Perfect Fluids: From Nano to Tera

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1 Perfect Fluids: From Nano to Tera Thomas Schaefer North Carolina State University 1

2 2

3 Perfect Fluids sqgp (T=180 MeV) Neutron Matter (T=1 MeV) Trapped Atoms (T=0.1 nev) 3

4 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

5 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

6 Elliptic Flow Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy Anisotropy Parameter v Hydro model π K p Λ PHENIX Data π + +π K +K p+p STAR Data 0 K S Λ+Λ b Transverse Momentum p T (GeV/c) source: U. Heinz (2005) 6

2 0.1 0 Hydro model π K p Λ PHENIX Data π + +π - + - K +K p+p STAR

7 Elliptic Flow II v 2 /ε 0.25 HYDRO limits E lab /A=11.8 GeV, E877 E lab /A=40 GeV, NA E lab /A=158 GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim (1/S) dn ch /dy Requires perfect fluidity (η/s < 0.1?) (s)qgp saturates (conjectured) universal bound η/s = 1/(4π)? 7

05 E lab /A=158 GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim.

8 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

9 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

= 3 4 s 0 Hawking temperature of black hole Hawking-Bekenstein entropy area of

10 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

horizon Graviton absorption cross section area of event horizon η s η s = 4πk B Son and

11 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

12 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

bound Scale (and conformally) invariant at unitarity E ( E ) A = ξ, OPE, power

13 Questions Equation of State, Quasi-Particles,... Critical Temperature Transport: Shear Viscosity,... Stressed Pairing 13

14 I. EOS, Quasi-Particles 14

15 Microscopic Effective Field Theory Effective field theory for pointlike, non-relativistic fermions L eff = ψ ( i M Match to effective range expansion ) ψ C 0 2 (ψ ψ) 2 + C [ 2 (ψψ) (ψ 2 ] ψ)+h.c Unitarity limit C 0 = 4πa M, C 2 = 4πa2 M C 0, C 2 0 r 2 15

to effective range expansion ) ψ C 0 2 (ψ ψ) 2 + C [ 2 (ψψ) (ψ 2 ]

16 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)

17 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)

ϕ: phase ψψ = e 2iϕ ψψ µ: chemical potential A 0 : gauge potential U(1) invariance Galilean

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 18

variables constrained by ϕ: phase ψψ = e 2iϕ ψψ µ: chemical potential A 0 : gauge

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 19

20 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

21 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

22 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

23 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

24 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 # mµ }

25 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

26 Low Energy Constants: Result Low energy constants c , c , c 2 0 Density functional in unitarity limit E(x) = n(x)5/3 m Compare: Free Fermions E(x) = n(x)5/3 m ( n(x)) mn(x) ( n(x)) mn(x) + O( 4 n) + O( 4 n) Volume energy reduced, surface energy increased 26

27 II. Transport Properties 27

28 Elliptic Flow Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy 28

29 Collective Modes Radial breathing mode Kinast et al. (2005) 2.00 B (Gauss) Frequency (ω/ω ) /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

30 Damping of Collective Excitations 60 (a) (<x 2 >) (1/2) ( µm) (b) (c) Damping rate (1/τω ) T ~ t hold (ms) 6 8 T/T F = (0.5,0.33,0.17) τω: decay time trap frequency Kinast et al. (2005) 30

31 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 η s = 3 4 ξ 1 2 (3N) 1 3 Γ ω ω N ω S see also Bruun, Smith, Gelman et T/T F al. 31

32 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

33 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

34 Low T: Phonons High T: Atoms Η s T T F η ( T ) 1 ) 3/2 log ( T ( TF ) 8 η + + s T s T F T F 34

35 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

Perfect Fluidity in Cold Atomic Gases?

Perfect Fluidity in Cold Atomic Gases? Thomas Schaefer North Carolina State University 1 Hydrodynamics Long-wavelength, low-frequency dynamics of conserved or spontaneoulsy broken symmetry variables τ