(Nearly) Scale invariant fluid dynamics for the dilute Fermi gas in two and three dimensions. Thomas Schaefer North Carolina State University

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1 (Nearly) Scale invariant fluid dynamics for the dilute Fermi gas in two and three dimensions Thomas Schaefer North Carolina State University

2 Outline I. Conformal hydrodynamics II. Observations (3d) III. Scale breaking (3d) IV. Two dimensional fluids V. Fluctuations

3 Two body scattering in 2d and 3d Consider zero range interaction L = ψ ( i M ) ψ C 0 2 (ψ ψ) 2 3d: Tune C 0 to unitarity. Two body scattering matrix T = 4π m 1 1/a ik T = 4π m 1 ik 2d: Classical scale invariance, broken by quantum effects T = 4π m 1 log[(ka 2d ) 2 ] iπ

4 I. Scale invariant fluid dynamics in 3d Many body system: Effective cross section σ tr n 2/3 (or σ tr λ 2 ) Systems remains hydrodynamic despite expansion

5 Scale and conformal symmetry Gallilean boosts 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

6 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 +... 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 [ Dπ ij + 53 ( v)πij ] = τ π ( π ij + ησ ij) +... Chao, Schaefer (2011)

7 Scaling ( Hubble ) expansion Why second order fluid dynamics? ρ(x i, t) = ρ 0 (b i (t)x i ), ideal stresses v i (x j, t) = α i (t)x i dissipative stresses Π 0 xx δ (1) Π xx Π 0 ii = P + ρα2 i x2 i δ (1) Π ii = η(α i 4 3 j α j) x x Ideal stresses propagate with speed c s, dissipative stresses propagate with infinite speed. Hydro always breaks down in the dilute corona. Solved by relaxation time τ π η P.

8 II. Scale invariant fluid dynamics: Observations Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy O Hara et al. (2002)

9 Elliptic flow: Breakdown of scale invariant hydrodynamics? switch from conformal hydro to scale breaking kinetics at scale factor b fr = 1,5,10,20 A R t [ms] no breakdown seen in the data

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

11 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)

12 III. Bulk viscosity and conformal symmetry breaking Conformal symmetry breaking (thermodynamics) 1 2E 3P = C 12πmaP 1 λ 6π nλ3 a Quasi-particles: 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 )

13 IV. Viscosity in two dimensions Kinetic theory with zero range interaction f p t + v x f p = C[f p ] C[f] = Shear viscosity in 2d and 3d Viscous damping η 2d = λ 2 π ([ log η 3d = 15 2πλ 3 16 Ė = 1 2 (5π a2 λ 2 )] 2 + π 2 ) ( π λ 2 ) a d 3 x η(x) (σ ij ) 2 = N 2 α n (σ ij ) 2

14 Comparison to collective mode data log(k F a) α n T/T F See arxiv: , data from Vogt et al. arxiv: Also: Bruun arxiv: , Baur et al. arxiv:

15 Hydrodynamic variables fluctuate V. Thermal 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 +...

16 Hydro Loops: Breakdown of second order hydro Response function G xyxy R = θ(t)[π xy,π xy ] ω,k Π xy = ρv x v y v T ρ ρ v T 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 )] 3d: Enhanced shear viscosity, divergent relaxation time δη T ( ) 2 ( ρ P η ρ ) 1/2 δ(ητ π ) 1 ( ) 3/2 ρ ω η Fluctuations large if bare viscosity is small

17 Fluctuation induced bound on η/s η/s η(ω)/s 0.4 fluctuations (η/s) min kinetic theory T/T F ω/t spectral function non-analytic ω term See arxiv: , also Kovtun, Moore, Romatschke (2011).

18 Fluctuations in 2d 2d: Logarithmic divergence in shear viscosity δη 1 16π mnt η log ( 2nT ηω ) Collective modes: Logarithmic dependence on number of particles ( η δ n) = log(n) 16π Power divergence in relaxation time hard to observe δ(ητ π ) 1 ω mnt η 2

19 Outlook Can we observe bulk viscosity away from unitarity in 3d, or near the crossover in 2d? What about bulk viscosity in the superfluid phase? Need local measurements of η/s in 2d and 3d. Requires second order hydrodynamics or hydro+kinetics calculations. Measurements of the viscous relaxation time (based on collective modes and elliptic flow?). QMC measurements of the viscosity spectral function. Can we see fluctuation effects? Quasi-particle behavior?

20 Scale breaking in 2d monopole frequency Scale invariance implies undamped monopole mode ω = 2ω 0. Frequency shift due to scale breaking Randeria, Taylor (2012) ω 2 = 1 d2 d d r γ d γ d = 2 + d 8 dd r nv (r) d P ρ P ρ 4ω 2 0 s High temperature limit: Virial expansion γ d = mz2 T 2 π ( EB b 2 (T) = ν T ( 2Tb 2 + T 2 b ) 2 ) E B = 1 ma 2 Ω Ω logk F a