  Recent case-by-case studies revealed that the dispersion of low energy
excitations in gapless frustration-free Hamiltonians is often quadratic or
softer. In this work, we argue that this is actually a general property of such
systems. By combining a previous study by Bravyi and Gosset and the min-max
principle, we prove this hypothesis for models with local Hilbert spaces of
dimension two that contains only nearest-neighbor interactions on cubic
lattice. This may be understood as a no-go theorem realizing gapless phases
with linearly dispersive excitations in frustration-free Hamiltonians. We also
provide examples of frustration-free Hamiltonians in which the plane-wave state
of a single spin flip does not constitute low energy excitations.
