1. Unitary Fermi gas in the  expansion Yusuke Nishida18 January 2007 Contents of this talk 1. Fermi gas at infinite scattering length 2. Formulation of expansions in terms of 4-d and d-2 3. LO & NLO results at zero T &  4. Summary and outlook

2. 1. Introduction 2. Two-body scattering in vacuum 3. Unitary Fermi gas around d=4 4. Phase structure of polarized Fermi gas 5. Fermions with unequal masses 6. Expansions around d=2 7. Matching of expansions at d=4 and d=2 8. Thermodynamics below T c 9. Thermodynamics above T c 10. Summary and concluding remarks Contents of thesis

3. 3/23 Introduction : Fermi gas at infinite scattering length

4. 4/23 Interacting Fermion systems AttractionSuperconductivity / Superfluidity Metallic superconductivity (electrons) Kamerlingh Onnes (1911), T c = ~9.2 K Liquid 3 He Lee, Osheroff, Richardson (1972), T c = 1~2.6 mK High-T c superconductivity (electrons or holes) Bednorz and Müller (1986), T c = ~160 K Atomic gases ( 40 K, 6 Li) Regal, Greiner, Jin (2003), T c ~ 50 nK Nuclear matter (neutron stars): ?, T c ~ 1 MeV Color superconductivity (quarks): ??, T c ~ 100 MeV BCS theory (1957)

5. 5/23 Feshbach resonance Attraction is arbitrarily tunable by magnetic field C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003) S-wave scattering length :  [0,  ] Strong coupling |a|  a<0 No bound state 40 K a (r Bohr ) Weak coupling |a|  0 a>0 Bound state formation Feshbach resonance

6. 6/23 BCS-BEC crossover  0 BCS state of atoms weak attraction: ak F  -0 BEC of molecules weak repulsion: ak F  +0 Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) Strong coupling limit : |a k F |   Maximal S-wave cross section Unitarity limit Threshold: E bound = 1/(2ma 2 )  0 -B Superfluid phase ? Strong interaction Fermi gas in the strong coupling limit a k F =  : Unitary Fermi gas

7. 7/23 Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge” r0r0 V 0 (a) k F -1 k F is the only scale ! Atomic gas : r 0 =10Å << k F -1 =100Å << |a|=1000Å Energy per particle 0  r 0 << k F -1 << a  cf. dilute neutron matter |a NN |~18.5 fm >> r 0 ~1.4 fm   is independent of systems 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?

8. 8/23 Mean field approx., Engelbrecht et al. (1996):  <0.59 Linked cluster expansion, Baker (1999):  =0.3~0.6 Galitskii approx., Heiselberg (2001):  =0.33 LOCV approx., Heiselberg (2004):  =0.46 Large d limit, Steel (’00)  Schäfer et al. (’05):  =0.44  0.5 Universal parameter  Simplicity of system   is universal parameter Difficulty for theory No expansion parameter Models Simulations 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). Systematic expansion for  and other observables ( ,T c,…) in terms of  (=4-d) Carlson et al., Phys.Rev.Lett. (2003):  =0.44(1) Astrakharchik et al., Phys.Rev.Lett. (2004):  =0.42(1) Carlson and Reddy, Phys.Rev.Lett. (2005):  =0.42(1) This talk

9. 9/23 Formulation of  expansion  =4-d <<1 : d=spatial dimensions

10. 10/23 T-matrix at arbitrary spatial dimension d Specialty of d=4 and d=2 2-component fermions local 4-Fermi interaction : iT =   (p 0,p)  2-body scattering in vacuum (  =0) 1 n “a  ” Scattering amplitude has zeros at d=2,4,… Non-interacting limits

11. 11/23 T-matrix at d=4-  (  <<1) T-matrix around d=4 and 2 iT = ig iD( p 0,p ) Small coupling b/w fermion-boson g = (8  2  ) 1/2 /m T-matrix at d=2+  (  <<1) iT = ig 2 Small coupling b/w fermion-fermion g = (2   /m) 1/2

12. 12/23 Boson’s kinetic term is added, and subtracted here. =0 in dimensional regularization Expand with Ground state at finite density is superfluid : Lagrangian for  expansion Hubbard-Stratonovish trans. & Nambu-Gor’kov field : Rewrite Lagrangian as a sum : L = L 0 + L 1 + L 2

13. 13/23 Feynman rules 1 L 0 : Free fermion quasiparticle  and boson  L 1 : Small coupling “g” between  and  (g ~  1/2 ) Chemical potential insertions (  ~  )

14. 14/23 + = O(    ) Feynman rules 2 L 2 : “Counter vertices” to cancel 1/  singularities in boson self-energies pp p+k k + = O(  ) pp p+k k 1. 2. O(  ) O(    )

