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Topological Quantum Field Theory (TQFT) is the low energy effective theory for the topological ordered states in real world, such as FQH states. In fact, the name "topological order" was motived from the term "Topological Quantum Field Theory". [See Topological Orders in Rigid States, Xiao-Gang Wen, Int. J. Mod. Phys. B4, 239 (1990) ...

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The trivial and non-trivial SPT states both are symmetric under on-site unitary symmetry transformations. The trivial and non-trivial SPT states can be mapped into each other by local unitary transformations (the $U$ in you question). Although such a $U$ is a local unitary transformation, 1. it is not on-site, 2. it is not the on-site unitary symmetry ...

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It seems that you thought $\Phi_0$ is a trivial SPT while $\Phi$ is nontrivial. This does not make sense without defining the symmetry transformation. The fact that $\Phi=U\Phi_0$ means both are product state ($U$ is a honest local unitary transformation). However, symmetry transformation is defined differently in the two states: for $\Phi$ the symmetry ...

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The first question is in a system supporting anyons, what are the universal/topologically invariant physical quantities? Our current theoretical understanding is that all such quantities are related in one way or another to anyon braiding, to be more precise, the $S$ and $T$ matrices. Here $T$ matrix is a diagonal matrix whose diagonal entries are the ...

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The pentagon and hexagon equations are consistency equations for the fusion and braiding of anyons. They should be satisfied by anyons realized in a gapped, local Hamiltonian. If you do not find solutions, then according to our current understanding of topological phase, yes, such anyons do not exist (in the sense that can not be realized by gapped, local ...

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There is a nice pedagogical review of the quantum Hall effect that can be found here. They explain (sketchily) how to derive an effective action describing the bulk of a quantum Hall fluid, which is a topological quantum field theory: Chern-Simons (CS) theory. The main physics which can be gleaned from this is that any defects in the bulk of the fluid could ...

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