1.  expansion in cold atoms Yusuke Nishida (INT, U. of Washington  MIT) in collaboration with D. T. Son (INT) 1. Fermi gas at infinite scattering length 2. New approach :   (=4-d, d-2)  expansions Idea and how to use  expansion LO & NLO results and interpolations 3. Summary and outlook September 16, 2008 @ Center for Ultracold Atoms

2. 2/20 Fermions at infinite scattering length

3. 3/20 40 K Feshbach resonance Attraction is arbitrarily tunable by magnetic field C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90 (2003) B (Gauss) V 0 (a) r0r0 a Weak attraction a<0 Strong attraction a>0 bound molecule S-wave scattering length : a (r Bohr ) Feshbach resonance zero binding energy |a| 

4. 4/20 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) Unitary Fermi gas Strong coupling limit : |a k F |  Atomic gas @ Feshbach resonance Superfluid phase ? Strong interaction a dd =0.6 a

5. 5/20 Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge” Atomic gas @ Feshbach resonance: 0  r 0 << k F -1 << a  spin-1/2 fermions interacting via a zero-range, infinite scattering length contact interaction Strong coupling limit Perturbation a k F =  Difficulty for theory No expansion parameter r0r0 V 0 (a) k F -1 Previous approaches … Mean field approximations Monte Carlo simulations  expansion ! Use spatial dimensions as a small parameter Universal properties, but

6. 6/20 Our approach from d≠3  BCS BEC Strong coupling Unitary regime d=4 d=2 g g d  4 : Weakly-interacting system of fermions & bosons, their coupling is g 2 ~(4-d) d  2 : 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”

7. 7/20  expansion

8. 8/20 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 Fermions at unitarity in d  4 are free bosons Normalization at unitarity a  At d  2, any attractive potential leads to bound states “a  ” corresponds to zero interaction Fermions at unitarity in d  2 are free fermions diverges at r  0 for d  4

9. 9/20 Simple example Energy of 2 fermions at unitarity in a harmonic potential     : fermions at unitarity become free fermions : fermions at unitarity forms pointlike bosons Exact result is known :

10. 10/20 Unitary Fermi gas in d  2 is a free Fermi gas Unitary Fermi gas in d  4 is a free Bose gas Ground state energy in d = 2 & 4 Ground state energy of unitary Fermi gas in d=3 !? J.Carlson and S.Reddy, Phys.Rev.Lett. 95, (2005) Cf. MC simulation in 3d Starting points for systematic expansions of  around d=4 & 2

11. 11/20 T-matrix at arbitrary spatial dimension d Field theoretical approach iT =   (p 0,p)  1 n “a  ” Scattering amplitude has zeros at d=2,4,… Non-interacting limits Spin-1/2 fermions with local 4-Fermi interaction : 2-body scattering at vacuum (  =0)

12. 12/20 T-matrix at d=4-  (  <<1) T-matrix around d=4 and 2 iT = Small coupling b/w fermion-boson g = (8  2  ) 1/2 /m T-matrix at d=2+  (  <<1) iT = ig Small coupling b/w fermion-fermion g = 2   /m ig iD( p 0,p )

13. 13/20 Results to next-to-leading order

14. 14/20 Calculation of pressure (NLO) O(1) O(  ) + + P (  0,  ) = Pressure and gap equation around d=4 + O(  2 ) O(1) O(  ) + P (  0,  ) = Pressure and gap equation around d=2 + O(  2 )

15. 15/20 Equation of state at T=0 Universal parameter  around d=4 and 2 Systematic expansion of  in terms of  ! Universal equation of state Density “N” is the only scale !  0 (d  4 : free Bose gas)  1 (d  2 : free Fermi gas)

16. 16/20 Quasiparticle spectrum - i  ( p ) = Fermion dispersion relation :  ( p ) Energy gap : Location of min. : LO self-energy diagrams 0 Expansion over 4-d Expansion over d-2 or O(  )

17. 17/20 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+ 

18. 18/20 d ♦=0.42 4d 2d Matching of two expansions in  Borel transformation + Padé approximants Interpolated results to 3d Expansion around 4d free Fermi gas free Bose gas 2d boundary condition  = E unitary / E free

19. 19/20 Critical temperature Critical temperature from d=4 and 2 NLO correction is small ~4 % Monte Carlo simulations Bulgac et al. (’05): T c /  F = 0.23(2) 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 d Tc / FTc / F 4d 2d Interpolated results to d=3

20. 20/20 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, T c 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 Very simple and useful starting points to understand the unitary Fermi gas in d=3 !

