Flat bands and Dirac cones in breathing lattices

In breathing pyrochlores and kagomes, couplings between neighbouring tetrahedra and triangles are free to differ. Breathing lattices thus offer the possibility to explore a different facet of the rich physics of these systems. Here we consider nearest-neighbour classical Heisenberg interactions, both ferromagnetic and antiferromagnetic, and study how the anisotropy of breathing lattices modifies the mode spectrum of pyrochlore and kagome systems. The nature and degeneracy of the flat bands are shown to be preserved for any value of the anisotropy. These flat bands can coexist with Dirac nodes at the $\Gamma$ point when the model becomes particle-hole symmetric. We also derive the nature of the ground state for the breathing kagome lattice, which bears a spontaneous chirality when neighbouring triangles are alternatively ferromagnetic and antiferromagnetic.