Dynamics and lifetime of geometric excitations in moir\'e systems

We show that spin-2 geometric excitations, known as graviton modes, generally exhibit vanishing lifetimes in lattice Chern bands, including in moir\'e systems. In contrast to the Landau levels, we first numerically demonstrate that the prominent graviton peaks in spectral functions diminish rapidly with increasing system sizes. We explore how the choice of interaction affects the strength of these peaks, with short-ranged interactions pushing the graviton mode far into the continuum of excitations, where it can be significantly scattered due to the increased density of states. We also analytically investigate the short lifetime of the graviton mode. In lattice systems, continuous rotational symmetry is broken, leading to highly anisotropic gapped excitations that mix different angular momentum or ``spins''. This is despite the surprising emergence of a ``guiding center" continuous rotational symmetry in the ground state, which is shared by the graviton mode. Consequently, the graviton mode in Chern bands can be strongly scattered by the anisotropic gapped excitations. However, the emergent rotational symmetry implies that gravitons can be robust in principle, and we propose experimental tuning strategies to lower the graviton mode energy below the continuum. We argue this is a necessary condition enabling the observation of graviton modes and geometric excitations in realistic moir\'e systems.