ǫ k Erf T ǫ k F D ) 2 ζ ( 1 2E 3P ( ǫk T ) ( ǫk ) 2 η T T )
21 Shear viscosity and conformal symmetry breaking Consider shear viscosity at a ( ) ( ) λ 2 zλ η = η 0 {1+O a 2 +O a } +... Medium effects at O(zλ/a): Self energy, in-medium scattering Π(P,q) = // // η/η t = 1 t = 2 t = /(k F a) Minimum shear viscosity achieved on BEC side Bluhm, Schaefer (2014)
22 Can be used to extrapolate Boltzmann result to T T F = IIIb. Quantum Field Theory The diagrammatic content of the Boltzmann equation is known: Kubo formula with Maki-Thompson + Azlamov-Larkin + Self-energy viscosity η [ h n] classical 2.77 T 3/2 viscosity η(ω=0) η(ω) [ h n] T=10 T= 5 T= 2 T= 1 T=0.5 T= T [T F ] ω [E F / h] Enss, Zwerger (2011), see also Levin (2014)
23 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)
24 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)
25 Schrödinger Metric Coordinates (u,v, x,r), periodic in v, x = (x,y) {[ ] ds 2 = r2 1 f(r) k(r) 2/3 4β 2 r 2 f(r) du 2 + β2 r+ 4 } r 4 dv 2 [1+f(r)] dudv +k(r) 1/3 { r 2 d x 2 + dr2 r 2 f(r) Fluctuations δg y x = e iωu χ(ω,r) satisfy (u = (r + /r) 2 ) χ (ω,u) 1+u2 f(u)u χ (ω,u)+ u f(u) 2w2 χ(ω,u) = 0 } Retarded correlation function G R (ω) = βr3 + v 4πG 5 f(u)χ (ω,u) uχ(ω, u). u 0 Adams et al. (2008), Herzog et al. (2008)
26 Spectral function η(ω)/s η(0)/s = 1/(4π) η(ω ) ω 1/ 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)
27 Quasi-normal modes Re w Re w Im w 4 Im w Sch 2 2 AdS 5 QNM s are stable, Imλ < 0. Pole at ω it limits convergence of fluid dynamics. Modes overdamped in Sch 2 2. Schaefer (2014), Starinets (2002), Heller (2012)
28 V. Experiments: Elliptic flow σ σ µ σ σ Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy O Hara et al. (2002)
29 Determination of η(n, T) Measurement of A R (t,e 0 ) determines η(n,t). But: fluid gas v(x) n(x) transition regime The whole cloud is not a fluid. Can we ignore this issue? heat current viscous stress δπ( x) No. Hubble flow & low density viscosity η T 3/2 lead to paradoxical fluid dynamics.
30 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 ǫ = P a 2τ E t + j ǫ = 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.
31 Anisotropic Hydrodynamics: Aspect ratio ideal hydrodynamics free streaming Navier-Stokes (α = 0.1) Navier-Stokes (α = 1) A-Hydro (α = 0.1) A-Hydro (α = 1) A-Hydro (α = 1000) A R Consider η = αn and α [0, ) Navier-Stokes: Ideal hydro very viscous hydro. A-hydro: Ideal hydro ballistic expansion. _ t AVH1 hydro code, M. Bluhm & T.S. (2015)
32 Anisotropic Hydrodynamics η = α n n η = α T (mt) 3/2 0 0 Π xx (A-Hydro) Π xx (Navier-Stokes) -P x (A-Hydro) Π xx / P Π xx / P Π xx (A-Hydro) Π xx (Navier-Stokes) -P x (A-Hydro) x / x 0 Π xx (Navier-Stokes) x / x 0 Π xx (A-Hydro) AVH1 hydro code, M. Bluhm & T.S. (2015)
33 Outlook Fluid dynamics as an E(F)T: Many interesting questions remain. Experiment: Main issue is temperature, density dependence of η/s. How to unfold? Need hydro codes that exit gracefully (anisotropic hydro, hydro+cascade, or LBE) Quasi-particles vs quasi-normal modes (kinetics vs holography) unresolved. Need better holographic models, improved lattice calculations.
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