  We show that a hierarchy of topological phases in one dimension -- a
topological Devil's staircase -- can emerge at fractional filling fractions in
interacting systems, whose single-particle band structure describes a
topological or a crystalline topological insulator. Focusing on a specific
example in the BDI class, we present a field-theoretical argument based on
bosonization that indicates how the system, as a function of the filling
fraction, hosts a series of density waves. Subsequently, based on a numerical
investigation of the low-lying energy spectrum, Wilczek-Zee phases, and
entanglement spectra, we show that they are symmetry protected topological
phases. In sharp contrast to the non-interacting limit, these topological
density waves do not follow the bulk-edge correspondence, as their edge modes
are gapped. We then discuss how these results are immediately applicable to
models in the AIII class, and to crystalline topological insulators protected
by inversion symmetry. Our findings are immediately relevant to cold atom
experiments with alkaline-earth atoms in optical lattices, where the band
structure properties we exploit have been recently realized.
