Neutral Excitations of Quantum Hall States: a Density Matrix Renormalization Group Study

We use the dynamical structure factors of the quantum Hall states at $\nu=1/3$ and $\nu=1/2$ in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low energy properties. At $\nu=1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasi-particle. At $\nu=1/2$, we find low energy modes with linear dispersion and the static structure factor $\bar s(q) \sim (q\ell)^3$ in the limit $q\ell \rightarrow 0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.