  The next-to-next-to-leading order spin-squared interaction potential for
generic compact binaries is derived for the first time via the effective field
theory for gravitating spinning objects in the post-Newtonian scheme. The
spin-squared sector is an intricate one, as it requires the consideration of
the point particle action beyond minimal coupling, and mainly involves the
spin-squared worldline couplings, which are quite complex, compared to the
worldline couplings from the minimal coupling part of the action. This sector
also involves the linear in spin couplings, as we go up in the nonlinearity of
the interaction, and in the loop order. Hence, there is an excessive increase
in the number of Feynman diagrams, of which more are higher loop ones. We
provide all the Feynman diagrams and their values. The beneficial
nonrelativistic gravitational fields are employed in the computation. This
spin-squared correction, which enters at the fourth post-Newtonian order for
rapidly rotating compact objects, completes the conservative sector up to the
fourth post-Newtonian accuracy. The robustness of the effective field theory
for gravitating spinning objects is shown here once again, as demonstrated in a
recent series of papers by the authors, which obtained all spin dependent
sectors, required up to the fourth post-Newtonian accuracy. The effective field
theory of spinning objects allows to directly obtain the equations of motion,
and the Hamiltonians, and these will be derived for the potential obtained here
in a forthcoming paper.
