How does the notion of topological order relate to the Landau-Ginzburg theory of phase transitions?

Question

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?

Answer 1

Topological order can not be described in Ginzburg-Landau symmetry breaking paradigm. It is actually fair to say that topological order are more or less the properties of (gapped) quantum phases that can not be captured by GL. One way to define it is to use the notion of adiabatic continuity: if two gapped phases of matter can be connected by adiabatically varying the Hamiltonian without gap closing, we consider them to be the same phase. So we are doing "homotopy" theory of quantum phases in some sense. One can demand the allowed Hamiltonians connecting different phases must preserve certain symmetries, then one get refined notions such as symmetry protected/enriched topological phases.

Generally speaking there are no local order parameters (sometimes a topologically ordered state also breaks certain symmetry, for example time-reversal symmetry, but this fact itself does not define topological order). And often times topologically ordered states do not break any microscopic symmetries (e.g. quantum spin liquid on a lattice).

In my opinion, two general "principles" for topological order are (1) emergent gauge structure: One can usually describe topological order and their phase transitions using some kind of emergent gauge theory. (2) Entanglement structure. If you ask for a mathematical framework, then modular tensor category theory captures the essence of two-dimensional topological order in an abstract but quite powerful way (up to some details about edge physics).