Oribtal Physics

Orbital is a degree of freedom independent of charge and spin, which plays an important role in physical properties in transition-metal-oxides, including superconductivity, magnetism, and transport. There are two fundamental features of orbital physics: orbital degeneracy and orbital anisotropy. Similar to the solid state orbital systems (e.g. transition-metal-oxides), optical lattices also possess orbital-band structures. The study of orbital physics with ultra-cold atoms, which has recently become an important research direction of ultra-cold atom physics.

My research has been focusing on novel features of orbital physics in optical lattices different from those known in solid-state orbital systems. Typically, the p-orbital active materials in solids are mostly semiconductors, in which interactions are typically weak. For example, we do not have many systems of p-orbital Mott insulators. In optical lattices, we do can have strongly correlated p-orbital systems. Furthermore, the p-orbitals exhibit the strongest anisotropy -- they are unidirectional. Hence in optical lattices, we can study novel p-orbital Mott insulators combing strong interaction and anisotropy. Our theory research has provided important guidance and support to various experiments.

The px and py-orbital-active honeycomb lattice

In the honeycomb lattice, the site symmetry group D3 only allows the on-site orbitals to be either non-degenerate, or, doubly degenerate. The former case is realized in the graphene systems, which is of single orbital and hence orbital inactive. In the latter case, each site has a pair of degenerate orbitals forming a 2D representation, which can be realized by px and py-orbitals. Each of them is anisotropic, and we need both of them to realize the lattice symmetry.

The band structure of the px(py)-orbital honeycomb lattice is markedly different from that of graphene, but inculdes both Dirac cones and flat bands Ref. [1] and Ref. [2] . Although the Dirac bands exhibit the same dispersion as in graphene, their wavefunctions are orbital-active, hence are fundamentally different. The flat-band can be constructed as superpositions of a set of degenerate localized eigenstates, which exists in each hexagonal plaquette. The band flatness dramatically amplifies the interaction effects, and results in non-perturbative strong correlation effects. We have found Wigner crystallization and flat-band ferromagetism inside the flat-band.

These band structures, including both the flat band and the Dirac cones, have been experimentally seen in the exciton-polariton systems with the honeycomb lattice structure in A. Amo's group (PRL 112, 116402(2014)).

120-degree model and orbital frustration

We investigate the general structure of orbital exchange physics in Mott-insulating states of p-orbital systems in optical lattices. [Ref. 3] . in the honeycomb lattice, the situation is particularly interesting, which is described by a novel quantum 120-degree model. It has a close connection to the Kitaev model in that each bond the exchange is essentially Ising-like, but the Ising axes are different along different bonds. Here, the Ising axes can be just choosen as the bond orientation by using a set of suitable basis, hence, this model is dubbed 120-degree model. In contrast, the Ising axes in the Kiatev model form an orthogonal triad. Imagine distorting the three Ising axes from orthogonal to be co-planer, we arrive from the Kitaev model to the 120-degree model.

The 120-degree model is no-longer exactly solvable, which is heavily frustrated. Its classical ground states are mapped into configurations of the fully-packed loop model with an extra U(1) rotation degree of freedom. Quantum orbital fluctuations select a six-site plaquette ground state ordering pattern in the semiclassical limit from the ``order from disorder'' mechanism. This effect arises from the appearance of a zero energy flat-band of orbital excitations. It is conceivable that the exotic properties in the Kitaev model can still survive to a large extent in such a realistic orbital system. This provides a promising system to realize the exotic orbital liquid state for the future research.

Very recently, it has been used by J. Cava and C. Broholm's groups to explain their neutral scattering data on NaNi2BiO. (arXiv:1807.02528).

Orbital-assisted f-wave pairing state

Although the Dirac band dispersion is similar to graphene, the Bloch wavefunctions carry different non-trivial orbital structures. Consider the pairing structure of spinless fermions. We have found that It is strongly influenced by the orbital structure, giving rise to the nodal lines of the f-wave symmetry [Ref. 4] . This is a novel mechanism for unconventional pairing independent from the strong correlation effect.

To our knowledge, the $f$-wave pairing has not been conclusively identified in spite of some evidence in UPt3. Its successful realization with cold atoms based on our proposal would benefit the research of unconventional Cooper pairing.

Orbital analogue of quantum anomalous Hall effect in p-band systems

I showed that the p-orbital system can exhibit the quantum anomalous Hall (QAH) effect by applying the shaking lattice method, which rotates each optical site around its own center Ref. [5] . The key idea is that both at the \Gamma point, and at the Dirac points $K$ and K^\prime, the wavefunctions are orbitally polarized in the configurations of px+ipy or px-ipy. Hence, the above experiment method splits the energy degeneracy between these two orbitals. The rotation angular velocity opens the topological gaps both at the \Gamma and Dirac points K and K^\prime with values equal to rotation angular velocity. As a result, the system becomes a quantum anomalous Hall insulator with a topological band structure, which is induced by the onsite orbital angular momentum polarization.

The same mechanism when applied to solid state states can be used to realize the large gap topological insulators to the order of atomic scale spin-orbit coupling. In that case, the onsite spin-orbit coupling Lz.\sigma_z plays the role of the onsite rotation, but a Kramers double of it. This mechanism has been realized in the Bismuthene system (please see Boost the topological gap in the px and py orbitals for details.)

References and talks

1. Congjun Wu, Doron Bergman, Leon Balents, and S. Das Sarma, "Flat bands and Wigner crystallization in the honeycomb optical lattice",
Phys. Rev. Lett. 99, 70401 (2007), see pdf file

2. Congjun Wu , and S. Das Sarma,
"The $p_{x,y}$-orbital counterpart of graphene: cold atoms in the honeycomb optical lattice",
Phys. Rev. B 77, 235107 (2008). See pdf file.

3. Congjun Wu, "Orbital Ordering and Frustration of p-Band Mott Insulators",
Phys. Rev. Lett. 100, 200406 (2008) , see pdf file .

4. Wei-cheng Lee, Congjun Wu, and S. Das Sarma, "$F$-wave pairing of cold atoms in optical lattices",
Phys. Rev. A 82, 053611 (2010), See pdf file .

5. Congjun Wu, "Orbital analogue of quantum anomalous Hall effect in $p$-band systems"
Phys. Rev. Lett. 101, 186807 (2008) , see pdf file .

Talk: "Novel p-orbital physics in the honeycomb optical lattice" .

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Last modified: July 15, 2007.