Strongly interacting quantum fluids: Transport theory

Transcription

1 Strongly interacting quantum fluids: Transport theory Thomas Schaefer North Carolina State University

2 Fluids: Gases, Liquids, Plasmas,... Hydrodynamics: Long-wavelength, low-frequency dynamics of conserved or spontaneously broken symmetry variables. τ τ micro τ λ Historically: Water (ρ, ǫ, π)

3 Simple non-relativistic fluid Simple fluid: Conservation laws for mass, energy, momentum ρ t + (ρ v) = 0 ǫ t + j ǫ = 0 t (ρv i) + Π ij = 0 x j Constitutive relations: 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 ) reactive dissipative 2nd order Expansion Π 0 ij δπ1 ij δπ2 ij

4 Regime of applicability Expansion parameter Re 1 = η( v) ρv 2 = η ρlv 1 Re = n η fluid property mvl flow property Kinetic theory estimate: η npl mfp Re 1 = v c s Kn Kn = l mfp L expansion parameter Kn 1

5 Relativistic hydrodynamics Energy momentum tensor of an ideal fluid T µν = (ǫ + P)u µ u ν + Pη µν, Energy-momentum conservation: µ T µν = 0 µ (su µ ) = 0 Du µ = 1 ǫ + P µ P D = u µ = µν ν µν = η µν + u µ u ν Viscous contribution δ (1) T µν = ησ µν ζ µν u σ µν = µα ( νβ α u β + β u α 2 ) 3 η αβ u.

6 Relativistic hydrodynamics: causality Linearized hydro: consider small fluctuations g i = δt 0i S L gg = 2sT Γ s ω 2 k 2 (ω 2 c 2 sk 2 ) 2 + (Γ s ωk 2 ) 2 Γ s = 4 3 η + ζ st S T gg = 2ηk 2 ω 2 + ( η st k2 ) 2 L/T channel: propagating sound mode, diffusive shear mode. Consider speed of shear wave v diff = ω k = η st k Find acausal behavior v diff > c for k > k cr. Occurs outside regime of validity of hydro. But: Causes numerical difficulties.

7 Second order hydrodynamics Causality can be restored by introducing a finite relaxation time η η(ω) 1 + iωτ π More formal approach: Second order hydrodynamics (BRSSS) [ δ (2) T µν = ητ II Dσ µν + 1 ] 3 σµν ( u) + λ 1 σ µ λ σν λ + λ 2 σ µ λ Ων λ + λ 3 Ω µ λ Ων λ A µν = 1 2 µα νβ A αβ + A βα 2 3 µν αβ A αβ Ω µν = 1 2 µα νβ αu β β uα Contains four new transport coefficients τ II, λ i Can be written as a relaxation equation for π µν δt µν π µν = ησ µν τ II Dπ µν +...

8 Shear viscosity Viscosity determines shear stress ( friction ) in fluid flow F = A η v x y Kinetic theory: conserved quantities carried by quasi-particles f p t + v x f p + F p f p = C[f p ] η 1 3 n p l mfp Dilute, weakly interacting gas: l mfp 1/(nσ) η 1 3 p σ independent of density!

9 Shear viscosity non-interacting gas (σ 0): η non-interacting and hydro limit (T ) limit do not commute η strongly interacting gas: n pl mfp but: kinetic theory not reliable! what happens if the gas condenses into a liquid? liquid gas Pair correlation function

10 Viscosity of a liquid is a dominant theme here, and you know Vicki s program of explaining everything in terms of fundamental constants. The viscosity of a liquid is a tough nut to crack... because when the stuff is cooled by 40 o its viscosity can change by a factor of Purcell anticipates Vicki s theory η exp(e/t) But it s more mysterious than that: The viscosities have a big range, but they all stop in the same place. I don t understand that. Eyring, Frenkel: η hn exp(e/t)

