Cooperative orders I am familiar with the more "traditional" types of order and their corresponding degrees of freedom one might find in a many-body system, such as crystalline (translational d.o.f.), ferroelectric (rotational d.o.f.) and magnetic (spin d.o.f.) ordering. However, I am reading a review on quantum materials and am having a hard time understanding more exotic classes, such as charge order and orbital order. What are the corresponding degrees of freedom for these types of order, and why are they of interest?
Furthermore, what does it mean to have different types of order "cooperate" with each other? For example, there supposedly exists a CE state of quarter-filled manganites with simultaneous spin, charge, and orbital order. What does this mean? And do materials with cooperative order possess desirable properties?
 A: The degrees of freedom in charge order and orbital order are exactly that; the charge and orbital degrees of freedom.
For example, a charge density wave would be an example of charge order.
For orbital order, you can imagine a situation where in the disordered phase, two electronic orbitals are degenerate, but then there is a spontaneous symmetry breaking transition where the energies and occupations of those orbitals become inequivalent. This is often called "ferro-orbital" order.
As far as cooperative order, very often symmetry will dictate that certain types of symmetry breaking always have to come together. This is actually at the heart of many of the big problems in unconventional superconductors.  For example, in the Fe based superconductors, there is a rotation symmetry breaking "nematic" phase that breaks the tetragonal symmetry of the crystal lattice. In principle, this could be driven by the lattice, charge, or spin degrees of freedom, and below the transition all of these "order" in the sense that once the symmetry is broken, it is broken for all.
I think if you want to think about cooperative order very generally, you can consider the Landau free energy with all of the order parameters and allowed couplings. If two orders are cooperative, then there coupling will have a negative sign, meaning that the system saves energy by having both orders simultaneously present.
