Permrtation group approach to heisenbnrg open spin chain with the third component of single and the nearest neighber interaction to construct energy matrices and the character of matrices given with the approach
Firstly, this article showed the knowledge of the permutation group operating complete basis vectors, and introduced the approach that the Hamiltonian operator of the third component of spin function complete basis vectors to form the energy matrix, then observed the character of energy matrices.
The Hamiltonian of one-dimensional integrable multi-impurity XXZ spin chain with open boundary cases is constructed by using the method of adding parameters to the vertical matrices. This is the first approach to add multi-impurity and next-nearest-neighbor interaction to the integrable anisotropic Heisenberg spin chain. It is found that all of the impurity parameters in the bulk must be pure imaginary numbers if the cross parameter is real for getting the hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interactions are introduced. The Hamiltonian can be also applied to describe the many body systems with two kinds of particles if all of the impurity parameters are equal.