# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 4 Conclusions  
In this paper, we review some results on the capability of hidden non-local orders, described by string and parity operators, of giving an exhaustive characterization of the insulating states induced by short range interaction in low-dimensional Fermionic materials. We have seen how, in the absence of disorder, in one dimension the classification of the insulating phases by non-local  
Figure 6: Behavior of the non-local correlation functions in Equation (5) and the Luttinger liquid parameter \(K_{\rm s}\) for the Hamiltonian in Equation (13) with Coulomb interaction (\(\alpha=1\), \(\lambda=0\)) truncated to third nearest neighbors (\(R_{T}=3\)) at \(t=1\) and \(U=4t\). The simulations were performed with finite DMRG for a chain of length \(L=32\), by keeping up to 1200 states and performing six sweeps. The string operator was computed in the middle of the chain \(r=L/2\), while the parity operators were averaged between the two central points \(L/2\) and \(L/2+1\).  
Figure 7: Behavior of the non-local correlation functions in Equation (5) and the Luttinger liquid parameter \(K_{\rm s}\) for the Hamiltonian in Equation (13) with screened Coulomb interaction (\(\alpha=1\), \(\lambda=0.6\)) truncated to fifth nearest neighbors (\(R_{T}=5\)) at \(t=1\) and \(U=4t\). The simulations were performed with finite DMRG for a chain of length \(L=24\), by keeping up to 1400 states and performing eight sweeps. The string operator was computed in the middle of the chain \(r=L/2\), while the parity operators were averaged between the two central points \(L/2\) and \(L/2+1\).  
orders is possible thanks to the characteristic spin-charge separation of the degrees of freedom involved. Moreover, we discuss how the insulating state associated to the parity operator, which is trivial from the topological point of view, persists in two dimensions and is still characterized by a generalized parity operator: the physics remains the same. In addition, we show that instead the insulating states associated to the ordering of the string operators are characterized in one dimension by non-trivial topological properties, notably the non-unity, degenerate, fractional and entangled edge charges.  
Finally, we applied the previous framework to a preliminary study of the effect of various long ranged interaction added to the extended Hubbard model. The study was performed at finite size and a representative value of the on-site interaction \(U=4t\), half-filling and zero magnetization. By means of DMRG numerical simulations, we investigated the changes in the behavior of the non-local parameters and phase transition lines, by varying the density-density interaction scale in the case of dipolar, screened and unscreened interactions, respectively. Our results suggest that all three phases of the EHM (Mott, BOW, and CDW) are robust in the presence of an interaction range beyond one lattice cell, both transitions still occurring in proximity of the value \(U=2V\). Moreover, the non-local parity and string charge operators are still capable of describing the transitions.  
We observed that, in the case of dipolar interaction, a further increase of the truncation distance \(R_{T}\) is not believed to affect sensibly the phase diagram. The same can be expected in the case of screened Coulomb interaction for suitable choices of \(\lambda\). On the contrary, different results may be observed if an unscreened long-range Coulomb repulsion is considered, upon including a sufficiently large number of neighbors. Further investigations go beyond the aim of this review.  
Both authors have contributed equally to the research. This research received no external funding. The authors declare no conflicts of interest.