  Symmetry protected topological (SPT) phases are one of the simplest, yet
nontrivial, gapped systems that go beyond the Landau paradigm. In this work, we
study an extension of the notion of SPT for gapless systems, namely, gapless
symmetry protected topological states. We construct several simple gapless-SPT
models using the decorated defect construction, which allow analytical
understanding of non-trivial topological features including the symmetry charge
under twisted boundary conditions, and boundary (quasi)-degeneracy under open
boundary conditions. We also comment on the stability of the gapless-SPT models
under symmetric perturbations, and apply small-scale exact diagonalization when
direct analytic understanding is not available.
