Scale invariant fluid dynamics for the dilute Fermi gas at unitarity

Transcription

1 Scale invariant fluid dynamics for the dilute Fermi gas at unitarity 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 1 Re = η n fluid property mvl flow property Consider mvl : Hydrodynamics requires η/( n) < 1

5 Shear viscosity in kinetic theory 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 pl mfp Weakly interacting gas: l mfp 1/(nσ) η 1 3 Strongly interacting gas: η(σ 0) η n pl mfp but: kinetic theory not reliable! p σ

6 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 ct 3 = 3 4 s(λ = 0) Gubser and Klebanov

7 Holographic duals: Transport properties Thermal (conformal) field theory AdS 5 black hole CFT entropy shear viscosity Strong coupling limit η s = 4πk B Hawking-Bekenstein entropy area of event horizon Graviton absorption cross section area of event horizon η s Son and Starinets (2001) h 4πk B Strong coupling limit universal? Provides lower bound for all theories? 0 g 2 N c Answer appears to be no; e.g. theories with higher derivative gravity duals.

8 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 ( TF T ) 3/2

9 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(d) AdS d+3 X d+3 t (ρv i)+ x j Π ij = 0 (ω < T)

10 Kinetics vs No-Kinetics AdS/CFT low viscosity goo gravitational dual η/s 1/(4π) kinetic liquid quasi-particles η/s > 1

11 Outline I. Conformal second order hydrodynamics II. Kinetic theory III. Transport coefficients from elliptic flow

12 I. Non-relativistic fermions in unitarity limit Consider simple square well potential a < 0 a =, ǫ B = 0 a > 0, ǫ B > 0

13 Non-relativistic fermions in unitarity limit Now take the range to zero, keeping ǫ B 0 Universal relations T = 1 ik +1/a ǫ B = 1 2ma 2 ψ B 1 ar exp( r/a)

14 Universal fluid dynamics Many body system: Effective cross section σ tr n 2/3 (or σ tr λ 2 ) Systems remains hydrodynamic despite expansion

15 Scale and conformal symmetry Gallilean boosts x = x+t t = t scale trafo x = e s x t = e 2s t conformal trafo x = x/(1+ct) 1/t = 1/t+c Ideal fluid dynamics Π 0 ij = Pδ ij +ρv i v j, P = 2 3 E First order viscous hydrodynamics ( δ (1) Π ij = ησ ij, σ ij = i v j + j v i 2 ) 3 δ ij( v), ζ = 0

16 Scale invariant superfluid hydrodynamics Momentum density: π i = ρ n v n,i +ρ s v s,i Stress tensor Π ij +δπ ij with δπ ij = η ( i v n,j + j v n,i 23 ) δ ij i v n,i δ ij (ζ 1 i (ρ s (v s,i v n,i ))+ζ 2 ( i v n,i ) ) Equation of motions for v s : v s (v2 s) = (µ+h) with H = ζ 3 i (ρ s (v s,i v n,i )) ζ 4 i v n,i Conformal symmetry: ζ 1 = ζ 2 = ζ 4 = 0 Son (2007)

17 Why second order fluid dynamics? Consider ideal expansion after release from a harmonic trap ρ(x i,t) = ρ 0 (b i (t)x i ), v i (x j,t) = α i (t)x i, α i (t) = ḃi(t)/b i (t) Compare ideal and dissipative stresses Π 0 ii = P +ρα i x 2 i, δ (1) Π ii = η ( α i 4 3 j α ) j Π 0 xx δ (1) Π xx x x Ideal stresses propagate with speed c s, dissipative stresses propagate with infinite speed. Hydro always breaks down in the dilute corona.

18 Second order conformal hydrodynamics Relaxation of shear stress is a second order hydro term. Complete list [ 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 +γ 1 i T j T +γ 2 i P j P +γ 3 i j T +... 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)

19 II. Linear response and kinetic theory Consider background metric g ij (t,x) = δ ij +h ij (t,x). Linear response δπ ij = δπeq ij h ij 1 δh ij 2 Gijkl R h kl Kubo relation: η(ω) = 1 ω ImGxyxy R (ω, 0) Kinetic theory: Boltzmann equation ( ( t + pi m x i g il ġ lj p j +Γ i jk p j p k m ) ) p i f(t,x,p) = C[f] C[f] =

20 Kinetic theory linearize f = f 0 +δf, solve for δf, δπ ij, G R, η(ω) η(ω) = η 1+ω 2 τ 2 R η = π (mt)3/2 τ R = η nt shear channel sound channel xyxy (,k)/( k F a= T T/T F =1, k=0.5 T/T F =1, k=1.0 T/T F =2, k=0.5 T/T F =2, k=1.0 T/T F =3, k=0.5 T/T F =3, k=1.0 zzzz (,k)/( k F a=100 T/T F =1, k=0.5 T/T F =1, k=1.0 T/T F =2, k=0.5 T/T F =2, k=1.0 T/T F =3, k=0.5 T/T F =3, k= T Braby, Chao, T.S. (2010)

21 Shear viscosity: Sum rules Randeria & Taylor proved the sum rules (corrected by Enss & Zwerger) [ ] 1 C dw η(ω) π 15π = E mω 3 C 10πma ( ) 1 1 C dw ζ(ω) = π 72πma 2 a 1 where C is Tan s contact, ρ(k) C/k 4. Sum rules constrain spectral function and euclidean correlator η(ω)/n T/T F = 0.5, 1, 2 G xyxy E (τ) G full E G tail E ω/t F τt F

22 Second order hydrodynamics from kinetic theory Boltzmann equation (BGK approximation) [ Dσ ij σij ( v) δ (2) Π ij = η2 P + η2 P [ σ i k σj k +σ i k Ωj k] +O(κη i j T) ] relaxation time τ R = η P η nt

23 III. Almost ideal fluid dynamics (cold Fermi gas) σ σ µ σ σ Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy O Hara et al. (2002)

24 Elliptic flow: High T limit Quantum viscosity η = η 0 (mt) 3/ Aspect Ratio η = η 0 (mt) 3/2 τ R = η/p Cao et al., Science (2010) Time After Release (µs) fit: η 0 = 0.33±0.04 theory: η 0 = π = 0.26

25 Elliptic flow: Freezeout? switch from hydro to (weakly collisional) kinetics at scale factor b fr = 1,5,10,20 A R t [ms] no freezeout seen in the data

26 Elliptic flow: Shear vs bulk viscosity Dissipative hydro with both η, ζ A R 1.5 ideal 1.0 η = ζ = t [ms]

27 Elliptic flow: Shear vs bulk viscosity Dissipative hydro with both η, ζ β η,ζ = (η,ζ) E F E 1 (3λN) 1/ β η 0.2 η ζ β ζ Dusling, Schaefer (2010)

28 Viscosity to entropy density ratio consider both collective modes (low T) and elliptic flow (high T) Cao et al., Science (2010) η/s 0.4

29 Outlook Experimental determination of transport properties: Collective modes and elliptic flow give η/s < 0.4. Local analysis requires second order hydro or hydro+kinetic. (I am working on this.) Shear viscous relaxation time can be measured by comparing collective modes and elliptic flow. Can we observe breaking of scale invariance and the return of bulk viscosity away from unitarity? Can we measure η and ζ 3 in the superfluid phase?

30 Note: Experiment (Helium vs Fermi gas at unitarity) η/s η [µp] T [K] T/T F