expansion in cold atoms
Yusuke Nishida (Univ. of Tokyo & INT)in collaboration with D. T. Son (INT)
1. Fermi gas at infinite scattering length
2. Formulation of (=4-d, d-2) expansions
3. LO & NLO results
4. Summary and outlook
ECT* workshop on “the interface on QGP and cold atoms”
[Ref: Phys. Rev. Lett. 97, 050403 (2006) cond-mat/0607835, cond-mat/0608321]
2/19Interacting Fermion systems
Attraction Superconductivity / Superfluidity
Metallic superconductivity (electrons)Kamerlingh Onnes (1911), Tc = ~9.2 K
Liquid 3HeLee, Osheroff, Richardson (1972), Tc = 1~2.6 mK
High-Tc superconductivity (electrons or holes)Bednorz and Müller (1986), Tc = ~160 K
Cold atomic gases (40K, 6Li)Regal, Greiner, Jin (2003), Tc ~ 50 nK
• Nuclear matter (neutron stars): ?, Tc ~ 1 MeV
• Color superconductivity (cold QGP): ??, Tc ~ 100 MeV
• Neutrino superfluidity: ??? [Kapusta, PRL(’04)]
BCS theory
(1957)
3/19Feshbach resonance
Attraction is arbitrarily tunable by magnetic field
C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003)
S-wave scattering length : [0, ]
Strong coupling|a|
a<0 No bound state
atoms40K
a (rBohr)
Weak coupling|a|0
a>0
Bound stateformation
molecules
Feshbach resonance
4/19BCS-BEC crossover
0
BCS state of atomsweak attraction: akF-0
BEC of moleculesweak repulsion: akF+0
Eagles (1969), Leggett (1980)Nozières and Schmitt-Rink (1985)
Strong coupling limit : |a kF|• Maximal S-wave cross section Unitarity limit• Threshold: Ebound = 1/(2ma2) 0
-B
Superfluidphase
?
Strong interaction
5/19Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge”
r0
V0(a)
kF-1
kF is the only scale !
Atomic gas : r0 =10Å << kF-1=100Å << |a|=1000Å
Energy per particle
0 r0 << kF-1 << a
cf. dilute neutron matter |aNN|~18.5 fm >> r0 ~1.4 fm
is independent of systems
spin-1/2 fermions interacting via a zero-range,
infinite scattering length contact interaction
• Strong coupling limit Perturbation a kF=
• Difficulty for theory No expansion parameter
6/19Unitary Fermi gas at d≠3
BECBCS Strong coupling
Unitary regime
d=4
d=2
g
g
• d4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d)1/2
• d2 : Weakly-interacting system of fermions, their coupling is g~(d-2)
Systematic expansions for and other observables (, Tc, …) in terms of “4-d” or “d-2”
7/19Specialty of d=4 and 2
2-body wave function
Z.Nussinov and S.Nussinov, cond-mat/0410597
Pair wave function is concentrated near its origin
Unitary Fermi gas for d4 is free “Bose” gas
Normalization at unitarity a
diverges at r0 for d4
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Unitary Fermi gas for d2 is free Fermi gas
8/19
T-matrix at arbitrary spatial dimension d
Field theoretical approach
iT =
(p0,p) 1 n
“a”
Scattering amplitude has zeros at d=2,4,…Non-interacting limits
2-component fermionslocal 4-Fermi interaction :
2-body scattering at vacuum (=0)
9/19
T-matrix at d=4- (<<1)
T-matrix around d=4 and 2
iT =ig ig
iD(p0,p)
Small coupling b/w fermion-boson
g = (82 )1/2/m
T-matrix at d=2+ (<<1)
iT =ig Small coupling
b/w fermion-fermiong = 2 /m
10/19Thermodynamic functions at T=0
O(1) O()
+ +Veff (0,) =
• Effective potential and gap equation around d=4
+ O(2)
O(1) O()
+Veff (0,) =
• Effective potential and gap equation around d=2
+ O(2)
is negligible
11/19Universal parameter
• Universal parameter around d=4 and 2
Systematic expansion of in terms of !
