1. Unitary Fermi gas in the e expansion
Yusuke Nishida (Univ. of Tokyo & INT)
in collaboration with D. T. Son
[Ref: Phys. Rev. Lett. 97, (2006), cond-mat/ , cond-mat/ ]
16 January, T.I.Tech

2. Unitary Fermi gas in the e expansion
Contents of this talk
Fermi gas at infinite scattering length
Formulation of expansions
in terms of 4-d and d-2
Results at zero/finite temperature
Summary and outlook

3. Introduction : Fermi gas at infinite scattering length

4. Interacting Fermion systems
Attraction Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Kamerlingh Onnes (1911), Tc = ~9.2 K
Liquid 3He
Lee, Osheroff, Richardson (1972), Tc = 1~2.6 mK
High-Tc superconductivity (electrons or holes)
Bednorz and Müller (1986), Tc = ~160 K
Atomic gases (40K, 6Li)
Regal, Greiner, Jin (2003), Tc ~ 50 nK
Nuclear matter (neutron stars): ?, Tc ~ 1 MeV
Color superconductivity (quarks): ??, Tc ~ 100 MeV
Neutrino superfluidity: ??? [Kapusta, PRL(’04)]
BCS theory
(1957)

5. Feshbach resonance Attraction is arbitrarily tunable by magnetic field
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
Attraction is arbitrarily tunable by magnetic field
S-wave scattering length :  [0, ]
a (rBohr)
Feshbach resonance
a>0
Bound state
formation
Strong coupling
|a|
a<0
No bound state
40K
Weak coupling
|a|0

6. ? BCS-BEC crossover - + BCS state of atoms weak attraction: akF-0
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
Strong interaction ?
Superfluid phase -B - +
BCS state of atoms
weak attraction: akF-0
BEC of molecules
weak repulsion: akF+0
Strong coupling limit : |a kF|
Maximal S-wave cross section Unitarity limit
Threshold: Ebound = 1/(2ma2)  0
Fermi gas in the strong coupling limit a kF= : Unitary Fermi gas

7. Unitary Fermi gas What are the ground state properties of
George Bertsch (1999),
“Many-Body X Challenge”
Atomic gas : r0 =10Å << kF-1=100Å << |a|=1000Å
What are the ground state properties of
the many-body system composed of
spin-1/2 fermions interacting via a zero-range,
infinite scattering length contact interaction?
0 r0 << kF-1 << a 
kF is the only scale ! r0
V0(a)
kF-1
Energy per particle
x is independent of systems
cf. dilute neutron matter
|aNN|~18.5 fm >> r0 ~1.4 fm

8. Universal parameter x Simplicity of system Difficulty for theory
x is universal parameter
Difficulty for theory
No expansion parameter
Models
Simulations
Experiments
Mean field approx., Engelbrecht et al. (1996): x<0.59
Linked cluster expansion, Baker (1999): x=0.3~0.6
Galitskii approx., Heiselberg (2001): x=0.33
LOCV approx., Heiselberg (2004): x=0.46
Large d limit, Steel (’00)Schäfer et al. (’05): x=0.440.5
Carlson et al., Phys.Rev.Lett. (2003): x=0.44(1)
Astrakharchik et al., Phys.Rev.Lett. (2004): x=0.42(1)
Carlson and Reddy, Phys.Rev.Lett. (2005): x=0.42(1)
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

9. Unitary Fermi gas at d≠3 d=4 BCS BEC - + d=2
d4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d)1/2
　Strong coupling
Unitary regime
BCS
BEC - + g
d2 : Weakly-interacting system of fermions, their coupling is g~(d-2)
d=2
Systematic expansions for x and other observables (D, Tc, …) in terms of “4-d” or “d-2”

10. Formulation of e expansion
e=4-d <<1 : d=spatial dimensions

11. Specialty of d=4 and d=2  = 
2-component fermions
local 4-Fermi interaction :
2-body scattering in vacuum (m=0)  iT
(p0,p)  = 1 n 
T-matrix at arbitrary spatial dimension d
“a”
Scattering amplitude has zeros at d=2,4,…
Non-interacting limits

