Higher-order-in-spin interaction Hamiltonians for binary black holes from Poincaré invariance

The fulfillment of the space-asymptotic Poincaré algebra is used to derive new higher-order-in-spin interaction Hamiltonians for binary black holes in the Arnowitt-Deser-Misner canonical formalism almost completing the set of the formally $1/c^4$ spin-interaction Hamiltonians involving nonlinear spin terms. To linear order in $G$, the expressions for the $S^{3}p$ and the $S^2{p}^2$ Hamiltonians are completed. It is also shown that there are no quartic nonlinear $S^4$ Hamiltonians to linear order in $G$.