  Versatile controllability of interactions in ultracold atomic and molecular
gases has now reached an unprecedented era where quantum correlations and
unconventional many-body phases can be studied with no corresponding analogs in
solid state systems. Recent experiments in Rydberg atomic gases have achieved
exquisite control over non-local interactions, allowing novel quantum phases
unreachable with the usual local interactions in atomic systems. Here, we study
Rydberg dressed atomic fermions in a three dimensional optical lattice
predicting the existence of hitherto unheard-of exotic mixed topological
density wave phases. We show that varying spatial range of the non-local
interaction leads to a rich phase diagram containing various bond density
waves, with unexpected spontaneous time-reversal symmetry breaking.
Quasiparticles in these chiral phases experience emergent gauge fields and form
three dimensional quantum Hall and Weyl semimetal states. Remarkably, certain
density waves even exhibit mixed topologies beyond the existing topological
classification. Experimental signatures of density waves and their topological
properties are predicted in time-of-flight measurements.
