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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). BCS-BEC crossover and unitarity limit Formulation of e (=4-d) expansion LO & NLO results Summary and outlook.

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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) • BCS-BEC crossover and unitarity limit • Formulation of e(=4-d) expansion • LO & NLO results • Summary and outlook 21COE WS “Strongly correlated many-body systems” 19/Jan/07

  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 • Atomic gases (40K, 6Li) • Regal, Greiner, Jin (2003), Tc ~ 50 nK • Nuclear matter (neutron stars):?, Tc ~ 1MeV • Color superconductivity (quarks):??, Tc ~ 100MeV 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) Superfluid phase - + 0 BCS state of atoms weak attraction:akF-0 BEC of molecules weak repulsion:akF+0 Strong interaction : |akF| • Maximal S-wave cross section Unitarity limit • Threshold: Ebound = 1/(ma2)  0 Fermi gas in the strong coupling limit : akF= Unitary Fermi gas

  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 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

  6. 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). Systematic expansion forx and other observables (D,Tc,…) in terms ofe(=4-d) This talk

  7. 2-body scattering around d=4 iT iT 2-component fermions local 4-Fermi interaction : 2-body scattering at vacuum (m=0)  (p0,p)  = n 1  T-matrix at d=4-e(e<<1) Coupling with boson g = (8p2e)1/2/m is SMALL !!! ig ig = iD(p0,p)

  8. 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

  9. Feynman rules 1 • L0 : • Free fermion quasiparticle  and boson  • L1 : Small coupling “g” between  and  (g~e1/2) Chemical potential insertions (m~e)

  10. 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)

  11. 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”

  12. Thermodynamic functions at T=0 • Universal equation of state • Effective potential and gap equation for 0 + O(e2) Veff (0,m) = + + O(e) O(1) • Universal numberxaround d=4 Systematic expansion !

  13. Quasiparticle spectrum p-k k-p -iS(p) = + p p p p k k • O(e) fermion self-energy • Fermion dispersion relation : w(p) Aroundminimum Expansion over 4-d Energy gap : Location of min. : 0

  14. 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 %

  15. 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

  16. Summary • Systematic expansions over e=4-d • Unitary Fermi gas around d=4 becomes • weakly-interacting system of fermions & bosons • LO+NLO results onx, D, e0 • 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

  17. Back up slides

  18. 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

  19. Specialty of d=4 and d=2 iT 2-component fermions local 4-Fermi interaction : 2-body scattering in 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

  20. 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)1/2 ig2 =

  21. 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”

  22. 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

  23. 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

  24. 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)

  25. Matching of two expansions (Tc) Tc/eF 4d 2d d • Borel + Padé approx. • Interpolated results to 3d

  26. 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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