Pairing and magnetism nearFeshbach resonance

$$ NSF, AFOSR MURI, DARPA, AROHarvard-MIT

David Pekker (Harvard/Caltech)Mehrtash Babadi (Harvard) Lode Pollet (Harvard/ETHZ)Rajdeep Sensarma (Harvard/JQI Maryland)Eric Vernier (Harvard/ENS) Nikolaj Zinner (Harvard/Niels Bohr Institute) Antoine Georges (Ecole Polytechnique)Martin Zwierlein (MIT)Eugene Demler (Harvard)

Thanks to W. Ketterle and other members of the MIT group

Outline

Competition of pairing and magnetism near Feshbach resonanceMotivated by experiments by Jo et al., Science (2009)

Shiba states in paired fermionic superfluidsMotivated by experiments in M. Zwierlein’s lab

Competition of pairing and magnetism near Feshbach resonance

Pekker, Babadi, Sensarma, Pollet, Zinner, Zwierlein, Demler arXiv:1005.2366

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); …

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?

Also earlier work by C. Salomon, D. Jin, others

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

-p

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

On-shell T-matrix. Universal low energy expression

CooperonTwo particle scattering in the presence of a Fermi sea

kp

-k

-p Cooperon equation

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

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 functiondescribe 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

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 instability

From wide to narrow resonances

Pairing instabilityat different temperatures

Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005

Magnetic instability

Stoner instability. Naïve theory

Spin response functionRelates induced spin polarizationto 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>Uc 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 particlescattering amplitude

= + + + …= + + + …

Stoner instabilityRPA 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

Summary of part I

Competition 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

Shiba states in fermionic superfluids

E. Vernier, D. Pekker, M. Zwierlein, E. Demler

Simplest example of interplay of magnetism and pairingMotivated by experiments in M. Zwierlein’s lab

Shiba states in superconductors

Classical spin impurity in SCquasiparitcle can make a localized in-gap state byaligning its spin with impurity

N(w)

w-D D

Local density of states

Gd on the surface of NbSTM measurements of LDOSYazdani et al., Science 1997

Ultracold atoms: Shiba states in paired Fermi superfluids

a aScattering lengths needto be computed includingimpurity confinement

Example

Shiba states may exist even when there are no Feshbach bound states

RF spectroscopy as a probe of Shiba states

Other possible probes: modulation type experiments on impurity confinement,interaction with fermions

Interesting open questions:• From Shiba states to midgap band• Gapless superconductivity• Suppression of pairing• Simplest example of interplay of magnetism and pairing

Why study Shiba states in paired Fermi superfluids?

Shiba states as a local probe(M. Zwierlein)• Local probe of pairing for FFLO, pseudogap, etc.• Probe of unconventional pairing

Zn impuritiesIn high HTSCDavis et al., 2003

SummaryCompetition 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

State dependent interaction with localized magneticimpurities should allow the study of Shiba states.

Simplest example of the interplay of pairing and magnetismLocal probe of pairing

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