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