From Trapped Atoms

To Liberated Quarks

Thomas Schaefer

North Carolina State University

1

BNL and RHIC

2

Heavy Ion Collision

Star TPC

3

Elliptic Flow

Hydrodynamic

expansion converts

coordinate space

anisotropy

to momentum space

anisotropy

b

2A

niso

trop

y P

aram

eter

v

(GeV/c)TTransverse Momentum p

0 2 4 6

0

0.1

0.2

0.3

-π++π 0SK-+K+Kpp+

Λ+Λ

STAR DataPHENIX DataHydro modelπKpΛ

source: U. Heinz (2005)

4

Elliptic Flow II

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

/dy ch (1/S) dN

ε/2 v HYDRO limits

/A=11.8 GeV, E877labE

/A=40 GeV, NA49labE

/A=158 GeV, NA49labE

=130 GeV, STAR NNs

=200 GeV, STAR Prelim. NNs

Requires “perfect” fluidity (η/s < 0.1 ?)

(s)QGP saturates (conjectured) universal bound η/s = 1/(4π)?

5

Designer Fluids

Atomic gas with two spin states: “↑” and “↓”

open

closed

V

r

Feshbach resonance a(B) = a0

(

1 +∆

B −B0

)

“Unitarity” limit a→ ∞ σ =4π

k2

6

Elliptic Flow

A)

300

200

100

σ x

, σ z

(µ m

)

2.01.51.00.50.0

Expansion Time (ms)

2.5

2.0

1.5

1.0

0.5

Aspect R

atio

(

σ x / σ z

)

2.01.51.00.50.0

Expansion Time (ms)

B)

Hydrodynamic

expansion converts

coordinate space

anisotropy

to momentum space

anisotropy

7

Perfect Liquids

sQGP (T=180 MeV)

Trapped Atoms (T=1 neV)

Neutron Matter (T=1 MeV)

8

Universality

What do these systems have in

common?

dilute: rρ1/3 ¿ 1strongly correlated: aρ1/3 À 1

a

r

k −1F

Feshbach Resonance in 6Li Neutron Matter

9

Questions

Equation of State

Critical Temperature

Transport: Shear Viscosity, ...

Stressed Pairing

10

I. Equation of State

11

Universal Equation of State

Consider limiting case (“Bertsch” problem)

(kFa) → ∞ (kF r) → 0

Universal equation of state

E

A= ξ

(E

A

)

0= ξ

3

5

( k2F2M

)

How to find ξ?

Numerical Simulations

Experiments with trapped fermions

Analytic Approaches

12

Effective Field Theory

Effective field theory for pointlike, non-relativistic neutrons

Leff = ψ†(

i∂0 +∇22M

)

ψ− C02

(ψ†ψ)2 +C216

[

(ψψ)†(ψ↔

∇2

ψ)+h.c.]

+ . . .

(a, r, . . .) ⇒ (C0, C2, . . .)

Partition Function (Hubbard-Stratonovich field s, Fermion matrix Q)

Z =

∫

Ds exp [−S′] , S′ = − log(det(Q)) + V (s)

C0 < 0 (attractive): det(Q) ≥ 0

13

Continuum Limit

Fix coupling constant at finite lattice spacing

M

4πa=

1

C0+

1

2

∑

~p

1

E~p

Take lattice spacing b, bτ to zero

µbτ → 0 n1/3b→ 0 n1/3a = const

Physical density fixed, lattice filling → 0

Consider universal (unitary) limit

n1/3a→ ∞

14

Lattice Results

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2

E/A

*(3/

5 E

F)-

1

T/TF

0.07

0.42

0.14

T = 4 MeVT = 3 MeVT = 2 MeV

T = 1.5 MeVT = 1 MeV

Canonical T = 0 calculation: ξ = 0.25(3) (D. Lee)

Not extrapolated to zero lattice spacing

15

Green Function Monte Carlo

10 20 30 40N

0

10

20E/

E FG

E = 0.44 N EFG

Pairing gap (∆) = 0.9 EFG

odd Neven N

Other lattice results: ξ = 0.4 (Burovski et al., Bulgac et al.)

Experiment: ξ = 0.27+0.12−0.09 [1], 0.51 ± 0.04 [2], 0.74 ± 0.07 [3][1] Bartenstein et al., [2] Kinast et al., [3] Gehm et al.

