From the alignment of magnets to the melting of ice, the transition between different phases of matter underpins our exploitation of materials. Both a quantum and a classical phase can undergo an instability into another state. In this thesis, we study the stability of matter in both contexts: topological states and crystalline solids.

We start with the stability of fractional quantum Hall states on a lattice, known as fractional Chern insulators. We investigate, using exact diagonalization, fractional Chern insulators in higher Chern bands of the Harper-Hofstadter model, and examine the robustness of their many-body energy gap in the effective continuum limit. We report evidence of stable states in this regime; comment on two cases associated with a bosonic integer quantum Hall effect; and find a modulation of the correlation function in higher Chern bands.

We next examine the stability of molecules using variational and diffusion Monte Carlo. By incorporating the matrix of force constants directly into the algorithms, we find that we are able to improve the efficiency and accuracy of atomic relaxation and eigenfrequency calculation. We test the performance on a diverse selection of case studies, with varying symmetries and mass distributions, and show that the proposed formalism outperforms existing restricted Hartree-Fock and density functional theory methods.

Finally, we analyze the stability of three-dimensional crystals. We note that for repulsive Coulomb crystals of point nuclei, cubic systems have a zero matrix of force constants at second order. We investigate this by constructing an analytical model in the tight-binding approximation, and present a phase diagram of the most stable crystal structures, as we tune core and valence orbital radii. We reconcile our results with calculations in the nearly free electron regime, as well as current research in condensed matter and plasma physics.