  Quantum many-body systems with fractonic excitations can realize fascinating
phases of matter. Here, we study the low-energy excitations of a constrained
Bose-Hubbard model in one dimension, which conserves the center of mass or,
equivalently, the dipole moment in addition to the particle number. This model
is known to realize fractonic phases, including a dipole Mott insulator, a
dipole Luttinger liquid, and a metastable dipole supersolid. We use tensor
network methods to compute spectral functions from the dynamical response of
the system and verify predictions from low-energy field theories of the
corresponding ground state phases. We demonstrate the existence of gapped
excitations compatible with strong coupling results in a dipole Mott insulator,
linear sound modes characteristic of a Luttinger liquid of dipoles, and soft
quadratic modes at both zero and finite momenta in a supersolid state with
charge density wave order and phase coherence at non-integer filling.
