slides of viscosity talk in Nancy

Viscosity and scale invariance
in the unitary Fermi gas
Searching for the perfect fluid
Tilman Enss (TU München)
Nancy, 7 February 2011
[Annals of Physics 326, 770 (2011); arXiv:1008.0007]
Outline
• shear viscosity and perfect fluidity
• transport equations for the unitary Fermi gas
• analytical and numerical results, universal high-frequency behavior
• scale invariance and universal dynamics
Shear viscosity: definition
[Schaefer, Teaney, Rep. Progr. Phys. 2009]
Measures of Perfection
Measures of Perfection
Measures
of Perfection
Measures
of
Perfection
• Viscosity determines shear stress, or friction, of fluid flow:
determines
shearinstress
(“friction”) in fluid flow
y determinesViscosity
shear stress
(“friction”)
fluid flow
Viscosity
determines
shear (“friction”)
stress (“friction”)
fluid flow
Viscosity
determines
shear stress
in fluidinflow
∂vx
F =Aη
∂vx
∂v
x
∂y
F = AFη = A η ∂y
∂y
Dimensionless
measure
of shear
stress: Reynolds number
onless measure
of shear stress:
Reynolds
number
Dimensionless
measure
shear stress:
Reynolds
number
• dimensionless
measure
ofofshear
Reynolds
number
Dimensionless
measure
shearofstress:
stress:
Reynolds
number
n
n
Re
=
×n
mvr
Re =
×
mvr n
η
×
mvr
η
Re =Re =
×
mvr
η
η
flow
fluid
flow fluid
property
fluid fluidproperty
flow flow
property
property
property
property
property
property
• [η/n] = !
• [η/n] = !
• [η/n]• s=[η/n]
! =!
s
• Relativistic
Re = × τ T s
• relativistic
systems:
T
• Relativistic
systems
Re = × τsystems
sRe = × τ T
η
•
Relativistic
systems
η
• Relativistic systems Re = × τ Tη
η
∂vx
F =Aη
∂y
Shear viscosity: estimates
• kinetic theory (Boltzmann equation) for dilute gas:
η=
1
3 n p̄ �mfp
,
�mfp
1
=
:
nσ
η�
√
mkB T
σ(T )
grows with T
• superfluid: ηSF = 0
but phonon contribution
[Landau, Khalatnikov 1949]
• minimum in between: at which temperature and viscosity?
η
∼ p̄ �mfp ≥ �
• uncertainty suggests
n
(careful!)
Insights from string theory
• conformal field theory (CFT) dual to AdS5 black hole:
shear viscosity
graviton absorption cross section
(~ area of event horizon)
CFT entropy
Hawking-Bekenstein entropy
(~ area of event horizon)
• specifically SU(N), N = 4 SYM theory (no confinement, no running cpl)
2
λ
=
g
N is dual to classical gravity:
in strong-coupling ‘t Hooft limit
η
�
≥
s
4πkB
[Policastro, Son, Starinets 2001;
Kovtun, Son, Starinets 2005]
• conjecture of universal lower bound
Perfect fluidity
[Schaefer, Teaney, Rep. Progr. Phys. 2009]
• Definition: “perfect fluid” saturates bound
η
�
=
s
4πkB
How to qualify?
