Topological Bogoliubov quasiparticles from Bose-Einstein condensate in a flat band system

For bosons with flat energy dispersion, condensation can occur in different symmetry sectors. Here, we consider bosons in a Kagome lattice with $\pi$-flux hopping, which in the presence of mean-field interactions exhibit degenerate condensates in the $\Gamma$- and the $K$-point. We analyze the excitation above both condensates and find strikingly different properties: For the $K$-point condensate, the Bogoliubov-de Gennes (BdG) Hamiltonian has broken particle-hole symmetry (PHS) and exhibits a topologically trivial quasiparticle band structure. However, band flatness plays a key role in breaking the time reversal symmetry (TRS) of the BdG Hamiltonian for a $\Gamma$-point condensate. Consequently, its quasiparticle band structure exhibits non-trivial topology, characterized by non-zero Chern numbers and the presence of edge states. Although quantum fluctuations energetically favor the $K$-point condensate, the interesting properties of the $\Gamma$-point condensate become relevant for anisotropic hopping. The topological properties of the $\Gamma$-point condensate get even richer in the presence of extended Bose-Hubbard interactions. We find a topological phase transition into a topological condensate characterized by high Chern number and also comment on the realization and detection of such excitations.