Exact renormalization in quantum spin chains

We introduce a real-space exact renormalization group method to find exactly solvable quantum spin chains and their ground states. This method allows us to provide a complete list for exact solutions within SU(2) symmetric quantum spin chains with $S\leq 4$ and nearest-neighbor interactions, as well as examples with S=5. We obtain two classes of solutions: One of them converges to the fixed points of renormalization group and the ground states are matrix product states. Another one does not have renormalization fixed points and the ground states are partially ferromagnetic states.