• bound is quantum mechanical
➡ need quantum-mechanical scattering mechanism
• bound is incompatible with weak coupling and kinetic theory:
�/τ � �qp � kB T ,
s ∼ nkB :
η
�qp τ
�
∼
�
s
kB
kB
➡ need strong interactions, no good quasi-particles
(above Quantum Critical Point: �/τ = CkB T [Sachdev])
200
4π η
hs
Helium 0.1MPa
Nitrogen 10MPa
Water 100MPa
150
100
50
Viscosity bound
0
1
10
T, K
100
1000
viscosity
helium,
FIG. 2 (color
online). for
The
viscosity-entropy ratio for some
[Kovtun, Son, Starinets 2005]
common substances:
nitrogen and water. The ratio is
nitrogen,helium,
and water
always substantially larger than its value in theories with gravity
Perfect Fluids: The co
Perfect fluids: the contenders
[Schaefer, Teaney 2009]
Perfect Fluids: The contenders
ct Fluids: The contenders
Liquid Helium
Liquid
Helium
QGP (T=180
MeV)
Liquid
Helium
(T=0.1 meV)
(T=0.1 meV)
Quark-Gluon
PlasmaMeV)
QGP (T=180
(T=180 MeV)
(T=0.1 meV)
−6
η
=
1.7
·
10
Pa · s
T=180 MeV) Trapped Atoms
(T=0.1 neV)
Trapped Atoms
oms
(T=0.1 neV)
V)
η = 1.7 · 10−15 Pa · s
η = 5 · 1011 Pa · s
Trapped Atoms
Consider ratios η/s
(T=0.1 neV)
min=0.5; 0.8; 0.4
The unitary Fermi gas
• two-component Fermi gas with zero-range interactions:
LE =
�
σ=↑,↓
ψσ†
�
�
�
g(Λ) † †
2
�∂τ −
∇ ψσ +
ψσ ψ−σ ψ−σ ψσ
2m
2
2
2
• renormalized coupling g(Λ) �→ g = 4π� a/m
−1
r
�
k
• ultracold atoms: interaction range 0
F � a scattering length
• Hubbard-Stratonovich transformation with pair field ϕ
�
�
1 ∗
L[ψ] �→ L[ψ, φ] = L0 + ψ↑ ψ↓ φ + h.c. − φ φ
g
∗
• ϕ is massless at infinite coupling g=∞ (unitarity)
roundness of the optical trap and its alignment with both
Evaporative cooling (achieved by reducing the laser power)
stirrer and the axes of the magnetic potential were cr
accompanied by a magnetic field ramp to 766 G on the BEC-side
6
3
deviation from cylindrical symmetry owing to misalignme
of the resonance typically produced a BEC of 3 £ 10 molecules .
aberrations, or gravity rapidly damped the rotation. The
Previous experiments studying the rotation of atomic BECs
27–31
of vortices in sodium was comparatively forgiving, and
. Because
employed magnetic traps operating at low bias fields
optimized before vortices in 6Li2 could be observed.
the Feshbach resonance in our system occurs between two high-field
seeking states that cannot be trapped magnetically, an optical dipole
Observation of vortex lattices
trap operating at high magnetic bias fields was necessary. Our set-up
6
2
In
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via the magnetic field. Thus, it is possible to choos
a power of 145 mW. Axial confinement
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trap frequency
SPECTRAL
ANDtuned
RF RESPONSE
OF…
magnetic fields to optimize the three steps involved in the
n z ¼ 23 Hz was provided by an applied magnetic field curvature
fermions in tw
a vortex lattice: stirring of the cloud (for 800 ms at a typi
that decreased the radial trap frequency to n r ¼ 57 Hz. The aspect
states: s-wave
frequency of 45 Hz), the subsequent equilibration
(typica
ratio of the trap was 2.5. In this trap, at a field of 766 G, condensates of
6
0.25in our experiment after and time-of-flight expansion for imaging.
1 £ 10 molecules (the typical number
belowTo
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c
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and 110 mm axially, a peak molecular density of 2.6 £ 10 cm23, a
0.2
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BCS
a non-interacting two-state
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0.15
mixture of 6Li atoms of mass m with total atom number N in a
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transport! [2, 3]. One contributing factor to the
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smooth
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Eq.
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ddby
Fa ismethods
Baym-Kadanoff (2PI) conserving approximation
7
!"#
at unitarity,
(0)
Tσ" (τ X1 X1! ) =
h̄
2
∇1 ∇!1
(5.8)
δX1 X1! δ(τ − τ1 ) ,
m
!
2ḡ(Λ)−1 δX1 X1! δ(τ − τ1 ) for # = 0 7
(0)
!
S" (τ X1 X1 ) =
0
for # = 2 .
(5.9)
• self-consistent equations for fermionic and bosonic self-energies:
!"#
(5.8)
=0
= 2.
(5.9)
externctions
s func-
(5.10)
(5.11)
nd line
vertex
The response of the grand potential to the external field in terms of the fermionic Green’s functions
Gσ,XX ! = #T ψσ (X)ψσ† (X ! )$ and bosonic Green’s func†
tions GB,XX ! = #T ψB (X)ψB
(X ! )$ is
Σ↑!$#
(K) =
�
Γ(Q)G↓ (Q + K)
δΩ = tr[(δUσ )Gσ ] + tr[(δUB )GB ]
=
Q
(0)
tr[Tσ" Gσ δh" (τ )]
−
(0)
tr[S" Γ
(5.10)
δh" (τ )
] (5.11)
(1 + 2h" (τ ))2
Σ↑ (X) = Γ(X)G↓ (−X)
where the trace includes the spin sum. In the second line
the
Green’s
is replaced
by the vertex
FIG.bosonic
1: [color
online]function
Diagrammatic
contributions
to the
2
!
