e expansion in cold atoms Yusuke Nishida (INT, Univ. of Washington) in collaboration with D.
Download ReportTranscript e expansion in cold atoms Yusuke Nishida (INT, Univ. of Washington) in collaboration with D.
e expansion in cold atoms Yusuke Nishida (INT, Univ. of Washington) in collaboration with D. T. Son (INT) [Ref: Phys. Rev. Lett. 97, 050403 (2006) Phys. Rev. A 75, 063617 & 063618 (2007)] 1. Fermi gas at infinite scattering length 2. Formulation of e (=4-d, d-2) expansions 3. LO & NLO results 4. Summary and outlook June 5, 2008 @ Grenoble BEC 2008 Workshop Interacting Fermion systems Attraction 2/16 Superconductivity / Superfluidity Metallic superconductivity (electrons) Kamerlingh Onnes (1911), Tc = ~4.2 K BCS Liquid 3He theory (1957) Lee, Osheroff, Richardson (1972), Tc = ~2.5 mK High-Tc superconductivity (electrons or holes) Bednorz and Müller (1986), Tc ~100 K Cold atomic gases (40K, 6Li) Regal, Greiner, Jin (2003), Tc ~ 50 nK • Nuclear matter (neutron stars): ?, Tc ~ 1 MeV • Color superconductivity (cold QGP): ??, Tc ~ 100 MeV Feshbach resonance C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90 (2003) 3/16 Attraction is arbitrarily tunable by magnetic field S-wave scattering length : a (rBohr) Strong attraction a>0 Feshbach resonance zero binding energy |a| Weak attraction a<0 bound molecule a 40K B (Gauss) V0(a) r0 BCS-BEC crossover Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) 4/16 Strong interaction add=0.6a ? Superfluid phase + BEC of molecules weak repulsion: akF+0 - 0 BCS state of atoms weak attraction: akF-0 Unitary Fermi gas • Strong coupling limit : |akF| • Atomic gas @ Feshbach resonance George Bertsch (1999), “Many-Body X Challenge” Unitary Fermi gas 5/16 Atomic gas @ Feshbach resonance: 0 r0 << kF-1 << a spin-1/2 fermions interacting via a zero-range, infinite scattering length contact interaction kF Universal properties, but -1 • Strong coupling limit Perturbation akF= r0 V0(a) Previous approaches … • Mean field approximations • Monte Carlo simulations • Difficulty for theory No expansion parameter e expansion ! • Use spatial dimensions as a small parameter 6/16 Our approach from d≠3 g d=4 BEC + Strong coupling Unitary regime g d=2 • d4 : Weakly-interacting system of fermions & bosons, their coupling is g2~(4-d) BCS - • d2 : Weakly-interacting system of fermions, their coupling is g~(d-2) Systematic expansions for various physical observables in terms of “4-d” or “d-2” Specialty of d=4 and 2 Z.Nussinov and S.Nussinov, cond-mat/0410597 2-body wave function Normalization at unitarity a diverges at r0 for d4 Pair wave function is concentrated near its origin Unitary Fermi gas at d4 is free Bose gas At d2, any attractive potential leads to bound states “a” corresponds to zero interaction Unitary Fermi gas at d2 is free Fermi gas 7/16 8/16 Field theoretical approach Spin-1/2 fermions with local 4-Fermi interaction : 2-body scattering at vacuum (m=0) (p0,p) iT = 1 n T-matrix at arbitrary spatial dimension d “a” Scattering amplitude has zeros at d=2,4,… Non-interacting limits T-matrix around d=4 and 2 9/16 T-matrix at d=4-e (e<<1) = iT ig ig iD(p0,p) Small coupling b/w fermion-boson g = (8p2 e)1/2/m T-matrix at d=2+e (e<<1) ig iT = Small coupling b/w fermion-fermion g = 2p e/m 10/16 Effective potential (NLO) • Effective potential and gap equation around d=4 Veff (0,m) = + + O(e2) + O(e) O(1) • Effective potential and gap equation around d=2 Veff (0,m) = + O(e2) + O(1) O(e) Equation of state at T=0 11/16 • Universal equation of state Density “N” is the only scale ! • Universal parameter x around d=4 and 2 Systematic expansion of x in terms of e ! x0 (d4 : free Bose gas) x1 (d2 : free Fermi gas) 12/16 Quasiparticle spectrum • Fermion dispersion relation : w(p) LO self-energy - i S(p) = diagrams or O(e) O(e) Expansion over 4-d Energy gap : Location of min. : Expansion over d-2 0 Extrapolation to d=3 from d=4-e 13/16 • Keep LO & NLO results and 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 % Matching of two expansions in x 14/16 • Borel transformation + Padé approximants x = Eunitary /Efree ♦=0.42 Expansion around 4d 2d boundary condition 2d 4d • Interpolated results to 3d d free Fermi gas free Bose gas 15/16 Critical temperature • Critical temperature from d=4 and 2 NLO correction is small ~4 % • Interpolated results to d=3 Tc / eF 4d Monte Carlo simulations • Bulgac et al. (’05): Tc/eF = 0.23(2) • 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 