I am studying a one-dimensional bosonic system with $ U(1) \rtimes Z_2^T$ symmetry numerically, which might has a symmetry-proteced toplogical(SPT) phase. I have several questions about the symmetry-protected topological state, particularly, the gapless edge excitations in an SPT state.
On the first page of paper http://arxiv.org/abs/1209.4399 (PRB 87, 144421 (2013)) by Cenke Xu, he gave three criteria for SPT phases. In particular, the second criteria requires that the Hamiltonian with open boundary has either gapless edge excitations or gapped but degenerate ground states. (See also wiki)
Q1) What is the difference between the "gapless edge excitations" and "gapped but degenerate ground states"?
I am confused about that because the bulk of SPT phase
is always gapped. Therefore, I think there is always
a gap above those gapless edge modes and these two
conditions should be equivalent in the thermodynamic limit.
Can someone give some examples to show the difference?
I prefer the examples in one dimension, for example,
the spin-1 Haldane chain with open boundary.
Q2) Since SPT phase is protected by some symmetry. How can I know the folds of degeneracy of the ground states from the symmetry without knowing the details of the Hamiltonian?
Q3) Is this a necessary condition for an SPT state? Or is this a necessary condiction in one dimension? If not, can you give several examples?
Q4) In one-dimensional bosonic systems, if an SPT phase is protected by $U(1)\rtimes Z_2^T$ symmetry, can I conclude that it is a Haldane phase?