Novel orbital physics with cold atoms in optical lattices Congjun Wu Department of Physics, UC San Diego C. Wu, arXiv:0805.3525, to appear in PRL. Shizhong Zhang and C. Wu, arXiv:0805.3031. V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301(2008). C. Wu, PRL 100, 200406 (2008). C. Wu, and S. Das Sarma, PRB 77, 235107 (2008). C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007). C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006). W. V. Liu and C. Wu, PRA 74, 13607 (2006). Good morning, everyone. It is my great pleasure to take this opportunity to discuss my future research plan with the committee and learn your advice. First let me thank all the committee for your time and energy for my interview and application. Oct 22, 2008, UCLA.

Collaborators L. Balents UCSB D. Bergman Yale S. Das Sarma Univ. of Maryland W. V. Liu Univ. of Pittsburg W. C. Lee, H. H. Hung UCSD I. Mondragon Shem Let me thank my collaborators. J. Moore Berkeley Shi-zhong Zhang UIUC Many thanks to I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C. Zhang for helpful discussions.

Outline Introduction to orbital physics. New directions of cold atoms: orbital physics in high-orbital bands; pioneering experiments. Bosons: exotic condensate, complex-superfluidity breaking time-reversal symmetry. Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices Orbital exchange, frustrations, order from disorder, orbital liquid?.

Bose-Einstein condensation Bosons in magnetic traps: dilute and weakly interacting systems. M. H. Anderson et al., Science 269, 198 (1995) In the past decade, there is a rapid progress cold atomic physics. The first landmark in the realization of the BEC using cold atoms in the magnetic traps. This is a dilute weakly interacting system.

New era of cold atom physics: optical lattices Strongly correlated systems. Interaction effects are tunable by varying laser intensity. After that, important progress has been made. In particular, optical lattices provide a wonderful opportunity to study strongly correlated systems. Optical lattices are artificial lattices made by several counter-progagating t : inter-site tunneling U: on-site interaction

Superfluid-Mott insulator transition t<<U t>>U After that, important progress has been made. In particular, optical lattices provide a wonderful opportunity to study strongly correlated systems. Greiner et al., Nature (2001).

Research focuses of cold atom physics Great success of cold atom physics in the past decade: BEC, superfluid-Mott insulator transition, fermion superfluidity and BEC-BCS crossover … … New focus: novel strong correlation phenomena which are NOT easily accessible in solid state systems. New physics of bosons and fermions in high-orbital bands. In short, the past decade has witnessed the great success of cold atom physics, including the. It is natural to expect even more important Good timing: pioneering experiments; square lattice (Mainz); double well lattice (NIST); quasi 1D polariton lattice (stanford). J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006); T. Mueller et al., Phys. Rev. Lett. 99, 200405 (2007); C. W. Lai et al., Nature 450, 529 (2007).

Orbital physics Orbital: a degree of freedom independent of charge and spin. Tokura, et al., science 288, 462, (2000). Orbital band degeneracy and spatial anisotropy. cf. transition metal oxides (d-orbital bands with electrons). Orbital physics plays an important role in condensed matter physics, in particular in the transition metal oxides. The orbital physics is characterized by the degeneracy of the Orbital bands, and spatial anisotropy. For example, for the d-orbitals, it has five bands Each of them has distinct spatial orientations. La1-xSr1+xMnO4 LaOFeAs

Advantages of optical lattice orbital system Solid state orbital systems: Jahn-Teller distortion quenches orbital degree of freedom; only fermions; correlation effects in p-orbitals are weak. Optical lattices orbital systems: rigid lattice free of distortion; both bosons (meta-stable excited states with long life time) and fermions; strongly correlated px,y-orbitals: stronger anisotropy s-bond p-bond Orbital physics plays an important role in condensed matter physics, in particular in the transition metal oxides. The orbital physics is characterized by the degeneracy of the Orbital bands, and spatial anisotropy. For example, for the d-orbitals, it has five bands Each of them has distinct spatial orientations.

Pumping bosons by Raman transition Long life-time: phase coherence. Quasi-1d feature in the square lattice. Our current knowledge does not go beyond the original simple argument by Stoner. Hubbard model for fermions was originally proposed to explain ferromagnetism. After Many years of numerical studies, it is now believed that the Hubbard model does Not have ferromagnetism. And Hubbard model is really the simplest example. When we take systems with more realistic long range interactions, we do not Really know when such mean-field argument applies. For example, there Is currently a lot of controversy and debates regarding possible ferromagnetism in 2d electron gas at low density. T. Mueller, I. Bloch et al., Phys. Rev. Lett. 99, 200405 (2007).

