  Recently, an {\it algebraic-dynamical theory} (ADT) for strongly interacting
many-body quantum Hamiltonians in W. Ding, arXiv: 2202.12082 (2022). By
introducing the complete operator basis set, ADT proposes a generic framework
for systematically constructing dynamical theories for interacting quantum
Hamiltonians, using quantum entanglement as the organizing principle. In this
work, we study exact ADT solutions of interacting two-spin problems which can
be used as "free theories" for perturbation study on relevant solvable limits.
Then we perform ADT perturbation calculations and obtain the correct ground
state under the perturbation, which shows that the ADT framework is capable of
constructing correct dynamical perturbation theories for strongly interacting
quantum spin models. We also discuss the implication for relevant lattice
models.
