1. Strongly interacting scale-free matter in cold atoms Yusuke Nishida March 12, 2009 @ MIT Faculty Lunch

2. 2/32 Fermions at infinite scattering length

3. 3/32 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 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 (quarks): T c ~ 100 MeV ?? Neutrino superfluidity ??? BCS theory (1957)

4. 4/32 r Feshbach resonance S-wave scattering length : E interatomic potential bound level       E=  B   40 K C.A.Regal and D.S.Jin, Phys.Rev.Lett. (2003)

5. 5/32 40 K S-wave scattering length : a (Gauss) a Weak attraction a<0 Strong attraction a>0 bound molecule zero binding energy : |a|  Attraction is arbitrarily tunable by magnetic field r   (r) r V0V0 r0r0 a <0 |a|  a >0

6. 6/32 strong attraction BCS-BEC crossover  0 weak attraction Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) superfluid phase B (gauss) scattering length : a BEC of molecules BCS state of atoms k F = (3  n) 1/3 Fermi momentum

7. 7/32 strong attraction BEC of molecules BCS-BEC crossover  0 weak attraction BCS state of atoms Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) superfluid phase M. Zwierlein et al. Nature (2005) Vortex lattices throughout BCS- BEC crossover 880  m

8. 8/32 BCS-BEC crossover a dd =0.6 a Bose gas with weak repulsion ak F << 1 Fermi gas with weak attraction |ak F | << 1 Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985) strong attraction BEC of molecules  weak attraction BCS state of atoms 0 Strong interaction |ak F | >> 1

9. 9/32 Unitary Fermi gas  0 weak BCS weak BEC strong interaction 40 K S-wave scattering length : a B (Gauss) |ak F | 

10. 10/32 Strong coupling limit : |ak F |  Maximal s-wave cross section Unitarity limit No perturbative expansion Challenge for theorists Scale invariant interaction a  & zero range r 0  0 Nonrelativistic CFT Universality Atomic gas @ Feshbach resonance Dilute neutron matter : |a NN | ~ 19 fm >> r 0 ~ 1 fm Unitary Fermi gas  0 weak BCS weak BEC strong interaction  expansion !

11. 11/32 d=4 d=2 New approach from d≠3  Strong coupling Unitary regime g d  4 : Weakly-interacting fermions & bosons with small coupling g 2 ~(4-d) d  2 : Weakly-interacting fermions with small coupling g~(d-2) Systematic expansions for various physical observables in terms of “ 4-d” or “d-2” weak BECweak BCS 0 g

12. 12/32  expansion

13. 13/32 Scale invariant interaction r   (r) V0V0 r0r0 rr V 0 ~ 1/(m r 0 2 ) 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 r   (r) r   

14. 14/32 Specialty of d=2 & 4 Z.Nussinov and S.Nussinov, cond-mat/0410597 4 2 3 d 2-body wave function in general dimensions “a  ” corresponds to zero interaction Fermions at unitarity in d  2 are free fermions Wave function  (r) becomes smooth at r  0 for d=2 ( Any attractive potential in d=2 leads to bound states )

15. 15/32 Specialty of d=2 & 4 Z.Nussinov and S.Nussinov, cond-mat/0410597 4 2 3 d 2-body wave function in general dimensions Pair wave function is concentrated near its origin Fermions at unitarity in d  4 are free bosons Normalization diverges at r  0 for d=4

16. 16/32 Ground state energy Ground state energy of unitary Fermi gas at T=0 Density “N” is the only scale  : fundamental quantity of unitary Fermi gas Mean field approx., Engelbrecht et al. (1996):  <0.59 Simulations Experiments Innsbruck(’04): 0.32(13), Duke(’05): 0.51(4), Rice(’05): 0.46(5), JILA(’06): 0.46(12), ENS(’07): 0.41(15) 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)

17. 17/32 Ground state energy in d = 2 & 4 Ground state energy of unitary Fermi gas 4 2 3 d Unitary Fermi gas in d  4 is a free Bose gas Unitary Fermi gas in d  2 is a free Fermi gas in d=3 !? J.Carlson and S.Reddy (2005) Cf. MC simulation in 3d

18. 18/32 Ground state energy in d = 2 & 4 Ground state energy of unitary Fermi gas 4 2 3 d Unitary Fermi gas in d  4 is a free Bose gas Unitary Fermi gas in d  2 is a free Fermi gas d=4 & d=2 are starting points for systematic expansions of 

19. 19/32 T-matrix in general dimensions 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 contact interaction : 2-body scattering at vacuum (  =0) Y.N. and D.T.Son PRL(’06) & PRA(’07)

