Intersections of nuclear physics and cold atom physics Thomas Schaefer North Carolina State University

Unitarity limit Consider simple square well potential

a 0, ǫB > 0

Unitarity limit Now take the range to zero, keeping ǫB ≃ 0

Universal relations 1 T = ik + 1/a

1 ǫB = 2ma2

1 ψB ∼ √ exp(−r/a) ar

Feshbach resonances Atomic gas with two spin states: “↑” and “↓” V closed

open

r

Feshbach resonance

“Unitarity” limit a → ∞

a(B) = a0

∆ 1+ B − B0

4π σ= 2 k

Universality Neutron Matter

What do these systems have in common? dilute: rρ1/3 ≪ 1

strongly correlated: aρ1/3 ≫ 1

Feshbach Resonance in 6 Li

r a k F−1

Dilute Fermi gas: field theory Non-relativistic fermions at low momentum 2 C0 † 2 ∇ † ψ− (ψ ψ) Leff = ψ i∂0 + 2M 2 Unitary limit: a → ∞, σ → 4π/k 2 (C0 → ∞) This limit is smooth: HS-trafo, Ψ = (ψ↑ , ψ↓† ) # " ~2 1 ∗ ∇ † † φ φ, Ψ + Ψ σ+ Ψφ + h.c. − L = Ψ i∂0 + σ3 2m C0 Low T (T < Tc ∼ µ): Pairing and superfluidity

Dilute Fermi gas: BCS-BEC crossover

Intersection I: Many body physics/equation of state Free fermi gas at zero temperature E 3 kF2 = N 5 2m

N kF3 = V 3π 2

Consider unitarity limit (a → ∞, r → 0) E 3 kF2 =ξ N 5 2m

kF ≡ (3π 2 N/V )1/3

Prize problem (George Bertsch, 1998): Determine ξ Similar problems: ∆ = αǫF ,

kB Tc = βǫF

Analytic work: Epsilon expansion

O(1)

O(1)

O(ǫ)

Green function MC

1 5/2 1 3/2 ξ = ǫ + ǫ ln ǫ 2 16 − 0.0246 ǫ5/2 + . . . ξ(ǫ = 1) = 0.475

Experiment

20

E/EFG

Pairing gap (∆) = 0.9 EFG

odd N even N

10

E = 0.44 N EFG

0

10

20

30

40

N

ξ = 0.40-0.44 (Carlson et al.)

ξ = 0.38(2) (Luo, Thomas)

Neutron matter with realistic interactions LO3 NLO3

1.2 1

E0/Efree 0

0.8

FP 1981 APR 1998 CMPR v6 2003 CMPR v8′ 2003 SP 2005 GC 2007 GIFPS 2008

0.6 0.4 0.2 0 0

50

100 150 kF (MeV)

200

Results close to unitary limit (for kF |a| > 10). Corrections tend to cancel (range effects, p-waves, 3-body).

Density Functionals Gradient terms (from epsilon expansion) 2

(∇n(x)) n(x)5/3 + 0.032 + O(∇4 n) E(x) = n(x)V (x) + 1.364 m mn(x) free Fermi gas: (1.364 → 2.871) consider V (x) = 12 mω 2 x2 p EN lim = ξ ≃ 0.63 0 N →∞ EN evidence for large surface effects

(0.032 → 0.014)

0.8 0 EN /E∞ 0.775

O(∇n)2

0.75 0.725

N −1/3 fit

0.7 0.675 0.65

ETFT 5

15

10

20

N Blume et al., see also Bulgac (SFLDA), Gandolfi et al.

Intersection II: Pairing Numerical results (Carlson & Reddy, Burovsky et al.) ∆ = 0.48EF

Tc = 0.15EF

Gap remarkably close to extrapolated BCS+Gorkov result π 8EF exp − ∆= 1/3 2 2kF |a| (4e) e ∆(a → ∞) = 0.49EF

=

=

Gorkov (induced interaction) crucial, reduces gap by ∼ 1/2

+

Pairing gap with realistic interactions -1

-1

kF [fm ]

kF [fm ] 0 0.5

0.1

0.2

0.3

0.4

0.5

0

0.6

BCS QMC s-wave QMC AV4

2.5 2

∆ (MeV)

∆ / EF

0.4 0.3

1.5

0.2

1

0.1

0.5

0

0

2

4

6

- kF a

8

10

12

0

0

0.1

0.2

0.3

0.4

0.5

0.6

BCS Chen 93 Wambach 93 Schulze 96 Schwenk 03 Fabrocini 05 Cao 06 dQMC 09 AFDMC 08 QMC AV4

2

4

6

- kF a

8

10

12

Range corrections important, ∆ smaller than in unitary limit. But: QMC gaps larger than previous estimates.

Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy

Anisotropy Parameter v 2

Intersection III: Elliptic flow (QGP) Hydro model π K p Λ

0.3

STAR Data K 0S Λ+Λ

0.2

0.1

0 0

b

PHENIX Data π++πK ++Kp+p

2

4

6

Transverse Momentum p T (GeV/c) source: U. Heinz (2005)

dN p0 3 = v0 (p⊥ ) (1 + 2v2 (p⊥ ) cos(2φ) + . . .) d p pz =0

Viscosity and elliptic flow Viscous effects increase with impact parameter and pT . 25

ideal η/s=0.03 η/s=0.08 η/s=0.16 PHOBOS

v2

0.06

0.04

15 10

0.02

0 0

ideal η/s=0.03 η/s=0.08 η/s=0.16 STAR

20

v2 (percent)

0.08

5

100

200 NPart

300

400

0 0

1

2 pT [GeV]

3

4

Romatschke (2007), see also Teaney (2003)

Many questions: Dependence on initial conditions, freeze out, etc. conservative bound

η < 0.4 s

Almost ideal fluid dynamics (cold gases) Hydrodynamic expansion converts

A) σx, σz (µm)

300

coordinate space anisotropy

200

100

0.0

0.5

1.0

1.5

2.0

Expansion Time (ms)

Aspect Ratio ( σx / σz )

B) 2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Expansion Time (ms)

O’Hara et al. (2002)

2.0

to momentum space anisotropy

Collective oscillations Ideal fluid hydrodynamics (P = 23 E)

Radial breathing mode

∂n ~ + ∇ · (n~v ) = 0 ∂t ~ ~ ∂~v ~ ∇P ∇V + ~v · ∇ ~v = − − ∂t mn m 60

(a)

50 40

50

0.0 0.5 1.0 1.5 2.0 2.5

60

(b)

50 40 0.0 0.5 1.0 1.5 2.0 2.5

50

2

()

(1/2)

( µm)

40

40 50

40 30

(c)

40

Hydro frequency at unitarity r 10 ω⊥ ω= 3

0.0 0.5 1.0 1.5 2.0 2.5

Damping small, depends on T /TF .

30 20 2

4

6

8

thold(ms)

experiment: Kinast et al. (2005)

Viscous hydrodynamics Energy dissipation (η, ζ, κ: shear, bulk viscosity, heat conductivity) 2 Z 1 2 3 ˙ E = − d x η(x) ∂i vj + ∂j vi − δij ∂k vk 2 3 Z Z 1 2 2 3 d3 x κ(x) (∂i T ) − d x ζ(x) (∂i vi ) − T Shear viscosity to entropy ratio (assuming ζ = κ = 0)

1.5

hη/si 1.0

1 Γ η E0 N = (3λN ) 3 s ω ⊥ EF S

Schaefer (2007), see also Bruun, Smith

0.5

0.0

0.2

0.4

0.6

0.8

T /TF T ≪ TF

T ≫ TF , τR ≃ η/P

Elliptic flow: High T limit (mT )3/2 Quantum viscosity η = η0 ~2

Aspect Ratio

1.5

1.0

η = η0 (mT )3/2 τR = η/P

0.5

0

200

400

600

800

1000

1200

1400

1600

Time After Release (µs)

fit: η0 = 0.33 ± 0.04 theory: η0 =

15 √ 32 π

= 0.26

Summary Unitary Fermi gas has become “the” benchmark problem for many body methods (equation of state, pairing, DFT) in nuclear physics (at least for pure neutron matter). Interesting (but maybe not quantitative) connections to the physics of quark matter and the quark gluon plasma, in particular nearly perfect fluidity. Many questions: Universality of nearly perfect fluidity? Quasi-particle picture?

More intersections Few body physics: Efimov effect, etc. Several species:

Three species (quark-hadron transition), four species (nuclear matter, SU(4) symmetry).

Finite polarization: critical δµ, LOFF phase (relevant to stressed color superconductivity). Rotating systems: Vortices (formation, pinning, etc.). New ideas:

gauge fields, role of dimensionality, AdS/NRCFT.