!
function
= −g̃ self-energies.
GB,XX . Hence, we obtain
fermionicΓand
XX bosonic
• Dyson equation:
" (0)
δΩ
!
!X
=
−
Tσ" (τ−1
XX
) GS̃
−
σ,X
−1
followδhfrom
scale
invariance.
T̃
and
the fermionic
"
" are
" (τ )
!
σXX response
↑
and bosonic
0 functions
↑ viscosity
"
1
(0)
S
(τ XX ! ) ΓX ! X . (5.12)
+
!
"
δG
2
σ,XX
!
T̃σ" (τ XX(1) +
= 2h" (τ )) XX
= !#T Π" (τ )ψσ (X)ψσ† (X ! )$ ,
δh" (τ )
In particular for a static scaling perturbation h"=0 (τ(5.16)
)≡h
!
δΓXX
† correct
we S̃recover
energy
formula
2
!
!the Tan
= −g̃
#T Π" (τ[27]
)ψBwith
(X)ψthe
" (τ XX ) =
B (X )$ .
δh" (τ )
UV regularization,
(5.17)
2 2#
$
"
C
dΩ
h̄ k
nkσ − 4 = 2#H$(0)= 2ε .
3p = #Πii $ = −
=2
Note that for the
shear viscosity
dhcase of the2m
k where S" =
kσ
0, the viscosity response function can be expressed
by
(5.13)
(0)
the L function [47] as T̃" = LT" , and the correlation
function
reduces
well-known
form
(τ )re=
The
Kubo(5.15)
formula
(3.1)tointhe
imaginary
time
canχ"be
G
(K) = G
(K) − Σ (K)
!$#
Σ (Q) =
�
G (K)G (Q − K)
FIG. 1: [color
b online] Diagrammatic
↑ contributions
↓ to the
fermionic and bosonic self-energies.
K
Σ (X) = G (X)G (X)
follow from scale invariance. T̃" and S̃" are the fermionic
b viscosity response
↑ functions↓
and bosonic
δGσ,XX !
= #T Π" (τ )ψσ (X)ψσ† (X ! )$ ,
δh" (τ )
(5.16)
δΓXX !
−1
−1
†
!
= −g̃ 2 #T Π" (τ )ψB (X)ψB
(X
S̃" (τ XX ! ) =
b )$ .
δh" (τ )
(5.17)
T̃σ" (τ XX ! ) =
Γ
(Q) = a
− Σ (Q)
• solve by iteration on logarithmic grid
(300
& 300
momenta)
Note that
for thefrequencies
case of the shear viscosity
where Sradial
=
0, the viscosity response function can be expressed by
(0)
the L function [47] as T̃" = LT" , and the correlation
function (5.15) reduces to the well-known form χ" (τ ) =
(0)
(0)
− tr[Tσ" (τ )LTσ" (0)].
Up to now the derivation of the correlation function
(5.15) has been completely general and exact, but the
challenge remains to evaluate the viscosity response functions T̃" and S̃" within the microscopic model. In the
following we shall do this within the T-matrix approxi-
• conserving: number, momentum current, scale invariance, ...
fulfills Tan energy formula & adiabatic relation exactly
(τ ) ≡ h
(5.12)
correct
(0)
(0)
(0)
"
Pair spectral function
Viscosity in linear response: Kubo formula
• viscosity from stress correlations (cf. hydrodynamics):
1
η(ω) =
ω
�
d3 x dt eiωt θ(t)
with stress tensor Π̂xy =
� p x py
p,σ
m
��
Π̂xy (�x, t), Π̂xy (0, 0)
†
cpσ cpσ
∂vx
(cf. Newton
)
∂y
• or, from transverse current correlations [Hohenberg, Martin 1965]
ηq 2
χ⊥ (ω, q) = 2
,
2
2
ω + (D⊥ q )
• equivalence via momentum balance
∂t (ρvi ) + ∂j Πij = 0
��
η = D ⊥ ρn
Transport equations
7
• Single-particle Green functions:
7
(5.8)
(5.8)
=0
=
= 02 .