20. 20/32 When d=4-  (  <<1) Field theoretical approach 4 2 3 d iT = ig iD( p 0,p ) Small coupling between fermions & boson g = (8  2  ) 1/2 /m T-matrix in general dimensions Y.N. and D.T.Son PRL(’06) & PRA(’07)

21. 21/32 Field theoretical approach 4 2 3 d iT = ig Small coupling between fermion & fermion g = 2   /m When d=2+  (  <<1) T-matrix in general dimensions Y.N. and D.T.Son PRL(’06) & PRA(’07)

22. 22/32 g fermions with small coupling g~(d-2) << 1 Systematic expansions 4 2 3 d O(1) O(  ) + + P (  ) = + O(  2 ) fermions & bosons with small coupling g 2 ~(4-d) << 1 g  =4-d &  =d-2

23. 23/32 Systematic expansions 4 2 3 d Carlson & Reddy (2005) Cf. MC simulation in 3d NLO correction is small ~5 % g fermions & bosons with small coupling g 2 ~(4-d) << 1 fermions with small coupling g~(d-2) << 1 g O(1) O(  ) + P (  ) = + O(  2 )  =4-d &  =d-2

24. 24/32 Systematic expansions 4 2 3 d g fermions & bosons with small coupling g 2 ~(4-d) << 1 fermions with small coupling g~(d-2) << 1 Carlson & Reddy (2005) Cf. MC simulation in 3d g NLO correction is small ~5 %  =4-d &  =d-2

25. 25/32 d ♦=0.42 4d 2d Matching of two expansions in  Padé approximants ( + Borel transformation) Interpolations to 3d free Fermi gas free Bose gas  = E unitary / E free

26. 26/32 d Tc / FTc / F 4d 2d Critical temperature Critical temperature from d=4 and 2 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 Interpolated results to d=3 Y.N., Phys. Rev. A (2007) free Fermi gas free Bose gas

27. 27/32 Few body aspects

28. 28/32 Correspondence Schrödinger equation in free space with E=0 Scaling solution Schrödinger equation in a harmonic potential S.Tan, cond-mat/0412764 F.Werner & Y.Castin, PRA (2006) Y.N. & D.T.Son, PRD (2007)  = anomalous dimension of operator in nonrelativistic CFT

29. 29/32 3 fermions in a harmonic potential 2d 4d Angular momentum l = 0 Angular momentum l = 1

30. 30/32 3 fermions in a harmonic potential 2d 4d Angular momentum l = 0 Angular momentum l = 1

31. 31/32 Summary Fermi gas at infinite scattering length = New strongly interacting matter in cold atoms Unitary Fermi gas around d=4 becomes weakly-interacting system of fermions & bosons Weakly-interacting system of fermions around d=2 Thermodynamics & Quasiparticle spectrum (Y.N. & D.T.Son 2006) Atom-dimer & dimer-dimer scatterings (G.Rupak 2006) Phase structure of polarized Fermi gas with (un)equal masses (Y.N. 2007, G.Rupak & T.Schafer & A.Kryjevski 2007) BCS-BEC crossover (J.W.Chen & E.Nakano 2007) Momentum distribution & condensate fraction (Y.N. 2007) Energy of a few atoms in a harmonic potential (Y.N. & D.T.Son 2007) Low-energy dynamics (A.Kryjevski 2008) Energy-density functional (G.Rupak & T.Schafer 2009) …

32. 32/32 Thermodynamics & Quasiparticle spectrum (Y.N. & D.T.Son 2006) Atom-dimer & dimer-dimer scatterings (G.Rupak 2006) Phase structure of polarized Fermi gas with (un)equal masses (Y.N. 2007, G.Rupak & T.Schafer & A.Kryjevski 2007) BCS-BEC crossover (J.W.Chen & E.Nakano 2007) Momentum distribution & condensate fraction (Y.N. 2007) Energy of a few atoms in a harmonic potential (Y.N. & D.T.Son 2007) Low-energy dynamics (A.Kryjevski 2008) Energy-density functional (G.Rupak & T.Schafer 2009) … Summary Very simple and useful starting points to understand the unitary Fermi gas in d=3 ! Fermi gas at infinite scattering length = New strongly interacting matter in cold atoms Unitary Fermi gas around d=4 becomes weakly-interacting system of fermions & bosons Weakly-interacting system of fermions around d=2

33. 33/32 Back up slides

34. 34/32 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

35. 35/32 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

36. 36/32 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???

37. 37/32 2 fermions in a harmonic potential T.Busch et.al., Found. Phys. (1998) T.Stoferle et al., Phys.Rev.Lett. (2006)

38. 38/32 2 fermions in a harmonic potential |a| 

39. 39/32 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(  )

40. 40/32 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+ 