Novel Orbital Phases in Optical Lattices – Unconventional BEC and Itinerant F erromagnetism

1. Novel Orbital Phases in Optical Lattices – Unconventional BEC and Itinerant Ferromagnetism Congjun Wu Department of Physics, UC San Diego 06/26/2016, Zhejiang Univ.

2. Collaborators: Yi Li (UCSD  Princeton JohnsHopkins) Shenglong Xu (UCSD) ZiCai (UCSD Innsbruck Shanghai JT U) Ellioit H. Lieb (Princeton) Thank S. Das Sarma, L. Balents, W. V. Liu for early collaborations, and G. W. Chern,H. H. Hung, R. Scalettar, C. W. Zhang, M. C. Zhang, S. Z. Zhang for collaboration on related projects. Supported by NSF, AFOSR, CAIQUE

3. Outline • Introduction: cold atoms in high orbital bands. What is orbital physics in the condensed matter context? Why orbital physics with cold atoms is interesting? • Orbital Bosons (unconventional symmetry): BECs beyond the “no-node” theorem – prediction already observed! • Orbital Fermions: Itinerant FM, a long-standing problem – non-perturbative studies, to be tested in experiments. 3

4. s-bond p-bond Electronic orbital physics in solids • Orbital: a degree of freedom independent of charge and spin. • Two features: orbital degeneracy and spatial anisotropy. p-orbitals: d-orbitals: crystal field splitting Tokura, et al., science 288, 462, (2000).

5. Transition-metal oxide (TMO) orbital systems • Orbitals play an important role in magnetism, superconductivity, and transport properties. orbital charge spin lattice Iron-pnictide: LaOFeAs Manganite: La1-xSr1+xMnO4

6. New physics of bosons and fermions in high-orbital bands in optical lattices. A new direction: orbital physics in optical lattices Here orbital refers to the different energy levels (e.g. s, p) of each optical site. Good timing: pioneering experiments on orbital-bosons. Square lattice (Mainz); double well lattice (NIST, Hamburg); polariton lattice (Stanford) p s 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).

7. What’s is new? --- orbital bosons • Solid states: orbital physics only of fermions . • Optical lattices: orbital physics of bosons. meta-stable excited states of bosons Novel properties not shown in boson ground states – beyond the constraint by the “no-node” theorem. Unconventional BEC (non-s-wave symmetries, (e.g. , spontaneous time-reversal symmetry breaking) --- already seen in experiments. C. Wu, Mod. Phys. Lett. 23, 1 (2009) (review).

8. s-bond What’s new? --- Strongly correlated p-orbitals • Strongly correlated orbital systems: d-orbital TMO; f-orbital rare-earth compounds. • p-orbitals are weakly correlated (e.g. semiconductors). Not many p-orbital Mott-insulators and ferromagnets. • New materials: strong correlation + strong anisotropyin p-orbitals. Itinerant FM, topological states, unconventional Cooper pairing, etc

9. Outline • Introduction: cold atoms in high orbital bands. • Orbital bosons (unconventional symmetry): BECs beyond the “no-node” theorem – already observed! • Orbital fermions: Itinerant FM, a long-standing problem – a non-perturbative study. 9 9

10. The “No-node” theorem • Many-body ground-state wavefunctions of bosons are positive-definite. • A general property of the ground states: • Laplacian kinetic energy (no rotation). • Arbitrary single-particle potential (with lattice or not) . • Coordinate-dependent interactions. R. P. Feynman

11. Proof (Perron-Frobenius) • Single-particle: harmonic oscillator, hydrogen atom, etc. • Many-body generalization to bosons. But NOT fermions!

12. The “no-node” consequences • Valid for superfluid, Mott-insulating, super-solid states, etc. • Strong constraint! • Complex-valued wavefunctions reduce to positive-definite distr. • Time-reversal (TR) symmetry cannot be spontaneously broken! TR: • Search for boson condensate beyond the “no-node” paradigm. C. Wu, Mod. Phys. Lett. 23, 1 (2009).

