Fluid dynamics for the unitary Fermi gas. Thomas Schaefer, North Carolina State University

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1 Fluid dynamics for the unitary Fermi gas Thomas Schaefer, North Carolina State University

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

3 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)

4 Fermi gas at unitarity: Field Theory Non-relativistic fermions at low momentum L eff = ψ ( i M ) ψ C 0 2 (ψ ψ) 2 Unitary limit: a (DR: C 0 ) This limit is smooth (HS-trafo, Ψ = (ψ, ψ ) [ ] L = Ψ 2 i 0 + σ 3 Ψ + ( Ψ σ + Ψφ + h.c. ) 1 φ φ, 2m C 0

5 Scale invariant fluid dynamics Many body system: Effective cross section σ tr n 2/3 (mt F ) 1 At high temperature σ tr λ 2 (mt) 1 Systems remains hydrodynamic despite expansion

6 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

7 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 Schr 2 d t (ρv i) + x j Π ij = 0 (ω < T)

8 Outline I. Conformal fluid dynamics II. Kinetic theory III. Quantum field theory IV. Holography V. Experiment

9 I. Conformal fluid dynamics 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 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 Son (2007)

10 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 + γ 1 i T j T + γ 2 i P j P + γ 3 i j T +... D = 0 + v A ij = 1 2 A ij + A ji 3 2 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)

11 Why second order fluid dynamics? Consider scaling ( Hubble ) expansion 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 xx δ (1) Π xx Π 0 ii = P + ρα ix 2 i δ (1) Π ii = η(α i 4 3 j α j) x x v(ideal stresses) c s v(dissipative stresses). Acausal behavior; hydro always breaks down in the dilute corona. Solved by relaxation time τ π η P.

12 Hydrodynamic variables fluctuate Hydrodynamic fluctuations 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 +...

13 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 Response function, Kubo limit vl v L v L ρ ρ ρ vt ρ G xyxy R = P + δp iω[η + δη] + ω 2 [ητ π + δ(ητ π )] δη T ( ) 2 ( ρ P η ρ ) 1/2 δ(ητ π ) 1 ( ) 3/2 ρ ω η

14 Hydro Loops: Breakdown of second order hydro δη T ( ) 2 ( ρ P η ρ ) 1/2 δ(ητ π ) 1 ( ) 3/2 ρ ω η Small shear viscosity enhances fluctuation corrections. Small η leads to large δη: There must be a bound on η/n. Relaxation time diverges: 2nd order hydro without fluctuations inconsistent.

15 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)

16 II. Linear response and kinetic theory Consider background metric g ij (t,x) = δ ij + h ij (t,x). Linear response δπ ij = 1 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] =

17 Kinetic theory Linearize f = f 0 + δf, solve for δf, δπ ij, G R, η(ω) η(ω) = η 1 + ω 2 τ 2 π η = π (mt)3/2 τ π = 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)

18 Second order hydrodynamics from kinetic theory Chapman-Enskog expansion f = f 0 + δf 1 + δf Knudsen expansion δf n = O(Kn n ) = O( n ) [ δ (2) Π ij = η2 Dσ ij + 2 ] P 3 σij ( v) + η2 P [ ] σ i k σj k σ i k Ωj k + O(κη i j T) relaxation time τ π = η P η nt Chao, Schaefer (2012), Schaefer (2014)

19 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). Im Σ(k) zt Bulk viscosity ζ = π λ 3 ( zλ a Re Σ(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 )

20 Can be used to extrapolate Boltzmann result to T T F = III. 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)

21 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)

22 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)

23 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 + k(r) 1/3 { r 2 d x 2 + Fluctuations δg y x = e iωu χ(ω, r) satisfy (u = (r + /r) 2 ) } dv 2 [1 + f(r)] du dv } dr2 r 2 f(r) 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 Adams et al. (2008), Herzog et al. (2008)

24 Spectral function η(ω)/s η(0)/s = 1/(4π) ω/(πt) η(ω ) ω 1/3 Relaxation time: 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 Schaefer (2014), BRSSS (2008)

25 Quasi-normal modes Re w Re w Im w 4 Im w Sch 2 2 AdS 5 QNM s are stable, Im λ < 0. Schrödinger metric has unpaired eigenvalues, similar to relaxation time Boltzmann. Schaefer (2014), Starinets (2002)

26 V. Experiments: Flow and Collective Modes Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy O Hara et al. (2002)

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

28 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

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

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

31 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

32 Lessons & Outlook Experiment: Main issue is temperature, density dependence of η/s. How to unfold? Need hydro codes that exit gracefully (LBE, anisotropic hydro, hydro+cascade) Quasi-particles vs quasi-normal modes (kinetics vs holography) unresolved. Need better holographic models, improved lattice calculations. 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?

33 4 Elliptic flow: Shear vs bulk viscosity Viscosity Coefficients E E F

34 Elliptic flow away from unitarity