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e expansion in cold atoms

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

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e expansion in cold atoms

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  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 = 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 ~ 1MeV • Color superconductivity (cold QGP):??, Tc ~ 100MeV • 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) a>0 Bound state formation molecules Strong coupling |a| a<0 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 : |akF| • Maximal S-wave cross section Unitarity limit • Threshold: Ebound = 1/(2ma2)  0

  5. Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge” kF-1 r0 V0(a) Atomic gas : r0=10Å << kF-1=100Å << |a|=1000Å spin-1/2 fermions interacting via a zero-range, infinite scattering length contact interaction 0r0 << kF-1<< a kF is the only scale ! Energy per particle x is independent of systems cf.dilute neutron matter |aNN|~18.5 fm >> r0~1.4 fm • Strong coupling limit • Perturbation akF= • Difficulty for theory • No expansion parameter

  6. Unitary Fermi gas at d≠3 g g d=4 • 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) d=2 Systematic expansions forx and other observables (D, Tc, …) in terms of “4-d” or “d-2”

  7. Specialty of d=4 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 iT 2-component fermions local 4-Fermi interaction : 2-body scattering at vacuum (m=0)  (p0,p)  = n 1  T-matrix at arbitrary spatial dimension d “a” Scattering amplitude has zeros at d=2,4,… Non-interacting limits

  9. T-matrix around d=4 and 2 iT iT T-matrix at d=4-e(e<<1) Small coupling b/w fermion-boson g = (8p2e)1/2/m ig ig = iD(p0,p) T-matrix at d=2+e(e<<1) Small coupling b/w fermion-fermion g = 2pe/m ig =

  10. Thermodynamic functions at T=0 • Effective potential and gap equation around d=4 + O(e2) Veff (0,m) = + + O(e) O(1) • Effective potential and gap equation around d=2 + O(e2) Veff (0,m) = + O(e) O(1) is negligible

  11. Universal parameterx • Universal equation of state • Universal parameterxaround d=4 and 2 Arnold, Drut, Son (’06) Systematic expansion of x in terms ofe !

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

  13. Extrapolation to d=3 from d=4-e • Keep LO & NLO resultsand extrapolate to e=1 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 %

  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 = + + + minsertions • Critical temperature from d=4 and 2 NLO correctionis small ~4 % Simulations : • Lee and Schäfer (’05): Tc/eF < 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) Tc/eF 4d 2d d • Borel + Padé approx. • Interpolated results to 3d

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

  18. Polarized Fermi gas around d=4 • 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

  19. Summary • 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 • LO+NLO results onx, D, e0, Tc • NLO corrections around d=4 are small • Extrapolations to d=3 agree with recent MC data • Future problems • Large order behavior + NN…LO corrections • More understanding Precise determination Picture of weakly-interacting fermionic & bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3

  20. Back up slides

  21. Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge” kF-1 r0 V0(a) Atomic gas : r0=10Å << kF-1=100Å << |a|=1000Å 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 << kF-1<< a kF is the only scale ! Energy per particle x is independent of systems cf.dilute neutron matter |aNN|~18.5 fm >> r0~1.4 fm

  22. Universal parameterx • Strong coupling limit • Perturbation akF= • Difficulty for theory • No expansion parameter Models Simulations Experiments • Mean field approx., Engelbrecht et al. (1996): x<0.59 • Linked cluster expansion, Baker (1999): x=0.3~0.6 • Galitskii approx., Heiselberg (2001): x=0.33 • LOCV approx., Heiselberg (2004): x=0.46 • Large d limit, Steel (’00)Schäfer et al. (’05): x=0.440.5 • Carlson et al., Phys.Rev.Lett. (2003): x=0.44(1) • Astrakharchik et al., Phys.Rev.Lett. (2004): x=0.42(1) • Carlson and Reddy, Phys.Rev.Lett. (2005): x=0.42(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

  23. Lagrangian for e expansion Boson’s kinetic term is added, and subtracted here. • Hubbard-Stratonovish trans. & Nambu-Gor’kov 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 k p p =O(e) + p+k k p p p+k =O(em) + • L2 : “Counter vertices” to cancel 1/e singularities in boson self-energies 1. 2. O(e) O(em)

  26. Power counting rule ofe • 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 ofe 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 ofe 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 T=0, D(T=0) ~m/e >> m 2 energy scales (i) Low : T~m << DT~m/e (ii) Intermediate : m < T < m/e (iii) High : T~m/e >> 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) • ConsiderTto be O(1) • Condensate vanishes at Tc ~m/e • Fermions and bosons are excited

  30. Large order behavior • d=2 and 4 are critical points free gas r0≠0 2 3 4 • Critical exponents of O(n=1) 4 theory (e=4-d1) • Borel transform with conformal mapping g=1.23550.0050 • Boundary condition (exact value at d=2) g=1.23800.0050 e expansion is asymptotic series but works well !

  31. eexpansion in critical phenomena Critical exponents of O(n=1) 4 theory (e=4-d1) • 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 e expansion is asymptotic series but works well ! How about our case???

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