= (5.9)
2.
(5.9)
exterexternctions
s funcnctions
s func(5.10)
(5.10)
(5.11)
(5.11)
nd line
vertex
nd
line
vertex
(5.12)
(5.12)
(τ ) ≡ h
correct
τ) ≡ h
correct
[Enss, Haussmann, Zwerger 2010]
!"#Response to shear perturbations:
!"#
!"#
!"#
correc
correc
T!MT
T!M
!$#
!$#
!$#
Maki-Thompson
Aslamazov-Larkin
!$#
T!A
T!A
The a
The a
FIG. 1: [color online] Diagrammatic contributions to the
fermionic
and bosonic
FIG.
1: [color
online]self-energies.
Diagrammatic contributions to the
fermionic and bosonic self-energies.
!"# !"#
follow
from scale invariance.
T̃ and S̃(Kubo
are the fermionic
• follow
correlation
function
formula): η(ω) =
and bosonic
viscosity
responseT̃ functions
from scale
invariance.
and S̃ are the fermionic
"
"
"
"
and bosonic viscosity response functions
δGσ,XX !
T̃σ" (τ XX ! ) =
= #T Π" (τ )ψσ (X)ψσ† (X ! )$ ,
δG
δhσ,XX
" (τ ) !
T̃σ" (τ XX ! ) =
= #T Π" (τ )ψσ (X)ψσ† (X ! )$
,
(5.16)
δh" (τ )
(5.16)
δΓXX !
†
= −g̃ 2 #T Π" (τ )ψB (X)ψB
(X ! )$ .
S̃" (τ XX ! ) =
δh
(τ )!
δΓ"XX
†
!
= −g̃ 2 #T Π" (τ )ψB (X)ψB
(X(5.17)
)$ .
S̃" (τ XX ! ) =
δh" (τ )
(5.17)
(0)
Note that for the case of the shear viscosity where S" =
(0)by
0, thethat
viscosity
function
can be expressed
Note
for theresponse
case of the
shear
viscosity
where
S
=
"
(0)
the
L
function
[47]
as
T̃
=
LT
,
and
the
correlation
0, the viscosity response" function
can be expressed by
"
(0)
function
(5.15) reduces
theLT
well-known
form χ" (τ ) =
the
L (0)
function
[47] as T̃to" =
" , and the correlation
(0)
FIG.modes:
2:FIG.
[color
online]online]
Diagrammatic
contributions
2: [color
Diagrammatic
contributionstotothe
the
• transport via fermionic and bosonic
renormalized
viscosity
response
functions
T! and
. (a)The
The
renormalized
viscosity
response
functions
T! and
S! .S!(a)
very efficient description, satisfiesfullconservation
laws
full fermionic
response
function
T! has
Maki-Thompsonand
and
fermionic
response
function
T! has
Maki-Thompson
• assumes no quasiparticles:
Aslamazov-Larkin
contributions,
in addition
to the
bareververAslamazov-Larkin
contributions,
in addition
to the
bare
(0)
(0)
T! . (b) The full bosonic response function S! has two
tex T! tex
. (b)
The full bosonic response function S! has two
contributions which are equivalent for equal fermion masses
contributions
which are equivalent for equal fermion
(0) masses
beyond
Boltzmann
and populations, in addition to the bare vertex(0)
S! . (c) The
and populations,
in addition
to theisbare
S! . (c)
The
viscosity correlation
function
givenvertex
by a skeleton
diagram
viscositywith
correlation
function
is given byviscosity
a skeleton
diagram
one bare and
one renormalized
response
vertex
S! (
S! (
isis give
give
(5.20)
(5.20)
SS!!(τ
The b
The
the pa
pa
the
bare b
b
bare
UV di
UV
di
which
which
The
The
100
T=10
T= 5
T= 2
T= 1
T=0.5
T=0.151
1
<
d(t) [h n]
10
0.1
∼
W τη−1
ω2
+
−2
τη
Cη
∼√
mω
0.01
0.001
0.1
1
10
100
1000
<
t [EF / h]
Viscosity spectral function
[Enss, Haussmann, Zwerger 2010]
Transport peak
[Enss, Haussmann, Zwerger 2010]
• fit to Drude peak and tail:
ω2
+
−2
τη
pressure p=2¡/3
peak weight W
10
1
0.1
0.1
1
T [TF]
�3/2 Cη ω(ω + τ −1 )
+ √
mω ω 2 + τ −2
visc. scattering time od [<h / EF]
peak weight W [EF n]
η(ω) =
W τη−1
10
10
classical 2.77 T1/2
scattering time od
1
0.1
• near Tc: �/τη ∼ 0.5 EF ∼ 3 kB T
decay rate larger than energy, no good quasiparticles!
