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The $SO(3)\times SO(3)\times U(1)$ Hubbard model on a square lattice in terms of $c$ and $αν$ fermions and deconfined $η$-spinons and spinons

In this paper a description of the energy eingenstates of the Hubbard model on the square lattice with nearest-neighbor transfer integral $t$, on-site repulsion $U$, and $N_a^2\gg 1$ sites in terms of occupancy configurations of charge $c$ fermions, spin-1/2 spinons, and $η$-spin-1/2 $η$-spinons is introduced. Such objects emerge from a suitable electron - rotated-electron unitary transformation. In chromodynamics the quarks have color but all quark-composite physical particles are color-neutral. Within our description the $η$-spinon (and spinons) that are not invariant under the electron - rotated-electron unitary transformation have $η$ spin 1/2 (and spin 1/2) but are part of $η$-spin-neutral (and spin-neutral) $2ν$-$η$-spinon (and $2ν$-spinon) composite $ην$ fermions (and $sν$ fermions). Here $ν=1,2,...$ is the number of $η$-spinon (and spinon) pairs. In turn, a well-defined number of independent spinons and independent $η$-spinons are invariant under the electron - rotated-electron unitary transformation. Simple occupancy configurations of (i) the $c$ fermions, (ii) independent spinons and $2ν$-spinon composite $sν$ fermions, and (iii) independent $η$-spinons and $2ν$-$η$-spinon composite $ην$ fermions generate an useful complete set of states. The configurations (i), (ii), and (iii) correspond to the state representations of the U(1), spin SU(2), and $η$-spin SU(2) symmetries, respectively, associated with the model $SO(3)\times SO(3)\times U(1) =[SU(2)\times SU(2)\times U(1)]/Z_2^2$ global symmetry.