Yangian symmetry in molecule {V6} and four-spin Heisenberg model

The symmetry operator Q=Y² is introduced to re-describe the Heisenberg spin triangles in the {V6} molecule, where Y stands for the Yangian operator which can be viewed as special form of Dzyaloshinsky-Moriya (DM) interaction for spin 1/2 systems. Suppose a parallelogram Heisenberg model that is comprised of four 12-spins commutes with Q, which mean that it possesses Yangian symmetry, we show that the ground state of the Hamiltonian H₄ for the model allows to take the total spin S=1 by choosing some suitable exchange constants in H₄. In analogy to the molecule {V6} where the two triangles interact through Yangian operator we then give the magnetization for the theoretical molecule "{V8}" model which is comprised of two parallelograms. Following the example of molecule {V15}, we give another theoretical molecule model regarding the four 12-spins system with total spin S=1 and predict the local moments to be 910μB,110μB,110μB,910μB, respectively.