There is a rich interplay of strong correlations, Hund's rule coupling, spin frustration, spin-orbit coupling, flat bands, and Dirac cone physics.

Possible ground states include some sort of Mott insulator, a Haldane phase, semi-metal, ...

A good place to start is the following paper

Low-energy effective theories of the two-thirds filled Hubbard model on the triangular necklace lattice

C. Janani, J. Merino, Ian P. McCulloch, and B. J. Powell

The figure below (taken from this paper) shows some of the molecular structure and some of the hopping integrals that are associated with an underlying decorated honeycomb lattice.

This model could be called kagomene, because it interpolates between the kagome lattice and the honeycomb lattice (graphene). The figure below is taken from this paper, which uses DFT and Wannier orbitals to estimate the tight-binding parameters and the spin-orbit coupling. Interaction driven topological insulator states are possible on this lattice.

There are a few things that are not "normal" about the physics, arising from the 4/3 band filling and the molecular orbitals that are delocalised over the triangles. Specifically, the orbital degeneracy does

**not**arise from atomic orbital degeneracy (cf. d orbitals, or t2g and eg), but rather the E representation associated with C3 symmetry of the triangles.

**Hund's rule coupling.**

This involves the E orbitals and arises purely from the Hubbard U on the non-degenerate orbital on a single lattice site.

**Spin-orbital coupling.**

This is Spin Molecular Orbital Coupling, where the electron spin couples to the angular momentum associated with motion around the triangle, not the angular momentum of degenerate atomic orbitals.

**Haldane phase.**

The associated spin-1's arise from the triplet ground state of four electrons on a triangle.

A DMRG study shows that this is the ground state of a three leg-ladder Hubbard model at 2/3 filling.

Many interesting and important open questions remain about the general phase diagram of the Hubbard model on the kagomene lattice. For example, the nature of the Mott insulator, different types of topological order, the possibility of superconductivity.....

Hopefully, these studies will stimulate new experimental studies and synthesis of new chemical compounds in this fascinating class of materials.

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