13. Conventional superconductivity (SC) and BEC • Cooper pair wavefunctions (WF): isotropic • Conventional SC: s-wave Cooper pair WF. • Conventional BEC: s-wave condensate WF --- “no-node” theorem

14. Analogy: unconventional superconductivity • High partial wave channel Cooper pairings (e.g. p, d-wave …). • d-wave: high Tccuprates. • p-wave: Sr2RuO4, 3He-A and B. i - +1 -1 + + - -i D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995); C. C. Tsuei et al., Rev. Mod. Phys. 72, 969 (2000).

15. Unconventional BEC (UBEC) – metastable states • Definition: The condensate possesses a non-s-wave symmetry  Nodal lines or points beyond “no-node”. • Complex-valued  spontaneous time-reversal symmetry breaking. • Example: The p-orbital bands with degenerate band minima. C. Wu, brief review, Mod. Phys. Lett. 23, 1(2009). W. V. Liu and C. Wu, PRA 2006. p+ip UBEC:

16. Ferro-orbital interaction (spinless) • A single-site problem: two orbitals px and py with two bosons. polar axial • Axial states are spatially more extended  lower energy! C. Wu, Mod. Phys. Lett. 23, 1(2009). W. V. Liu and C. Wu, PRA 2006.

17. polar v.s. axial Bosonic version for the 2nd Hund’s rule • Maximize orbital angular momentum. • The axial state (e.g. p+ip) NOT the polar state (e.g. px) • c.f. Electronsfill in degenerate atomic orbitals. • 1st Hund’s rule  maximize total spin; • 2nd Hund’s rule  maximize total orbital angular momentum.

18. Already observed! Double-well lattice experiment • Condensate wavecters (: half values of reciprocal lattice vectors. Wirth, Oelschlaeger, Hemmerich, Nature Physics 7, 147 (2011).

19. Experiment double-well lattice – shallow (weak interaction) ZiCai, C. Wu, PRA, 84,033635 (2011) • Energy minima --- halves of reciprocal lattice vectors. • Real space: are standing waves (real) with nodal lines. • Nodal lines pass the centers of the deep A-sites (p-orbital). • The anti-node regions are on the shallow B-site (s-orbital).

20. Nodal points (complex) v.s.nodal lines (real) • Real : nodal lines • Complex : nodal points at crossings  more uniform (favored by interaction) • Phase winding: vortex-anti-vortex lattice. • Spontaneous TR symmetry breaking.

21. Lattice asymmetry and interaction • Asymmetry tuned by  favor real condensate or . interaction symmetric point • Real (nodal but TR invariant) v. s. complex condensates (TR breaking).

22. Quantum Monte-Carlo Phase Diagram (strong interaction) • Time-reversal symmetry and U(1) phase symmetry. time-reversal  staggered angular momentum (SAM) U(1) phase superfluidity/Mott, normal phase • The staggered orbital angular momentum order can survive in the Mott state. (TR invariant) (comp. BEC) (real BEC) Herbert, Cai, Rousseau Wu, Scallater, Batrouni, PRB 87, 224505 (2013) anisotropy

23. Symmetric (complex) to asymmetric (real, nodal) p-BECs Wirth, Oelschlaeger, Hemmerich, Nature Physics 7, 147 (2011). asymmetric symmetric Z. Cai and C. Wu, PRA 2011.

