Tunable Band Topology in Gyroscopic Lattices

Gyroscopic metamaterials --- mechanical structures composed of interacting spinning tops --- have recently been found to support one-way topological edge excitations. In these structures, the time reversal symmetry breaking that enables their topological behavior emerges directly from the lattice geometry. Here we show that variations in the lattice geometry can therefore give rise to more complex band topology than has been previously described. A `spindle' lattice (or truncated hexagonal tiling) of gyroscopes possesses both clockwise and counterclockwise edge modes distributed across several band gaps. Tuning the interaction strength or twisting the lattice structure along a Guest mode opens and closes these gaps and yields bands with Chern numbers of $|C| > 1$ without introducing next-nearest-neighbor interactions or staggered potentials. A deformable honeycomb structure provides a simple model for understanding the role of lattice geometry in constraining the effects of time reversal symmetry and inversion symmetry breaking. Lastly, we find that topological band structure generically arises in gyroscopic networks, and a simple protocol generates lattices with topological excitations.