Hubbard model(s) Eugene Demler Harvard University $$ NSF, AFOSR, MURI, DARPA, Collaborations with...
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Transcript of Hubbard model(s) Eugene Demler Harvard University $$ NSF, AFOSR, MURI, DARPA, Collaborations with...
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)
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 antiferromagnetismin 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
Q=(,)
Fermi surface for n=1 Analysis of spin instabilities.Random Phase Approximation
Nesting of the Fermi surface leads to singularity
BCS-type instability for weak interaction
Hubbard model at half filling
U
TN
paramagneticMott phase
Paramagnetic Mott phase:
one fermion per sitecharge fluctuations suppressedno spin order
BCS-typetheory applies
Heisenbergmodel 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-waveScalapino et al, PRB (1986)
k’
k
-k’
-kspinfluctuation
Pairing of holes in the Hubbard model
Q
BCS equation for pairing amplitude
k’
k
-k’
-kspinfluctuation
Systems close to AF instability:
(Q) is large and positive
k should change sign for k’=k+Q
++-
-
dx2-y2
Stripe phases in the Hubbard model
Stripes:Antiferromagnetic domainsseparated by hole rich regions
Antiphase AF domainsstabilized by stripe fluctuations
First evidence: Hartree-Fock calculations. Schulz, Zaannen (1989)
Stripe phases in ladders
DMRG study of t-J model on laddersScalapino, White, PRL 2003
t-J model
Possible Phase Diagram
doping
T
AF
D-SCSDW
pseudogap
n=1
After several decades we do not yet know the phase diagram
AF – antiferromagneticSDW- Spin Density Wave(Incommens. Spin Order, Stripes)D-SC – d-wave paired
Antiferromagnetic and superconducting Tc of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro Kelvin temperatures
Fermionic Hubbard modelFrom high temperature superconductors to ultracold atoms
t
U
t
Fermionic atoms in optical lattices
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 measurementsU. Schneider et al., Science (2008)
Fermions in optical lattice. Next challenge: antiferromagnetic state
TN
U
Mott
currentexperiments
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 absorbedby 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.
Energy carried by
spin excitations ~ J =4t2/U
Relaxation requires creation of ~U2/t2
spin excitations
Relaxation of doublon hole pairs in the Mott state
Relaxation rate
Very slow Relaxation
Energy U needs to be absorbed by spin excitations
Doublon decay in a compressible state
Excess energy U isconverted to kineticenergy of single atoms
Compressible state: Fermi liquid description
Doublon can decay into apair of quasiparticles with many particle-hole pairs
Up-p
p-h
p-h
p-h
Doublon decay in a compressible state
To calculate the rate: consider processes which maximize the number of particle-hole excitations
Perturbation theory to order n=U/tDecay probability
Doublon decay in a compressible state
Doublon decay
Doublon-fermion scattering
Doublon
Single fermion hopping
Fermion-fermion scattering due toprojected hopping
Doublon decay in a compressible state
Doublon decay with generation of particle-hole pairsTheory: R. Sensarma, D. Pekker, et. al.
SummaryBose 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