Symmetry and spontaneous symmetry breaking of the Hubbard model on a square lattice ground state

In this paper we consider the simplified form that a recently introduced general operator description of the Hubbard model on the square lattice with $N_a^2\gg 1$ sites, effective transfer integral $t$, and onsite repulsion $U$ has in a suitable one- and two-electron subspace. Such an operator description is that consistent with the model exact global symmetry recently extended to $SO(3)\times SO(3)\times U(1)$. There is a large consensus that in the thermodynamic limit $N_a^2\to \mathrm{\infty }$ long-range antiferromagnetic order occurs in the spin-density $m=0$ ground state of the half-filled Hubbard model on the square lattice. Here we find that the corresponding spontaneous symmetry breaking lowers the symmetry of such a state from $SO(3)\times SO(3)\times U(1)$for$N_a^2\gg 1$largebutfiniteto$[U(2)\times U(1)]/Z_2^2 \equiv [SO(3)\times U(1)\times U(1)]/Z_2$for$N_a^2\rightarrow\infty$. Moreover, we argue that the spin effective lattice being identical to the original lattice is a necessary condition for the occurrence of ground-state long-range antiferromagnetic order in the limit $N_a^2\to \mathrm{\infty }$. Consistently, strong evidence is provided that for very small hole concentration $0<x\ll 1$ the ground state has a short-range incommensurate-spiral spin order. Our results are of interest both for condensed-matter systems and ultra-cold fermionic atoms on an optical square lattice.