Unitary Fermi gas in the e expansion Yusuke Nishida (Univ. of Tokyo & INT) in collaboration with D. T. Son [Ref: Phys. Rev. Lett. 97, 050403 (2006), cond-mat/0607835, cond-mat/0608321] 16 January, 2007 @ T.I.Tech

Unitary Fermi gas in the e expansion Contents of this talk Fermi gas at infinite scattering length Formulation of expansions in terms of 4-d and d-2 Results at zero/finite temperature Summary and outlook

Introduction : Fermi gas at infinite scattering length

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 ~ 1 MeV Color superconductivity (quarks): ??, Tc ~ 100 MeV Neutrino superfluidity: ??? [Kapusta, PRL(’04)] BCS theory (1957)

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

? BCS-BEC crossover - + BCS state of atoms weak attraction: akF-0 Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) Strong interaction ? Superfluid phase -B - + BCS state of atoms weak attraction: akF-0 BEC of molecules weak repulsion: akF+0 Strong coupling limit : |a kF| Maximal S-wave cross section Unitarity limit Threshold: Ebound = 1/(2ma2)  0 Fermi gas in the strong coupling limit a kF= : Unitary Fermi gas

Unitary Fermi gas What are the ground state properties of George Bertsch (1999), “Many-Body X Challenge” 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 ! r0 V0(a) kF-1 Energy per particle x is independent of systems cf. dilute neutron matter |aNN|~18.5 fm >> r0 ~1.4 fm

Universal parameter x Simplicity of system Difficulty for theory 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). No systematic & analytic treatment of unitary Fermi gas

Unitary Fermi gas at d≠3 d=4 BCS BEC - + d=2 d4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d)1/2 　Strong coupling Unitary regime BCS BEC - + g d2 : Weakly-interacting system of fermions, their coupling is g~(d-2) d=2 Systematic expansions for x and other observables (D, Tc, …) in terms of “4-d” or “d-2”

Formulation of e expansion e=4-d <<1 : d=spatial dimensions

Specialty of d=4 and d=2  =  2-component fermions local 4-Fermi interaction : 2-body scattering in vacuum (m=0)  iT (p0,p)  = 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 Small coupling b/w fermion-boson T-matrix at d=4-e (e<<1) Small coupling b/w fermion-boson g = (8p2 e)1/2/m iT ig ig = iD(p0,p) T-matrix at d=2+e (e<<1) Small coupling b/w fermion-fermion g = (2p e/m)1/2 iT ig2 =

Lagrangian for e expansion 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.

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

Feynman rules 2 = O(e) + = O(e m) + “Counter vertices” to cancel 1/e singularities in boson self-energies p p+k k + = O(e) 1. 2. O(e) p p+k k + = O(e m) O(e m)

Power counting rule of e 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 of e 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”

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 of e will be Ng/2

Results at zero/finite temperature Leading and next-to-leading orders

Thermodynamic functions at T=0 Effective potential : Veff = vacuum diagrams p q p-q k k Veff (0,m) = + O(e2) + + O(e) O(1) Gap equation of 0 C=0.14424… Assumption is OK ! Pressure : with the solution 0(m)

Systematic expansion of x in terms of e ! Universal parameter x Universal equation of state Universal parameter x around d=4 and 2 Arnold, Drut, Son (’06) Systematic expansion of x in terms of e !

Quasiparticle spectrum Fermion dispersion relation : w(p) O(e) p k p-k p k k-p Self-energy diagrams - i S(p) = + Expansion over 4-d Energy gap : Location of min. : Expansion over d-2

Extrapolation to d=3 from d=4-e 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 Borel transformation + Padé approximants Expansion around 4d x ♦=0.42 2d boundary condition 2d Interpolated results to 3d 4d d

NLO correction is small ~4 % Critical temperature Gap equation at finite T Veff = + + + m insertions Critical temperature from d=4 and 2 NLO correction is 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)

Matching of two expansions (Tc) d Tc / eF 4d 2d Borel + Padé approx. Interpolated results to 3d Tc / eF P / eFN E / eFN m / eF S / N NLO e1 0.249 0.135 0. 212 0.180 0.698 2d + 4d 0.183 0.172 0.270 0.294 0.642 Bulgac et al. 0.23(2) 0.27 0.41 0.45 0.99 Burovski et al. 0.152(7) 0.207 0.31(1) 0.493(14) 0.16(2)

Picture of weakly-interacting fermionic & Summary 1 e expansion for unitary Fermi gas Systematic expansions over 4-d and 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 on x, D, e0, Tc (P,E,m,S) NLO corrections around d=4 are small Naïve extrapolation from d=4 to d=3 gives good agreement with recent MC data Picture of weakly-interacting fermionic & bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3

e expansion for unitary Fermi gas Summary 2 e expansion for unitary Fermi gas Matching of two expansions around d=4 and d=2 NLO 4d + NLO 2d Borel transformation and Padé approximants Results are not too far from MC simulations Future Problems More understanding on e expansion Large order behavior + NN…LO corrections Analytic structure of x in “d” space Precise determination of universal parameters Other observables, e.g., Dynamical properties

Back up slides

Specialty of d=4 and 2 2-body wave function 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

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

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

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

NNLO correction for x x O(e7/2) 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 NNLO 4d d

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 ~ m Tc ~ m/e Similar power counting m/T ~ O(e) Consider T to be O(1) Condensate vanishes at Tc ~ m/e Fermions and bosons are excited

e expansion is asymptotic series but works well ! 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) O(1) +e1 +e2 +e3 +e4 +e5 Lattice g 1 1.167 1.244 1.195 1.338 0.892 1.239(3) 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 !

e expansion in critical phenomena Critical exponents of O(n=1) 4 theory (e=4-d  1) O(1) +e1 +e2 +e3 +e4 +e5 Lattice Exper. g 1 1.167 1.244 1.195 1.338 0.892 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) 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???