Fractons and subsystem symmetries of superfluid field theories and quantum crystals

Fractons represent an intriguing class of emergent quasiparticles that exhibit restricted mobility and fractionalized excitations. While traditionally associated with topological phases, recent studies have unveiled the presence of fractonic behavior in certain superfluid systems. The goal of this thesis is to explore vortices in superfluid systems, which behaves like fraction excitations, as well as fractons in quantum crystals. The first task is to review the basic properties of fractonic excitations and their restricted mobility properties in (2+1)-dimensional models. This, via a systematic exploration of the relevant literature, will provide the basic knowledge of the quantum mechanics and classical field theories techniques useful for the study of fractons. The second objective consists of studying conserved charges and momenta of superfluid field theories in the presence of vortex configurations leading to conserved charge and dipole moment. In addition, it would be very interesting to explore the relation of conservation of charge and dipole moment to subsystem symmetries. The third task is to review fractons in tensor gauge theories, and relate this to the superfluid case via the particle-vortex duality for (2+1)-dimensional systems. An optional extra task would be to study the relation of fractons in quantum crystals between the lattice model and the low-energy continuum limit, where the crystal is modelled by an effective field theory similar, in some sense, to the superfluid case.