An Analysis of Solutions of the One-Dimensional Hubbard Model

We discuss the one-dimensional Hubbard model of N sites half-filled chain, on the example of a ring with four nodes, in context of the diagonalization process. We consider translational and unitary symmetry of the problem, given by the CN, and the direct product of two unitary groups SU(2) × SU(2), respectively. We construct nine of the sixteen eigenstates which are not dependent on parameters U and t using the irreducible basis of the appropriate groups, and study the seven remaining states using graphical representations for different ranges of the coefficients. We discuss the different and often conflicting behaviors of the kinetic and potential parts of the Hubbard Hamiltonian in different areas of a parameter space.