Arnold, Drut, Son (’06)
• Universal equation of state
12/19Quasiparticle spectrum
- i (p) =
• Fermion dispersion relation : (p)
Energy gap :
Location of min. :
NLOself-energydiagrams
0
Expansion over 4-d
Expansion over d-2
or
O() O()
13/19Extrapolation to d=3 from d=4-• Keep LO & NLO results and extrapolate to =1
J.Carlson and S.Reddy,
Phys.Rev.Lett.95, (2005)
Good agreement with recent Monte Carlo data
NLOcorrectionsare small5 ~ 35 %
NLO are 100 %cf. extrapolations from d=2+
14/19Matching of two expansions in • Borel transformation + Padé approximants
• Interpolated results to 3d
2d boundary condition
d
♦=0.42
4d
2d
Expansion around 4d
15/19Critical temperature
Veff = + + + insertions
• Gap equation at finite T
• Critical temperature from d=4 and 2NLO correctionis small ~4 %
Simulations : • Lee and Schäfer (’05): Tc/F < 0.14• Burovski et al. (’06): Tc/F = 0.152(7)• Akkineni et al. (’06): Tc/F 0.25
• Bulgac et al. (’05): Tc/F = 0.23(2)
16/19
d
Tc / F
4d
2d
Matching of two expansions (Tc)
• Borel + Padé approx.
• Interpolated results to 3d
Tc / F P / FN E / FN / F S / N
NLO 1 0.249 0.135 0. 212 0.180 0.698
2d + 4d 0.183 0.172 0.270 0.294 0.642
Bulgac et al. 0.23(2) 0.27 0.41 0.45 0.99
Burovski et al. 0.152(7) 0.207 0.31(1) 0.493(14) 0.16(2)
17/19Comparison with ideal BEC
• Unitarity limit
1 of 9 pairs is dissociated all pairs form molecules
• Ratio to critical temperature in the BEC limit
• BEC limit
Boson and fermion contributions to fermion density at d=4
18/19
unitarity
BCS BEC
Gapped superfluid
1-plane waveFFLO : O(6)
Polarized normal state
Polarized Fermi gas around d=4• Rich phase structure near unitarity point in the plane of and : binding energy
Stable gapless phases (with/without spatially varying condensate) exist on the BEC side of unitarity point
Gapless superfluid
19/19
1. Systematic expansions over =4-d or d-2• Unitary Fermi gas around d=4 becomes weakly-interacting system of fermions & bosons
• Weakly-interacting system of fermions around d=2
2. LO+NLO results on , , 0, Tc
• NLO corrections around d=4 are small• Extrapolations to d=3 agree with recent MC data
3. Future problems• Large order behavior + NN…LO correctionsMore understanding Precise determination
Summary
Picture of weakly-interacting fermionic &bosonic quasiparticles for unitary Fermi gas
may be a good starting point even at d=3
20/19
Back up slides
21/19Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge”
r0
V0(a)
kF-1
kF is the only scale !
Atomic gas : r0 =10Å << kF-1=100Å << |a|=1000Å
Energy per particle
0 r0 << kF-1 << a
cf. dilute neutron matter |aNN|~18.5 fm >> r0 ~1.4 fm
is independent of systems
What are the ground state properties ofthe many-body system composed of
spin-1/2 fermions interacting via a zero-range,
infinite scattering length contact interaction?
22/19
• Mean field approx., Engelbrecht et al. (1996): <0.59• Linked cluster expansion, Baker (1999): =0.3~0.6• Galitskii approx., Heiselberg (2001): =0.33• LOCV approx., Heiselberg (2004): =0.46• Large d limit, Steel (’00)Schäfer et al. (’05): =0.440.5
Universal parameter
Models
Simulations
Experiments Duke(’03): 0.74(7), ENS(’03): 0.7(1), JILA(’03): 0.5(1),
Innsbruck(’04): 0.32(1), Duke(’05): 0.51(4), Rice(’06): 0.46(5).
