The Lieb nano-ribons with the spin-orbit coupling, the lattice modulation and the magnetic field are exactly studied. They are constructed from the Lieb lattice with two open boundaries in a direction. The interplay between the spin-orbit coupling, the lattice modulation and the magnetic field emerges various exotic ground states. With certain conditions of the spin-orbit coupling, the lattice modulation, the magnetic field and filling the ground state becomes half metallic or half topological. In the half metallic ground state, one spin component is metallic, while the other spin component is insulating. In the half topological ground state, one spin component is topological, while the other spin component is topological trivial. The model exhibits very rich phase diagram.
Using the Bogoliubov variational method we study the magnetic competition in a minimal model proposed for kagome magnets. The minimal model consists of itinerant electrons with their spin-orbit coupling and localized electrons with their anisotropic spin exchange. A spin exchange between localized and itinerant electrons is also included into the model, and it is anisotropic. At half filling and in the region of stable in-plane antiferromagnetism, a magnetic competition between antiferromagnetic phases with different spin chiralities is found. Depending on the sign of the hopping integral and the spin-orbit coupling, either the 1 × 1 or
3
×
3
in-plane antiferromagnetism is established. These in-plane antiferromagnetic states are characterized by distinguishable spin chiralities.
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