Lattice polariton condensation in the p-orbital C. W. Lai et al, Nature 450, 529 (2007) Quasi 1D polariton lattice by deposing metallic strips. Condensates at both s-orbital (k=0) and p-orbital (k=p) states.

Outline Introduction. Bosons: complex-superfluidity breaking time-reversal symmetry. New condensates different from Feynman’s argument of the positive-definitiveness of ground state wavefunctions. C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006). W. V. Liu and C. Wu, PRA 74, 13607 (2006). Other group’s related work: V. W. Scarola et. al, PRL, 2005; A. Isacsson et. al., PRA 2005; A. B. Kuklov, PRL 97, 2006; C. Xu et al., cond-mat/0611620 . This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects Orbital exchange, frustrations, order from disorder, orbital liquid?.

Feynman’s celebrated argument The many-body ground state wavefunctions (WF) of boson systems in the coordinate-representation are positive-definite in the absence of rotation. This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices Strong constraint: complex-valued WF  positive definite WF; time-reversal symmetry cannot be broken. Feynman’s statement applies to all of superfluid, Mott-insulating, super-solid, density-wave ground states, etc.

New states of bosons beyond Feynman’s argument How to bypass this argument to obtain complex-valued many-body wavefunctions of bosons and spontaneous breaking of time-reversal symmetry? Solution 1: Excited (meta-stable) states of bosons: bosons in higher orbital bands ---- orbital physics of bosons. Solution 2: Feynman’s argument does not apply to Hamiltonians linearly dependent on momentum. Ground states of spinful bosons with spin-orbit coupling (e.g. excitons). This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices C. Wu and I. Mondragon-Shem, ariXiv:0809.3532.

Ferro-orbital interaction for spinless p-orbital bosons A single site problem: two orbitals px and py with two spinless bosons. polar This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices axial

Orbital Hund’s rule for p-orbital spinless bosons Bosons go into one axial state to save repulsive interaction energy, thus maximizing orbital angular momentum. Axial states (e.g. p+ip) are spatially more extended than polar states (e.g. px or py). polar axial This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices cf. Hund’s for electrons; p+ip superconductors.

“Complex” superfluidity with time-reversal symmetry breaking Staggered ordering of orbital angular momentum moment in the square lattice. Next I will first prove this symmetry and present its physical consequences. Time of flight (zero temperature): Bragg peaks located at fractional values of reciprocal lattice vectors. W. V. Liu and C. Wu, PRA 74, 13607 (2006).

“Complex” superfluidity with time-reversal symmetry breaking Stripe ordering of orbital angular momentum moment in the triangular lattice. Next I will first prove this symmetry and present its physical consequences. Each site behaves like a vortex with long range interaction in the superfluid state. Stripe ordering to minimize the global vorticity. C. Wu, W. V. Liu, J. Moore and S. Das Sarma, Phy. Rev. Lett. 97, 190406 (2006).

Strong coupling analysis Ising variable for vortex vorticity: The minimum of the effective flux per plaquette is . The stripe pattern minimizes the ground state vorticity. cf. The same analysis also applies to p+ip Josephson junction array.

Staggered plaquette orbital moment cf. d-density-wave state in high Tc cuprate: Sudip Chakravarty, R. B. Laughlin, Dirk K. Morr, and Chetan Nayak, Phys. Rev. B 63, 094503 (2001)  

Digression: Exciton with Rashba SO coupling in 2D quantum well Excitons: bound states of electrons and holes (heavy). Effective bosons at low densities below the Mott-limit. The SO coupling still survives in the center-of-mass motion. Let us consider the sector with fixed heavy hole spin. Next I will first prove this symmetry and present its physical consequences. Kramer doublet:

Ground state condensates: Kramer doublet A harmonic trap: SO length scale: Half-quantum vortex and spin skyrmion texture configuration in the strong SO coupling limit. Next I will first prove this symmetry and present its physical consequences.

Outline Introduction. Bosons: Complex-superfluidity beyond Feynman’s argument. Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects. Shizhong Zhang and C. Wu, arXiv:0805.3031. C. Wu, and S. Das Sarma, PRB 77, 235107 (2008). C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007). This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices Orbital exchange, frustrations, order from disorder, orbital liquid?.

p-orbital fermions in honeycomb lattices cf. graphene: a surge of research interest; pz-orbital; Dirac cones. pxy-orbital: flat bands; interaction effects dominate. C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 70401 (2007).

px, py orbital physics: why optical lattices? pz-orbital band is not a good system for orbital physics. isotropic within 2D; non-degenerate. Interesting orbital physics in the px, py-orbital bands. 1s 2s 2p 1/r-like potential However, in graphene, 2px and 2py are close to 2s, thus strong hybridization occurs. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) In optical lattices, px and py-orbital bands are well separated from s. p s

Honeycomb optical lattice with phase stability Three coherent laser beams polarizing in the z-direction. G. Grynberg et al., Phys. Rev. Lett. 70, 2249 (1993).

