Topological phases and fermionic superfluidity on the Lieb lattice
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The Lieb lattice is an interesting lattice structure that can host interesting topological phases and novel fermionic superfluids. The fact that this two-dimensional lattice has three sites per unit cell, leads to three bands in the spectrum for tight-binding models. An intrinsic spin-orbit coupling term opens gaps between the three bands, which in addition, host a quantum spin Hall effect. The intricate nature of the multi-gap structure opens up the possibility to realize topological phase transitions driven by the next-nearest neighbour hopping t', which are accompanied by a change of spin Chern numbers of the various bands. We envisage a realization of such phenomena in optical lattices under shaking, where the ratio of t' to the next-nearest hopping t can be tuned to, in principle, any value. Finally, we will consider interaction-driven instabilities towards fermionic superfluidity on the Lieb lattice, with a special emphasis on systems with a point-like Fermi surface. Various types of possible superfluids will then be discussed.