1. Copenhagen, June 15, 2006 Unitary Polarized Fermi Gases Erich J. Mueller Cornell University Sourish Basu Theja DeSilva NSF, Sloan, CCMR Outline: Interesting Questions What goes wrong with mean field theory and LDA Speculations about the phase diagram Collective modes [no time, but ask me about them]

2. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions2 of 18 Questions 1.Nature of normal fluid at zero T at unitarity? 2.Existence of exotic phases? a)Modulated Order parameters [FFLO] b)Deformed Fermi surfaces c)Polarized s-wave superfluids [Sarma, Breeched gap] 3.Properties? a)Equation of state b)How to probe?

3. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions3 of 18 Questions 1.Nature of normal fluid at zero T at unitarity? 2.Existence of exotic phases? a)Modulated Order parameters [FFLO] b)Deformed Fermi surfaces c)Polarized s-wave superfluids [Sarma, Breeched gap] 3.Properties? a)Equation of state b)How to probe? Simplest assumptions (ex. Fermi-liquid) -- calculate properties Controlled approximations: Limit n  =0

4. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions4 of 18 Questions 1.Nature of normal fluid at unitarity? 2.Existence of exotic phases? a)Modulated Order parameters [FFLO] b)Deformed Fermi surfaces c)Polarized s-wave superfluids [Sarma, Breeched gap] 3.Properties? a)Equation of state b)How to probe? Homogeneous: Mean Field Phase Diagram Fluctuations and Stability Scaling Arguments Trap: Surface tension Collective modes How to explain experiments

5. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions5 of 18 Summary Hulet’s “wings” can be explained by surface tension in the normal-superfluid interface Hulet’s low polarization data is a mystery Preliminary theoretical studies indicate: Partially polarized normal phase at unitarity May be partially polarized superfluid phase at unitarity  Probably breaks translation/rotation symmetry [eg. FFLO]

6. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions6 of 18 Homogeneous Phases Mean Field Theory: a>0 a>0 [near resonance] a<0 Red: 1st order transition Normal-FFLO: Continuous SF-FFLO: Discontinuous (Schematic)

7. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions7 of 18 Application to Trap LDA P. Pieri, and G.C. Strinati, PRL 96, 150404 (2006) ; W. Yi, and L. -M. Duan, cond-mat/0604558; M. Haque and H.T.C. Stoof, cond-mat/0601321; Zheng-Cheng Gu, Geoff Warner and Fei Zhou, cond-mat/0603091 ; C.-H. Pao, S.-K. Yip, J. Phys.: Condens. Matter 18 (2006) 5567;Theja N. De Silva, Erich J. Mueller, Phys. Rev. A 73, 051602(R) (2006) BECBCS

8. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions8 of 18 Beyond Mean Field Theory + LDA Discrepancies with experiments: P Data: Rice/MIT

9. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions9 of 18 LDA in harmonic trap: Integrate: Axial density should be monotonic [in LDA] Beyond Mean Field Theory + LDA Wings -- A violation of LDA

10. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions10 of 18 Simple Explanation: Beyond Mean Field Theory + LDA Large Aspect Ratio Trap: LDA: Surface Tension: z Unitarity constrains form:

11. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions11 of 18 Calculating  : Beyond Mean Field Theory + LDA  x E  Gradient expand quadratic termTake rest to be local, but go to all order in  MFT  minimizes  Equivalent to approximate solution of BdG eqns. Ansatz: Find  by minimizing E(  )

12. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions12 of 18 Surface tension result Beyond Mean Field Theory + LDA  =0.9 10 -3 at unitarity P= 0.14 P= 0.53 P= 0.72 Theja De-Silva and EJM, Cond-mat/0603068 Data: Hulet

13. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions13 of 18 Beyond Mean Field Theory + LDA Discrepancies with experiments: P Data: MIT Experimentalist interpretation: Evidence of partially polarized normal phase Crude theorist argument: Possibly existence of partially polarized superfluid phase [Vortex experiments seem to not be consistent with this] a>0 [near resonance]

14. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions14 of 18 Normal State at T=0 Start fully polarized: Energy to add single spin down =   Ignore Pauli blocking [Leggett ~1999] Include Pauli Blocking   =   (0,0)= + + + … (Generalization of Hartree to include beyond-Born scattering) [Part of a systematic self-consistent theory -- but appears to be important bit] -- NSR+approximate self-consistency

15. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions15 of 18 Normal State at T=0 2 0 1.04 Thouless criterion: Normal state is unstable to pairing with q=0.6k  [k  =0.44k  (n  /n  =0.09)] 1.18 Fully Polarized Normal [Exact] 1.885 Unpolarized Superfluid [Monte-Carlo] ?????

16. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions16 of 18 Experimental Clues Chevy [cond-mat/0605751] Bulgac and Forbes [cond-mat/0606043] LDA: Upper bound from self-energy

17. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions17 of 18 Normal State at T=0 2 0 1.885 1.04 Thouless criterion: Normal state is unstable to pairing with q=0.6k  [k  =0.44k  (n  /n  =0.09)] 1.18 Transition to unpolarized superfluid must be right of this line

18. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions18 of 18 Alternative Pictures Gubbels, Romans and Stoof, cond-mat/0606330 Finds MIT data is consistent with Finite T + polarized superfluid Explains temp dep of Rice critical polarization

19. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions19 of 18 Differences MIT Resolves 3 shells No distortion of aspect ratio No “critical polarization” for phase separation –Compares lowest T data Superfluid-normal transition at P=0.7 Rice Resolves 2 shells –May be 3 shells Interacting normal gas looks a lot like superfluid –May be in different regime [above tricritical point] Sees distortion of aspect ratio –Surface energies important: Large aspect ratio + small particle number Observes “critical polarization” –Increases with increasing T

20. Copenhagen June 15, 2006 Erich Mueller -- Polarized Fermions20 of 18 Summary Surface Tension: –Another illustration of strong interactions in such dilute gases Phase Diagram –Argument that there may be a polarized superfluid -- normal state is only stable for quite low polarizations. –Normal state instability is at finite q [FFLO?]