Ground states of center-of-mass preserving Hamiltonians

We investigate the energy spectra for a class of spinless fermion Hamiltonians that, in addition to the conservation of center-of-mass momentum, also conserve the center-of-mass position. Such models were first proposed in the context of featureless Mott insulators. At several filling factors p/q, where p and q are coprime integers, the low-energy spectrum can be described by a simpler nearest-neighbor effective Hamiltonian with a smaller Hilbert space. We find examples of both gapped and gapless effective Hamiltonians at various filling factors. We obtain the exact gapless ground state at half-filling, which is expected to fall outside the usual Luttinger liquid paradigm due to center-of-mass preservation, and we compute its properties.