1
T [TF]
10
Contact coefficient
• generically, short-distance (UV) behavior depends on non-universal details of
interaction potential
−1
kF )
• for zero-range interaction ( r0 �
this becomes universal:
at most two particles within distance r0 , all others far away (medium)
• two-particle density matrix for r0 < r �
�
−1
kF :
many-body
few-body
�
�2
�
r †
r
r
r
1 1
†
3
d R ψ↑ (R + )ψ↓ (R − )ψ↓ (R − )ψ↑ (R + ) = C
−
2
2
2
2
r
a
�
• Tan contact C: probability of finding up and down close together
(property of strongly coupled medium)
[Tan 2005; Braaten, Platter 2008; Zhang, Leggett 2009;
Combescot et al. 2009; Haussmann et al. 2009;
Schneider, Randeria 2010; Son, Thompson 2010;
Werner, Castin; Braaten 2010...]
formula.
Subject Areas: Atomic and Molecular Physics
A Viewpoint on:
Verification of Universal Relations in a Strongly Interacting Fermi Gas
J. T. Stewart, J. P. Gaebler, T. E. Drake and D. S. Jin
Phys. Rev. Lett. 104, 235301 (2010) – Published June 7, 2010
Contact coefficient (2)
Some of the most vexing present-day problems in
physics center on understanding the many-body properties and phases of strongly interacting fermions. Part
of the difficulty arises from the fact that while the behavior of interacting fermions is understood in certain welldefined regions of parameter space, this understanding
p
often breaks down4 in the correlated strongly interactp→∞
ing regime. Achieving this understanding of strongly
interacting fermions would have wide-ranging implications: the systems of interest include high-temperature
superconductors, high-density quark matter, and ultracold atomic fermion gases. The latter is the subject of a
et al.John
2010]
recent experimental investigation[Stewart
by John Stewart,
Gaebler, Tara Drake, and Debbie Jin at JILA in the US,
appearing in Physical Review Letters[1]. Stewart et al.
succeeded in confirming the validity of certain nonperturbative theoretical formulas, known as the Tan relaFIG. 1: Schematic plot of the typical momentum distribution
tions [2], which describe the properties of fermions with
n(k ) of a fermion gas with short-range interactions, showshort-range interactions. The verification of these relaing the characteristic behavior of n(k ) approaching a constant
tions shows that cold-fermion experiments are particvalue below unity at k → 0 and exhibiting a rapid variation
ularly suited as a testing laboratory for studying the
near the Fermi wave vector kF (that would be a sharp disgeneral problem of strongly interacting fermion systems
continuity at T → 0 for a Fermi liquid). At asymptotically
large k, as shown theoretically by Tan and confirmed experi[3, 4].