24. Phase-sensitive measurement (high ) • Corner-Josephson -junction for D. Van Harlingen, RMP (1995) YBCO - + + - YBCO FS - FS + + -

25. See the “i” -- Matter-wave interference Hemmerichgroup PRL 114, 115301 (2015). 25ms expansion Interference along the z-axis Upper space Z Lower space XY

26. Nature Physics , June 2011.. Solid state systems: real but nodal BECs with the symmetry.

27. Outline • Introduction: cold atoms in high orbital bands. • Orbital bosons (unconventional symmetry): px+ipy BECs beyond the “no-node” theorem – already observed! • Orbital Fermions (strong correlation): Itinerant FM, a long-standing problem – a Non-perturbative study. 27

28. The magnetic stone attracts iron. ---- Guiguzi (鬼谷子), (4th century BC) The early age of ferromagnetism (FM) 慈 (ci) 石(shi) 召(zhao) 铁(tie) 慈 (loving, kind): the original Chinese character for magnetism 磁: magnetism, magnetic World’s first compass司南: magnetic spoon: 1 century AD Thales says that a stone (lodestone) has a soul because it causes movement to iron. ----De Anima, Aristotle (384-322 BC) Magnetic declination： “Slightly eastward, not directly south(常微偏东,不全南也)”- Shen Kuo (沈括)(1031-1095)

29. Absence of magnetism in classic equilibrium states Bohr-van Leeuwen theorem: Landau diamagnetism Earth magnetic field: Non-equilibrium convection of electric current inside the earth. ? 29

30. Stoner criterion: E. C. Stoner U – average interaction strength; N0 – density of states at the Fermi level Origin of itinerant FM – fermion exchange • Fermi statistics  Slater determinant-like wavefunction direct exchange.

31. Fe Co Ni FM and superconductivity, which one is rarer? FM elements: Dy Gd gadolinium, dysprosium

32. Correlations! • Electrons with oppositespinscan still avoid each other!! • Correlated WFs (unpolarized)  less kinetic energy cost. • Need non-perturbative treatment! 1. Two electrons: always non-magnetic ground states. singlet triplet nodal (ground state) nodeless 2. Absence of FM in 1D – Lieb & Mattis theorem. 32

33. Itinerant FM of cold atoms? Not yet. • Two-component fermions . • Feshbach resonances to enhance repulsive interaction (s-wave scattering). At , roughly equivalent to the Stoner criterion  kinetic energy increases and 3-body loss decreases. • Consistent with strongly correlated paramagnet! • No evidence of magnetic domains • Our proposal: p-orbital band in optical lattices. G. B. Jo, W. Ketterleet al., Science 325, 1521(2009).

34. c.f. Hund’s coupling in transition metals 3d 3d 3d 3d • 1st Hund’s rule -- Electron spins add up when filling in degenerate orbitals due to exchange interactions. • Hund’s rule is local physics. Usually it cannot polarize the entire lattice. • Our result: Hund’s rule + quasi-1D bands  2D and 3D FM in the strong interaction regime. 34

35. -orbital bands • FM in p-orbital optical lattices through Guzwiller projection method. Wang et al, PRA 78, 023603(2008) • BiS2-based SC (-orbitals). Exp: [Bi4O4S3] Y. Mizuguchi et al, PRB 86, 220510 (2012), [LaO1-xFxBiS2] Y. Mizuguchi et al, J. Phys. Soc. Jpn. 81 (2012) 35

36. p-orbital bands (square lattice) s-bond p-bond • Quasi-1D band structure: ONLY keep -bonding . • c.f. the and -orbitals in 2D transition metal oxide layers. Yi Li, E. H. Lieb, C. Wu, Phys. Rev. Lett. 112, 217201 (2014) 36

37. Multi-orbital onsite (Hubbard) interactions • Intra-orbital repulsion . Intra-orbitalsinglet projected out • Inter-orbital Hund’s coupling J>0, and repulsion V. Inter-orbital singlet 3-fold triplet

38. The Orbital-assisted Itinerant FM • Theorem: The above models in the 2D square and 3D cubic lattices exhibit FM ground states at (fully polarized and unique up to 2Stot+1-fold spin degeneracy). • An entire FM phase: valid at any generic filling (0<n<2 for square lattice), any value for J>0, and V. Serve as a concrete reference point for the analytic and numeric study of FM in multi-orbital systems • Free of quantum Monte-Carlo (QMC) sign problemat any filling – a rare case for fermions. Yi Li, E. H. Lieb, Congjun Wu, Phys. Rev. Lett. 112, 217201 (2014) 38

