Abstract
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.
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