Collapse of the vacuum in hexagonal graphene quantum dots: a comparative study between the tight-binding and the mean-field Hubbard models

In this paper, we perform a systematic study on the electronic, magnetic, and transport properties of the hexagonal graphene quantum dots (GQDs) with armchair edges in the presence of a charged impurity using two different configurations: (1) a central Coulomb potential and (2) a positively charged carbon vacancy. The tight binding (TB) and the half-filled extended Hubbard models are numerically solved and compared with each other in order to reveal the effect of electron interactions and system sizes. Numerical results point out that off-site Coulomb repulsion leads to an increase in the critical coupling constant to $\beta_{\text{c}}$ = 0.6 for a central Coulomb potential. This critical value of the $\beta$ is found to be independent of GQD size, reflecting its universality even in the presence of electron-electron interactions. In addition, a sudden downshift in the transmission peaks shows a clear signature of the transition from subcritical $\beta$ $<$ $\beta_{\text{c}}$ to supercritical $\beta$ $>$ $\beta_{\text{c}}$ regime. On the other hand, for a positively charged vacancy, the collapse of the lowest bound state occurs at $\beta_{\text{c}}$ = 0.7 for the interacting case. Interestingly, the local magnetic moment, induced by a bare carbon vacancy, is totally quenched when the vacancy is subcritically charged, whereas the valley splittings in electron and hole channels continue to exist in both regimes.