e ?????? ?????

1
e ????????????????????
?? ?? (INT, ???????)

• ????? ???????????
• ??e ?????? (e 4-d, d-2)
• LO NLO ????
• ??????

e expansion for a Fermi gas at infinite
scattering length Y. N. D. T. Son, Phys.
Rev. Lett. 97, 050403 (2006) Phys. Rev. A 75,
063617 (2007) Y. N., Phys. Rev. A 75, 063618
(2007)
2
??????????????
?????? ??????

• ????? (??)
• Kamerlingh Onnes (1911), Tc 9.2 K
• ??????3
• Lee, Osheroff, Richardson (1972), Tc 12.6 mK
• ????? (?? or ???)
• Bednorz and Müller (1986), Tc 160 K
• ?????? (40K, 6Li)
• Regal, Greiner, Jin (2003), Tc 50 nK
• ??? (???) ?, Tc 1 MeV
• ?????? (????) ??, Tc 100 MeV

BCS theory (1957)
3
??????????
C.A.Regal and D.S.Jin, Phys.Rev.Lett. (2003)
???? ???????????? (??????????)
?????????????????????????
S???? a ? 0, ?
a (rBohr)
?????????? ????????? a??
agt0 ???? ???
alt0 ??????
40K
??? a?0 ??? a?-0
B (Gauss)
4
BCS-BEC ???????
Eagles (1969), Leggett (1980) Nozières and
Schmitt-Rink (1985)
?
?????
?
-?
0
????
?????BCS?? ???? akF?-0
?????BEC?? ???? akF?0

• ???????????
• ????? a kF??
• ???? _at_ ??????????

??????????? (?????????????) ????
???????????????????
5
???????????
George Bertsch (1999), Many-Body X Challenge
???? _at_ ?????????? r0 ltlt kF-1 ltlt a
??????? (a ??) ? ????????? (r0 ?0) ???
???1/2???????
????? a kF? ??????? ????????????
????????

• ??????????
• ?????(???)
• ????????????????
• ????????????? e ??

6
e ?????????? (e 4-d, d-2)
d4

• d?4 ???????? ???? ??????? ??????
  g(4-d)1/2

???? ???????
BEC
BCS
?
-?

• d?2 ???????? ???????? ?????? g(d-2)

d2
???????????4-d ???? d-2 ? ??????????????????
???
7
???????????
??????????? ???1/2???????
?????2??? (???? m0)
?
(p0,p) ?

n
1
?
??????? d ??T??
a??
????? d2,4, ???????
???????????
8
d4, d2 ????????
d4-e (eltlt1) ????T??
???? ??????????? g (8p2 e)1/2/m

ig
ig
iD(p0,p)
d2e (eltlt1) ????T??
???????????? g 2p e/m
ig

9
???1 ????????(NLO)

• d4 ????????????????????

Veff (?0,m)
O(e2)


O(e)
O(1)

• d2 ????????????????????

Veff (?0,m)
O(e2)

O(e)
O(1)
10
???????????

• ?????????????????

1???????????

• d4, d2 ??????????? x

?????? e (4-d, d-2) ?????????? !
11
???2 ????????

• ???????????? w(p)

NLO ???????
or
- i S(p)
O(e)
O(e)
e 4-d ?????
????????? ???????
e d-2 ?????
0
12
d4-e ??????3????

• LO NLO ???? e1 ???

NLO?? LO????? ??? 5 35
????????????????????????
J.Carlson and S.Reddy, Phys.Rev.Lett.95, (2005)
cf. d2e ?? d3 ???
NLO??LO??100
13
4-d ??? d-2 ????????

• ????? ?????

4-d ?????
x
?0.42
d-2 ???????
2d

• d3 ???????

4d
d
14
???????????

• 4-d, d-2 ?????????

NLO???????? 4
Tc / eF

• d3 ???????

4d
??????????

• Bulgac et al. (05) Tc/eF 0.23(2)
• Lee and Schäfer (05) Tc/eF lt 0.14
• Burovski et al. (06) Tc/eF 0.152(7)
• Akkineni et al. (06) Tc/eF ? 0.25

2d
d
15
??????

• ??????????????? e 4-d, d-2 ????????????!
• d4 ????????????????? ???????
• d2 ?????????????????????
• ?????? (x, D, e0, Tc ,) ? LONLO ???
• 4-d ??? NLO?? LO??????????
• d3 ?????????MC?????????????
• ?????
• ???????? e ???????????
• ????????????

