• Fermions in optical lattices
Magnetism
Pairing in systems with repulsive interactions
Current experiments: Paramagnetic Mott state
• Experiments on nonequilibrium fermion dynamics
Lattice modulation experiments
Doublon decay
• Stoner instability

U t t
Experiments with fermions in optical lattice, Kohl et al., PRL 2005

Quantum simulations with ultracold atoms
Atoms in optical lattice
Antiferromagnetic and superconducting Tc of the order of 100 K
Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
-

Fermionic Hubbard model
Phenomena predicted
Superexchange and antiferromagnetism (P.W. Anderson)
Itinerant ferromagnetism. Stoner instability (J. Hubbard)
Incommensurate spin order. Stripes (Schulz, Zaannen,
Emery, Kivelson, Fradkin, White, Scalapino, Sachdev, …)
Mott state without spin order. Dynamical Mean Field Theory
(Kotliar, Georges, Metzner, Vollhadt, Rozenberg, …) d-wave pairing
(Scalapino, Pines,…) ddensity wave (Affleck, Marston, Chakravarty, Laughlin,…)

Superexchange and antiferromagnetism in the Hubbard model. Large U limit
Singlet state allows virtual tunneling and regains some kinetic energy
Triplet state: virtual tunneling forbidden by Pauli principle
Effective Hamiltonian:
Heisenberg model

Hubbard model for small U.
Antiferromagnetic instability at half filling
Fermi surface for n=1 Analysis of spin instabilities.
Random Phase Approximation
Q=(p,p)
Nesting of the Fermi surface leads to singularity
BCS-type instability for weak interaction

T
N
BCS-type theory applies paramagnetic
Mott phase
Paramagnetic Mott phase: one fermion per site charge fluctuations suppressed no spin order
U
Heisenberg model applies

SU(N) Magnetism with
Ultracold Alkaline-Earth Atoms
A. Gorshkov, et al., Nature Physics 2010
Ex: 87 Sr (I = 9/2)
|e
>
= 3 P
0
Alkaline-Earth atoms in optical lattice:
698 nm
150 s ~ 1 mHz
|g
>
= 1 S
0
Nuclear spin decoupled from electrons SU(N=2I+1) symmetry
→ SU(N) spin models ⇒ valence-bond-solid & spin-liquid phases
• orbital degree of freedom ⇒ spin-orbital physics
→ Kugel-Khomskii model [transition metal oxides with perovskite structure]
→ SU(N) Kondo lattice model [for N=2, colossal magnetoresistance in manganese oxides and heavy fermion materials]

Attraction between holes in the Hubbard model
Loss of superexchange energy from 8 bonds
Single plaquette: binding energy
Loss of superexchange energy from 7 bonds

Pairing of holes in the Hubbard model
Leading istability: d-wave
Scalapino et al, PRB (1986)
Non-local pairing of holes k’ k spin fluctuation
k’
-k

-
+
d x2-y2
+
Q
Pairing of holes in the Hubbard model
BCS equation for pairing amplitude k’ k’ k spin fluctuation
-k
Systems close to AF instability: c (Q) is large and positive
D k should change sign for k’=k+Q

Stripe phases in the Hubbard model
Stripes:
Antiferromagnetic domains separated by hole rich regions
Antiphase AF domains stabilized by stripe fluctuations
First evidence: Hartree-Fock calculations. Schulz, Zaannen (1989)
Numerical evidence for ladders: Scalapino, White (2003);
For a recent review see Fradkin arXiv:1004.1104

Possible Phase Diagram
T
AF – antiferromagnetic
SDW- Spin Density Wave
(Incommens. Spin Order, Stripes)
D-SC – d-wave paired
AF pseudogap
SDW
D-SC n=1 doping
After several decades we do not yet know the phase diagram

Quantum simulations with ultracold atoms
Atoms in optical lattice
Antiferromagnetic and superconducting Tc of the order of 100 K
Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Quantum noise analysis of TOF images is more than HBT interference

Second order interference from the BCS superfluid
Theory: Altman et al., PRA 70:13603 (2004) n(k) n(r’) k
F
BCS
BEC k n(r) r n
, )

Momentum correlations in paired fermions
Greiner et al., PRL 94:110401 (2005)

Second Order Interference
In the TOF images
Normal State
Superfluid State measures the Cooper pair wavefunction
One can identify unconventional pairing

T
N
Mott current experiments
U

Signatures of incompressible Mott state of fermions in optical lattice
Suppression of double occupancies
R. Joerdens et al., Nature (2008)
Compressibility measurements
U. Schneider et al., Science (2008)

T
N
Mott current experiments
U

t t
|U|
Phase diagram:
Micnas et al.,
Rev. Mod. Phys.,1990

Hackermuller et al., Science 2010
Constant Entropy
Constant Particle
Number
Constant
Confinement
13.04.2020
27

13.04.2020
28

Probing the Mott state of fermions
Sensarma, Pekker, Lukin, Demler, PRL (2009)
Related theory work: Kollath et al., PRL (2006)
Huber, Ruegg, PRB (2009)
Orso, Iucci, et al., PRA (2009)

Lattice modulation experiments
Probing dynamics of the Hubbard model
Modulate lattice potential
Measure number of doubly occupied sites
Main effect of shaking: modulation of tunneling
Doubly occupied sites created when frequency w matches Hubbard U

