Based on what I have learned so far, Haldane phases are a nontrivial SPT for 1D spin-1 chains. The trivial phase acts as an "identity" under the group of SPT phases ( with stacking as the group operation ). If my understanding is correct, can anyone please tell me the inverse of Haldane phase? Is the Haldane phase an inverse of itself by any chance ?
If my question ( or my understanding ) is incorrect, any insight or hint will be appreciated. Thanks.
The phases of spin chains under SO(3) symmetry (as well as other symmetries which typically stabilize the Haldane phase) are classified by $\mathbb Z_2$. Here, the non-trivial element of $\mathbb Z_2$ corresponds to the Haldane phase. Thus, indeed, the Haldane phase is its own inverse.
One way to see this is to note that the Haldane is characterized by the fact that the entanglement spectrum transforms as a half-integer spin. Thus, juxtaposing two Haldane chains gives an entanglement spectrum which transforms as the product of two half-integer representations, that is, an integer spin representation, which is exactly what characterizes a chain in the trivial phase.