39. Hund’s rule  global FM at generic fillings • Intra-chain physics at : infinite degeneracy regardless of spin configurations. • Inter-chain physics (J): Hund’s coupling lifts the degeneracy by aligning spins  global FM. ? • 2D FM coherence in spite of 1D band structure (the total spin in each chain is not conserved). 39

40. Open question: Thermodynamics & phase transitions • Curie-Weiss metal at . Local-moment-like behavior. Natural for Ising model, but not for metals with Fermi surfaces. • The paramagnetic phase is NOT simple: domain fluctuations! FS • Non-perturbative study – QMC simulations, asymptotically exact. (Aesthetics of brutal force：大巧不工)

41. The Curie-Weiss metal V=0, J=2 0.40 0.68 1.00 1.44 1.80 V=0, J=2 • Local moment-like: Curie-Weise (spin incoherent). • Metalic (itinerancy): satures at , is roughly the kinetic energy scale. 41

42. QMC: Curie-Weiss temperature v.s filling (V=0) V=0, J=2 Error bar Overestimate Tc • 0 at both n0 (particle vaccum), and n 2 (hole vaccum). • reaches the maximum at n~ 1 : .

43. Deviation from the Curie-Weiss law (critical region) • SU(2): thermal fluctuations suppress the 2D long-range FM ordering -- Mermin-Wagner theorem. • No long-range order at fintie T (Mermin-Wagner theorem) • If we neglect itineracy and quantum effects: 2D O(3) class • O(3) NLσ-model: FM directional fluctuations In the critical regime, • As , crosses over into an exponential growth. Itinerant FS 1st or 2nd or phase transition 1D system, landau damping in p-h continuum Alpha log Beta b = 3.1± 0.3

44. Fermi distribution – non-pertubative result Paramagnetic state close to • At k0, • far from the saturated value of 2 even at k0 . . • Large Entropy in the k-space Reference: polarized fermion with • The paramagnetic phase is strongly correlated 0.5 • real space • for fluctuating real space magnetic domains. =+ 0.5

45. Summary: orbital physics with cold atoms • Novel orbital physics not easily accessible in solid state systems. • Unconventional BEC beyond the “no-node” theorem. • A novel system for itinerant ferromagnetism – a non-perturbative study.

46. Other work: orbital physics in the honeycomb lattice • Strong correlation from band flatness: ferromagnetism and Wigner crystal. • Novel mechanism for f-wave Cooper pairing; • Mott-insulator: a new type of frustrated magnet-like model. • Band insulator (topological): quantum anomalous Hall effect.

47. Other selected work • Mott-insulator: quantum model in the honeycomb lattice. A new frustrated magnet-like model. The Kitaev model C. Wu, PRL 100, 200406 (2008). • Band insulator (topological): quantum anomalous Hall effect. C. Wu, PRL 101, 168807 (2008). S. Chu et al. rotation of optical lattices.

48. Z. Cai and C. Wu, arxiv1106.1121. Ginsburg-Landau analysis • and do not couple at quadratic level, but couple at quartic level. The g4 term is due to lattice momentum conservation. • g4 fixes the phases of and : • g4>0 favors phase differences . • In the weak Mott insulating or normal states due to phase fluctuations. The relative phases are pinned at . • Thus the staggered current order still survives  TR breaking normal states.

49. Cold atom orbital systems (I): no Jahn-Teller distortion • Solid state orbital systems: lattice is not rigid. Jahn-Teller distortion lifts the orbital degeneracy and quenches the orbital degree of freedom. • Cold atom orbital systems: atoms in external optical lattices. • Rigid lattice free of distortion; orbital degeneracy is robust.

50. Z. Cai and C. Wu, PRA 2011. Ginsburg-Landau analysis (I) • and do not couple at quadratic level, but couple at quartic level. The g4 term is due to lattice momentum conservation. • g4 >0 pins the relative phases of and : • g4 favors phase differences .