  The simultaneous interplay of strong electron-electron correlations,
topological zero-energy states, and disorder is yet an unexplored territory but
of immense interest due to their inevitable presence in many materials. Copper
oxide high-temperature superconductors (cuprates) with pair breaking edges host
a flat band of topological zero-energy states, making them an ideal playground
where strong correlations, topology, and disorder are strongly intertwined.
Here we show that this interplay in cuprates generates a new phase of matter: a
fully gapped ``phase crystal" state that breaks both translational and
time-reversal invariance, characterized by a modulation of the $d$-wave
superconducting phase co-existing with a modulating extended $s$-wave
superconducting order. In contrast to conventional wisdom, we find that this
phase crystal state is remarkably robust to omnipresent disorder, but only in
the presence of strong correlations, thus giving a clear route to its
experimental realization.
