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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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I assume you refer to something like a BHZ-$\mathbb{Z}_2$-insulator (with the experimental realization of a 2d-$\text{HgTe}$-quantum well). This model can be understood as two Chern insulators. Chern insulators have a chiral edge mode (that is, an edge mode, that propagates only in one turning sense, thus breaking time reversal). This is analogous to the ...

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The zero energy states are localized at the boundary, and their wave functions decay exponentially in the bulk. So they are boundary modes (or edge states), which do not count as the bulk states, and do not contribute to the bulk gap closing. The only way to close the bulk gap in the SSH model is to tune $\delta t$ to zero. As long as $\delta t\neq 0$, the ...

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A topological invariant in condensed matter systems is a number that doesn't change under a smooth deformation of the Hamiltonian. I like to think this as stretching a material that would change for instance the hopping constants. Now the way that one can take such a number may vary and I don't think there is a limitation in the definition neither that there ...

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