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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,
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
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
U
t
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
Bose Hubbard model. Mean-field phase diagram
 U
M.P.A. Fisher et al.,
PRB40:546 (1989)
N=3 Mott
n 1
4
0 N=2
2
N=1
Mott
Superfluid
Mott
0
Superfluid phase
Weak interactions
Mott insulator phase
Strong interactions
Optical lattice and parabolic potential
 U
N=3
n 1
4
N=2 MI
2
N=1
MI
0
Jaksch et al.,
PRL 81:3108 (1998)
SF
Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)

U
Mott insulator
Superfluid
n 1
t/U
Shell structure in optical lattice
S. Foelling et al., PRL 97:060403 (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
Extended Hubbard model
Charge 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
Hebert et al., PRB 65:14513 (2002)
Sengupta et al., PRL 94:207202 (2005)
Dipolar bosons in optical lattices
Goral et al., PRL88:170406 (2002)
Two component
Bose Hubbard model.
Magnetism
Two component Bose mixture in optical lattice
Example:
. Mandel et al., Nature 425:937 (2003)
t
t
Two component Bose Hubbard model
Quantum magnetism of bosons in optical lattices
Kuklov and Svistunov, PRL (2003)
Duan et al., PRL (2003)
• Ferromagnetic
• Antiferromagnetic
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
Altman et al., NJP 5:113 (2003)
Hysteresis
1st order
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
Observation of superexchange in a double well potential
Theory: A.M. Rey et al., PRL 99:140601 (2008)
J
J
Use magnetic field gradient to prepare a state
Observe oscillations between
and
states
Preparation and detection of Mott states
of atoms in a double well potential
Comparison to the Hubbard model
Experiments: 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
Ferromagnetic Interactions for
Antiferromagnetic Interactions for
Antiferromagnetic spin F=1 atoms in optical lattices
Hubbard Hamiltonian
Demler, Zhou, PRL (2003)
Symmetry constraints
Nematic Mott Insulator
Spin Singlet Mott Insulator
Nematic insulating phase for N=1
Effective S=1 spin model
When
Imambekov et al., PRA (2003)
the ground state is nematic in d=2,3.
One dimensional systems are dimerized: Rizzi et al., PRL (2005)
Fermionic Hubbard model
P.W. Anderson (1950)
J. Hubbard (1963)
U
t
t
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,…)
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
Hubbard model at half filling
TN
Paramagnetic Mott phase:
paramagnetic
Mott phase
one fermion per site
charge fluctuations suppressed
no spin order
U
BCS-type
theory applies
Heisenberg
model applies
Doped Hubbard model
Attraction between holes
in the Hubbard model
Loss of superexchange
energy from 8 bonds
Loss of superexchange
energy from 7 bonds
Pairing of holes
in the Hubbard model
Non-local
pairing
of holes
Leading istability:
d-wave
Scalapino et al, PRB (1986)
-k’
k’
k
spin
fluctuation
-k
Pairing of holes
in the Hubbard model
BCS equation for pairing amplitude
Q
-k’
k’
-
+
+
-
dx2-y2
k
spin
fluctuation
-k
Systems close to AF instability:
c(Q) is large and positive
Dk 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)
Stripe phases in ladders
t-J model
DMRG study of
t-J model on ladders
Scalapino, White, PRL 2003
Possible Phase Diagram
T
AF – antiferromagnetic
SDW- Spin Density Wave
(Incommens. Spin Order, Stripes)
D-SC – d-wave paired
AF
pseudogap
SDW
n=1
D-SC
doping
After several decades we do not yet know the phase diagram
Fermionic Hubbard model
From high temperature superconductors to ultracold atoms
Atoms in optical lattice
Antiferromagnetic and
superconducting Tc
of the order of 100 K
Antiferromagnetism and
pairing at sub-micro Kelvin
temperatures
Fermionic atoms in optical lattices
U
t
t
Noninteracting fermions in optical lattice, Kohl et al., PRL 2005
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)
Fermions in optical lattice. Next challenge:
antiferromagnetic state
TN
current
experiments
Mott
U
Nonequilibrium dynamics
of the Hubbard model.
Decay of repulsively bound pairs
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
Fermions in optical lattice.
Decay of repulsively bound pairs
Experimets: T. Esslinger 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
=4t2/U
 Relaxation requires
creation of ~U2/t2
spin excitations
Relaxation rate
Very slow Relaxation
Doublon decay in a compressible state
Excess energy U is
converted to kinetic
energy of single atoms
Compressible state: Fermi liquid description
p -h
p -h
Doublon can decay into a
pair of quasiparticles with
many particle-hole pairs
U
p -h
p -p
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
Doublon decay in a compressible state
Doublon
Single fermion
hopping
Doublon decay
Doublon-fermion
scattering
Fermion-fermion
scattering due to
projected hopping
Doublon decay in a compressible state
Doublon decay with generation of particle-hole pairs
Theory: R. Sensarma, D. Pekker, et. al.
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