mentally by Stewart et al., one has n(k ) ∼ C/k4 (characterizThe JILA experiments studied gaseous, trapped
ing the fermions as they undergo collisions, seen in the inset),
potassium-40 atoms in two different internal states,
with the contact C governing thermodynamic quantities like
spin-up and spin-down (Fig. 1, inset). The effective inthe total energy of the gas. (Illustration: Alan Stonebraker)
teractions between these states are experimentally adjustable via a Feshbach resonance—a resonance of a
many-body system in which a bound state is achieved
scattering length that is much larger than the interpartiunder certain conditions—that can be tuned with a magcle spacing (| a|3 n >> 1, with n the fermion density), denetic field. For this experiment, the detailed structure
spite the fact that the effective range r0 of the interatomic
of the Feshbach resonance scattering is less important
• determine C:
C
lim n =
p
• intuitively: absorb external perturbation with large energy/momentum far
away from coherent peak of a single particle
➡ need to hit 2 particles close together to give energy+momentum to both
➡ absorption rate ~C
• access strong coupling and arbitrary temperature
via perturbation theory
3
[Enss, Haussmann, Zwerger 2010]
�3/2 C
√
η(ω → ∞) =
15π mω
• viscosity sum rule
2
π
�
∞
0
0.1
4
• analytical expression for tail
Contact coefficient C [kF ]
Viscosity tail
Tan contact C
visc. tail 15/Cd
0.01
0.1
�2 C
dω [η(ω) − tail] = p −
4πma
provides non-perturbative check
[Enss, Haussmann, Zwerger 2010;
cf. Taylor, Randeria 2010]
1
T [TF]
10
20
5
<
d/s [h / kB]
10
2
1
phonon
classical
full Kubo
∼ T 3/2
classical limit
phonon
contribution
∼ T −8
0.5
Tc
0.2
0.1 0.2
0.5
1
2
T [TF]
Shear viscosity/entropy
of the unitary Fermi gas
5
10
20
[Enss, Haussmann, Zwerger 2010]
Scaling Flows and Dissipation in the Dilute Fermi Gas at Unitarity
25
1.5
!η/s"
1.0
0.5
0.0
0.2
0.4
0.6
0.8
T /TF
Fig. 11 Trap average !!s " = !" /s" extracted from the damping of the radial breathing mode. The
data points were obtained using equ. (91) to analyze the data published by Kinast et al. [9]. The
thermodynamic quantities (S/N) and E0 /EF were taken from [22]. The solid red and blue lines
[fit: Schaefer, Chafin 2009;
show the expected low and high temperature limits. Both exp:
theory
curves include relaxation time
Kinast et al. 2004, Luo et al. 2009]
effects. The blue dashed curve is a phenomenological two-component model explained in the text.
shear viscosity
for trapped atoms
Scale invariance
• Unitary Fermi gas: particle spacing only length scale, scale invariance
�
d3 x Π̂ii (x, t) = 2Ĥ = 2(T̂ + V̂ ) ➡ 3p = 2� [Ho 2004]
�
�
�
�
�
1
d3 x dt eiωt θ(t) Π̂ii (x, t), Π̂jj (0, 0) ≡ 0
• bulk viscosity ζ(ω) =
9ω
[Nishida, Son 2007]
• pressure 3p̂ =
• response of Green’s functions to scale transformation: Ward identities
[Enss, Haussmann, Zwerger 2010]
ωT [Πii ] = (� + µ)G(p, �) − (� + ω + µ)G(p, � + ω)
ωTbos [Πii ] = (Ω + ω + 2µ)Γ(p, Ω) − (Ω + 2µ)Γ(p, Ω + ω)
• WI provides solution of transport equation:
conserving approximation
Beyond viscosity: universal dynamics
• viscosity determines several correlation functions at unitarity (small q):
[cf. Taylor, Randeria 2010]
mom.
conserv.
Im �[Πxy , Πxy ]�q,ω = ω η(ω)
charge
conserv.
T
Im �[Ji , Jk ]�q,ω
L
Im �[Ji , Jk ]�q,ω
Im �[ρ, ρ]�q,ω
(generally)
q2
=
η(ω)
ω �
at
unitarity
�
2
q
4
=
η(ω) + ζ(ω)
ω 3
q4 4
= 3 η(ω)
ω 3
�
qi qk
qi qk
L
�[Ji , Jk ]� = 2 �[Ji , Jk ]� + δik − 2
q
q
�
�[Ji , Jk ]�T
(generally)
(generally)
(at unitarity)
Conclusion and Outlook
• are cold atoms (unitary Fermi gas) the perfect fluid?
20
➡ most perfect real non-relativistic fluid known
(factor ≈7 above string theory bound)
d/s [<h / kB]
10
phonon
classical
full Kubo
5
2
1
0.5
➡ strongest contender with quark-gluon plasma
0.2
0.1 0.2
0.5
1
2
T [TF]
5
10
20
• transport calculation of η(ω) beyond Boltzmann (tail, large scattering rate)
• conserves scale invariance and fulfills sum rules
10
d(t) [<h n]
100
1
• universal dynamic properties of unitary Fermi gas
T=10
T= 5
T= 2
T= 1
T=0.5
T=0.151
0.1
0.01
0.001
0.1
1
10
t [EF / <h]
100
1000