On the Transfer Matrix in the Hubbard Model

We discuss the one-dimensional Hubbard model for a finite chain of atoms, that is within the single band approximation, in the context of the transfer matrix. We show two ways of constructing the finite Hilbert space, one with the basis of electron configurations, and the second created as the antisymmetric part of the orbital and spin function tensor product. We classify the set of all electron configurations according to the action of the symmetric group. We present the Bethe Ansatz (BA) solution for the example of three electrons moving on a ring with N = 6 nodes. After that, we concentrate on giving the motivation for exploring the properties of the transfer matrix. We discuss the scattering and the winding operators in the tensor product space, resulting from the physical and the auxiliary spaces. We consider the transfer matrix as the partial trace over the auxiliary space from the winding operator for the auxiliary particle. After that, we derive the explicit expressions for some elements of the transfer matrix in the basis of the spin electron configurations.