Topological phase transitions represent a new paradigm beyond conventional Landau-Ginzburg symmetry breaking. Even in the absence of symmetry breaking, the symmetries of a Hamiltonian play an important role in stabilizing topological signatures. In this thesis, we will study the interplay between symmetry and topology in many-body systems, with three distinct physical setups in mind.

First, we investigate a two-dimensional system of ultracold bosons which is condensed in the p-band of a triangular lattice. Certain smooth interaction profiles result in non-Abelian symmetry generators of the order parameter manifold, leading to anomalous physical consequences. These include a lack of Berezinskii-Kosterlitz-Thouless transition due to unstable vortex configurations, additional Goldstone modes, and a marginally divergent normal fluid density which implies lack of superfluidity in the thermodynamic limit. Our results suggest possible connections to solid state experiments on 4He films.

We then show that disorder which keeps a Hamiltonian in the same Altland-Zirnbauer symmetry class is capable of inducing a topological transition. This is achieved by considering the normalizability of Majorana edge modes in a topological superconductor, and corroborated with entanglement metrics. While degeneracies of the many-body spectrum are robust with respect to weak perturbations, strong disorder can both promote and destroy a topological index.

Lastly, we study topological edge modes and degeneracies in atomic, molecular, and optical systems which evolve non-unitarily in time or space due to non-equilibrium effects, including: condensate instabilities after a quench, decoherence channels from an external environment, and dissipation into a medium. Mathematically, the non-unitary evolution is generated by effective Hamiltonians which break Hermiticity. These systems can possess uniquely non-Hermitian symmetries which are absent from equilibrium counterparts. We present a symmetry-based classification, and investigate several systems which can be solved exactly.