Integrable systems and spin gap

We work within an algebraic construction to map a specific Hamiltonian equation, which describes spin ladder, and spin tube models. The system could be called a spin chain because it is a continuous line of arranged atoms with periodic boundary conditions.For our specific case, we use SU (4) and SU (8) Lie algebra and introduce free parameters to preserve their integrability. These systems have an energy gap, an important feature of the internal structure of some materials, cuprates, and polymers have an internal geometry molecularly ladder. The interest of study is directed to solve the Hamiltonian that describes this type of systems, through an algebraic method known as Ansatz de Bethe, which will provide us with a mathematical resource to be able to find the energy of the system.