1. Pairing and magnetism near Feshbach resonance $$ NSF, AFOSR MURI, DARPA, ARO Harvard-MIT David Pekker (Harvard/Caltech) Mehrtash Babadi (Harvard) Lode Pollet (Harvard/ETHZ) Rajdeep Sensarma (Harvard/JQI Maryland) Eric Vernier (Harvard/ENS) Nikolaj Zinner (Harvard/Niels Bohr Institute) Antoine Georges (Ecole Polytechnique) Martin Zwierlein (MIT) Eugene Demler (Harvard) Thanks to W. Ketterle and other members of the MIT group

2. Outline Competition of pairing and magnetism near Feshbach resonance Motivated by experiments by Jo et al., Science (2009) Shiba states in paired fermionic superfluids Motivated by experiments in M. Zwierlein’s lab

3. Competition of pairing and magnetism near Feshbach resonance Pekker, Babadi, Sensarma, Pollet, Zinner, Zwierlein, Demler arXiv:1005.2366

4. Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion U N(0) = 1 U – interaction strength N(0) – density of states at Fermi level Theoretical proposals for observing Stoner instability with cold gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); … Recent work on hard sphere potentials: Pilati et al. (2010); Chang et al. (2010)

5. Experiments were done dynamically. What are implications of dynamics? Why spin domains could not be observed? Also earlier work by C. Salomon, D. Jin, others

6. Is it sufficient to consider effective model with repulsive interactions when analyzing experiments? Feshbach physics beyond effective repulsive interaction

7. Feshbach resonance Interactions between atoms are intrinsically attractive Effective repulsion appears due to low energy bound states Example: scattering length V(x) V 0 tunable by the magnetic field Can tune through bound state

8. Feshbach resonance Two particle bound state formed in vacuum BCS instability Stoner instability Molecule formation and condensation This talk: Prepare Fermi state of weakly interacting atoms. Quench to the BEC side of Feshbach resonance. System unstable to both molecule formation and Stoner ferromagnetism. Which instability dominates ?

9. Pair formation

10. Two-particle scattering in vacuum k-k p -p Lippman-Schwinger equation For positive scattering length bound state at appears as a pole in the T-matrix k k -k k -p’-p p pk p p’ -p On-shell T-matrix. Universal low energy expression

11. Cooperon Two particle scattering in the presence of a Fermi sea k p -k -p Cooperon equation k k -k k -p’-p p pk p p’ -p

12. Cooper channel response function Linear response theory Induced pairing field Response function Poles of the Cooper channel response function are given by

13. Cooper channel response function Linear response theory Time dependent dynamics When the mode frequency has imaginary part, the system is unstable to formation of paired state Poles of the response function describe collective modes

14. Pairing instability regularized BCS side Instability rate coincides with the equilibrium gap (Abrikosov, Gorkov, Dzyaloshinski) Instability to pairing even on the BEC side Related work: Lamacraft, Marchetti, 2008

15. Pairing instability Intuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance. Energy and momentum conservation laws can not be satisfied. This argument applies in vacuum. Fermi sea prevents formation of real Feshbach molecules by Pauli blocking. Molecule Fermi sea

16. Pairing instability From wide to narrow resonances Pairing instability at different temperatures Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005

17. Magnetic instability

18. Stoner instability. Naïve theory Spin response function Relates induced spin polarization to external Zeeman field Spin collective modes are given by the poles of response function Imaginary frequencies correspond to magnetic instability

19. Quench dynamics across Stoner instability For U>U c unstable collective modes Magnetic Stoner instability Unphysical divergence at unitarity

20. Stoner instability Divergence in the scattering amplitude arises from bound state formation. Bound state is strongly affected by the Fermi sea. Stoner instability is determined by two particle scattering amplitude =+++ … =++

21. Stoner instability RPA spin susceptibility Interaction = Cooperon

22. Stoner instability Pairing dominates over magnetic instability If ferromagnetic domains form, they form at large q

23. Relation to experiments

24. Pairing instability vs experiments

25. Summary of part I Competition of pairing and ferromagnetism near Feshbach resonance Dynamics of competing orders is important for understanding experiments Simple model with repulsive interactions may not be sufficient Strong suppression of Stoner instability by Feshbach resonance physics + Pauli blocking Alternative interpretation of experiments based on pair formation

26. Shiba states in fermionic superfluids E. Vernier, D. Pekker, M. Zwierlein, E. Demler Simplest example of interplay of magnetism and pairing Motivated by experiments in M. Zwierlein’s lab

27. Shiba states in superconductors Classical spin impurity in SC quasiparitcle can make a localized in-gap state by aligning its spin with impurity N(w) w -DD Local density of states Gd on the surface of Nb STM measurements of LDOS Yazdani et al., Science 1997

28. Ultracold atoms: Shiba states in paired Fermi superfluids a Scattering lengths need to be computed including impurity confinement Example Shiba states may exist even when there are no Feshbach bound states

29. RF spectroscopy as a probe of Shiba states Other possible probes: modulation type experiments on impurity confinement, interaction with fermions

30. Interesting open questions: From Shiba states to midgap band Gapless superconductivity Suppression of pairing Simplest example of interplay of magnetism and pairing Why study Shiba states in paired Fermi superfluids? Shiba states as a local probe (M. Zwierlein) Local probe of pairing for FFLO, pseudogap, etc. Probe of unconventional pairing Zn impurities In high HTSC Davis et al., 2003

31. Summary Competition of pairing and ferromagnetism near Feshbach resonance Dynamics of competing orders is important for understanding experiments. Simple model with repulsive interactions may not be sufficient. Strong suppression of Stoner instability by Feshbach resonance physics + Pauli blocking. Alternative interpretation of experiments based on pair formation State dependent interaction with localized magnetic impurities should allow the study of Shiba states. Simplest example of the interplay of pairing and magnetism Local probe of pairing Harvard-MIT