15. 15/23 1. Assume justified later and consider to be O(1) 2. Draw Feynman diagrams using only L 0 and L 1 3. If there are subdiagrams of type add vertices from L 2 : 4. Its powers of  will be N g /2 + N  5. The only exception is= O(1) O(  ) Power counting rule of  or Number of  insertions Number of couplings “g ~  1/2 ”

16. 16/23 Expansion over  = d-2 1. Assume justified later and consider to be O(1) 2. Draw Feynman diagrams using only L 0 and L 1 3. If there are subdiagrams of type add vertices from L 2 : 4. Its powers of  will be N g /2 Lagrangian Power counting rule of 

17. 17/23 Results at zero temperature Leading and next-to-leading orders

18. 18/23 Assumption is OK ! Thermodynamic functions at T=0 k k p q p-q O(1) O(  ) + + V eff (  0,  ) = Effective potential : V eff = vacuum diagrams + O(  2 ) Gap equation of  0 C=0.14424… Pressure : with the solution  0 (  )

19. 19/23 Universal parameter  Universal parameter  around d=4 and 2 Systematic expansion of  in terms of  ! Arnold, Drut, Son (’06) Universal equation of state

20. 20/23 Quasiparticle spectrum + - i  ( p ) = p p k p-k p p k k-p Fermion dispersion relation :  ( p ) Energy gap : Location of min. : Self-energy diagrams 0 Expansion over 4-d Expansion over d-2 O(  )

21. 21/23 Extrapolation to d=3 from d=4-  Keep LO & NLO results and extrapolate to  =1 J.Carlson and S.Reddy, Phys.Rev.Lett.95, (2005) Good agreement with recent Monte Carlo data NLO corrections are small 5 ~ 35 % NLO are 100 % cf. extrapolations from d=2+ 

22. 22/23 Matching of two expansions in  Borel transformation + Padé approximants Interpolated results to 3d 2d boundary condition d  ♦=0.42 4d 2d Expansion around 4d

23. 23/23 1. Systematic expansions over  =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 , ,  0 NLO corrections around d=4 are small Extrapolations to d=3 agree with recent MC data 3. Future problems Large order behavior + NN…LO corrections More understanding Precise determination Summary Picture of weakly-interacting fermionic & bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3

24. 24/23 Back up slides

25. 25/23 Specialty of d=4 and 2 2-body wave function Z.Nussinov and S.Nussinov, cond-mat/0410597 Pair wave function is concentrated near its origin Unitary Fermi gas for d  4 is free “Bose” gas Normalization at unitarity a  diverges at r  0 for d  4 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

26. 26/23 Unitary Fermi gas at d≠3  BEC BCS Strong coupling Unitary regime d=4 d=2 g g d  4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d) 1/2 d  2 : Weakly-interacting system of fermions, their coupling is g~(d-2) Systematic expansions for  and other observables ( , T c, …) in terms of “ 4-d” or “d-2”

27. 27/23 NNLO correction for  O(  7/2 ) correction for  Arnold, Drut, and Son, cond-mat/0608477 Borel transformation + Padé approximants d  NLO 4d NLO 2d NNLO 4d Interpolation to 3d NNLO 4d + NLO 2d cf. NLO 4d + NLO 2d

28. 28/23 Critical temperature V eff = + + +  insertions Gap equation at finite T Critical temperature from d=4 and 2 NLO correction is small ~4 % Simulations : Lee and Schäfer (’05): T c /  F < 0.14 Burovski et al. (’06): T c /  F = 0.152(7) Akkineni et al. (’06): T c /  F  0.25 Bulgac et al. (’05): T c /  F = 0.23(2)

29. 29/23 d Tc / FTc / F 4d 2d Matching of two expansions (T c ) Borel + Padé approx. Interpolated results to 3d Tc / FTc / F P / FNP / FNE / FNE / FN  / F / F S / NS / N NLO  1 0.2490.1350. 2120.1800.698 2d + 4d 0.1830.1720.2700.2940.642 Bulgac et al. 0.23(2)0.270.410.450.99 Burovski et al. 0.152(7)0.2070.31(1)0.493(14)0.16(2)

30. 30/23 Borel summation with conformal mapping  =1.2355  0.0050 &  =0.0360  0.0050 Boundary condition (exact value at d=2)  =1.2380  0.0050 &  =0.0365  0.0050  expansion in critical phenomena O(1)    2  3  4  5 LatticeExper.  11.1671.2441.1951.3380.8921.239(3) 1.240(7) 1.22(3) 1.24(2)  000.01850.03720.02890.05450.027(5) 0.016(7) 0.04(2) Critical exponents of O(n=1)  4 theory (  =4-d  1)  expansion is asymptotic series but works well ! How about our case???