No systematic & analytic treatment of unitary Fermi gas
• Carlson et al., Phys.Rev.Lett. (2003): =0.44(1)• Astrakharchik et al., Phys.Rev.Lett. (2004): =0.42(1)• Carlson and Reddy, Phys.Rev.Lett. (2005): =0.42(1)
• Strong coupling limit Perturbation a kF=
• Difficulty for theory No expansion parameter
23/19
Boson’s kinetic term is added,
and subtracted here.
=0 in dimensional regularization
Expand with
Ground state at finite density is superfluid :
Lagrangian for expansion
• Hubbard-Stratonovish trans. & Nambu-Gor’kov field :
• Rewrite Lagrangian as a sum : L = L0+ L1+ L2
24/19Feynman rules 1
• L0 :
Free fermion quasiparticle and boson
• L1 :
Small coupling “g” between and
(g ~ 1/2)
Chemical potential insertions ( ~ )
25/19
+ = O()
Feynman rules 2
• L2 :
“Counter vertices” to cancel 1/ singularitiesin boson self-energies
p p
p+k
k+ = O()
p p
p+k
k
1.
2.
O()
O()
26/19
1. Assume justified later
and consider to be O(1)
2. Draw Feynman diagrams using only L0 and L1
3. If there are subdiagrams of type
add vertices from L2 :
4. Its powers of will be Ng/2 + N
5. The only exception is = O(1) O()
Power counting rule of
or
or
Number of insertions
Number of couplings “g ~ 1/2”
27/19Expansion over = d-2
1. Assume justified later
and consider to be O(1)
2. Draw Feynman diagrams using only L0 and L1
3. If there are subdiagrams of type
add vertices from L2 :
4. Its powers of will be Ng/2
Lagrangian
Power counting rule of
28/19NNLO correction for • O(7/2) correction for
Arnold, Drut, and Son, cond-mat/0608477
• Borel transformation + Padé approximants
d
NLO 4dNLO 2d
NNLO 4d
Interpolation to 3d
• NNLO 4d + NLO 2d
cf. NLO 4d + NLO 2d
29/19
(i) Low : T ~ << T ~ /
(ii) Intermediate : < T < /
(iii) High : T ~ / >> ~ T
• Fermion excitations are suppressed
• Phonon excitations are dominant
Hierarchy in temperature
T
(T)
0Tc ~ /
(i) (ii) (iii)
At T=0, (T=0) ~ / >> 2 energy scales
• Condensate vanishes at Tc ~ /• Fermions and bosons are excited
Similar power counting• /T ~ O()• Consider T to be O(1)
~
30/19Large order behavior
• d=2 and 4 are critical pointsfree gas r0≠02 3 4
• Borel transform with conformal mapping=1.23550.0
050
• Boundary condition (exact value at d=2)=1.23800.0
050
O(1) 2 3 4 5 Lattice
1 1.167 1.244 1.195 1.338 0.892 1.239(3)
• Critical exponents of O(n=1) 4 theory (=4-d 1)
expansion is asymptotic series but works well !
31/19
• Borel summation with conformal mapping=1.23550.0050 & =0.03600.0050
• Boundary condition (exact value at d=2)=1.23800.0050 & =0.03650.0050
expansion in critical phenomena
O(1) 2 3 4 5 Lattice Exper.
1 1.167 1.244 1.195 1.338 0.892 1.239(3)
1.240(7) 1.22(3) 1.24(2)
0 0 0.0185 0.0372 0.0289 0.0545 0.027(5) 0.016(7) 0.04(2)
Critical exponents of O(n=1) 4 theory (=4-d 1)
expansion isasymptotic seriesbut works well !
How about our case???