As per Landau-Ginzburg (LG) theory, we write down a theory (Hamiltonian) with all possible interactions/operators (in terms of some order parameter) that respects certain symmetries. The ground state (which varies with the tunable couplings/parameters in the theory) might partially break some of the symmetry spontaneously, and the behaviour of correlation functions (observables) depends on the spectrum of fluctuations about the ground state. So, we characterize the ground state and the spectrum of fluctuations by some local order parameter which tells us about the qualitative behaviour of the system (aka "phase").
How does the notion of "topological order" and "quantum states of matter at zero temperature" fit into this picture of matter and phases? I would appreciate if someone could place this in context. What are the observables we use to characterize states/phases? Is it talking about a different understanding of the same phenomena as LG, or does is aim to explain completely different phenomena? If it has a broader scope than LG, then does LG fit into this theory in some manner? Is there some overarching principle here, like my description above for LG theory?