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with L$_{x}$ unit cells. We study several lengths of the cluster with up to L$_{x}$ = 8 ( 78 sites) and extrapolate the finite-size results to the thermodynamic limit L$_{x}$ → ∞ . The single-particle gap is thus given by Δ = lim$_{L}$ x $_{→∞}$Δ( L$_{x}$ ) with Δ( L$_{x}$ ) = E ( N$_{f}$ + 1 , L$_{x}$, L$_{y}$ ) + E ( N$_{f}$ − 1 , L$_{x}$, L$_{y}$ ) − 2 E ( N$_{f}$, L$_{x}$, L$_{y}$ ) , where E ( N$_{f}$, L$_{x}$, L$_{y}$ ) is the ground-state energy of the cor- responding cluster with N$_{f}$ fermions. The upper panels of Figure 6 show the extrapolated values of Δ as a function of V in units of | t | . The critical points are estimated as V$_{c}$ ∼ 2 . 6 and 4 . 0 for t > 0 and t < 0 , respectively. The behaviors of the gap opening are quite different: the gap for t > 0 in- creases almost linearly, while it rises gradually for t < 0 like the Berezinskii-Kosterlitz-Thouless transition. 

In order to gain further insight, we also calculate the or- der parameter corresponding to the CO pattern III. The CO phenomenon is observed as a state with a broken transla- tional symmetry: actually, there are two or more degenerate ground states and one configuration of the degenerate states is picked out as the ground state by an initial condition of the DMRG calculation, as a consequence of the cylindri- cal boundary conditions [14]. The order parameter is now defined as the amplitude of the charge-density modulation η ( L$_{x}$ ) = n$_{1}$ − 1 $_{2}$( n$_{2}$ + n$_{3}$ ) for a L$_{x}$ × 3 cluster, where n$_{1}$ and n$_{2}$ ( = n$_{3}$ ) are the charge densities of fermion-rich and fermion-poor sites on a triangle at the center of the cluster. In the lower panels of Figure 6, we show the extrapolated re- sults of order parameter η to the thermodynamic limit, i.e., η = lim$_{L}$ x $_{→∞}$η ( L$_{x}$ ) , as a function of V . We find that the estimated critical points are almost the same as those obtained from Δ for both signs of t . As seen similarly for Δ ., the be- havior of η differs markedly depending on the sign of t. For t > 0 , the transition appears to be nearly discontinuous, which might be analogous to that in a melting MIT observed in hard- core bosons on the triangular lattice [15]; for t < 0 , it looks continuous, which looks like a weakly first-order phase tran - sition for hard-core bosons on the Kagom´e lattice [8]. Hence, we conclude that the criticality for positive t is in stark con- trast to that for negative t . 

Now, we comment on the difference in values of the MIT critical points between t > 0 and t < 0 . As mentioned above, all the triangles are only allowed to be singly occupied in the 

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CO state. Therefore, our system can be mapped into a single- band half-filled Hubbard model on honeycomb lattice with on-site Coulomb interaction O ( V ) if we regard each trian- gle as a single site. The single-particle gap is then expected to behave like Δ ≈ V − W$_{eff}$ in the large- V region, where W$_{eff}$ is the effective bandwidth. Actually, as seen in Fig. 6, the gap behaves like Δ ∼ V − 5 t and Δ ∼ V − 8 | t | for t > 0 and t < 0 , respectively. In general, the MIT critical strength is roughly scaled by the bandwidth, i.e., V$_{c}$ ∝ W$_{eff}$ . The as- sumption of the effective bandwidths W$_{eff}$ = 5 t for t > 0 and 8 | t | for t < 0 are compatible with the obtained critical points. The narrower bandwidth for t > 0 might be interpreted as a remnant of the flat band in the noninteracting case [16]. 

Random phase approximation — In addition, the MIT is in- vestigated analytically using the RPA [17]. The charge sus- ceptibility is given by the Kubo formula as 

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