  With the ground breaking work of the Fulde, Ferell, Larkin, and Ovchinnikov
(FFLO), it was realized that superconducting order can also break translational
invariance; leading to a phase in which the Cooper pairs develop a coherent
periodic spatially oscillating structure. Such pair density wave (PDW)
superconductivity has become relevant in a diverse range of systems, including
cuprates, organic superconductors, heavy fermion superconductors, cold atoms,
and high density quark matter. Here we show that, in addition to charge density
wave (CDW) order, there are PDW ground states that induce spin density wave
(SDW) order when there is no applied magnetic field. Furthermore, we show that
PDW phases support topological defects that combine dislocations in the induced
CDW/SDW order with a fractional vortex in the usual superconducting order.
These defects provide a mechanism for fluctuation driven non-superconducting
CDW/SDW phases and conventional vortices with CDW/SDW order in the core.
