Perfect Fluids: From Nano to Tera Thomas Schaefer North Carolina State University 1
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Perfect Fluids sqgp (T=180 MeV) Neutron Matter (T=1 MeV) Trapped Atoms (T=0.1 nev) 3
Hydrodynamics Long-wavelength, low-frequency dynamics of conserved or spontaneoulsy broken symmetry variables. τ τ micro τ λ 1 Historically: Water (ρ, ǫ, π) 4
Example: Simple Fluid Continuity equation ρ t + (ρ v) = 0 Euler (Navier-Stokes) equation t (ρv i) + Π ij = 0 x j Energy momentum tensor Π ij = Pδ ij + ρv i v j + η reactive ( i v j + j v i 23 δ ij k v k ) dissipative +... 5
Elliptic Flow Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy Anisotropy Parameter v 2 0.3 0.2 0.1 0 Hydro model π K p Λ PHENIX Data π + +π - + - K +K p+p STAR Data 0 K S Λ+Λ b 0 2 4 6 Transverse Momentum p T (GeV/c) source: U. Heinz (2005) 6
Elliptic Flow II v 2 /ε 0.25 HYDRO limits 0.2 0.15 0.1 E lab /A=11.8 GeV, E877 E lab /A=40 GeV, NA49 0.05 E lab /A=158 GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim. 0 0 5 10 15 20 25 30 35 (1/S) dn ch /dy Requires perfect fluidity (η/s < 0.1?) (s)qgp saturates (conjectured) universal bound η/s = 1/(4π)? 7
Viscosity Bound: Rough Argument Kinetic theory estimate of shear viscosity η 1 3 n p l mfp (Note : l mfp 1/(nσ)) Entropy density: s k B n. Uncertainty relation implies η s n p l mft k B n k B Validity of kinetic theory as pl mfp? Why η/s? Why not η/n? 8
Holographic Duals at Finite Temperature Thermal (conformal) field theory AdS 5 black hole CFT temperature CFT entropy Strong coupling limit s = π2 2 N2 c T 3 = 3 4 s 0 Hawking temperature of black hole Hawking-Bekenstein entropy area of event horizon s weak coupling strong coupling Gubser and Klebanov λ = g 2 N 9
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 h 4πk B Strong coupling limit universal? Provides lower bound for all theories? 0 g 2 N c 10
Designer Fluids Atomic gas with two spin states: and V closed open r Feshbach resonance a(b) = a 0 ( 1 + B B 0 ) Unitarity limit a 11 σ = 4π k 2
Why are these systems interesting? System is intrinsically quantum mechanical cross section saturates unitarity bound Scale (and conformally) invariant at unitarity E ( E ) A = ξ, OPE, power laws,... A 0 System is strongly coupled but dilute (k F a) (k F r) 0 Strong hydrodynamic elliptic flow observed experimentally 12
Questions Equation of State, Quasi-Particles,... Critical Temperature Transport: Shear Viscosity,... Stressed Pairing 13
I. EOS, Quasi-Particles 14
Microscopic Effective Field Theory Effective field theory for pointlike, non-relativistic fermions L eff = ψ ( i 0 + 2 2M Match to effective range expansion ) ψ C 0 2 (ψ ψ) 2 + C [ 2 (ψψ) (ψ 2 ] ψ)+h.c. +... 16 Unitarity limit C 0 = 4πa M, C 2 = 4πa2 M C 0, C 2 0 r 2 15
Effective Lagrangian Fermions are paired ψψ 0. Energy gap ω E F Low energy degrees of freedom: phase of condensate ψψ = e 2iϕ ψψ Effective lagrangian ( L = f 2 ϕ 2 v 2 ( ϕ) 2) +... 16
Effective Lagrangian Low energy (ω < E F ) effective lagrangian L = c 0 m 3/2 X 5/2 + c 1 m 1/2 ( X) 2 X + c 2 m [ ( 2 ϕ ) 2 9m 2 A 0 ] X X = µ A 0 ϕ ( ϕ) 2 2m variables constrained by ϕ: phase ψψ = e 2iϕ ψψ µ: chemical potential A 0 : gauge potential U(1) invariance Galilean invariance Scale invariance Conformal invariance 17 Greiter et al. (1989); Son, Wingate (2005)
Effective Lagrangian Low energy (ω < E F ) effective lagrangian L = c 0 m 3/2 X 5/2 + c 1 m 1/2 ( X) 2 X + c 2 m [ ( 2 ϕ ) 2 9m 2 A 0 ] X X = µ A 0 ϕ ( ϕ) 2 2m variables constrained by ϕ: phase ψψ = e 2iϕ ψψ µ: chemical potential A 0 : gauge potential U(1) invariance Galilean invariance Scale invariance Conformal invariance 18
Effective Lagrangian Low energy (ω < E F ) effective lagrangian L = c 0 m 3/2 X 5/2 + c 1 m 1/2 ( X) 2 X + c 2 m [ ( 2 ϕ ) 2 9m 2 A 0 ] X X = µ A 0 ϕ ( ϕ) 2 2m variables constrained by ϕ: phase ψψ = e 2iϕ ψψ µ: chemical potential A 0 : gauge potential U(1) invariance Galilean invariance Scale invariance Conformal invariance 19
