  Topology in quantum matter is typically associated with gapped phases. For
example, in symmetry protected topological (SPT) phases, the bulk energy gap
localizes edge modes near the boundary. In this work we identify a new
mechanism that leads to topological phases which are not only gapless but where
the absence of a gap is essential. These `intrinsically gapless SPT phases'
have no gapped counterpart and are hence also distinct from recently discovered
examples of gapless SPT phases. The essential ingredient of these phases is
that on-site symmetries act in an anomalous fashion at low energies.
Intrinsically gapless SPT phases are found to display several unique properties
including (i) protected edge modes that are impossible to realize in a gapped
system with the same symmetries, (ii) string order parameters that are likewise
forbidden in gapped phases, and (iii) constraints on the phase diagram obtained
upon perturbing the phase. We verify predictions of the general theory in a
specific realization protected by $\mathbb Z_4$ symmetry, the one dimensional
Ising-Hubbard chain, using both numerical simulations and effective field
theory. We also discuss extensions to higher dimensions and possible
experimental realizations.