16

Upper and lower critical dimension

Zero energy bound state for arbitrary d

ψ′′(r) +d− 1r

ψ′(r) = 0 (r > r0)

d=2: Arbitrarily weak attractive

potential has a bound state

ξ(d=2) = 1

d=4: Bound state wave function

ψ ∼ 1/rd−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)

17

Epsilon Expansion

EFT version: Compute scattering amplitude (d = 4 − ²)

'

iDig ig

T =1

Γ(

1 − d2

)

(m

4π

)−d/2 (

−p0 +²p2

)1−d/2

' 8π2²

m2i

p0 +²p2

+ iδ

g2 ≡ 8π2²

m2D(p0, p) =

i

p0 +²p2

+ iδ

Weakly interacting bosons and fermions

18

Epsilon Expansion

Effective potential

O(1) O(1) O(�)

ξ =1

2²3/2 +

1

16²5/2 ln ²

− 0.0246 ²5/2 + . . .

ξ(²=1) = 0.475

Problem: Higher order corrections large (∼ 100 %)!

Combine d = 4 − ² and d = 2 + ²̄ (and Pade)

ξ = (0.3 − 0.35)

19

Quark Gluon Plasma Equation of State (Lattice)

0

1

2

3

4

5

6

7

1.0 1.5 2.0 2.5 3.0 3.5 4.0

T/T0

p/T4

pSB/T4

nf=(2+1),Nτ=4Nτ=6

nf=2, Nτ=4Nτ=6

0

2

4

6

8

10

12

14

16

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

T/Tc

ε/T4 εSB/T4

p4, Nτ=4, nf=3 asqtad, Nτ=6, nf=2+1

15%

Compilation by F. Karsch (SciDAC)

20

Holographic Duals at Finite Temperature

Thermal (conformal) field theory ≡ AdS5 black hole

CFT temperature ⇔ Hawking temperature ofblack hole

CFT entropy ⇔ Hawking-Bekenstein entropy=area of event horizon

s =π2

2N2c T

3 =3

4s0

Gubser and Klebanov

s

λ = g N2

weak coupling

strong coupling

Extended to transport properties by Policastro, Son and Starinets

η =π

8N2c T

3

21

II. How Large Can Tc Get?

22

Critical Temperature: From BCS to BEC

TcTF

BCS

(k a) −1F

BEC

TBCSc =4 · 21/3eγe7/3π

²F exp

(

− π|kFa|

)

Tc(a→ ∞) = 0.28²F

TBECc = 3.31

(

n2/3

m

)

Tc = 0.21²F +O(aBn1/3)

23

Experimental Results

0.1 1T/TF

0.01

0.1

1

10

E(t

heat

)/E

0 -1

0 0.2 0.8√1+β ~T

0

0.2

0.8

T/T

F

T /TF =0.27

Tc

Kinast et al. (2005)

Lattice results: Tc/TF = 0.15 (UMass)

24

Quark Matter: Color Superconductivity

Weak coupling result

TcTF

=beγ

πexp

(

− 3π2

√2g

)

b = 512π4g−5(2/Nf )52 e−

π2+4

8

= +

=

=

Maximum Tc/TF = 0.025. Strong coupling?

0.0 0.5 1.0 1.5 2.0 2.5nb [fm

−3]

0.00

0.05

0.10

∆ 0,T

c [G

eV]

µc=0.5GeV

µc=0.4GeV

Find Tc/TF ' 0.2

Note: Transition to χSB

Consider Nc = 2 QCD?

25

Importance of Tc/TF : Heavy Ion Collisions at Fair

26

III. Transport Properties

27

Collective Modes

Radial breathing mode Kinast et al. (2005)

2.00

1.95

1.90

1.85

1.80

1.75

Freq

uenc

y (ω

/ω⊥)

2 1 0 -1 -2

1/kFa

680 730 834 1114 B (Gauss)

Ideal fluid hydrodynamics, equation of state P ∼ n5/3

∂n

∂t+ ~∇ · (n~v) = 0

∂~v

∂t+(

~v · ~∇)

~v = − 1mn

~∇ (P + nV )ω =

√

10

3ω⊥

28

Damping of Collective Excitations

50

40

30

20

8642

4030

2.52.01.51.00.50.0

60

50

40

(<x2

>)(

1/2)