12. T-matrix around d=4 and 2 Small coupling b/w fermion-boson
T-matrix at d=4-e (e<<1)
Small coupling b/w fermion-boson
g = (8p2 e)1/2/m iT ig ig =
iD(p0,p)
T-matrix at d=2+e (e<<1)
Small coupling b/w fermion-fermion
g = (2p e/m)1/2 iT
ig2 =

13. Lagrangian for e expansion
Hubbard-Stratonovish trans. & Nambu-Gor’kov field :
=0 in dimensional regularization
Ground state at finite density is superfluid :
Expand with
Rewrite Lagrangian as a sum : L = L0+ L1+ L2
Boson’s kinetic term is added,
and subtracted here.

14. Small coupling “g” between  and 
Feynman rules 1
L0 :
Free fermion quasiparticle  and boson 
L1 :
Small coupling “g” between  and 
(g ~ e1/2)
Chemical potential
insertions (m ~ e)

15. Feynman rules 2 = O(e) + = O(e m) + “Counter vertices” to
cancel 1/e singularities
in boson self-energies p
p+k k +
= O(e) 1. 2.
O(e) p
p+k k +
= O(e m)
O(e m)

16. Power counting rule of e
Assume justified later
and consider to be O(1)
Draw Feynman diagrams using only L0 and L1
If there are subdiagrams of type
add vertices from L2 :
Its powers of e will be Ng/2 + Nm
The only exception is = O(1) O(e) or or
Number of m insertions
Number of couplings “g ~ e1/2”

17. Expansion over e = d-2 Lagrangian Power counting rule of 
Assume justified later
and consider to be O(1)
Draw Feynman diagrams using only L0 and L1
If there are subdiagrams of type
add vertices from L2 :
Its powers of e will be Ng/2

18. Results at zero/finite temperature
Leading and next-to-leading orders

19. Thermodynamic functions at T=0
Effective potential : Veff = vacuum diagrams p q
p-q k k
Veff (0,m) =
+ O(e2) + +
O(e)
O(1)
Gap equation of 0
C= …
Assumption is OK !
Pressure : with the solution 0(m)

20. Systematic expansion of x in terms of e !
Universal parameter x
Universal equation of state
Universal parameter x around d=4 and 2
Arnold, Drut, Son (’06)
Systematic expansion of x in terms of e !

21. Quasiparticle spectrum
Fermion dispersion relation : w(p)
O(e) p k
p-k p k
k-p
Self-energy diagrams
- i S(p) = +
Expansion over 4-d
Energy gap :
Location of min. :
Expansion over d-2

22. Extrapolation to d=3 from d=4-e
Keep LO & NLO results and extrapolate to e=1
NLO
corrections
are small
5 ~ 35 %
Good agreement with recent Monte Carlo data
J.Carlson and S.Reddy,
Phys.Rev.Lett.95, (2005)
cf. extrapolations from d=2+e
NLO are 100 %

23. Matching of two expansions in x
Borel transformation + Padé approximants
Expansion around 4d x
♦=0.42
2d boundary condition 2d
Interpolated results to 3d 4d d

24. NLO correction is small ~4 %
Critical temperature
Gap equation at finite T
Veff = m insertions
Critical temperature from d=4 and 2
NLO correction is small ~4 %
Simulations :
Lee and Schäfer (’05): Tc/eF < 0.14
Burovski et al. (’06): Tc/eF = 0.152(7)
Akkineni et al. (’06): Tc/eF  0.25
Bulgac et al. (’05): Tc/eF = 0.23(2)

25. Matching of two expansions (Tc)d
Tc / eF 4d 2d
Borel + Padé approx.
Interpolated results to 3d
Tc / eF
P / eFN
E / eFN
m / eF
S / N
NLO e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)

26. Picture of weakly-interacting fermionic &
Summary 1
e expansion for unitary Fermi gas
Systematic expansions over 4-d and d-2
Unitary Fermi gas around d=4 becomes
weakly-interacting system of fermions & bosons
Weakly-interacting system of fermions around d=2
LO+NLO results on x, D, e0, Tc (P,E,m,S)
NLO corrections around d=4 are small
Naïve extrapolation from d=4 to d=3 gives
good agreement with recent MC data
Picture of weakly-interacting fermionic &
bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3

