1. Competing instabilities in ultracold Fermi gases $$ NSF, AFOSR MURI, DARPA Motivated by experiments of G.-B. Jo et al., Science (2009) Harvard-MIT David Pekker (Harvard), Mehrtash Babadi (Harvard), Rajdeep Sensarma (Harvard/Maryland), Nikolaj Zinner (Harvard/Niels Bohr Institute), Antoine Georges (Ecole Polytechnique), Eugene Demler (Harvard)

2. Outline Introduction. Stoner instability Possible observation of Stoner instability in MIT experiments. G.B. Jo et al., Science (2009) Spin domains. Nonequilibrium dynamics across Stoner transition Competition of molecule formation and Stoner instability (motivated by discussions with Sandro Stringari)

3. 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 ultracold Fermi gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); … Kanamori’s counter-argument: renormalization of U then

4. Magnetic domains could not be resolved. Why? T.L. Ho (2009)

5. Stoner Instability New feature of cold atoms systems: non-adiabatic crossing of U c Screening of U (Kanamori) occurs on times 1/E F Two timescales in the system: screening and magnetic domain formation Magnetic domain formation takes place on much longer time scales: critical slowing down

6. Quench dynamics across Stoner instability Find collective modes Unstable modes determine characteristic lengthscale of magnetic domains For U>U c unstable collective modes

7. For MIT experiments domain sizes of the order of a few l F Quench dynamics in D=3 Dynamics of magnetic domain formation near Stoner transition Moving across transition at a finite rate 0u u*u* domains coarsen slow growth domains freeze Growth rate of magnetic domains Domain size Domains freeze when Domain size at “freezing” point M. Babadi et al. (2009)

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

9. Feshbach resonance Review: Duine and Stoof, 2004 Chin et al., 2009 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 ?

10. Many-body instabilities Imaginary frequencies of collective modes Magnetic Stoner instability Pairing instability

11. Change from bare interaction to the scattering length Instability to pairing even on the BEC side Pairing instability

12. 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

13. Pairing instability Time dependent variational wavefunction Time dependence of u k (t) and v k (t) due to D BCS (t) For small D BCS (t):

14. Pairing instability From wide to narrow resonances

15. Stoner vs pairing Does Stoner instability really exceed molecule formation rate?

16. 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

17. Stoner instability Spin susceptibility

18. Growth rate of pairing instability Growth rate of magnetic Stoner instability RPA with bare scattering length RPA with Cooperon Changing from scattering length to T-matrix gives appreciable suppression of the Stoner instability Additional suppression due to Pauli blocking Stoner instability

19. Stoner vs pairing G.B. Jo et al., Science (2009)

20. Stoner vs pairing Increase in the kinetic energy: consistent with pairing. In the BCS state kinetic energy goes up and the interaction energy goes down

21. Conclusions Competition of pairing and Stoner instabilities New features due to dynamical character of experiments Simple model with contact repulsive interactions may not be sufficient to understand experiments Strong suppression of Stoner instability by Fechbach resonance physics + Pauli blocking Interesting questions beyond linear instability analysis.