( µm

)

5040

2.52.01.51.00.50.0

(c)

(b) 5040

2.52.01.51.00.50.0

60

50

40

(a)

thold(ms)

0.10

0.05

0.00

Dam

ping

rate

(1/τ

ω⊥)

1.41.21.00.80.60.40.20.0

T~

T/TF = (0.5, 0.33, 0.17) τω: decay time × trap frequency

Kinast et al. (2005)

29

Viscous Hydrodynamics

Energy dissipated due to viscous effects is

Ė = −η2

∫

d3x

(

∂ivj + ∂jvi −2

3δij∂kvk

)2

− ζ∫

d3x (∂ivi)2,

η, ζ: shear, bulk viscosity

Shear viscosity to entropy ratio (ζ=0)

η

s∼ ci ×

Γ

ω⊥× µω⊥

× NS

ci determined by Hydro solution

Bruun, Smith, Gelman et al.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

η/s

T/TF

Problems: Scaling with N ; T dependence below Tc

30

IV. Stressed Pairing

31

Polarized Fermions: From BEC to BCS

Response of paired state

to pair breaking stress

(e.g. Zeeman field)

Lext = δµψ†σ3ψ

?

BEC limit: Tightly bound bosons, no polarization for δµ < ∆

δµ > ∆: Mixture of Fermi and Bose liquid, no phase separation

BCS limit: No homogeneous mixed phase

δµ > δµc1: LOFF pairing ∆(x) = eiqx∆

32

Inhomogeneous pairing

Onset? Consider EFT for gapless fermions interacting with GB’s

L = ψ†(

i∂0 − ²(−i~∂) − (~∂ϕ) ·↔

∂

2m

)

ψ +f2t2ϕ̇2 − f

2

2(~∂ϕ)2

Free energy of state with non-zero current vs = ∂ϕ/m

F (vs) =1

2nmv2s

+

∫

d3p

(2π)3²v(~p)Θ (−²v(~p))

Unstable for BCS-type dispersion relation

0.2 0.4 0.6 0.8 1

-0.04

-0.02

0.02

0.04Ω/C

x

h = −0.08

h = −0.05

x ∼ ∆

h ∼ δµ− δµc∆

33

Minimal Phase Diagram

1 2

1/a1/a*

∆

homogenoeous superfluid

gaplesssuperfluid

supercurrent

LOFF

1

δµ

∆∆ ∆

p

(p)ε

E

E

E

E

δµ

∆

gaplessBCS

homog. BCS

LOFF

1

2

δµ

∆homog. superfluid

supercurrentstate

Son & Stephanov (2005)

34

Experimental Situation

Zwierlein et al.(MIT group)

35

Color Superconductivity in QCD: Response to ms 6= 0QCD with three degenerate flavors: CFL pairing

〈qai qbj〉 = (δai δbj − δaj δbi )φ

〈ud〉 = 〈us〉 = 〈ds〉Pair breaking stress due to µs = m

2s/(2pF ) 6= 0

pF

µs

d

u

s

e

2δ p = m

2pF

sF

p1

p2

∆

kinematics + electric neutrality + weak equilibrium

36

Phase Structure of CFL Quark Matter

How does CFL (〈ud〉 = 〈ds〉 = 〈su〉) pairing responds to ms?

20 40 60 80 100

20

40

60

80

��� ��� � �� �� � �

� �� � �

��� ��� �� � �

��� ��� �� � �

� �� ��� �� � ��

�� �

Excitation energy of fermions

Gapless modes appear at

µs(crit) ∼ 0.75∆

20 40 60 80 100

-20

-15

-10

-5

�

��� � � �!" # $

%&

'( )

'( )*

+,- '( )*

Energy density of superfluid phases

µs(K − cond) ∼ m2/3u ∆4/3/µµs(GB − cur) ∼ 0.75∆

Figures: Kryjevski & Schäfer (2004) Schäfer (2005)

37

Trapped atoms provide interesting model system

equation of state of strongly correlated systems

(neutron matter, sQGP)

viscosity of strongly correlated systems (sQGP?)

superfluidity at strong coupling (Tc/TF , response to

pair breaking fields, precursor phenomena)

38