11 And now for something completely different...

12 Gauge theory at strong coupling: Holographic duality The AdS/CFT duality relates large N c (conformal) gauge theory in 4 dimensions correlation fcts of gauge invariant operators exp dx φ 0 O = string theory on 5 dimensional Anti-de Sitter space S 5 boundary correlation fcts of AdS fields Z string [φ( AdS) = φ 0 ] The correspondence is simplest at strong coupling g 2 N c strongly coupled gauge theory classical string theory

13 Holographic duals at finite temperature Thermal (conformal) field theory AdS 5 black hole CFT temperature Hawking temperature of black hole CFT entropy Hawking-Bekenstein entropy area of event horizon s weak coupling strong coupling λ = g 2 N s(λ ) = π2 2 N2 c T 3 = 3 4 s(λ = 0) Gubser and Klebanov

14 Holographic duals: Transport properties Thermal (conformal) field theory AdS 5 black hole T µν = 1 g δs δg µν g µν = g 0 µν + γ µν CFT entropy shear viscosity Hawking-Bekenstein entropy area of event horizon Graviton absorption cross section area of event horizon

15 Holographic duals: Transport properties Thermal (conformal) field theory AdS 5 black hole CFT entropy shear viscosity Strong coupling limit Hawking-Bekenstein entropy area of event horizon Graviton absorption cross section area of event horizon η s η s = 4πk B Son and Starinets (2001) h 4πk B Strong coupling limit universal? Provides lower bound for all theories? 0 g 2 N c

16 Viscosity bound: Common fluids 4π η h s Helium 0.1MPa Nitrogen 10MPa Water 100MPa Viscosity bound T, K

17 Kinetics vs No-Kinetics AdS/CFT low viscosity goo pqcd kinetic plasma

18 Effective theories for fluids (Here: Weak coupling QCD) L = q f (id/ m f )q f 1 4 Ga µνg a µν f p t + v x f p = C[f p ] (ω < T) t (ρv i) + x j Π ij = 0 (ω < g 4 T)

19 Effective theories (Strong coupling) L = λ(iσ D)λ 1 4 Ga µνg a µν +... S = 1 2κ 2 5 d 5 x gr +... t (ρv i) + x j Π ij = 0 (ω < T)

20 Kinetics vs No-Kinetics Spectral function ρ(ω) = ImG R (ω, 0) associated with T xy 1 ρ xyxy (ω) s 2ω 0.6 1/g 4 (ω/t) AdS/CFT π(ω/2πt) /s ρ xyxy (ω)/2ω g η/s=1/4π g 4 T g 2 T gt T weak coupling QCD ω ω/2πt strong coupling AdS/CFT transport peak vs no transport peak

21 Transport coefficients, theory 1. Kinetic theory 2. Kubo formula, lattice 3. Dynamic universality 4. Holography

22 Kinetic theory Quasi-Particles (γ ω): introduce distribution function f p (x, t) N = d 3 p f p T ij = d 3 p p i p j f p, Boltzmann equation f p t + v x f p + F p f p = C[f p ] Collision term C[f p ] = C gain C loss C loss = dp dqdq f p f p w(p, p ;q, q ) C gain =... p q p q p q p q p q p q

23 Linearized theory (Chapman-Enskog): f p = f 0 p(1 + χ p /T) RHS = C[f p ] f0 p T C pχ p Linear response to flow gradient linear collision operator f p = exp( (E p p v(x))/(kt)) Drift term proportional to driving term (v ij = i v j + j v i trace) LHS = f p t + v x f p f0 p T X Boltzmann equation X p ip j v ij C p χ p = X χ p g p p i p j v ij (χ p ) ij v ij

24 compute T ij [f 0 p + δf p ] T 0 ij + ηv ij η X χ X χ = d 3 p f 0 p (p i p j χ ij p ) Use Boltzmann equation C p χ p = X: η χ C p χ Variational principle χ var C p χ var χ C p χ χ var C p χ 2 = χ var X 2 Best bound for g p p α (α 0.1) η χ var X 2 χ var C χ var η = 0.34T 3 α 2 s log(1/α s ) log(α s ) from dynamic screening Baym et al. (1990)