27. e expansion for unitary Fermi gas
Summary 2
e expansion for unitary Fermi gas
Matching of two expansions around d=4 and d=2
NLO 4d + NLO 2d
Borel transformation and Padé approximants
Results are not too far from MC simulations
Future Problems
More understanding on e expansion
Large order behavior + NN…LO corrections
Analytic structure of x in “d” space
Precise determination of universal parameters
Other observables, e.g., Dynamical properties

28. Back up slides

29. Specialty of d=4 and 2 2-body wave function
Z.Nussinov and S.Nussinov, cond-mat/
2-body wave function
Normalization at unitarity a
diverges at r0 for d4
Pair wave function is concentrated near its origin
Unitary Fermi gas for d4 is free “Bose” gas
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Unitary Fermi gas for d2 is free Fermi gas

30. “Naïve” power counting of e
Feynman rules 2
L2 :
“Counter vertices” of boson 
“Naïve” power counting of e
Assume justified later
and consider to be O(1)
Draw Feynman diagrams using only L0 and L1 (not L2)
Its powers of e will be Ng/2 + Nm
Number of m insertions
Number of couplings “g ~ e1/2”
But exceptions
Fermion loop integrals produce 1/e in 4 diagrams

31. Exceptions of power counting 1
1. Boson self-energy naïve O(e) p
p+k k +
= O(e)
Cancellation with L2 vertices to restore naïve counting
2. Boson self-energy with m insertion naïve O(e2) p
p+k k
= O(e2) +

32. Exceptions of power counting 2
3. Tadpole diagram with m insertion p k
= O(e1/2) naïve O(e3/2)
Sum of tadpoles = Gap equation for 0 +
+ · · · = 0
O(e1/2)
O(e1/2)
4. Vacuum diagram with m insertion k
= O(1) O(e) Only exception !

33. NNLO correction for x x O(e7/2) correction for x
Arnold, Drut, and Son, cond-mat/
O(e7/2) correction for x
Borel transformation + Padé approximants x
Interpolation to 3d
NNLO 4d + NLO 2d
cf. NLO 4d + NLO 2d
NLO 4d
NLO 2d
NNLO 4d d

34. Hierarchy in temperature
At T=0, D(T=0) ~ m/e >> m
2 energy scales
(i) Low : T ~ m << DT ~ m/e
(ii) Intermediate : m < T < m/e
(iii) High : T ~ m/e >> m ~ DT
D(T)
Fermion excitations are suppressed
Phonon excitations are dominant
(i) (ii) (iii) T
~ m
Tc ~ m/e
Similar power counting
m/T ~ O(e)
Consider T to be O(1)
Condensate vanishes at Tc ~ m/e
Fermions and bosons are excited

35. e expansion is asymptotic series but works well !
Large order behavior
d=2 and 4 are critical points
free gas
r0≠0
Critical exponents of O(n=1) 4 theory (e=4-d  1)
O(1)
+e1
+e2
+e3
+e4
+e5
Lattice g 1
1.167
1.244
1.195
1.338
0.892
1.239(3)
Borel transform with conformal mapping g=1.23550.0050
Boundary condition (exact value at d=2) g=1.23800.0050
e expansion is asymptotic series but works well !

36. e expansion in critical phenomena
Critical exponents of O(n=1) 4 theory (e=4-d  1)
O(1)
+e1
+e2
+e3
+e4
+e5
Lattice
Exper. g 1
1.167
1.244
1.195
1.338
0.892
1.239(3)
1.240(7)
1.22(3)
1.24(2) 
0.0185
0.0372
0.0289
0.0545
0.027(5)
0.016(7)
0.04(2)
Borel summation with conformal mapping
g=1.2355 & =0.03600.0050
Boundary condition (exact value at d=2)
g=1.2380 & =0.03650.0050
e expansion is
asymptotic series
but works well !
How about our case???