  The leading order finite size effects due to spin, namely that of the cubic
and quartic in spin interactions, are derived for the first time for generic
compact binaries via the effective field theory for gravitating spinning
objects. These corrections enter at the third and a half and fourth
post-Newtonian orders, respectively, for rapidly rotating compact objects.
Hence, we complete the leading order finite size effects with spin up to the
fourth post-Newtonian accuracy. We arrive at this by augmenting the point
particle effective action with new higher dimensional nonminimal coupling
worldline operators, involving higher-order derivatives of the gravitational
field, and introducing new Wilson coefficients, corresponding to constants,
which describe the octupole and hexadecapole deformations of the object due to
spin. These Wilson coefficients are fixed to unity in the black hole case. The
nonminimal coupling worldline operators enter the action with the electric and
magnetic components of the Weyl tensor of even and odd parity, coupled to even
and odd worldline spin tensors, respectively. Moreover, the non relativistic
gravitational field decomposition, which we employ, demonstrates a coupling
hierarchy of the gravito-magnetic vector and the Newtonian scalar, to the odd
and even in spin operators, respectively, which extends that of minimal
coupling. This observation is useful for the construction of the Feynman
diagrams, and provides an instructive analogy between the leading order
spin-orbit and cubic in spin interactions, and between the leading order
quadratic and quartic in spin interactions.
