  Quantum materials display rich and myriad types of magnetic, electronic, and
structural ordering, often with these ordering modes either competing with one
another or 'intertwining', that is, reinforcing one another. Low dimensional
quantum materials, influenced strongly by competing interactions and/or
geometric frustration, are particularly susceptible to such ordering phenomena
and thus offer fertile ground for understanding the consequent emergent
collective quantum phenomena. Such is the case of the quasi-2D materials
R4Ni3O10(R=La, Pr), in which intertwined charge-and spin-density waves (CDW and
SDW) on the Ni sublattice have been identified and characterized. Not
unexpectedly, these density waves are largely 2D as a result of weak coupling
between planes, compounded with magnetic frustration. In the case of R=Pr,
however, we show here that exchange coupling between the transition metal and
rare earth sublattices upon cooling overcomes both obstacles, leading to a
dimensional crossover into a fully 3D ordered and coupled SDW state on both
sublattices, as an induced moment on notionally nonmagnetic Pr3+ opens exchange
pathways in the third dimension. In the process, the structure of the SDW on
the Ni sublattice is irreversibly altered, an effect that survives reheating of
the material until the underlying CDW melts. This 'bootstrapping' mechanism
linking incommensurate SDWs on the two sublattices illustrates a new member of
the multitude of quantum states that low-dimensional magnets can express,
driven by coupled orders and modulated by frustrated exchange pathways.