???????????? ??????? ?????????????????? ???
d3 ?????????????
16
Back up slides
17
d4 ? d2 ????
Z.Nussinov and S.Nussinov, cond-mat/0410597
2??(s?)??????
??????? ( a?? ) ??????? ? d?4 ??? ??
(r?0) ???
2??????????????
???????????? d?4 ????????????
d?2 ??????????????????????
a?? ??????????
d?2 ?????????????
18
e?????(NLO)???
?????????? MC ??? ???? ?? 0.42 0.591
??????????????????? 1.2 1.16
????? 0.58 0.699
???? 0.25 0.500
0.475
1.31
0.661
0.249
??????????? ?????????
19
NNLO correction for x

• NNLO correction for x

Arnold, Drut, Son, Phys.Rev.A (2006)
x
Nishida, Ph.D. thesis (2007)
Fit two expansions using Padé approximants

• Interpolations to 3d
• NNLO 4d NNLO 2d
• cf. NLO 4d NLO 2d

d
20
Unitary Fermi gas
George Bertsch (1999), Many-Body X Challenge
Atomic gas r0 10Å ltlt kF-1100Å ltlt
a1000Å
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 ltlt kF-1 ltlt a ??
kF is the only scale !
Energy per particle
x is independent of systems
cf. dilute neutron matter aNN18.5 fm gtgt r0
1.4 fm
21
Universal parameter x

• Strong coupling limit
• Perturbation a kF?

• Difficulty for theory
• No expansion parameter

• Mean field approx., Engelbrecht et al.
  (1996) xlt0.59
• Linked cluster expansion, Baker
  (1999) x0.30.6
• Galitskii approx., Heiselberg (2001) x0.33
• LOCV approx., Heiselberg (2004) x0.46
• Large d limit, Steel (00)?Schäfer et al.
  (05) x0.44?0.5

Models Simulations Experiments

• Carlson et al., Phys.Rev.Lett.
  (2003) x0.44(1)
• Astrakharchik et al., Phys.Rev.Lett.
  (2004) x0.42(1)
• Carlson and Reddy, Phys.Rev.Lett.
  (2005) x0.42(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
22
Lagrangian for e expansion

• Hubbard-Stratonovish trans. Nambu-Gorkov
  field

0 in dimensional regularization
Ground state at finite density is superfluid
Expand with

• Rewrite Lagrangian as a sum L L0 L1 L2

23
Feynman rules 1

• L0
• Free fermion quasiparticle ? and boson ?

• L1

Small coupling g between ? and ? (g
e1/2) Chemical potential insertions (m e)
24
Feynman rules 2

• L2

Counter vertices to cancel 1/e
singularities in boson self-energies
1. 2.
O(e)
O(e m)
25
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
26
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

27
Hierarchy in temperature
At T0, D(T0) m/e gtgt m
2 energy scales
(i) Low T m ltlt DT m/e (ii)
Intermediate m lt T lt m/e (iii) High T
m/e gtgt m DT
D(T)

• Fermion excitations are suppressed
• Phonon excitations are dominant

(i) (ii) (iii)
T
0
Tc m/e
m

• Similar power counting
• m/T O(e)
• Consider T to be O(1)

• Condensate vanishes at Tc m/e
• Fermions and bosons are excited

28
Large order behavior

• d2 and 4 are critical points

free gas
r0?0
2 3 4

• Critical exponents of O(n1) ?4 theory (e4-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
  g1.2355?0.0050
• Boundary condition (exact value at d2)
  g1.2380?0.0050

e expansion is asymptotic series but works well !
29
e expansion in critical phenomena
Critical exponents of O(n1) ?4 theory (e4-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 0 0.0185 0.0372 0.0289 0.0545 0.027(5) 0.016(7) 0.04(2)

• Borel summation with conformal mapping
• g1.2355?0.0050 ?0.0360?0.0050
• Boundary condition (exact value at d2)
• g1.2380?0.0050 ?0.0365?0.0050

e expansion is asymptotic series but works well !
How about our case???
30
Critical temperature

• Gap equation at finite T

Veff
m insertions

• Critical temperature from d4 and 2

NLO correctionis small 4
Simulations

• Lee and Schäfer (05) Tc/eF lt 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)

31
Matching of two expansions (Tc)

• 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)
32
Comparison with ideal BEC

• Ratio to critical temperature in the BEC limit

Boson and fermion contributions to fermion
density at d4

• Unitarity limit

• BEC limit

1 of 9 pairs is dissociated
all pairs form molecules
33
Polarized Fermi gas around d4

• Rich phase structure near unitarity point
• in the plane of and

binding energy
Polarized normal state
Gapless superfluid
1-plane wave FFLO O(e6)
Gapped superfluid
BCS
BEC
unitarity
Stable gapless phases (with/without spatially
varying condensate) exist on the BEC side of
unitarity point