Single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus

We analyze the collective magnetoroton excitations of bosonic Laughlin v = 1/2 fractional quantum Hall (FQH) states on the torus and of their analog on the lattice, the fractional Chern insulators (FCIs). We show that, by applying the appropriate mapping of momentum quantum numbers between the two systems, the magnetoroton mode can be identified in FCIs and that it contains the same number of states as in the FQH case. Further, we numerically test the single-mode approximation to the magnetoroton mode for both the FQH and FCI cases. This proves particularly challenging for the FCI because its eigenstates have a lower translational symmetry than the FQH states. In spite of this, we construct the FCI single-mode approximation such that it carries the same momenta as the FQH states, allowing for a direct comparison between the two systems. We show that the single-mode approximation captures well a dispersive subset of the magnetoroton excitations both for the FQH and the FCI cases. We find remarkable quantitative agreement between the two systems. For example, the many-body excitation gap extrapolates to almost the same value in the thermodynamic limit.