Are interacting symmetry protected topological (SPT) phases and symmetry enriched topological (SET) phases must be gapped?

Question

I wonder are interacting SPT and SET phases gaped? Can we have a SET or interacting SPT phase in a semi metal?

Answer 1

A gap is part of definition, although sometimes it's enough to have a gapped sector (spin or charge gap, usuall). Note also that in a crystal there are always phonons, which are gapless.

To see the importance of a gap, let's consider some 1d Majorana models, with this figure from Verresen et al:

As usual, the dots indicate Majorana operators $c_j$ and the bonds are pairing terms $ic_1 c_2$ so that you can read each picture as a Hamiltonian.

Consider placing a chain with Hamiltonian $H_1$ (the Kitaev wire) on top of one with $H_0 + H_1$ (massless free Majorana field, a 1d semimetal). Then it's possible to absorb the edge modes of $H_1$ into those of $H_0 + H_1$ with a local term.

Thus, the free Majorana is a gapless theory that can absorb an SPT, so one needs to be really careful in extending the SPT concept to gapless systems.