Competing instabilities in ultracold Fermi gases
$$ NSF, AFOSR MURI, DARPA AROHarvard-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
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
Competing instabilities in strongly correlated electron systems
Organic materials.Bechgaard salts
doping
tem
pera
ture
(K
)
0
100
200
300
400
High Tc superconductorsHeavy fermion
materials
This talk is also about competition between pairing and magnetism/CDW
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
then
Stoner model of ferromagnetismSpontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectrons
Mean-field criterion
U N(0) = 1
U – interaction strengthN(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)
Experiments weredone dynamically.What are implicationsof dynamics?Why spin domains could not be observed?
Earlier work by C. Salomon et al., 2003
Is it sufficient to consider effective model with repulsive interactions when analyzing experiments?
Feshbach physics beyond effective repulsive interaction
Feshbach resonanceInteractions between atoms are intrinsically attractiveEffective repulsion appears due to low energy bound states
Example:
scattering lengthV(x)
V0 tunable by the magnetic fieldCan tune through bound state
Feshbach resonanceTwo particle bound state formed in vacuum
BCS instabilityStoner instability
Molecule formationand 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 ?
Lippman-Schwinger equation
For positive scattering length bound state atappears as a pole in the T-matrix
kk
-k
k
-k -p’-p -p
pp k pp’
-p
T-matrix
On-shell T-matrix. Universal low energy expression
CooperonTwo 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
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
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
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
Pairing instabilityIntuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance.Energy and momentum conservation laws can notbe satisfied.
This argument applies in vacuum. Fermi sea preventsformation of real Feshbach molecules by Pauli blocking.
Molecule Fermi sea
Pairing instabilityTime dependent variational wavefunction
Time dependence of uk(t) and vk(t) due to DBCS(t)
For small DBCS(t):
Pairing instability
From wide to narrow resonances
Effects of finite temperature
Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005
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
RPA analysis for Stoner instability
Self-consistent equation on response function
RPA expression for the spin response function
Spin susceptibility for non-interacting gas
Quench dynamics across Stoner instability
Unstable modes determine characteristic lengthscale of magnetic domains
For U>Uc unstable collective modes
Stoner criterion
Stoner quench dynamics in D=3
Growth rate of magnetic domains
Domain size
Unphysical divergenceof the instability rate at unitarity
Scaling near transition
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 particlescattering amplitude
= + + + …= + + + …
Stoner instability
Pairing instability always dominates over pairing
If ferromagnetic domains form, they form at large q
Conclusions to part ICompetition of pairing and ferromagnetism near Feshbach resonance
Dynamics of competing orders is important for understanding experiments
Simple model with repulsive interactionsmay not be sufficient
Strong suppression of Stoner instability by Feshbach resonance physics + Pauli blocking
Alternative interpretation of experiments based on pair formation
Superfluidity and Dimerization in a Multilayered System of Fermionic
Dipolar Molecules
A. Potter, E. Berg, D.W. Wang, B. Halperin, and E. Demler
Pairing in a multilayer system
extE
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); …
Pairing in a multilayer system Dimerization
…
paired
unpaired
……
…
paired
unpaired
Interplay of two orders: superfluidity in individual bilayers and dimerization
Effective lattice model: Ising degrees of freedom
Effective lattice model: XY phase degrees of freedom
Light-Scattering Detection…
…Dimerization Order Parameter:
Finite Confinement Strength
New Bragg Peaks @:
Transverse Displacement:
Correlation Measurements:
Correlations:
Summary Competition between pairing and ferromagnetic
instabilities in ultracold Fermi gases near Feshbach resonances
D. Pekker et al., arXiv:1005.2366 Motivated by experiments of Jo et al., Science (2009)
Superfluidity and Dimerization in a Multilayered System of Fermionic Dipolar MoleculesA. Potter, E. Berg, D.W. Wang, B.I. Halperin, E. Demler
$$ NSF, AFOSR MURI, DARPAHarvard-MIT