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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), Eugene Demler (Harvard) Special thanks to W. Ketterle, G.B. Jo, and other members of the MIT group Details in arXiv:1005.2366

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Superfluidity and Dimerization in a Multilayered System of Fermionic Dipolar Molecules A. Potter, E. Berg, D.W. Wang, B. Halperin, and E. Demler If time permits

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Competing instabilities in strongly correlated electron systems Organic materials. Bechgaard salts doping temperature (K) 0 100 200 300 400 High Tc superconductors Heavy fermion materials This talk is also about competition between pairing and magnetism/CDW

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Outline Introduction. Stoner instability Possible observation of Stoner instability in MIT experiments. G.B. Jo et al., Science (2009) Dynamics of molecule formation near Feshbach resonance Dynamical crossing of Stoner transition Comparison of two instabilities Interplay of Superfluidity and Dimerization in a multilayered system of fermionic dipolar molecules

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

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then 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); … Kanamori’s counter-argument: renormalization of U. Recent work on hard sphere potentials: Pilati et al. (2010); Chang et al. (2010)

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Experiments were done dynamically. What are implications of dynamics? Why spin domains could not be observed? Earlier work by C. Salomon et al., 2003

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

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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

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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 ?

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Pair formation

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Two-particle scattering in vacuum k-k p -p Microscopic Hamiltonian Schrödinger equation

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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 T-matrix On-shell T-matrix. Universal low energy expression

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Cooperon Two particle scattering in the presence of a Fermi sea k p -k -p Need to make sure that we do not include interaction effects on the Fermi liquid state in scattered state energy

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Cooperon Grand canonical ensemble Define Cooperon equation

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Cooperon vs T-matrix k k -k k -p’-p p pk p p’ -p

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

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Cooper channel response function Poles of the response function,, describe collective modes Linear response theory Time dependent dynamics When the mode frequency has negative imaginary part, the system is unstable

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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

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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

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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):

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Pairing instability From wide to narrow resonances Effects of finite temperature Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005

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Magnetic instability

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Stoner instability. Naïve theory Linear response theory Spin response function Spin collective modes are given by the poles of response function Negative imaginary frequencies correspond to magnetic instability

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RPA analysis for Stoner instability Self-consistent equation on response function RPA expression for the spin response function Spin susceptibility for non-interacting gas

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

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Stoner quench dynamics in D=3 Growth rate of magnetic domains Domain size Unphysical divergence of the instability rate at unitarity Scaling near transition

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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 =+++ … =++

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Stoner instability RPA spin susceptibility Interaction = Cooperon

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

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Relation to experiments

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Pairing instability vs experiments

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Conclusions to 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

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Superfluidity and Dimerization in a Multilayered System of Fermionic Dipolar Molecules A. Potter, E. Berg, D.W. Wang, B. Halperin, and E. Demler

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++ - - Ultracold polar molecules Experiments on polar molecules: Innsbruck, Yale, Harvard, UConn,…

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Instability of Unstructured Systems

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Pairing in a multilayer system d Earlier theoretical work on polar molecules in layered systems: Shlyapnikov et al. (2003); Wang et al (2006); Santos et al. (2007); Collath et al. (2008); …

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Pairing in a multilayer system Dimerization … paired unpaired … … … paired unpaired Interplay of two orders: superfluidity in individual bilayers and dimerization

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Dimerization at mean-field level z z+1

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Effective Lattice Model: degrees of freedom σ= +1 σ= - 1 Phase of the order parameter Ising variable for dimerization

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Effective Lattice Model Physical Layers Lattice Site & L

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Effective lattice model: Ising degrees of freedom Effective lattice model: XY phase degrees of freedom

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Effective Ising/XY Lattice Model: Lattice model: generic phase diagram Mean-field

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Phase diagram If similar for layered system:

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Light-Scattering Detection … … Dimerization Order Parameter: Finite Confinement Strength New Bragg Peaks @: Transverse Displacement: Correlation Measurements: Correlations:

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Summary Competition between pairing and ferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances D. Pekker et al., arXiv:1005.2366arXiv:1005.2366 Motivated by experiments of Jo et al., Science (2009) Superfluidity and Dimerization in a Multilayered System of Fermionic Dipolar Molecules A. Potter, E. Berg, D.W. Wang, B.I. Halperin, E. Demler $$ NSF, AFOSR MURI, DARPA Harvard-MIT

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Summary of part II

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