1. Hubbard model(s) Eugene Demler Harvard University Collaboration with
E. Altman (Weizmann), R. Barnett (Caltech),
A. Imambekov (Yale), A.M. Rey (JILA), D. Pekker,
R. Sensarma, M. Lukin, and many others
Collaborations with experimental groups of
I. Bloch, T. Esslinger
$$ NSF, AFOSR, MURI, DARPA,

2. Outline Bose Hubbard model. Superfluid and Mott phases
Extended Hubbard model: CDW and Supersolid states
Two component Bose Hubbard model: magnetic superexchange
interactions in the Mott states
Bose Hubbard model for F=1 bosons: exotic spin states
Fermi Hubbard model: competing orders
Hubbard model beyond condensed matter paradigms:
nonequilibrium many-body quantum dynamics

3. Bose Hubbard model Atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001);
Greiner et al., Nature (2001);
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004);
many more …

4. Bose Hubbard model U t tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well

5. Bose Hubbard model. Mean-field phase diagram
M.P.A. Fisher et al.,
PRB40:546 (1989)
N=3
Mott 4
Superfluid
N=2
Mott 2
N=1
Mott
Superfluid phase
Weak interactions
Mott insulator phase
Strong interactions

6. Optical lattice and parabolic potential4
N=2 MI SF 2
N=1 MI
Jaksch et al.,
PRL 81:3108 (1998)

7. Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)
t/U
Superfluid
Mott insulator
What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition.
Explain experiment.
SF well described by a wavefunction in which the zero momentum state is macroscopically occupied.
Mott state is rather well described by definite RS occupations.

8. Shell structure in optical lattice
S. Foelling et al., PRL 97: (2006)
Observation of spatial
distribution of lattice sites
using spatially selective
microwave transitions
and spin changing collisions
n=1
n=2
superfluid regime
Mott regime

9. Extended Hubbard model Charge Density Wave and Supersolid phases

10. Extended Hubbard Model
- nearest neighbor repulsion
- on site repulsion
Checkerboard phase:
Crystal phase of bosons.
Breaks translational symmetry

11. Extended Hubbard model. Mean field phase diagram
van Otterlo et al., PRB 52:16176 (1995)
Hard core bosons.
Supersolid – superfluid phase with broken translational symmetry

12. Extended Hubbard model. Quantum Monte Carlo study
Hebert et al., PRB 65:14513 (2002)
Sengupta et al., PRL 94: (2005)

13. Dipolar bosons in optical lattices
Goral et al., PRL88: (2002)

14. Two component Bose Hubbard model. Magnetism

15. Two component Bose mixture in optical lattice
Example: Mandel et al., Nature 425:937 (2003) t t
Two component Bose Hubbard model

16. Quantum magnetism of bosons in optical lattices
Kuklov and Svistunov, PRL (2003)
Duan et al., PRL (2003)
Ferromagnetic
Antiferromagnetic

17. Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state

18. Two component Bose mixture in optical lattice
Two component Bose mixture in optical lattice. Mean field theory + Quantum fluctuations
Altman et al., NJP 5:113 (2003)
Hysteresis
1st order
2nd order line

19. using cold atoms in an optical lattice
Realization of spin liquid
using cold atoms in an optical lattice
Duan et al. PRL 91:94514 (2003)
Kitaev model
H = - Jx S six sjx - Jy S siy sjy - Jz S siz sjz
Ground state has topological order
Excitations are Abelian or non-Abelian anyons

20. Observation of superexchange in a double well potential
Theory: A.M. Rey et al., PRL 99: (2008) J J
Use magnetic field gradient to prepare a state
Observe oscillations between and states

21. Preparation and detection of Mott states
of atoms in a double well potential

22. Comparison to the Hubbard model
Experiments: S. Trotzky et al., Science 319:295 (2008)

23. Spin F=1 bosons in optical lattices
Spin exchange interactions.
Exotic spin orders (nematic, valence bond solid)

24. Spinor condensates in optical traps
Spin symmetric interaction of F=1 atoms
Ferromagnetic Interactions for
Antiferromagnetic Interactions for

25. Antiferromagnetic spin F=1 atoms in optical lattices
Hubbard Hamiltonian
Demler, Zhou, PRL (2003)
Symmetry constraints
Nematic Mott Insulator
Spin Singlet Mott Insulator

26. Nematic insulating phase for N=1
Effective S=1 spin model
Imambekov et al., PRA (2003)
When
the ground state is nematic in d=2,3.
One dimensional systems are dimerized: Rizzi et al., PRL (2005)

27. Fermionic Hubbard model
P.W. Anderson (1950)
J. Hubbard (1963) U t

28. 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, White, Scalapino, Sachdev, …)
Mott state without spin order. Dynamical Mean Field Theory
(Kotliar, Georges,…)
d-wave pairing
(Scalapino, Pines,…)
d-density wave (Affleck, Marston, Chakravarty, Laughlin,…)

29. 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

30. 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

31. 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

32. Doped Hubbard model

33. Attraction between holes in the Hubbard model
Loss of superexchange
energy from 8 bonds
Loss of superexchange
energy from 7 bonds

34. 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)

35. 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

36. 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)

37. Stripe phases in ladders
t-J model
DMRG study of
t-J model on ladders
Scalapino, White, PRL 2003

38. 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

39. Fermionic Hubbard model
From high temperature superconductors to ultracold atoms
Antiferromagnetic and superconducting Tc
of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and
pairing at sub-micro Kelvin
temperatures

40. Fermionic atoms in optical latticesU
Noninteracting fermions in optical lattice, Kohl et al., PRL 2005

41. 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)

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

43. Nonequilibrium dynamics of the Hubbard model
Nonequilibrium dynamics of the Hubbard model. Decay of repulsively bound pairs

44. Relaxation of repulsively bound pairs in the Fermionic Hubbard model
U >> t
For a repulsive bound pair to decay,
energy U needs to be absorbed
by other degrees of freedom in the system
Relaxation timescale is important for quantum
simulations, adiabatic preparation

45. Fermions in optical lattice. Decay of repulsively bound pairs
Experimets: T. Esslinger et. al.

46. 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 Relaxation

47. 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

48. Doublon decay in a compressible state
Perturbation theory to order n=U/t
Decay probability
To calculate the rate: consider processes which maximize the number of particle-hole excitations

49. Doublon decay in a compressible state
Single fermion
hopping
Doublon decay
Doublon-fermion
scattering
Fermion-fermion
scattering due to
projected hopping

50. Doublon decay in a compressible state
Doublon decay with generation of particle-hole pairs
Theory: R. Sensarma, D. Pekker, et. al.

51. Summary Bose Hubbard model. Superfluid and Mott phases
Extended Hubbard model: CDW and Supersolid states
Two component Bose Hubbard model: magnetic superexchange
interactions in the Mott states
Bose Hubbard model for F=1 bosons: exotic spin states
Fermi Hubbard model: competing orders
Hubbard model beyond condensed matter paradigms:
nonequilibrium many-body quantum dynamics
Harvard-MIT