Unitarity limit Consider simple square well potential

a 0, ǫB > 0

Unitarity limit Now take the range to zero, keeping ǫB ≃ 0

Universal relations 1 T = ik + 1/a

1 ǫB = 2ma2

1 ψB ∼ √ exp(−r/a) ar

Feshbach resonances Atomic gas with two spin states: “↑” and “↓” V closed

open

r

Feshbach resonance

“Unitarity” limit a → ∞

a(B) = a0

∆ 1+ B − B0

4π σ= 2 k

Universality Neutron Matter

What do these systems have in common? dilute: rρ1/3 ≪ 1

strongly correlated: aρ1/3 ≫ 1

Feshbach Resonance in 6 Li

r a k F−1

Dilute Fermi gas: field theory Non-relativistic fermions at low momentum 2 C0 † 2 ∇ † ψ− (ψ ψ) Leff = ψ i∂0 + 2M 2 Unitary limit: a → ∞, σ → 4π/k 2 (C0 → ∞) This limit is smooth: HS-trafo, Ψ = (ψ↑ , ψ↓† ) # " ~2 1 ∗ ∇ † † φ φ, Ψ + Ψ σ+ Ψφ + h.c. − L = Ψ i∂0 + σ3 2m C0 Low T (T < Tc ∼ µ): Pairing and superfluidity

Dilute Fermi gas: BCS-BEC crossover

Intersection I: Many body physics/equation of state Free fermi gas at zero temperature E 3 kF2 = N 5 2m

N kF3 = V 3π 2

Consider unitarity limit (a → ∞, r → 0) E 3 kF2 =ξ N 5 2m

kF ≡ (3π 2 N/V )1/3

Prize problem (George Bertsch, 1998): Determine ξ Similar problems: ∆ = αǫF ,

kB Tc = βǫF

Analytic work: Epsilon expansion

O(1)

O(1)

O(ǫ)

Green function MC

1 5/2 1 3/2 ξ = ǫ + ǫ ln ǫ 2 16 − 0.0246 ǫ5/2 + . . . ξ(ǫ = 1) = 0.475

Experiment

20

E/EFG

Pairing gap (∆) = 0.9 EFG

odd N even N

10

E = 0.44 N EFG

0

10

20

30

40

N

ξ = 0.40-0.44 (Carlson et al.)

ξ = 0.38(2) (Luo, Thomas)

Neutron matter with realistic interactions LO3 NLO3

1.2 1

E0/Efree 0

0.8

FP 1981 APR 1998 CMPR v6 2003 CMPR v8′ 2003 SP 2005 GC 2007 GIFPS 2008

0.6 0.4 0.2 0 0

50

100 150 kF (MeV)

200

Results close to unitary limit (for kF |a| > 10). Corrections tend to cancel (range effects, p-waves, 3-body).

Density Functionals Gradient terms (from epsilon expansion) 2

(∇n(x)) n(x)5/3 + 0.032 + O(∇4 n) E(x) = n(x)V (x) + 1.364 m mn(x) free Fermi gas: (1.364 → 2.871) consider V (x) = 12 mω 2 x2 p EN lim = ξ ≃ 0.63 0 N →∞ EN evidence for large surface effects

(0.032 → 0.014)

0.8 0 EN /E∞ 0.775

O(∇n)2

0.75 0.725

N −1/3 fit

0.7 0.675 0.65

ETFT 5

15

10

20

N Blume et al., see also Bulgac (SFLDA), Gandolfi et al.

Intersection II: Pairing Numerical results (Carlson & Reddy, Burovsky et al.) ∆ = 0.48EF

Tc = 0.15EF

Gap remarkably close to extrapolated BCS+Gorkov result π 8EF exp − ∆= 1/3 2 2kF |a| (4e) e ∆(a → ∞) = 0.49EF

=

=

Gorkov (induced interaction) crucial, reduces gap by ∼ 1/2

+

Pairing gap with realistic interactions -1

-1

kF [fm ]

kF [fm ] 0 0.5

0.1

0.2

0.3

0.4

0.5

0

0.6

BCS QMC s-wave QMC AV4

2.5 2

∆ (MeV)

∆ / EF

0.4 0.3

1.5

0.2

1

0.1

0.5

0

0

2

4

6

- kF a

8

10

12

0

0

0.1

0.2

0.3

0.4

0.5

0.6

BCS Chen 93 Wambach 93 Schulze 96 Schwenk 03 Fabrocini 05 Cao 06 dQMC 09 AFDMC 08 QMC AV4

2

4

6

- kF a

8

10

12

Range corrections important, ∆ smaller than in unitary limit. But: QMC gaps larger than previous estimates.

Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy

Anisotropy Parameter v 2

Intersection III: Elliptic flow (QGP) Hydro model π K p Λ

0.3

STAR Data K 0S Λ+Λ

0.2

0.1

0 0

b

PHENIX Data π++πK ++Kp+p

2

4

6

Transverse Momentum p T (GeV/c) source: U. Heinz (2005)

dN p0 3 = v0 (p⊥ ) (1 + 2v2 (p⊥ ) cos(2φ) + . . .) d p pz =0

Viscosity and elliptic flow Viscous effects increase with impact parameter and pT . 25

ideal η/s=0.03 η/s=0.08 η/s=0.16 PHOBOS

v2

0.06

0.04

15 10

0.02

0 0

ideal η/s=0.03 η/s=0.08 η/s=0.16 STAR

20

v2 (percent)

0.08

5

100

200 NPart

300

400

0 0

1

2 pT [GeV]

3

4

Romatschke (2007), see also Teaney (2003)

Many questions: Dependence on initial conditions, freeze out, etc. conservative bound

η < 0.4 s

Almost ideal fluid dynamics (cold gases) Hydrodynamic expansion converts

A) σx, σz (µm)

300

coordinate space anisotropy

200

100

0.0

0.5

1.0

1.5

2.0

Expansion Time (ms)

Aspect Ratio ( σx / σz )

B) 2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Expansion Time (ms)

O’Hara et al. (2002)

2.0

to momentum space anisotropy

Collective oscillations Ideal fluid hydrodynamics (P = 23 E)

Radial breathing mode

∂n ~ + ∇ · (n~v ) = 0 ∂t ~ ~ ∂~v ~ ∇P ∇V + ~v · ∇ ~v = − − ∂t mn m 60

(a)

50 40

50

0.0 0.5 1.0 1.5 2.0 2.5

60

(b)

50 40 0.0 0.5 1.0 1.5 2.0 2.5

50

2

()

(1/2)

( µm)

40

40 50

40 30

(c)

40

Hydro frequency at unitarity r 10 ω⊥ ω= 3

0.0 0.5 1.0 1.5 2.0 2.5

Damping small, depends on T /TF .

30 20 2

4

6

8

thold(ms)

experiment: Kinast et al. (2005)

Viscous hydrodynamics Energy dissipation (η, ζ, κ: shear, bulk viscosity, heat conductivity) 2 Z 1 2 3 ˙ E = − d x η(x) ∂i vj + ∂j vi − δij ∂k vk 2 3 Z Z 1 2 2 3 d3 x κ(x) (∂i T ) − d x ζ(x) (∂i vi ) − T Shear viscosity to entropy ratio (assuming ζ = κ = 0)

1.5

hη/si 1.0

1 Γ η E0 N = (3λN ) 3 s ω ⊥ EF S

Schaefer (2007), see also Bruun, Smith

0.5

0.0

0.2

0.4

0.6

0.8

T /TF T ≪ TF

T ≫ TF , τR ≃ η/P

Elliptic flow: High T limit (mT )3/2 Quantum viscosity η = η0 ~2

Aspect Ratio

1.5

1.0

η = η0 (mT )3/2 τR = η/P

0.5

0

200

400

600

800

1000

1200

1400

1600

Time After Release (µs)

fit: η0 = 0.33 ± 0.04 theory: η0 =

15 √ 32 π

= 0.26

Summary Unitary Fermi gas has become “the” benchmark problem for many body methods (equation of state, pairing, DFT) in nuclear physics (at least for pure neutron matter). Interesting (but maybe not quantitative) connections to the physics of quark matter and the quark gluon plasma, in particular nearly perfect fluidity. Many questions: Universality of nearly perfect fluidity? Quasi-particle picture?

More intersections Few body physics: Efimov effect, etc. Several species:

Three species (quark-hadron transition), four species (nuclear matter, SU(4) symmetry).

Finite polarization: critical δµ, LOFF phase (relevant to stressed color superconductivity). Rotating systems: Vortices (formation, pinning, etc.). New ideas:

gauge fields, role of dimensionality, AdS/NRCFT.