Competing instabilities in ultracold Fermi gases $$ NSF, AFOSR MURI, DARPA ARO Harvard-MIT David Pekker (Harvard) Mehrtash Babadi (Harvard) Lode Pollet (Harvard) Rajdeep Sensarma (Harvard/JQI Maryland) Nikolaj Zinner (Harvard/Niels Bohr Institute) Antoine Georges (Ecole Polytechnique) Martin Zwierlein (MIT) Eugene Demler (Harvard) Special thanks to W. Ketterle, G.B. Jo, and other members of the MIT group Details in arXiv:

Outline Introduction. Stoner instability Possible observation of Stoner instability in MIT experiments. G.B. Jo et al., Science (2009) Dynamics vs equilibrium: pairing and magnetism Dynamics of molecule formation near Feshbach resonance Magnetic Stoner instability near Feshbach resonance Comparison of two instabilities and relation to experiments

Stoner instability E. Stoner, Phil. Mag. 15:1018 (1933)

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)

Experiments were done dynamically. What are implications of dynamics? Why spin domains could not be observed?

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

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

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 ?

Pair formation

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

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

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

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

Pairing instability regularized BCS side Instability rate coincides with the equilibrium gap (Abrikosov, Gorkov, Dzyaloshinski) Instability to pairing even on the BEC side

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

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

Magnetic instability

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

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

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 =+++ … =++

Stoner instability RPA spin susceptibility Interaction = Cooperon

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

Relation to experiments

Pairing instability vs experiments

Pairing and magnetism in strongly correlated systems. Quantum dynamics

Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Same microscopic model Quantum simulations with ultracold atoms

Nonequilibrium dynamics in quantum many-body systems of ultracold atoms Long intrinsic time scales - Interaction energy and bandwidth ~ 1kHz - System parameters can be changed over this time scale Decoupling from external environment - Long coherence times Can achieve highly non equilibrium quantum many-body states Equilibrium properties of many-body systems. Many open questions but known paradigms: order parameters, universal fixed points (e.g. Fermi liquid) Nonequilibrium properties of many-body systems. We do not even have paradigms or understanding of universality

Competing instabilities in strongly correlated electron systems Organic materials. Bechgaard salts doping temperature (K) High Tc superconductors Heavy fermion materials This talk is also about competition between pairing and magnetism. Instabilities rather than ground states.

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 Harvard-MIT