Effective Lagrangian Low energy (ω < E F ) effective lagrangian L = c 0 m 3/2 X 5/2 + c 1 m 1/2 ( X) 2 X + c 2 m [ ( 2 ϕ ) 2 9m 2 A 0 ] X X = µ A 0 ϕ ( ϕ) 2 2m variables constrained by ϕ: phase ψψ = e 2iϕ ψψ µ: chemical potential A 0 : gauge potential U(1) invariance Galilean invariance Scale invariance Conformal invariance 20
Effective lagrangian determines Coupling to external fields Energy density functional Phonon interactions Superfluid hydrodynamics Non-perturbative physics in c 0, c 1, c 2,... Use epsilon (ǫ = d 4) expansion 21
Upper and lower critical dimension Zero energy bound state for arbitrary d ψ (r) + d 1 r ψ (r) = 0 (r > r 0 ) d=2: Arbitrarily weak attractive potential has a bound state ξ(d=2) = 1 d=4: Bound state wave function ψ 1/r d 2. Pairs do not overlap ξ(d=4) = 0 Conclude ξ(d=3) 1/2? Try expansion around d = 4 or d = 2? Nussinov & Nussinov (2004) 22
Epsilon Expansion EFT version: Compute scattering amplitude (d = 4 ǫ) ig id ig T = 1 Γ ( 1 d 2 ) ( m d/2 ( p 0 + 4π) ǫ ) 1 d/2 p 8π 2 ǫ 2 m 2 i p 0 + ǫ p 2 + iδ g 2 8π2 ǫ m 2 D(p 0, p) = i p 0 + ǫ p 2 + iδ Weakly interacting bosons and fermions 23
Matching Calculations Effective potential P = #(2m) d/2 µ d/2+1 O(1) O(1) O(ǫ) Phonon Propagator 1 = 1 Π Π = ω = c s p { ( ) p 2 1 + # mµ } +... 24
Matching (continued) Static susceptibility χ(q) = d 3 x e iqx ψ ψ(x)ψ ψ(0) = χ(0) { ( ) q 2 1 # mµ } +... Nishida, Son (2007) Rupak, Schaefer (2008) χ(q) + 25
Low Energy Constants: Result Low energy constants c 0 0.116, c 1 0.021, c 2 0 Density functional in unitarity limit E(x) = 1.364 n(x)5/3 m Compare: Free Fermions E(x) = 2.871 n(x)5/3 m ( n(x))2 + 0.032 mn(x) ( n(x))2 + 0.014 mn(x) + O( 4 n) + O( 4 n) Volume energy reduced, surface energy increased 26
II. Transport Properties 27
Elliptic Flow Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy 28
Collective Modes Radial breathing mode Kinast et al. (2005) 2.00 B (Gauss) 680 730 834 1114 Frequency (ω/ω ) 1.95 1.90 1.85 1.80 1.75 2 1 0-1 -2 1/k F a Ideal fluid hydrodynamics, equation of state P n 5/3 n t + (n v) = 0 v ( t + v ) v = 1 P mn 1 V m ω = 10 3 ω 29
Damping of Collective Excitations 60 (a) 50 40 50 0.0 0.5 1.0 1.5 2.0 2.5 (<x 2 >) (1/2) ( µm) 40 60 50 40 50 40 30 (b) 50 40 (c) 40 30 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.5 2.5 Damping rate (1/τω ) 0.10 0.05 0.00 0.0 0.2 0.4 0.6 T ~ 0.8 1.0 1.2 1.4 20 2 4 t hold (ms) 6 8 T/T F = (0.5,0.33,0.17) τω: decay time trap frequency Kinast et al. (2005) 30
Viscous Hydrodynamics Energy dissipation (η, ζ, κ: shear, bulk viscosity, heat conductivity) Ė = η d 3 x ( i v j + j v i 23 ) 2 2 δ ij k v k ζ d 3 x ( i v i ) 2 κ d 3 x ( i T) 2 T Shear viscosity to entropy ratio (assuming ζ =κ=0) η/s 1.6 1.4 1.2 η s = 3 4 ξ 1 2 (3N) 1 3 Γ ω ω N ω S see also Bruun, Smith, Gelman et 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 T/T F al. 31
Damping dominated by shear viscosity? Study dependence on flow pattern Study particle number scaling viscous hydro: Γ N 1/3 Boltzmann: Γ N 1/3 Role of thermal conductivity? suppressed for scaling flows: δt T(δn/n) const (δt) = 0 32
Kinetic Theory Quasi-Particles: Kinetic Theory T ij = d 3 p p ip j E p f p, Boltzmann equation f p t + v x f p + F p f p = C[f p ] Linearized theory (Chapman-Enskog): f p = f 0 p(1 + χ p /T) η χ X 2 χ C χ χ X = d 3 p f 0 p χ p p ij v ij v ij = v 2 δ ij 3v i v j 33
Low T: Phonons High T: Atoms 5 2 1 0.5 Η s 0.2 0.1 0.2 0.4 0.6 0.8 1 1.2 T T F η ( T ) 1 ) 3/2 log ( T ( TF ) 8 η + + s T s T F T F 34
Outlook Transport Theory near T c : Relation to Viscosity Bound Conjecture? Other experimental constraints: Observation of irrotational flow? Other uses of conformal symmetry? OPE? Braaten, Platter (2008) AdS/Cold Atom correspondence? Son (2008), Balasubramanian & McGreevy (2008) 35