Unitary Fermi gas in the e expansion

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, 050403 (2006), cond-mat/0607835,
cond-mat/0608321 10 January, 2007 _at_ Univ. of
Tokyo
2
Contents of thesis

• Introduction
• Two-body scattering in vacuum
• Unitary Fermi gas around d4
• Phase structure of polarized Fermi gas
• Fermions with unequal masses
• Expansions around d2
• Matching of expansions at d4 and d2
• Thermodynamics below Tc
• Thermodynamics above Tc
• Summary and concluding remarks

3
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

4
Introduction Fermi gas at infinite scattering
length
5
Interacting Fermion systems
Attraction Superconductivity / Superfluidity

• Metallic superconductivity (electrons)
• 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
• 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)
6
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
Strong coupling a??
alt0 No bound state
40K
Weak coupling a?0
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BCS-BEC crossover
Eagles (1969), Leggett (1980) Nozières and
Schmitt-Rink (1985)
Strong interaction
?
Superfluidphase
-B
1/(akF)-?
1/(akF)?
1/(akF)0
BCS state of atoms weak interaction akF?-0
BEC of molecules weak interaction 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
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Unitary Fermi gas
George Bertsch (1999), Many-Body X Challenge
Atomic gas a1000Å gtgt kF-1100Å gtgt r0
10Å
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
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Universal parameter x

• Simplicity of system
• x is universal parameter

• 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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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
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Formulation of e expansion
e4-d ltlt1 dspatial dimensions
12
Specialty of d4 and d2
2-component fermions local 4-Fermi interaction
2-body scattering in vacuum (m0)
?
(p0,p) ?

n
1
?
T-matrix at arbitrary spatial dimension d
a??
Scattering amplitude has zeros at d2,4,
Non-interacting limits
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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)1/2
ig2

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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

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Feynman rules 1

• L0
• Free fermion quasiparticle ? and boson ?

• L1

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

• L2

Counter vertices to cancel 1/e singularities
in boson self-energies with/out m insertion
1. 2.
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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)

or
or
Number of m insertions Number of couplings g
e1/2
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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

19
Results at zero/finite temperature
Leading and next-to-leading orders
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Thermodynamic functions at T0

• Effective potential Veff vacuum diagrams

Veff (?0,m)
O(e2)


O(e)
O(1)

• Gap equation of ?0

C0.14424
Assumption is OK !

• Pressure P(m)-Veff (?0,m) with solution of
  gap eq. ?0

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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 !
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Quasiparticle spectrum

• Fermion dispersion relation w(p)

O(e)
Self-energydiagrams
- i S(p)

Expansion over 4-d
Energy gap Location of min.
Expansion over d-2
0
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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
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Matching of two expansions in x

• Borel transformation Padé approximation

2d boundary condition

• Interpolated results to 3d

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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)

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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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Summary 1
e expansion for unitary Fermi gas

• Systematic expansions over 4-d and 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 (P,E,m,S)
• NLO corrections around d4 are small
• Naïve extrapolation from d4 to d3 gives
• good agreement with recent MC data

Picture of weakly-interacting fermionic bosonic
quasiparticle for unitary Fermi gas may be a
good starting point even at d3
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Summary 2
e expansion for unitary Fermi gas

• Matching of two expansions around d4 and d2
• NLO 4d NLO 2d
• Borel transformation and Padé approximation
• Results are not too far from MC simulations

Future Problems

• More understanding on e expansion
• Large order behavior NNLO corrections
• Analytic structure of x in d space

Precise determination of universal parameters
Other observables, e.g., Dynamical properties
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Back up slides
30
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
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Feynman rules 2

• L2

Counter verticesof 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
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Exceptions of power counting 1
1. Boson self-energy naïve O(e)
Cancellation with L2 vertices to restore naïve
counting
2. Boson self-energy with m insertion naïve O(e2)
O(e2)

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Exceptions of power counting 2
3. Tadpole diagram with m insertion
O(e1/2) naïve O(e3/2)
Sum of tadpoles 0 Gap equation for ?0
O(e1/2)
O(e1/2)
4. Vacuum diagram with m insertion
O(1) O(e) Only exception !
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Effective potential Veff (?0)

• Leading order O(1)

Bosons 1-loops and vanish at T0

• Next-to-leading order O(e)

C0.14424
35
Universal parameter x

• Gap equation of ?0
• Fermion density and Fermi energy
• Universal parameter x m/eF

Systematic expansion of x in terms of e !
36
NNLO correction for x
Arnold, Drut, and Son, cond-mat/0608477

• O(e7/2) correction for x

• Borel transformation Padé approximation

x

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

NLO 4d
NLO 2d
d
NNLO 4d
37
Quasiparticle spectrum

• Fermion 1-loop self-energy O(e)

p-k
k-p

- i S(p)
p
p
p
p
k
k

• Fermion dispersion relation w(p)

Around minimum
NLO
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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

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Comparisons of Tc
e expansion(LONLO)
Simulations Experiment

• Wingate (05) Tc/eF 0.04
• Lee and Schäfer (05) Tc/eF lt 0.14
• Bulgac et al. (05) Tc/eF 0.23(2)
• Burovski et al. (06) Tc/eF 0.152(7)
• Kinast et al. (05) Tc/eF 0.27(2)
• cf. BEC limit TBEC/eF 0.218

Ideal BEC at d4-e (convergent if e ? 2)
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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 !
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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???