Unitary Polarized Fermi Gases

1. 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. Questions • Nature of normal fluid at zero T at unitarity? • Existence of exotic phases? • Modulated Order parameters [FFLO] • Deformed Fermi surfaces • Polarized s-wave superfluids [Sarma, Breeched gap] • Properties? • Equation of state • How to probe? Erich Mueller -- Polarized Fermions

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

4. Questions • Nature of normal fluid at unitarity? • Existence of exotic phases? • Modulated Order parameters [FFLO] • Deformed Fermi surfaces • Polarized s-wave superfluids [Sarma, Breeched gap] • Properties? • Equation of state • How to probe? • Homogeneous: • Mean Field Phase Diagram • Fluctuations and Stability • Scaling Arguments • Trap: • Surface tension • Collective modes • How to explain experiments Erich Mueller -- Polarized Fermions

5. 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] Erich Mueller -- Polarized Fermions

6. a>0 [near resonance] Normal-FFLO: Continuous SF-FFLO: Discontinuous (Schematic) Homogeneous Phases Mean Field Theory: a>0 a<0 Red: 1st order transition Erich Mueller -- Polarized Fermions

7. Application to Trap BCS BEC 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) Erich Mueller -- Polarized Fermions

8. Beyond Mean Field Theory + LDA • Discrepancies with experiments: P Data: Rice/MIT Erich Mueller -- Polarized Fermions

9. LDA in harmonic trap: Integrate: Axial density should be monotonic [in LDA] Beyond Mean Field Theory + LDA • Wings -- A violation of LDA Erich Mueller -- Polarized Fermions

10. Beyond Mean Field Theory + LDA • Simple Explanation: Surface Tension: z Large Aspect Ratio Trap: LDA: Unitarity constrains form: Erich Mueller -- Polarized Fermions

11. Beyond Mean Field Theory + LDA • Calculating s: D E x D D0 MFT: D minimizes Gradient expand quadratic term Take rest to be local, but go to all order in D Ansatz: Find x by minimizing E(x) Equivalent to approximate solution of BdG eqns. Erich Mueller -- Polarized Fermions

12. Beyond Mean Field Theory + LDA • Surface tension result P= 0.14 h=0.9 10-3 at unitarity P= 0.53 P= 0.72 Data: Hulet Theja De-Silva and EJM, Cond-mat/0603068 Erich Mueller -- Polarized Fermions

13. 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] Beyond Mean Field Theory + LDA • Discrepancies with experiments: P Data: MIT Erich Mueller -- Polarized Fermions

14. Normal State at T=0 Start fully polarized: Energy to add single spin down = m Ignore Pauli blocking [Leggett ~1999] Include Pauli Blocking + m=S(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 Erich Mueller -- Polarized Fermions

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

16. Upper bound from self-energy Experimental Clues Chevy [cond-mat/0605751] Bulgac and Forbes [cond-mat/0606043] LDA: Erich Mueller -- Polarized Fermions

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

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 Erich Mueller -- Polarized Fermions

19. 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 Differences Erich Mueller -- Polarized Fermions

20. 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?] Erich Mueller -- Polarized Fermions