Transcript Competing instabilities in strongly correlated electron

Pairing and magnetism near
Feshbach resonance
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
Harvard-MIT
$$ NSF, AFOSR MURI, DARPA, ARO
Outline
Competition of pairing and magnetism
near Feshbach resonance
Motivated by experiments by Jo et al., Science (2009)
Shiba states in paired fermionic superfluids
Motivated 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 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?
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 resonance
Interactions between atoms are intrinsically attractive
Effective repulsion appears due to low energy bound states
Example:
V(x)
V0 tunable by the magnetic field
Can tune through bound state
scattering length
Feshbach resonance
Two particle bound state
formed in vacuum
Stoner instability
BCS 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
p
k
Lippman-Schwinger equation
-k
-p
k
k
-k
p
-p
k
p
-k
-p
k
p’
-p’
p
-p
On-shell T-matrix. Universal low energy expression
For positive scattering length bound state at
appears as a pole in the T-matrix
Cooperon
p
k
Two particle scattering in the
presence of a Fermi sea
-k
-p
Cooperon equation
k
k
-k
p
-p
k
p
-k
-p
k
p’
-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
Poles of the response function
describe collective modes
Time dependent dynamics
When the mode frequency has imaginary part,
the system is unstable to formation of paired state
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 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
Pairing instability
at different temperatures
Three body recombination
as in Shlyapnikov et al., 1996;
Petrov, 2003; Esry 2005
From wide to
narrow resonances
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
Magnetic Stoner instability
For U>Uc unstable collective modes
Unphysical divergence at unitarity
Stoner instability
=
+
+
Stoner instability is determined by two particle
scattering amplitude
Divergence in the scattering amplitude arises
from bound state formation. Bound state is
strongly affected by the Fermi sea.
+ …
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
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 interactions
may 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
Simplest example of interplay of magnetism and pairing
Motivated by experiments in M. Zwierlein’s lab
E. Vernier, D. Pekker, M. Zwierlein, E. Demler
Shiba states in superconductors
Classical spin impurity in SC
quasiparitcle can make a
localized in-gap state by
aligning its spin with impurity
Local density
of states
N(w)
w
-D
D
Gd on the surface of Nb
STM measurements of LDOS
Yazdani et al., Science 1997
Ultracold atoms: Shiba states
in paired Fermi superfluids
Example
a
a
Scattering lengths need
to be computed including
impurity confinement
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
Why study Shiba states
in paired Fermi superfluids?
Interesting open questions:
• From Shiba states to midgap band
• Gapless superconductivity
• Suppression of pairing
• Simplest example of interplay of
magnetism and pairing
Shiba states as a local probe
(M. Zwierlein)
• Local probe of pairing for
FFLO, pseudogap, etc.
• Probe of unconventional pairing
Zn impurities
In high HTSC
Davis et al., 2003
Summary
Harvard-MIT
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
State dependent interaction with localized magnetic
impurities should allow the study of Shiba states.
Simplest example of the interplay of pairing and magnetism
Local probe of pairing