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

Stoner instabilityE. Stoner, Phil. Mag. 15:1018 (1933)

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 ?

Pair formation

Two-particle scattering in vacuum

k -kp

-pMicroscopic Hamiltonian

Schrödinger equation

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

CooperonGrand canonical ensemble

Define

Cooperon equation

Cooperon vs T-matrix

kk

-k

k

-k -p’-p -p

pp k pp’

-p

Cooper channel response functionLinear 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

Magnetic instability

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 instabilityRPA spin susceptibility

Interaction = Cooperon

Stoner instability

Pairing instability always dominates over pairing

If ferromagnetic domains form, they form at large q

Relation to experiments

Pairing instability vs experiments

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

++-

-Ultracold polar molecules

Experiments on polar molecules: Innsbruck, Yale, Harvard, UConn,…

Instability of Unstructured Systems

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

Dimerization at mean-field level

z

z+1

Effective Lattice Model

Physical Layers

Lattice Site

&

L

Effective lattice model: Ising degrees of freedom

Effective lattice model: XY phase degrees of freedom

Effective Ising/XY Lattice Model:

Lattice model: generic phase diagram

Mean-field

Phase diagram

If similar for layered system:

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

Summary of part II