The one-dimensional Hubbard model is discussed assuming periodic boundary conditions and the half-filling case.The Schur-Weyl duality is applied to the chain consisting of six nodes with dual actions of the unitary and symmetric groups taken separately in the spin and pseudo-spin space.
We propose combinatorial approach to the representation of Schur-Weyl duality in physical systems on the example of one-dimensional spin chains. Exploiting the Robinson-Schensted-Knuth algorithm, we perform decomposition of the dual group representations into irreducible representations in a fully combinatorial way. As representation space, we choose the Hilbert space of the spin chains, but this approach can be easily generalized to an arbitrary physical system where the Schur-Weyl duality works.
The one-dimensional attractive Hubbard model (U 0) is discussed for the chains of N nodes and the same number of electrons, where N -1 of them have the same spin projection, assuming periodic boundary conditions and the half-lling case.Based on the analysis of the eigenvalue problem we provided the general analytical expression for the eigenvalues, for any number N .This formula implies the existence of two elementary particles with mutually dependent momenta on the ring with N sites the same number of electrons including N -1 of the same spin projection.
Abstract We discuss the one-dimensional Hubbard model, on finite sites spin chain, in context of the action of the direct product of two unitary groups SU(2)×SU(2). The symmetry revealed by this group is applicable in the procedure of exact diagonalization of the Hubbard Hamiltonian. This result combined with the translational symmetry, given as the basis of wavelets of the appropriate Fourier transforms, provides, besides the energy, additional conserved quantities, which are presented in the case of a half-filled, four sites spin chain. Since we are dealing with four elementary excitations, two quasiparticles called “spinons”, which carry spin, and two other called “holon” and “antyholon”, which carry charge, the usual spin-SU(2) algebra for spinons and the so called pseudospin-SU(2) algebra for holons and antiholons, provide four additional quantum numbers.
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.