In the Landau classification scheme, phases of matter differ in terms of symmetry. However, we know of many instances where this classification scheme does not apply. I have often heard topological insulators and even the different plateaus in the integer quantum hall effect referred to as distinct phases of matter. What does this mean? Do these states really qualify as genuine phases? I am perfectly aware that there may not be consensus on this issue, but I'd like to know the different viewpoints on this issue.
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$\begingroup$ For the Fractionsl Hall effect the answer seems to be yes, see this answer: physics.stackexchange.com/a/673660/16713 $\endgroup$– KF GaussCommented Oct 31, 2021 at 4:54
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Yes, integer quantum Hall phases are distinct phases of matter with invertible topological order. Any phase of matter can be distinguished from another by the presence of a phase transition, where a smooth change in parameter(s) leads to a discontinuous change in some observable quantity. In the case of the integer quantum Hall phases, different phases can be distinguished by their transverse electromagnetic and thermal responses. In spite of the fact that integer quantum Hall phases contain no non-trivial topological excitations, they are topological in that they cannot be smoothly connected to unentangled product states by local unitary transformations (that don't grow in size with the system size).
There may not be a consensus on details like whether IQH phases are long range or short range entangled, and I am not sure if everyone agrees about whether or not they are SPT phases (without any symmetries), but I don't think there is a lack of agreement about the fact that they are distinct phases of matter.
