  We study a system involving a single quantum degree of freedom per site of
the lattice interacting with a few neighbors (up to second neighbors), with the
interactions chosen as to produce frustration. At zero temperature, this system
undergoes several quantum phase transitions from both gapped-to-gapless and
gapless-to-gapless phases, providing a very rich phase structure with
disordered, homogeneous and modulated ordered phases meeting in a quantum
Lifshitz point. The gapless phases spontaneously break spatial lattice
translations as well as internal symmetries of the form $U(1)^{\mathsf{N}_c}$,
where $\mathsf{N}_c$ is the number of independent pitch vectors that arise in
the homogeneous and modulated ordered phases. We carry out a detailed analysis
of the quantum critical behavior discussing the mechanism leading to the phase
transitions. We also discuss a proper characterization of all the gapless
phases as well as the nature of the Goldstone excitations. We study the
behavior of the correlation functions and identify regions in the phase diagram
where the system exhibits generalized symmetries, such as polynomial shift
symmetry. This type of symmetry plays an important role in the so-called
fractonic phases, which is an exotic form of matter recently discovered.

Statistical Mechanics;Strongly Correlated Electrons;High Energy Physics - Theory