The Kennedy-Tasaki (KT) transformation was used to construct the gapped
symmetry protected topological (SPT) phase from the symmetry breaking phase
with open boundary condition, and was generalized in our proceeding work [L. Li
et al. arXiv:2301.07899] on a ring by sacrificing the unitarity, and should be
understood as a non-invertible duality transformation. In this work, we further
apply the KT transformation to systematically construct gapless symmetry
protected topological phases. This construction reproduces the known examples
of (intrinsically) gapless SPT where the non-trivial topological features come
from the gapped sectors by means of decorated defect constructions. We also
construct new (intrinsically) purely gapless SPTs where there are no gapped
sectors, hence are beyond the decorated defect construction. This construction
elucidates the field theory description of the various gapless SPTs, and can
also be applied to analytically study the stability of various gapless SPT
models on the lattice under certain symmetric perturbations