e expansion in cold atoms

1
e expansion in cold atoms
Yusuke Nishida (Univ. of Tokyo INT) in
collaboration with D. T. Son (INT)
Ref Phys. Rev. Lett. 97, 050403 (2006)
cond-mat/0607835, cond-mat/0608321

• Fermi gas at infinite scattering length
• Formulation of e (4-d, d-2) expansions
• LO NLO results
• Summary and outlook

ECT workshop on the interface on QGP and cold
atoms
2
Interacting Fermion systems
Attraction Superconductivity / Superfluidity

• Metallic superconductivity (electrons)
• Kamerlingh Onnes (1911), Tc 9.2 K
• Liquid 3He
• Lee, Osheroff, Richardson (1972), Tc 12.6 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
Feshbach resonance
C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003)
Attraction is arbitrarily tunable by magnetic
field
S-wave scattering length ? 0, ?
Feshbach resonance
a (rBohr)
agt0 Bound state formation molecules
Strong coupling a??
alt0 No bound state atoms
40K
Weak coupling a?0
4
BCS-BEC crossover
Eagles (1969), Leggett (1980) Nozières and
Schmitt-Rink (1985)
Strong interaction
?
Superfluidphase
-B
-?
?
0
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

5
Unitary Fermi gas
George Bertsch (1999), Many-Body X Challenge
Atomic gas r0 10Å ltlt kF-1100Å ltlt
a1000Å
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

• Strong coupling limit
• Perturbation a kF?

• Difficulty for theory
• No expansion parameter

6
Unitary Fermi gas at d?3
d4

• d?4 Weakly-interacting system of fermions
  bosons, their coupling is g(4-d)1/2

Strong coupling Unitary regime
BEC
BCS
?
-?

• d?2 Weakly-interacting system of fermions,
  their coupling is g(d-2)

d2
Systematic expansions for x and other observables
(D, Tc, ) in terms of 4-d or d-2
7
Specialty of d4 and 2
Z.Nussinov and S.Nussinov, cond-mat/0410597
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
8
Field theoretical approach
2-component fermions local 4-Fermi interaction
2-body scattering at vacuum (m0)
?
(p0,p) ?

n
1
?
T-matrix at arbitrary spatial dimension d
a??
Scattering amplitude has zeros at d2,4,
Non-interacting limits
9
T-matrix around d4 and 2
T-matrix at d4-e (eltlt1)
Small coupling b/w fermion-boson g (8p2 e)1/2/m
ig
ig

iD(p0,p)
T-matrix at d2e (eltlt1)
Small coupling b/w fermion-fermion g 2p e/m
ig

10
Thermodynamic functions at T0

• Effective potential and gap equation around d4

Veff (?0,m)
O(e2)


O(e)
O(1)

• Effective potential and gap equation around d2

O(e2)
Veff (?0,m)

O(e)
O(1)
is negligible
11
Universal parameter x

• Universal equation of state

• Universal parameter x around d4 and 2

Arnold, Drut, Son (06)
Systematic expansion of x in terms of e !
12
Quasiparticle spectrum

• Fermion dispersion relation w(p)

NLO self-energy diagrams
- i S(p)
or
O(e)
O(e)
Expansion over 4-d
Energy gap Location of min.
Expansion over d-2
0
13
Extrapolation to d3 from d4-e

• Keep LO NLO results and extrapolate to e1

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 d2e
NLO are 100
14
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
15
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)

16
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)
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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
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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
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Summary

• Systematic expansions over e4-d or d-2
• Unitary Fermi gas around d4 becomes
• weakly-interacting system of fermions
  bosons
• Weakly-interacting system of fermions around d2
• LONLO results on x, D, e0, Tc
• NLO corrections around d4 are small
• Extrapolations to d3 agree with recent MC data
• Future problems
• Large order behavior NNLO corrections
• More understanding Precise determination

Picture of weakly-interacting fermionic bosonic
quasiparticles for unitary Fermi gas may be a
good starting point even at d3
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Back up slides
21
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
22
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
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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

24
Feynman rules 1

• L0
• Free fermion quasiparticle ? and boson ?

• L1

Small coupling g between ? and ? (g
e1/2) Chemical potential insertions (m e)
25
Feynman rules 2

• L2

Counter vertices to cancel 1/e
singularities in boson self-energies
1. 2.
O(e)
O(e m)
26
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
27
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

28
NNLO correction for x
Arnold, Drut, and Son, cond-mat/0608477

• 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
d
NNLO 4d
29
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

30
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 !
31
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???