competing instabilities in ultracold fermi gases
DESCRIPTION
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) . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/1.jpg)
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
![Page 2: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/2.jpg)
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
![Page 3: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/3.jpg)
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
![Page 4: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/4.jpg)
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
![Page 5: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/5.jpg)
Stoner instabilityE. Stoner, Phil. Mag. 15:1018 (1933)
![Page 6: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/6.jpg)
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)
![Page 7: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/7.jpg)
Experiments weredone dynamically.What are implicationsof dynamics?Why spin domains could not be observed?Earlier work by C. Salomon et al., 2003
![Page 8: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/8.jpg)
Is it sufficient to consider effective model with repulsive interactions when analyzing experiments?
Feshbach physics beyond effective repulsive interaction
![Page 9: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/9.jpg)
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
![Page 10: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/10.jpg)
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 ?
![Page 11: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/11.jpg)
Pair formation
![Page 12: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/12.jpg)
Two-particle scattering in vacuum
k -kp
-pMicroscopic Hamiltonian
Schrödinger equation
![Page 13: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/13.jpg)
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
![Page 14: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/14.jpg)
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
![Page 15: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/15.jpg)
CooperonGrand canonical ensemble
Define
Cooperon equation
![Page 16: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/16.jpg)
Cooperon vs T-matrix
kk
-k
k
-k -p’-p -p
pp k pp’
-p
![Page 17: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/17.jpg)
Cooper channel response functionLinear response theory
Induced pairing field
Response function
Poles of the Cooper channel response function are given by
![Page 18: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/18.jpg)
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
![Page 19: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/19.jpg)
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
![Page 20: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/20.jpg)
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
![Page 21: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/21.jpg)
Pairing instabilityTime dependent variational wavefunction
Time dependence of uk(t) and vk(t) due to DBCS(t)
For small DBCS(t):
![Page 22: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/22.jpg)
Pairing instability
From wide to narrow resonances
Effects of finite temperature
Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005
![Page 23: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/23.jpg)
Magnetic instability
![Page 24: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/24.jpg)
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
![Page 25: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/25.jpg)
RPA analysis for Stoner instability
Self-consistent equation on response function
RPA expression for the spin response function
Spin susceptibility for non-interacting gas
![Page 26: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/26.jpg)
Quench dynamics across Stoner instability
Unstable modes determine characteristic lengthscale of magnetic domains
For U>Uc unstable collective modes
Stoner criterion
![Page 27: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/27.jpg)
Stoner quench dynamics in D=3
Growth rate of magnetic domains
Domain size
Unphysical divergenceof the instability rate at unitarity
Scaling near transition
![Page 28: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/28.jpg)
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
= + + + …= + + + …
![Page 29: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/29.jpg)
Stoner instabilityRPA spin susceptibility
Interaction = Cooperon
![Page 30: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/30.jpg)
Stoner instability
Pairing instability always dominates over pairing
If ferromagnetic domains form, they form at large q
![Page 31: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/31.jpg)
Relation to experiments
![Page 32: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/32.jpg)
Pairing instability vs experiments
![Page 33: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/33.jpg)
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
![Page 34: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/34.jpg)
Superfluidity and Dimerization in a Multilayered System of Fermionic
Dipolar Molecules
A. Potter, E. Berg, D.W. Wang, B. Halperin, and E. Demler
![Page 35: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/35.jpg)
++-
-Ultracold polar molecules
Experiments on polar molecules: Innsbruck, Yale, Harvard, UConn,…
![Page 36: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/36.jpg)
Instability of Unstructured Systems
![Page 37: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/37.jpg)
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); …
![Page 38: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/38.jpg)
Pairing in a multilayer system Dimerization
…
paired
unpaired
……
…
paired
unpaired
Interplay of two orders: superfluidity in individual bilayers and dimerization
![Page 39: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/39.jpg)
Dimerization at mean-field level
z
z+1
![Page 40: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/40.jpg)
Effective Lattice Model
Physical Layers
Lattice Site
&
L
![Page 41: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/41.jpg)
Effective lattice model: Ising degrees of freedom
Effective lattice model: XY phase degrees of freedom
![Page 42: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/42.jpg)
Effective Ising/XY Lattice Model:
Lattice model: generic phase diagram
Mean-field
![Page 43: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/43.jpg)
Phase diagram
If similar for layered system:
![Page 44: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/44.jpg)
Light-Scattering Detection…
…Dimerization Order Parameter:
Finite Confinement Strength
New Bragg Peaks @:
Transverse Displacement:
Correlation Measurements:Correlations:
![Page 45: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/45.jpg)
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
![Page 46: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/46.jpg)
Summary of part II
![Page 47: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/47.jpg)
![Page 48: Competing instabilities in ultracold Fermi gases](https://reader034.vdocuments.net/reader034/viewer/2022050809/568165e7550346895dd90735/html5/thumbnails/48.jpg)