Study Of Methods For Searching The Tetrahedral Lattice Spin States

When studying the novel magnetic properties about spin-frustrated systems, study-ing spin configurations to be degenerated is one of important research directions. In re-cent years, physics scientists have found that several kinds of spinel ACr2O4(Zn,Cd,Hg) where the magnetic Cr3+ions are from a network of corner-sharing tetrahedra, lead to numerous degenerations and have frustrated Heisenberg antiferromagnets by spin-lattice coupling and further-neighbor interactions. Owing to the research of the mag-netic properties, it is necessary to find all the spin states. Therefore, this paper is based on this kind of spin lattice, discussing problems of solving the whole spin states in collinear(Ising Model) and noncollinear(Heisenberg Model) magnetic orderings. In Ising Model’s problem, we firstly prove that the complexity of the problem is NPC and the number of solutions equals to the permanent of relative0-1matrix. We also do some work on estimating the number, and obtain the formula of solutions about one kind structure with its substructure. Finally we design a fine traversal algorithm to find all the solutions. On the other hand, we prove that the problems on Heisenberg Models have a more complex solution manifold without singular points and gain the formula of number of dimensions. Moreover, based on the method to solve the problem with a sole tetrahedra structure, we spread this method to solve multi-tetrahedra structure problems with one perfect matching and homotopy algorithm. Finally, we show some results and methods which can solve these problems on other structures.