21. 21/20 Back up slides

22. 22/20 NNLO correction for  Arnold, Drut, Son, Phys.Rev.A (2006) Fit two expansions using Padé approximants d  Interpolations to 3d NNLO 4d + NNLO 2d cf. NLO 4d + NLO 2d Nishida, Ph.D. thesis (2007) ♦=0.40

23. 23/20 unitarity BCS BEC Gapped superfluid 1-plane wave FFLO : O(  6 ) Polarized normal state Polarized Fermi gas around d=4 Rich phase structure near unitarity point in the plane of and : binding energy Stable gapless phases (with/without spatially varying condensate) exist on the BEC side of unitarity point Gapless superfluid

24. 24/20 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???

25. 25/20 Comparison with ideal BEC Unitarity limit at T c 1 of 9 pairs is dissociated all pairs form molecules Ratio to critical temperature in the BEC limit BEC limit at T c Boson and fermion contributions to fermion density at d=4

26. 26/20 3 fermions in a harmonic potential 2d 4d

27. 27/20 Interacting Fermion systems AttractionSuperconductivity / Superfluidity Metallic superconductivity (electrons) Kamerlingh Onnes (1911), T c = ~4.2 K Liquid 3 He Lee, Osheroff, Richardson (1972), T c = ~2.5 mK High-T c superconductivity (electrons or holes) Bednorz and Müller (1986), T c ~100 K Cold atomic gases ( 40 K, 6 Li) Regal, Greiner, Jin (2003), T c ~ 50 nK Nuclear matter (neutron stars): ?, T c ~ 1 MeV Color superconductivity (cold QGP): ??, T c ~ 100 MeV BCS theory (1957)

28. 28/20 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?

29. 29/20 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  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). No systematic & analytic treatment of unitary Fermi gas 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) Strong coupling limit Perturbation a k F =  Difficulty for theory No expansion parameter

30. 30/20 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

31. 31/20 Feynman rules 1 L 0 : Free fermion quasiparticle  and boson  L 1 : Small coupling “g” between  and  (g ~  1/2 ) Chemical potential insertions (  ~  )

32. 32/20 + = 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(    )

33. 33/20 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 ”

34. 34/20 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 

35. 35/20 (i) Low : T ~  <<  T ~  /  (ii) Intermediate :  < T <  /  (iii) High : T ~  /  >>  ~  T Fermion excitations are suppressed Phonon excitations are dominant Hierarchy in temperature T  (T) 0 T c ~  /  (i)(ii) (iii) At T=0,  (T=0) ~  /  >>  2 energy scales Condensate vanishes at T c ~  /  Fermions and bosons are excited Similar power counting  /T ~ O(  ) Consider T to be O(1) ~  ~ 

36. 36/20 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)

37. 37/20 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)

38. 38/20 Large order behavior d=2 and 4 are critical points free gasr0≠0r0≠0 2 3 4 Borel transform with conformal mapping  =1.2355  0.0050 Boundary condition (exact value at d=2)  =1.2380  0.0050 O(1)    2  3  4  5 Lattice  11.1671.2441.1951.3380.8921.239(3) Critical exponents of O(n=1)  4 theory (  =4-d  1)  expansion is asymptotic series but works well !

39. 39/20 Simple application Energy of 3 fermions in a harmonic potential    triplet statesinglet state in d=3 !? Cf. exact results in 3d :          : fermions at unitarity are free fermions : fermions at unitarity are free bosons + fermion