  We describe here a quantum simulator of extended bipartite Hubbard model with
broken sublattice symmetry. The simulator consists of a structured lateral gate
confining two dimensional electrons in a quantum well into artificial minima
arranged in a hexagonal lattice. The sublattice symmetry breaking is generated
by forming an artificial triangular graphene quantum dot (ATGQD) with zigzag
edges. The resulting extended Hubbard model generates tunable ratio of
tunneling strength to electron-electron interactions and of sublattice symmetry
with control over shape. The validity of the simulator is confirmed for small
systems using mean-field and exact diagonalization many-body approaches which
show that the ground state changes from a metallic to an antiferromagnetic (AF)
phase by varying the distance between sites or depth of the confining
potential. The one-electron spectrum of these triangular dots contains a
macroscopically degenerate shell at the Fermi level. The shell persists at the
mean-field level for weak interactions (metallic phase) but disappears for
strong interactions, in the AF phase. We determine the effects of
electron-electron interactions on the ground state, the total spin, and the
excitation spectrum as a function of filling of the ATGQD. We find that the
half-filled charge neutral shell leads to a partially spin polarized state in
both metallic and AF regimes in accordance with Liebs theorem. In both regimes
a relatively large gap separates the spin polarized ground state to the first
excited many-body state at half filling of the degenerate shell. By adding or
removing an electron, this gap drops dramatically, and alternate total spin
states emerge with energies nearly degenerate to a spin polarized ground state.
