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]

1. Fermi gas at infinite scattering length

2.

Formulation of

e

(=4-d, d-2) expansions

3.

LO & NLO results

4.

Summary and outlook

ECT * workshop on “the interface on QGP and cold atoms”

Interacting Fermion systems

2/19 Attraction Superconductivity / Superfluidity  Metallic superconductivity (electrons)  Kamerlingh Onnes (1911), T c = ~9.2 K BCS Liquid 3 He Lee, Osheroff, Richardson (1972), T c theory = 1~2.6 mK (1957)  High-T c superconductivity (electrons or holes) Bednorz and M üller (1986), T c = ~160 K 

Cold atomic gases ( 40 K, 6 Li)

Regal, Greiner, Jin (2003), T c ~ 50 nK • Nuclear matter (neutron stars): ?, T c ~ 1 MeV • Color superconductivity (cold QGP): ??, T c ~ 100 MeV • Neutrino superfluidity: ???

[Kapusta, PRL(’04)]

Feshbach resonance

C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003)

3/19 Attraction is arbitrarily tunable by magnetic field S-wave scattering length :  [0,  ] a (r Bohr ) Feshbach resonance a>0 Bound state formation molecules Strong coupling |a|  40 K a<0 No bound state atoms Weak coupling |a|  0

BCS-BEC crossover

Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985)

4/19 Strong interaction - BCS state of atoms weak attraction: ak F  -0 0 ?

Superfluid phase -B + BEC of molecules weak repulsion: ak F  +0 Strong coupling limit : |ak F |   • Maximal S-wave cross section • Threshold: E bound = 1/(2ma 2 )  0 Unitarity limit

Unitary Fermi gas

George Bertsch (1999), “Many-Body X Challenge”

Atomic gas : r 0 =10 Å << k F -1 =100 Å << |a|=1000Å 5/19

spin-1/2 fermions interacting via a zero-range, infinite scattering length contact interaction

0  r 0 << k F -1 << a  k F -1 r 0 V 0 (a) • Strong coupling limit Perturbation ak F =  k F is the only scale !

Energy per particle x is independent of systems cf. dilute neutron matter |a NN |~18.5 fm >> r 0 ~1.4 fm • Difficulty for theory No expansion parameter

Unitary Fermi gas at d ≠3

g 6/19 d=4 • d  4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d) 1/2 - BCS Strong coupling Unitary regime BEC + g d=2 • d  2 : Weakly-interacting system of fermions, their coupling is g~(d-2) Systematic expansions for observables ( x and other D , T c , …) in terms of “ 4 d” or “d-2”

Specialty of d=4 and 2

Z.Nussinov and S.Nussinov, cond-mat/0410597

7/19 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

Field theoretical approach

2-component fermions local 4-Fermi interaction : 2-body scattering at vacuum ( m =0)  (p 0 ,

p

) 

iT

=  T-matrix at arbitrary spatial dimension d 1 n “a  ” Scattering amplitude has zeros at d=2,4,… Non-interacting limits 8/19

T-matrix around d=4 and 2

T-matrix at d=4 e ( e <<1) 9/19

iT

= T-matrix at d=2+ e ( e <<1) ig iD( p 0 ,

p

) ig Small coupling b/w fermion-boson g = (8 p 2 e ) 1/2 /m

iT

= ig Small coupling b/w fermion-fermion g = 2 p e /m

Thermodynamic functions at T=0

• Effective potential and gap equation around d=4 10/19 V eff (  0 , m ) = + + + O( e 2 ) O(1) O( e ) • Effective potential and gap equation around d=2 V eff (  0 , m ) = + + O( e 2 ) O(1) O( e ) is negligible

Universal parameter

x • Universal equation of state 11/19 • Universal parameter x around d=4 and 2 Arnold, Drut, Son (’06) Systematic expansion of x in terms of e !

Quasiparticle spectrum

• Fermion dispersion relation : w ( p ) NLO self-energy diagrams i S ( p ) = Expansion over 4-d Energy gap : O( e ) Location of min. : Expansion over d-2 0 or O( e ) 12/19

Extrapolation to d=3 from d=4-

e • Keep LO & NLO results and extrapolate to e =1 13/19 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 %

Matching of two expansions in

x 14/19 • Borel transformation + Padé approximants Expansion around 4d x 2d boundary condition • Interpolated results to 3d 2d ♦ =0.42

4d d

Critical temperature

• Gap equation at finite T 15/19 V eff = + + + m insertions • Critical temperature from d=4 and 2 NLO correction is small ~4 % Simulations : • Bulgac et al. (’05): T c / e F = 0.23(2) • Lee and Schäfer (’05): T c / e F • Burovski et al. (’06): T c / e F < 0.14

= 0.152(7) • Akkineni et al. (’06): T c / e F  0.25

Matching of two expansions (T

c

)

• Borel + Padé approx.

T c / e F 4d 16/19 • Interpolated results to 3d NLO e 1 2d + 4d Bulgac et al.

Burovski et al.

