Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
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