be seen in Table \[4x4\], the $\mathcal{PQGT}'$ results are about $1\%$ below the exact result. As expected, just imposing the $\mathcal{PQG}$ constraints does not give satisfactory results. We also compare our v2DM results to those obtained by Anderson [*et al.*]{} in [@nakata_2dhub], and it is shown that they correspond.

In Table \[6x6\] the $\mathcal{PQGT}'$ results for the ground-state energy per particle on a $6\times6$ lattice for different filling factors and on-site repulsion $U$ are given for future reference. They provide rigourous lower bounds on the energy. For the $6\times6$ lattice with 36, 34 and 30 particles we can compare with a variational upper bound on the energy obtained with a two-site $SU(2) \otimes U(1)$ spin- and particle number-adapted DMRG code developed by one of us [@sebastian], with 7500 reduced renormalized basis states at each boundary. Note that this corresponds to a much larger number of effective renormalized basis states, as in a spin-adapted DMRG code only one basis state per multiplet is retained.

In Figures \[N36\], \[N34\], and \[N30\] the v2DM results obtained with both the $\mathcal{PQG}$ and $\mathcal{PQGT}'$ conditions are compared to the DMRG results. We notice that for half filling, the $\mathcal{PQGT}'$ and DMRG results are very close. 