Lattice modulation experiments
Probing dynamics of the Hubbard model
R. Joerdens et al., Nature 455:204 (2008)

Mott state
Regime of strong interactions U>>t.
Mott gap for the charge forms at
Antiferromagnetic ordering at
“High” temperature regime
All spin configurations are equally likely.
Can neglect spin dynamics.
“Low” temperature regime
Spins are antiferromagnetically ordered or have strong correlations

d
“Low” Temperature
Schwinger bosons Bose condensed
Propagation of holes and doublons strongly affected by interaction with spin waves h
Assume independent propagation of hole and doublon (neglect vertex corrections)
Self-consistent Born approximation
Schmitt-Rink et al (1988), Kane et al. (1989)
= +
Spectral function for hole or doublon
Sharp coherent part: dispersion set by t
2
/U , weight by t/U
Incoherent part: dispersion
Oscillations reflect shake-off processes of spin waves

“Low” Temperature
Rate of doublon production
• Sharp absorption edge due to coherent quasiparticles
• Broad continuum due to incoherent part
• Spin wave shake-off peaks

d h
“High” Temperature
Retraceable Path Approximation
Brinkmann & Rice, 1970
Consider the paths with no closed loops
Original Experiment:
R. Joerdens et al.,
Nature 455:204 (2008)
Theory:
Sensarma et al.,
PRL 103, 035303 (2009)
Spectral Fn. of single hole

Temperature dependence
Reduced probability to find a singlet on neighboring sites
Radius
Radius
D. Pekker et al., upublished

Ref: N. Strohmaier et al., PRL 2010

Doublons – repulsively bound pairs
What is their lifetime?
Direct decay is not allowed by energy conservation
Excess energy U should be converted to kinetic energy of single atoms
Decay of doublon into a pair of quasiparticles requires creation of many particle-hole pairs

Experiments: N. Strohmaier et. al.

Relaxation of doublon- hole pairs in the Mott state
Energy U needs to be absorbed by spin excitations
 Energy carried by spin excitations
~ J =4t 2 /U
Relaxation rate
 Relaxation requires creation of ~U 2 /t 2 spin excitations
Very slow, not relevant for ETH experiments

Doublon decay in a compressible state
Excess energy U is converted to kinetic energy of single atoms
Compressible state: Fermi liquid description
Doublon can decay into a pair of quasiparticles with many particle-hole pairs
U p h p h p h p p

Doublon decay in a compressible state
Perturbation theory to order n=U/6t
Decay probability
Doublon Propagator Interacting “Single” Particles

Doublon decay in a compressible state
N. Strohmaier et al., PRL 2010
Expt: ETHZ
Theory: Harvard
To calculate the rate: consider processes which maximize the number of particle-hole excitations

Prototype of decay processes with emission of many interacting particles .
Example: resonance in nuclear physics: (i.e. delta-isobar)
Analogy to pump and probe experiments in condensed matter systems
Response functions of strongly correlated systems at high frequencies . Important for numerical analysis.
Important for adiabatic preparation of strongly correlated systems in optical lattices

Expansion of interacting fermions in optical lattice
U. Schneider et al., arXiv:1005.3545
New dynamical symmetry: identical slowdown of expansion for attractive and repulsive interactions

arXiv:1005.2366
D. Pekker, M. Babadi, R. Sensarma, N. Zinner,
L. Pollet, M. Zwierlein, E. Demler

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
Existence of Stoner type ferromagnetism in a single band model is still a subject of debate
Theoretical proposals for observing Stoner instability with ultracold Fermi gases:
Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit,
Simons (2009); LeBlanck et al. (2009); …

Experiments were done dynamically.
What are implications of dynamics?
Why spin domains could not be observed?

Is it sufficient to consider effective model with repulsive interactions when analyzing experiments?
Feshbach physics beyond effective repulsive interaction

Interactions between atoms are intrinsically attractive
Effective repulsion appears due to low energy bound states
Example:
V(x) scattering length
V
0 tunable by the magnetic field
Can tune through bound state

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 ?

Imaginary frequencies of collective modes
Magnetic Stoner instability
Pairing instability
= + + + …

Many body instabilities near Feshbach resonance: naïve picture
Pairing (BCS) Stoner (BEC)
E
F
=
Pairing (BCS)
Stoner (BEC)

bubble is
UV divergent
To keep answers finite, we must tune together: upper momentum cut-off interaction strength U
Change from bare interaction to the scattering length
Instability to pairing even on the BEC side

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

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.

RPA spin susceptibility
Interaction = Cooperon

Pairing instability always dominates over pairing
If ferromagnetic domains form, they form at large q

• Introduction. Magnetic and optical trapping of ultracold atoms.
• Cold atoms in optical lattices.
• Bose Hubbard model. Equilibrium and dynamics
• Bose mixtures in optical lattices
Quantum magnetism of ultracold atoms.
• Detection of many-body phases using noise correlations
• Experiments with low dimensional systems
Interference experiments. Analysis of high order correlations
• Fermions in optical lattices
Magnetism and pairing in systems with repulsive interactions.
Current experiments: paramgnetic Mott state, nonequilibrium dynamics.
• Dynamics near Fesbach resonance. Competition of
Stoner instability and pairing
Emphasis of these lectures : • Detection of many-body phases
• Nonequilibrium dynamics