2d d Summary 16/16 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, e0, Tc • NLO corrections around d=4 are small • Extrapolations to d=3 agree with recent MC data Picture of weakly-interacting fermionic & bosonic quasiparticles for unitary Fermi gas may be a good starting point even in d=3 3. Future problems • Large order behavior + NN…LO corrections More understanding Precise determination NNLO correction for x 18/16 • NNLO correction for x Arnold, Drut, Son, Phys.Rev.A (2006) Nishida, Ph.D. thesis (2007) x Fit two expansions using Padé approximants Interpolations to 3d • NNLO 4d + NNLO 2d cf. NLO 4d + NLO 2d d e expansion in critical phenomena 19/16 Critical exponents of O(n=1) 4 theory (e=4-d 1) O(1) g 1 0 +e1 1.167 0 +e2 +e3 +e4 +e5 Lattice Exper. 1.239(3) 1.240(7) 1.22(3) 1.24(2) 0.0185 0.0372 0.0289 0.0545 0.027(5) 0.016(7) 0.04(2) 1.244 e expansion is 1.195 1.338 0.892 • Borel summation with conformal mapping g=1.23550.0050 & =0.03600.0050 asymptotic series but works well ! • Boundary condition (exact value at d=2) g=1.23800.0050 & =0.03650.0050 How about our case??? Comparison with ideal BEC 20/16 • Ratio to critical temperature in the BEC limit Boson and fermion contributions to fermion density at d=4 • Unitarity limit at Tc 1 of 9 pairs is dissociated • BEC limit at Tc all pairs form molecules 21/16 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 Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge” 22/16 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-1 kF is the only scale ! Energy per particle r0 x is independent of systems V0(a) cf. dilute neutron matter |aNN|~18.5 fm >> r0 ~1.4 fm Universal parameter x • Strong coupling limit Perturbation akF= 23/16 • Difficulty for theory No expansion parameter Models • Mean field approx., Engelbrecht et al. (1996): x<0.59 • Linked cluster expansion, Baker (1999): • Galitskii approx., Heiselberg (2001): • LOCV approx., Heiselberg (2004): • Large d limit, Steel (’00)Schäfer 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=0.3~0.6 x=0.33 x=0.46 x=0.440.5 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 24/16 • 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 Boson’s kinetic term is added, and subtracted here. Feynman rules 1 • L0 : Free fermion quasiparticle and boson • L1 : Small coupling “g” between and (g ~ e1/2) Chemical potential insertions (m ~ e) 25/16 26/16 Feynman rules 2 • L2 : “Counter vertices” to cancel 1/e singularities in boson self-energies 1. k p O(e) 2. p + = O(e) p + = O(em) p+k k p O(em) p+k Power counting rule of 1. Assume and consider e 27/16 justified later to be O(1) 2. Draw Feynman diagrams using only L0 and L1 3. If there are subdiagrams of type or add vertices from L2 : 4. Its powers of or e will be Ng/2 + Nm 5. The only exception is Number of m insertions Number of couplings “g ~ e1/2” = O(1) O(e) Expansion over e = d-2 28/16 Lagrangian Power counting rule of e 1. Assume and consider justified later to be O(1) 2. Draw Feynman diagrams using only L0 and L1 3. If there are subdiagrams of type add vertices from L2 : 4. Its powers of e will be Ng/2 29/16 Hierarchy in temperature At T=0, D(T=0) ~ m/e >> m 2 energy scales (i) Low : T ~ m << DT ~ m/e D(T) • Fermion excitations are suppressed • Phonon excitations are dominant (i) (ii) (iii) T 0 (ii) Intermediate : m < T < m/e (iii) High : T ~ m/e >> m ~ DT • Condensate vanishes at Tc ~ m/e • Fermions and bosons are excited ~m Tc ~ m/e Similar power counting • m/T ~ O(e) • Consider T to be O(1) 30/16 Critical temperature • Gap equation at finite T Veff = + + + m insertions • Critical temperature from d=4 and 2 NLO correction is small ~4 % Simulations : • Bulgac et al. (’05): Tc/eF = 0.23(2) • 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 31/16 Matching of two expansions (Tc) Tc / eF • Borel + Padé approx. 4d • Interpolated results to 3d 2d d NLO e1 2d + 4d Bulgac et al. Burovski et al. Tc / eF P / eFN E / eFN m / eF S/N 0.249 0.135 0. 212 0.180 0.698 0.183 0.172 0.270 0.294 0.642 0.23(2) 0.27 0.41 0.45 0.99 0.152(7) 0.207 0.31(1) 0.493(14) 0.16(2) 32/16 Large order behavior • d=2 and 4 are critical points free gas 2 3 4 r0≠0 • Critical exponents of O(n=1) 4 theory (e=4-d 1) g O(1) +e1 +e2 +e3 +e4 +e5 Lattice 1 1.167 1.244 1.195 1.338 0.892 1.239(3) • Borel transform with conformal mapping g=1.23550.0050 • Boundary condition (exact value at d=2) g=1.23800.0050 e expansion is asymptotic series but works well !