1. Outline of these lectures
Introduction. Systems of ultracold atoms.
Cold atoms in optical lattices.
Bose Hubbard model
Bose mixtures in optical lattices
Detection of many-body phases using noise correlations
Experiments with low dimensional systems
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

2. Ultracold fermions in optical lattices

3. Fermionic atoms in optical latticesU
Experiments with fermions in optical lattice, Kohl et al., PRL 2005

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

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

6. 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,…)
d-density wave (Affleck, Marston, Chakravarty, Laughlin,…)

7. Superexchange and antiferromagnetism in the Hubbard model
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

8. Hubbard model for small U
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

9. Hubbard model at half fillingTN
paramagnetic
Mott phase
Paramagnetic Mott phase:
one fermion per site
charge fluctuations suppressed
no spin order U
BCS-type
theory applies
Heisenberg
model applies

10. Ultracold Alkaline-Earth Atoms
SU(N) Magnetism with
Ultracold Alkaline-Earth Atoms
A. Gorshkov, et al., Nature Physics 2010
Ex: 87Sr (I = 9/2)
|g> = 1S0
|e> = 3P0
698 nm
150 s ~ 1 mHz
Alkaline-Earth atoms in optical lattice:
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]

11. Doped Hubbard model

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

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

14. Pairing of holes in the Hubbard model
BCS equation for pairing amplitude Q k’ k
-k’ -k
spin
fluctuation - + + -
Systems close to AF instability:
c(Q) is large and positive
Dk should change sign for k’=k+Q
dx2-y2

15. 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:

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

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

18. How to detect fermion pairing
Quantum noise analysis of TOF images
is more than HBT interference

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

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

21. Fermion pairing in an optical lattice
Second Order Interference
In the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing

22. Current experiments

23. Fermions in optical lattice. Next challenge: antiferromagnetic stateTN U
Mott
current
experiments

24. 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)

25. Fermions in optical lattice. Next challenge: antiferromagnetic stateTN U
Mott
current
experiments

26. Negative U Hubbard model
Phase diagram:
Micnas et al.,
Rev. Mod. Phys.,1990

27. Hubbard model with attraction: anomalous expansion
Hackermuller et al., Science 2010
Constant Entropy
Constant Particle Number
Constant Confinement

28. Anomalous expansion: Competition of energy and entropy

29. Lattice modulation experiments with fermions in optical lattice.
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)

30. 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

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

32. 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

33. “Low” Temperature d h = + Self-consistent Born approximation
Schwinger bosons Bose condensed d h
Propagation of holes and doublons strongly
affected by interaction with spin waves
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 t2/U, weight by t/U
Incoherent part:
dispersion
Oscillations reflect shake-off
processes of spin waves

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

35. “High” Temperature d h
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, (2009)
Spectral Fn. of single hole

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

37. Fermions in optical lattice. Decay of repulsively bound pairs
Ref: N. Strohmaier et al., PRL 2010

38. Fermions in optical lattice. Decay of repulsively bound pairs
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
Magnestism is a goal but we have problems
Decay of doublon into a pair of quasiparticles requires creation of many particle-hole pairs

39. Fermions in optical lattice. Decay of repulsively bound pairs
Experiments: N. Strohmaier et. al.

40. Relaxation of doublon- hole pairs in the Mott state
Energy U needs to be
absorbed by
spin excitations
Relaxation requires
creation of ~U2/t2
spin excitations
Energy carried by
spin excitations
~ J =4t2/U
Relaxation rate
Very slow, not relevant for ETH experiments

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

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

43. 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

44. Why understanding doublon decay rate is important
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

45. Surprises of dynamics
in the Hubbard model

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

47. Competition between pairing and ferromagnetic instabilities in
ultracold Fermi gases near
Feshbach resonances
arXiv:
D. Pekker, M. Babadi, R. Sensarma, N. Zinner,
L. Pollet, M. Zwierlein, E. Demler

48. 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
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); …

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

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

51. Feshbach resonance
Interactions between atoms are intrinsically attractive
Effective repulsion appears due to low energy bound states
Example:
scattering length
V(x)
V0 tunable by the magnetic field
Can tune through bound state

52. 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 ?

53. Many-body instabilities
Imaginary frequencies of collective modes
Magnetic Stoner instability
Pairing instability = +
+ …

54. Many body instabilities near Feshbach resonance: naïve picture
Pairing (BCS)
Stoner (BEC)
EF=

55. Pairing instability regularized
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

56. 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

57. 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.

58. Stoner instability
RPA spin susceptibility
Interaction = Cooperon

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

60. Pairing instability vs experiments

61. Summary
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

63. 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
Kanamori’s counter-argument: renormalization of U.
then
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)

64. Experiments were
done dynamically.
What are implications
of dynamics?
Why spin domains could
not be observed?
Earlier work by C. Salomon et al., 2003

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

66. Feshbach resonance
Interactions between atoms are intrinsically attractive
Effective repulsion appears due to low energy bound states
Example:
scattering length
V(x)
V0 tunable by the magnetic field
Can tune through bound state

67. 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 ?

68. Pair formation

69. Two-particle scattering in vacuumk -k p -p
Microscopic Hamiltonian
Schrödinger equation

70. T-matrix Lippman-Schwinger equationk -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

71. Cooperon Two particle scattering in the presence of a Fermi sea p k -k
Need to make sure that we do not
include interaction effects on the Fermi
liquid state in scattered state energy

72. Cooperon
Grand canonical ensemble
Define
Cooperon equation

73. Cooperon vs T-matrix k -k
-p’ -p p p’

74. Cooper channel response function
Linear response theory
Induced pairing field
Response function
Poles of the Cooper channel response function
are given by

75. Cooper channel response function
Linear response theory
Poles of the response function, ,
describe collective modes
Time dependent dynamics
When the mode frequency has negative imaginary part,
the system is unstable

76. 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

77. 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

78. Pairing instability Time dependent variational wavefunction
Time dependence of uk(t) and vk(t) due to DBCS(t)
For small DBCS(t):

79. Pairing instability From wide to Effects of finite narrow resonances
temperature
Three body recombination
as in Shlyapnikov et al., 1996;
Petrov, 2003; Esry 2005

80. Magnetic instability

81. 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

82. RPA analysis for Stoner instability
Self-consistent equation on response function
Spin susceptibility for
non-interacting gas
RPA expression for
the spin response function

83. Quench dynamics across Stoner instability
Stoner criterion
For U>Uc unstable collective modes
Unstable modes determine
characteristic lengthscale
of magnetic domains

84. Stoner quench dynamics in D=3
Unphysical divergence
of the instability rate at unitarity

85. 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.

86. Stoner instability
RPA spin susceptibility
Interaction = Cooperon

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

88. Relation to experiments

89. Pairing instability vs experiments

90. Schwinger bosons and Slave Fermions
Constraint :
Singlet Creation
Boson Hopping

91. Schwinger bosons and slave fermions
Fermion hopping
Propagation of holes and doublons is coupled to spin excitations.
Neglect spontaneous doublon production and relaxation.
Doublon production due to lattice modulation perturbation
Second order perturbation theory. Number of doublons

92. Propogation of doublons and holes
Spectral function:
Oscillations reflect shake-off processes of spin waves
Comparison of Born approximation and exact diagonalization: Dagotto et al.
Hopping creates string of altered spins: bound states