Hubbard model(s) Eugene Demler Harvard University $$ NSF, AFOSR, MURI, DARPA, llaborations with experimental groups of Bloch, T. Esslinger llaboration with Altman (Weizmann), R. Barnett (Caltech), Imambekov (Yale), A.M. Rey (JILA), D. Pekker, Sensarma, M. Lukin, and many others
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Hubbard model(s) Eugene Demler Harvard University $$ NSF, AFOSR, MURI, DARPA, Collaborations with experimental groups of I. Bloch, T. Esslinger Collaboration.
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Hubbard model(s)
Eugene Demler Harvard University
$$ NSF, AFOSR, MURI, DARPA,
Collaborations with experimental groups of I. Bloch, T. Esslinger
Collaboration with E. Altman (Weizmann), R. Barnett (Caltech),A. Imambekov (Yale), A.M. Rey (JILA), D. Pekker, R. Sensarma, M. Lukin, and many others
OutlineBose 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
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 …
Bose Hubbard model
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
U
t
4
Bose Hubbard model. Mean-field phase diagram
1n
U
02
0
M.P.A. Fisher et al.,PRB40:546 (1989)
MottN=1
N=2
N=3
Superfluid
Superfluid phase
Mott insulator phase
Weak interactions
Strong interactions
Mott
Mott
Optical lattice and parabolic potential
41n
U
2
0
N=1
N=2
N=3
SF
MI
MI
Jaksch et al., PRL 81:3108 (1998)
Superfluid to Insulator transitionGreiner et al., Nature 415:39 (2002)
U
1n
t/U
SuperfluidMott insulator
Shell structure in optical latticeS. Foelling et al., PRL 97:060403 (2006)
Observation of spatial distribution of lattice sites using spatially selective microwave transitions and spin changing collisions
superfluid regime Mott regime
n=1
n=2
Extended Hubbard modelCharge Density Wave and Supersolid phases
Extended Hubbard Model
- on site repulsion - nearest neighbor repulsion
Checkerboard phase:
Crystal phase of bosons. Breaks translational symmetry
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
Extended Hubbard model. Quantum Monte Carlo study
Sengupta et al., PRL 94:207202 (2005)Hebert et al., PRB 65:14513 (2002)
Dipolar bosons in optical lattices
Goral et al., PRL88:170406 (2002)
Two component Bose Hubbard model.
Magnetism
t
t
Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)
Two component Bose Hubbard model
Quantum magnetism of bosons in optical lattices
Duan et al., PRL (2003)
• Ferromagnetic• Antiferromagnetic
Kuklov and Svistunov, PRL (2003)
Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
Two component Bose mixture in optical lattice.Mean field theory + Quantum fluctuations
2 nd order line
Hysteresis
1st order
Altman et al., NJP 5:113 (2003)
Ground state has topological orderExcitations are Abelian or non-Abelian anyons
Realization of spin liquid using cold atoms in an optical lattice Duan et al. PRL 91:94514 (2003)
H = - Jx ix j
x - Jy iy j
y - Jz iz j
z
Kitaev model
J
J
Use magnetic field gradient to prepare a state
Observe oscillations between and states
Observation of superexchange in a double well potentialTheory: A.M. Rey et al., PRL 99:140601 (2008)
Preparation and detection of Mott statesof atoms in a double well potential
Comparison to the Hubbard modelExperiments: S. Trotzky et al., Science 319:295 (2008)
Spin F=1 bosons in optical lattices
Spin exchange interactions. Exotic spin orders (nematic, valence bond solid)
Spinor condensates in optical traps
Spin symmetric interaction of F=1 atoms
Antiferromagnetic Interactions for
Ferromagnetic Interactions for
Antiferromagnetic spin F=1 atoms in optical lattices
Hubbard Hamiltonian
Symmetry constraints
Demler, Zhou, PRL (2003)
Nematic Mott Insulator
Spin Singlet Mott Insulator
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)
Fermionic Hubbard model
U
tt
P.W. Anderson (1950)J. Hubbard (1963)
Fermionic Hubbard modelPhenomena predicted
Superexchange and antiferromagnetism (P.W. Anderson)