  We introduce exactly solvable gapless quantum systems in $d$ dimensions that
support symmetry protected topological (SPT) edge modes. Our construction leads
to long-range entangled, critical points or phases that can be interpreted as
critical condensates of domain walls "decorated" with dimension $(d-1)$ SPT
systems. Using a combination of field theory and exact lattice results, we
argue that such gapless SPT systems have symmetry-protected topological edge
modes that can be either gapless or symmetry-broken, leading to unusual surface
critical properties. Despite the absence of a bulk gap, these edge modes are
robust against arbitrary symmetry-preserving local perturbations near the
edges. In two dimensions, we construct wavefunctions that can also be
interpreted as unusual quantum critical points with diffusive scaling in the
bulk but ballistic edge dynamics.
