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 <= 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.