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An e expansion for Fermi gas at infinite scattering length

An e expansion for Fermi gas at infinite scattering length. Yusuke Nishida (Univ. of Tokyo & INT) Y. N. and D. T. Son, arXiv: cond-mat/0604500 15 May, 2006 @ INT Seminar. An e expansion for Fermi gas at infinite scattering length. Contents of this talk

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An e expansion for Fermi gas at infinite scattering length

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  1. An e expansion for Fermi gas at infinite scattering length Yusuke Nishida (Univ. of Tokyo & INT) Y. N. and D.T. Son, arXiv: cond-mat/0604500 15 May, 2006 @ INT Seminar

  2. An e expansion for Fermi gas at infinite scattering length • Contents of this talk • Fermi gas at infinite scattering length • Formulation ofe(=4-d) expansion • Results at zero temperature • Results at finite temperature • Summary

  3. Strongly interacting Fermi gas

  4. Interacting Fermion systems Attraction Superconductivity/Superfluidity • Metallic superconductivity (electrons) • 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 • Neutrino superfluidity ???[Kapusta, PRL(’04)] BCS theory (1957)

  5. Feshbach resonance C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003) Interaction is arbitrarily tunable by magnetic field scattering length :[0,] Feshbach resonance a (rBohr) Repulsive (a>0) Bound state formation Strong coupling |a| Attractive (a<0) No bound state 40K Weak coupling |a|0

  6. BEC-BCS crossover Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) Superfluidphase B 1/(akF)=+ 1/(akF)=- 1/(akF)=0 BEC of molecules weak interaction:akF+0 BCS state of atoms weak interaction:akF-0 Strong interaction : |akF| • Maximal S-wave cross section Unitarity limit • Threshold: Ebound = 1/(2ma2)  0

  7. Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge” kF-1 r0 V0(a) Atomic gas : |a|=1000Å >> kF-1=100Å >> r0=10Å 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 nuclear matter |aNN|~19 fm >> r0~1 fm

  8. Universal parameterx • Simplicity of system • x is universal parameter • 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 in terms ofe (=4-d)

  9. Formulation ofe expansion e=4-d <<1 : d=spatial dimensions

  10. Specialty at d4 rd-2R(r) R(r) : pair wave function r : separation of 2 particles r0 : range of potential a C r00 r V0 Z.Nussinov and S.Nussinov, cond-mat/0410597 2-body scattering at zero energy limit Tune the attractive potential V0 at threshold (a/r0 ) Normalization of wave func. diverges at r0 for d4 Pair wave function is concentrated near its origin r0 Finite density system at unitarity for d4is weakly-interacting Bose gas But they never developed an expansion around d=4

  11. Field theoretical approach T T Universality allows 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 !!! g g = D(p0,p)

  12. Lagrangian 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

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

  14. Feynman rules 2 • L2 : “Counter terms”of boson  “Naïve” power counting ofe • Assume justified later • and consider to be O(1) • Draw Feynman diagrams using only L0 and L1 (not L2) • Its powers ofe will be Ng/2 + Nm Number of m insertions Number of couplings “g” (g~e1/2) But exceptions Fermion loop integrals produce 1/ein 4 diagrams

  15. Exceptions of power counting 1 k p p =O(e) + p+k k p p p+k 1. Boson self-energy naïve O(e) Cancellation with L2 vertices to restore naïve counting 2. Boson self-energy with m insertion naïve O(e2) =O(e2) +

  16. Exceptions of power counting 2 p k + · · · = 0 + k 3. Tadpole diagram with m insertion =O(e1/2) naïve O(e3/2) Gap equation of0 Sum of tadpoles = 0 O(e1/2) O(e1/2) 4. Vacuum diagram with m insertion =O(1) O(e)Only exception !

  17. “Revised” power counting 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 or or = O(1)

  18. Results at zero temperature Leading and next-to-leading orders

  19. Effective potential Veff (0) k k p p-q q • Leading order O(1) Boson’s 1-loops and vanish at T=0 • Next-to-leading order O(e) C=0.14424…

  20. Universal parameterx Assumption is OK ! • Gap equation of0 • Fermion density and Fermi energy • Universal parameter : x = m/eF Systematic expansion of x in terms ofe !

  21. Quasiparticle spectrum LO : Energy gap : Location of min. : • Fermion 1-loop self-energy O(e) p-k k-p -iS(p) = + p p p p k k • Fermion dispersion relation : w(p) Around minimum : NLO

  22. Extrapolation to d=3 • 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) Experiments : Duke(’05): x=0.51(4), Rice(’06): x=0.46(5)

  23. Results at finite temperature preliminary

  24. 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

  25. Critical temperature • Leading & next-to-leading orders Veff = + + + minsertions • Gap equation : • Fermion density : Critical temperature & energy density NLOcorrectionsare small 4~9 %

  26. Comparisons of Tc e expansion(LO+NLO) Simulations : Experiment : • Wingate (’05): Tc/eF = 0.04 • Lee and Schäfer (’05): Tc/eF < 0.14 • Bulgac et al. (’05): Tc/eF = 0.23(2) • Burovski et al. (’06): Tc/eF = 0.152(7) • Kinast et al. (’05): Tc/eF = 0.27(2) • cf. BEC limit : TBEC/eF = 0.218… Ideal BEC at d=4-e(convergent if |e|2)

  27. 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???

  28. Summary e expansion for unitary Fermi gas • The only systematic expansion at T=0 for now • LO+NLO results onx, D, e0, Tc, E • NLO corrections are small compared to LO • Extrapolations to d=3 give good agreement • with MC simulations and experiments (infinite) Future Problems • NNLO corrections + Resummation • Thermodynamics • Finite polarization • BCS-BEC crossover (finite a) • Vortex structure • Dynamical properties

  29. Back up slides

  30. Feshbach resonance C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003) repulsive strongcoupling r 40K attractive Interaction is arbitrarily tunable by magnetic field scattering length :[0,] S-wave m+Dm m m E DE=DmB bound level interatomic potential

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