Artificial graphene in optical lattices Band Hamiltonian (s-bonding) for spin- polarized fermions. A B Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte)

Flat bands in the entire Brillouin zone! Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) Flat band + Dirac cone. localized eigenstates. If p-bonding is included, the flat bands acquire small width at the order of . Realistic band structures show p-bond

Realistic Band structure with the sinusoidal optical potential Excellent band flatness. px,y-orbital bands s-orbital bands Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte)

Hubbard model for spinless fermions: Exact solution: Wigner crystallization in Hubbard gapped state Close-packed hexagons; avoiding repulsion. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) The crystalline ordered state is stable even with small . Particle statistics is irrelevant. The result is also good for bosons.

Orbital ordering with strong repulsions Various orbital ordering insulating states at commensurate fillings. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) Dimerization at <n>=1/2! Each dimer is an entangled state of empty and occupied states.

Flat-band ferromagnetism (FM) FM requires strong repulsion to overcome kinetic energy cost, and thus has no well-defined weak coupling picture. Hubbard-type models cannot give FM unless with the flat band structure. However, flat band models suffer the stringent condition of fine-tuned long range hopping, thus are difficult to realize. A. Mielke and H. Tasaki, Comm. Mat. Phys 158, 341 (1993). In spite of its importance, FM has not been paid much attention in cold atom community because strong repulsive interaction renders system unstable to the dimer-molecule formation. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) We propose the realistic flat-band ferromagnetism in the p-orbital honeycomb lattices. Interaction amplified by the divergence of DOS. Realization of FM with weak repulsive interactions in cold atom systems. Shizhong Zhang and C. Wu, arXiv:0805.3031; .

Outline Introduction. Bosons: new states of matter beyond Feynman’s argument of the positive-definitiveness of ground state wavefunctions. Complex-superfluidity breaking time-reversal symmetry. Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices Orbital exchange and frustrations, order from disorder; orbital liquid? C. Wu, PRL 100, 200406 (2008).

Mott-insulators with orbital degrees of freedom: orbital exchange of spinless fermion Pseudo-spin representation. No orbital-flip process. Exchange is antiferro-orbital Ising. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte)

Hexagon lattice: quantum model For a bond along the general direction . : eigen-states of After a suitable transformation, the Ising quantization axes can be chosen just as the three bond orientation. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) cf. Kitaev model.

Large S picture: heavy-degeneracy of classic ground states Ground state constraint: the two t-vectors have the same projection along the bond orientation. or Ferro-orbital configurations. Oriented loop config: t-vectors along the tangential directions. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte)

Global rotation degree of freedom Each loop config remains in the ground state manifold by a suitable arrangement of clockwise/anticlockwise rotation patterns. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) Starting from an oriented loop config with fixed loop locations but an arbitrary chirality distribution, we arrive at the same unoriented loop config by performing rotations with angles of .

“Order from disorder”: 1/S orbital-wave correction Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte)

Zero energy flat band orbital fluctuations Each un-oriented loop has a local zero energy model up to the quadratic level. The above config. contains the maximal number of loops, thus is selected by quantum fluctuations at the 1/S level. Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information. Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows Computer drives capable of storing a trillion bytes (terabyte) Project under investigation: the quantum limit (s=1/2)? A very promising system to arrive at orbital liquid state?

Summary Orbital Hund’s rule and complex superfludity of spinless bosons px,y-orbital counterpart of graphene: flat band and Wigner crystalization. Orbital frustration. This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices

More work Orbital analogue of anomalous quantum Hall effect – topological insulators in the p- band. C. Wu, arXiv:0805.3525, to appear in PRL. Incommensurate superfluidity in the double-well lattice of NIST. This the outline of the topic. First I will give an introduction to the rapid progress of the experiments of cold atoms in optical lattices. Then I will show that currently it is a good time to further study the orbital physics in the optical lattices. I will also give a brief introduction to orbital There are already several important pioneering experiment Doing this. Orbital physics in optical lattices exhibit many new features compare Next we will present our theoretical work in this direction. Including fermions in the p_{xy} orbital bands of honeycomb lattices V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301 (2008).