25 pqcd and psym: weak versus strong coupling Arnold, Dogan, Moore (2006) Huot, Jeon, Moore (2006)

26 Kinetic Theory: Quasiparticles low temperature high temperature unitary gas phonons atoms + + helium phonons, rotons atoms + + QCD pions quarks, gluons

27 Theory Summary η/s unitary gas η/s 6 4 He T/T F T[K] η/s 5 4 QCD T[MeV ]

28 What if the coupling is strong? Kubo Formula Linear response theory provides relation between transport coefficients and Green functions G R (ω, 0) = dt d 3 x e iωt Θ(t) [T xy (t, x), T xy (0,0)] 1 η = lim ω 0 ω G R(ω, 0) This result is hard to use for quantum fluids, but there are some heroic efforts by lattice QCD theorists, e.g. Meyer (2007).

29 ρ(ω) K(x 0 =1/2T,ω)/T 4 T=1.65T c T=1.24T c ω/t T 1.02 T c 1.24 T c 1.65 T c η/s 0.102(56) 0.134(33) ζ/s 0.73(3) 0.065(17) 0.008(7)

30 Dynamic Universality Continuous phase transition: Dynamics of low energy modes universal Universality for transport coefficients Universal theory: Hydro (diffusive modes), order parameters (time dependent LG), stochastic forces (Langevin) [ t (ρv i) = Pij η 0 2 δh δ(ρv j ) + w 0( j φ) δh ] δφ + ζ j Model H of Hohenberg and Halperin η ξ x η (x η 0.06) ζ ξ x ζ (x ζ 2.8)

31 Anti-DeSitter Space Consider a hyperboloid embedded in 6-d euclidean space R 2 = x 2 i x 2 0 x 2 5 i=1,4 This is a space of constant negative curvature, and a solution of the Einstein equation R µν 1 2 g µνr = 1 2 g µνλ with negative cosmological constant. Isometries of AdS 5 : SO(4,2) conformal group in d = 3 + 1

32 metric of AdS 5 S 5 (note that (L/l s ) 4 = g 2 N c ) ds 2 = r2 L 2 ( dt 2 + dx 2) + L2 r 2 dr2 + L 2 dω 2 5 r boundary of AdS 5 Finite temperature: AdS 5 black hole solution ds 2 = r2 L 2 ( f(r)dt 2 + dx 2) + L2 f(r)r 2 dr2, where f(r) = 1 (r 0 /r) 4. Hawking temperature T H = r 0 /π. Compute induced stress tensor on the boundary 2 δs T µν = lim ǫ 0 g δg µν Find ideal fluid with T µν = diag(ǫ, P, P, P), bd ǫ 3 = P = N2 c 8π 2 (πt)4

33 Eddington-Finkelstein coordinates Hydrodynamics from AdS/CFT ds 2 = 2dv dr + r2 L 2 [ f(r)dv 2 + r 2 dx 2] Introduce local rest frame u µ = (1,0), scale parameter b ds 2 = 2u µ dx µ dr + r2 L 2 [ f(br)uµ u ν dx µ dx ν + r 2 P µν dx µ dx ν] Pµν = uµuν + ηµν promote u µ (x) and b(x) to fields determine metric order by order in gradients compute induced stress

34 leading order: ideal fluid dynamics with ǫ = 3P T µν 0 = N2 c 8π 2 (πt)4 (η µν + 4u µ u ν ) next-to-leading order: Navier-Stokes with η/s = 1/(4π) δ (1) T µν = N2 c 8π 2 (πt)3 σ µν next-to-next-to-leading order: second order conformal hydro [ δ (2) T µν = ητ II Dσ µν + 1 ] 3 σµν ( u) + λ 1 σ µ λ σν λ + λ 2 σ µ λ Ων λ + λ 3 Ω µ λ Ων λ relaxation times τ Π = 2 ln 2 πt λ 1 = 2η πt λ 2 = 2η ln 2 πt λ 3 = 0