# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 3 Insulating Phases of the 1D Hubbard Model Extended by Long-Range Interaction
### Results of DMRG Simulations  
We applied the DMRG method [33,34] to observe the behavior of NL correlators in the Hamiltonian in Equation (13). In particular, we considered the case of the dipolar interaction (\(\alpha=3\), \(\lambda=0\)) and that of the Coulomb interaction (\(\alpha=1\)), both in the presence (\(\lambda\
eq 0\)) and in the absence (\(\lambda=0\)) of screening. The numerical simulations were performed on a chain of \(L=32\) sites with periodic boundary conditions, by keeping up to 1200 DMRG states and performing six sweeps to achieve convergence with a truncation error of \(5\times 10^{-6}\). Moreover, the long range interaction was truncated to the distance \(R_{T}=3\). The results, in the three cases, are shown in Figures 4-6, respectively, for fixed \((t,U)=(1,4t)\) and changing \(V\). In particular, the left side of each figure shows the charge correlators, while the right side refers to the spin channel. Here, we do not display the spin string,  
Figure 3: Phase diagram of the extended Hubbard model. which was always zero; instead, in addition to the spin parity \(\mathcal{C}_{\mathcal{P}}^{\rm s}\), we computed the Luttinger liquid parameter \(K_{\rm s}\) defined as  
\[K_{\rm s}=\frac{4\pi}{L}\lim_{q\to 0}\frac{1}{q}\sum_{i,j}e^{iq(i-j)}\langle s_{ si}s_{sj}\rangle\quad. \tag{14}\]  
The charge string correlator \(\mathcal{C}_{\mathcal{S}}^{c}(r)\) (first panel) was computed at the maximum distance, i.e., in the middle of the chain \(r=L/2\). Instead, because of a possible staggered behavior, the parity correlators were averaged over the sites \(L/2\) and \(L/2+1\). In all cases, the results for the NL correlators support the presence of the three already mentioned phases, namely Mott, BOW and CDW, which can be inferred from the results in Figures 4-6, by taking into account the classification outlined in Table 1. The computation of \(K_{\rm s}\) is helpful for identifying the transitions. In particular, the Mott-BOW transition, which involves the opening of the spin gap, is signaled by \(K_{s}=1\). We observed that, in all the figures, the BOW phase appears slightly shifted from the point \(U\approx 2V\) towards higher values of \(V\); thus, the long-range interaction basically does not alter the phase diagram of the EHM. We noticed that, in the case of screened Coulomb repulsion, the amplitude of the interaction must be scaled by the factor \(e^{-\lambda}\) in order to compare the result with the EHM. We chose \(\lambda=0.7\) in such a way that the contribution from the fourth nearest neighbors, which we neglect, is of the same order as the contribution neglected for dipolar interaction. Such a choice should make the screened Coulomb interaction comparable to the case \(\alpha>1\), for which it is known that the long-range interaction does not qualitatively modify the phase diagram [9]. We verified that the results are barely affected by the truncation of the long range interaction. To this end, we considered the screened Coulomb interaction truncated to \(R_{T}=5\). In that case, we assumed, with the same criterion, \(\lambda=0.6\). Data are shown in Figure 7. We performed the computation for a system of size \(L=24\), with up to 1400 DMRG states and eight sweeps. We observed that the results are comparable with those obtained for the case \(R_{T}=3\). In fact, the BOW phase appears closer to the point \(U=2V\); however, this small difference could also be caused either by finite size effects or by the difficulty of DMRG in reaching convergence despite the eight sweeps. On the other hand, in the case of dipolar interaction, one may infer, by comparison with the work in [35], where up to five neighbors are included, that the transition lines are not much affected by adding further neighbors to the interaction or by increasing the system size from \(L=32\) to the thermodynamic limit. These findings complement those reported in the previous literature [36; 37].