The comparison of ground-state energy E_g (left), double occupancy d (middle), and bond order of physical fermions χ (right) of the 2-site Hubbard model. (a)–(c) Slave-spin mean-field (MF) method (green circle) and analytical solution (red square) in half-filling situation. Besides, the bond order of pseudo-fermions (cyan circle) is also demonstrated in panels (g)–(i). (d)–(f) Slave-spin MF method (green circle) and analytical solution (red square) without particle–hole symmetry. (g)–(i) Slave-spin method with gauge constraints (GC) (blue triangle) and analytical solution (red square) without particle–hole symmetry. (j)–(l) Slave-spin method without GC (yellow diamond) and analytical solution (red square) without particle–hole symmetry.