T c / e F 0.249

0.183

0.23(2) 0.152(7) P / e F N 0.135

0.172

0.27

0.207

2d E / e F N 0. 212 0.270

0.41

0.31(1) m / e F 0.180

0.294

0.45

0.493(14) S / N 0.698

0.642

0.99

0.16(2) d

Comparison with ideal BEC

• Ratio to critical temperature in the BEC limit 17/19 Boson and fermion contributions to fermion density at d=4 • Unitarity limit • BEC limit 1 of 9 pairs is dissociated all pairs form molecules

Polarized Fermi gas around d=4

18/19 • Rich phase structure near unitarity point in the plane of and : binding energy Polarized normal state Gapless superfluid 1-plane wave FFLO : O( e 6 ) Gapped superfluid BCS BEC unitarity Stable gapless phases (with/without spatially varying condensate) exist on the BEC side of unitarity point

Summary

1. Systematic expansions over e =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 x , D , e 0 , T c • NLO corrections around d=4 are small • Extrapolations to d=3 agree with recent MC data 19/19 Picture of weakly-interacting fermionic & bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3 3. Future problems • Large order behavior + NN…LO corrections More understanding Precise determination

Back up slides

20/19

Unitary Fermi gas

George Bertsch (1999), “Many-Body X Challenge”

Atomic gas : r 0 =10 Å << k F -1 =100 Å << |a|=1000Å 21/19

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  r 0 << k F -1 << a  k F -1 k F is the only scale !

Energy per particle r 0 V 0 (a) x is independent of systems cf. dilute neutron matter |a NN |~18.5 fm >> r 0 ~1.4 fm

Universal parameter

x 22/19 • Strong coupling limit Perturbation ak F =  • Difficulty for theory No expansion parameter Models • Mean field approx., Engelbrecht et al. (1996): • Linked cluster expansion, Baker (1999): • Galitskii approx., Heiselberg (2001): • LOCV approx., Heiselberg (2004): • Large d limit, Steel (’00)  Sch äfer et al. (’05): Simulations • Carlson et al., Phys.Rev.Lett. (2003): • Astrakharchik et al., Phys.Rev.Lett. (2004): • Carlson and Reddy, Phys.Rev.Lett. (2005): x <0.59

x =0.3~0.6

x =0.33

x =0.46

x =0.44

 0.5

x =0.44(1) x =0.42(1) x =0.42(1) 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

Lagrangian for

e

expansion

• Hubbard-Stratonovish trans. & Nambu-Gor’kov field : 23/19 =0 in dimensional regularization Ground state at finite density is superfluid : Expand with • Rewrite Lagrangian as a sum : L = L 0 + L 1 + L 2 Boson’s kinetic term is added, and subtracted here.

Feynman rules 1

• L 0 : Free fermion quasiparticle  and boson  24/19 • L 1 : Small coupling “g” between  and  (g ~ e 1/2 ) Chemical potential insertions ( m ~ e )

Feynman rules 2

• L 2 : “Counter vertices” to cancel 1/ e singularities in boson self-energies 1.

O( e ) p k p+k p 2.

O( em ) p k p+k p + + 25/19 = O( e ) = O( em )

Power counting rule of

e 1.

Assume justified later and consider to be O(1) 2.

Draw Feynman diagrams using only L 0 and L 1 3.

If there are subdiagrams of type or 26/19 add vertices from L 2 : or 4.

Its powers of e will be N g /2 + N m Number of m insertions Number of couplings “g ~ e 1/2 ” 5.

The only exception is = O(1) O( e )

Expansion over

e

= d-2

Lagrangian Power counting rule of e 1.

Assume and consider to be O(1) justified later 2.

Draw Feynman diagrams using only L 0 and L 1 3.

If there are subdiagrams of type add vertices from L 2 : 4.

Its powers of e will be N g /2 27/19

NNLO correction for

x • O( e 7/2 ) correction for x

Arnold, Drut, and Son, cond-mat/0608477

28/19 • Borel transformation + Padé approximants x Interpolation to 3d • NNLO 4d + NLO 2d NLO 4d NLO 2d cf. NLO 4d + NLO 2d NNLO 4d d

Hierarchy in temperature

At T=0, D (T=0) ~ m

/

e >> m (i) Low : T ~ m << D T ~ m / e • Fermion excitations are suppressed D (T) 29/19 2 energy scales • Phonon excitations are dominant (ii) Intermediate : m < T < m / e (iii) High : T ~ m / e >> m ~ D T • Condensate vanishes at T c ~ m / e • Fermions and bosons are excited • (i) 0 ~ m (ii) T c ~ (iii) m / Similar power counting m /T ~ O( e ) e • Consider T to be O(1) T

Large order behavior

• d=2 and 4 are critical points free gas 2 3 4 r 0 ≠0 30/19 • Critical exponents of O(n=1)  4 theory ( e =4-d  1) g O(1) 1 +e 1 1.167

+e 2 1.244

+e 3 1.195

+e 4 1.338

+e 5 0.892

Lattice 1.239(3) • Borel transform with conformal mapping • Boundary condition (exact value at d=2) g =1.2355

 0.0050

g =1.2380

 0.0050

e expansion is asymptotic series but works well !

e

expansion in critical phenomena

Critical exponents of O(n=1)  4 theory ( e =4-d  1) 31/19 g  O(1) 1 0 +e 0 1 +e 2 1.167

1.244

+e 3 1.195

+e 4 1.338

+e 5 Lattice 0.892

1.239(3) 0.0185 0.0372 0.0289

0.0545

0.027(5) Exper.

1.240(7) 1.22(3) 1.24(2) 0.016(7) 0.04(2) e expansion is asymptotic series but works well ! • Borel summation with conformal mapping g =1.2355

 0.0050 &  =0.0360

 0.0050

• Boundary condition (exact value at d=2) g =1.2380

 0.0050 &  =0.0365

 0.0050

How about our case???