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???