What are the differences between Haldane phase, non-interacting topological insulator/superconductor, and SPT order?

Question

Haldane phase, and non-interacting topological insulator/superconductor are often regarded as examples of symmetry protected topological (SPT) orders.

Answer 1

The essence of Haldane phase is its energy gap. It is known that AF spin-1/2 chain has quasi long rang order and is gapless. This leads people to think that integer spin chain, with weaker quantum fluctuations, should also has quasi long rang order and is gapless. Haldane's discovery that integer spin chain is gapped comes as a surprise. So the gapped integer spin chain is named Haldane phase.

The essence of non-interacting topological insulator/superconductor is the topological twist in band structure. The Z-class twist described by Chern number is well known. The Z2-class twist in the presence of U(1) and time reversal symmetry is a surprise. With different symmetries, there are many different kinds of topological twist in band structure.

The essence of SPT order is the short-range many-body entanglement with a twisted symmetry action. In the presence of symmetry, even short-range entangled states can be in different phases, which is a surprise.

Haldane phase is different from SPT order. In fact only odd-spin Haldane phase has SPT order while even-spin Haldane phase has no SPT order. Similarly, non-interacting topological insulator/superconductor is also different from fermionic SPT order. In particular, some non-trivial non-interacting topological insulators/superconductors actually have trivial SPT order. So, to be precise, Haldane phase, non-interacting topological insulator/superconductor, and SPT order are not really the same.