Hubbard models Eugene Demler Harvard University Collaboration with

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 M. P. A. Fisher et al. , PRB 40: 546 (1989) N=3 Mott 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 N=3 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) Mott insulator Superfluid 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. , PRL 88: 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 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 1 st order 2 nd or de rl in e

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’ + + - dx 2 -y 2 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 v. Energy carried by spin excitations ~J =4 t 2/U v Relaxation requires creation of ~U 2/t 2 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 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