html5-img
1 / 31

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

gali
Download Presentation

e expansion in cold atoms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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???

More Related