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In this paper a description of the energy eingenstates of the Hubbard modelon the square lattice with nearest-neighbor transfer integral $t$, on-siterepulsion $U$, and $N_a^2\gg 1$ sites in terms of occupancy configurations ofcharge $c$ fermions, spin-1/2 spinons, and $\eta$-spin-1/2 $\eta$-spinons isintroduced. Such objects emerge from a suitable electron - rotated-electronunitary transformation. In chromodynamics the quarks have color but allquark-composite physical particles are color-neutral. Within our descriptionthe $\eta$-spinon (and spinons) that are not invariant under the electron -rotated-electron unitary transformation have $\eta$ spin 1/2 (and spin 1/2) butare part of $\eta$-spin-neutral (and spin-neutral) $2\nu$-$\eta$-spinon (and$2\nu$-spinon) composite $\eta\nu$ fermions (and $s\nu$ fermions). Here$\nu=1,2,...$ is the number of $\eta$-spinon (and spinon) pairs. In turn, awell-defined number of independent spinons and independent $\eta$-spinons areinvariant under the electron - rotated-electron unitary transformation. Simpleoccupancy configurations of (i) the $c$ fermions, (ii) independent spinons and$2\nu$-spinon composite $s\nu$ fermions, and (iii) independent $\eta$-spinonsand $2\nu$-$\eta$-spinon composite $\eta\nu$ fermions generate an usefulcomplete set of states. The configurations (i), (ii), and (iii) correspond tothe state representations of the U(1), spin SU(2), and $\eta$-spin SU(2)symmetries, respectively, associated with the model $SO(3)\times SO(3)\timesU(1) =[SU(2)\times SU(2)\times U(1)]/Z_2^2$ global symmetry.

annals of physics

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