# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 3 Insulating Phases of the 1D Hubbard Model Extended by Long-Range Interaction  
### The Extended Long-Range Hubbard Model  
The one-dimensional extended Hubbard Model (EHM) is a lattice Hamiltonian describing the competition of kinetic energy and diagonal on-site (\(U\)) and nearest neighboring sites (\(V\)) density-density Coulomb interaction. Its applications range from high-\(T_{c}\) superconductors [25], to conducting polymers [26], organic charge-transfer salts [27], and ultracold atomic gases ([10] and references therein).  
The full EHM phase diagram has been studied for specific fillings by means of different techniques [28; 29; 30; 31; 32]. In particular, the repulsive interaction regime (\(U,V>0\)) has been intensively investigated due its physical relevance, and has a very rich structure, reported qualitatively in Figure 3 at half-filling. In this case, it amounts to three different insulating phases: a Mott insulator (Mott) characterized by dominant spin density waves, which takes place for \(U\) sufficiently greater than \(2V\); a phase with BOW order [32], which appears for \(U\approx 2V\); and a CDW phase, for \(U\) much smaller  
Figure 2: Cartoon of the 2D Mott insulator.  
than \(2V\). While both BOW and CDW phases are fully gapped, and are thus associated to the ordering of appropriate local operators, the Mott insulating phase has no local order parameter. Remarkably, as shown in [5,14], NL operators are able to capture accurately and distinguish all three insulating regimes, as well as the transitions among them.  
It is thus natural to ask how including terms further than nearest neighbors in the interaction, such as in the case of dipolar or Coulomb interaction, will affect the behavior of NL operators, as well as whether they will still behave as order parameters for the three distinct phases. To this end, we consider a generalized one-dimensional extended Hubbard model with \(L\) Fermions trapped in \(L\) lattice sites (half-filling), described by the following Hamiltonian  
\[H = -t\sum_{j,\sigma}\left(c^{\dagger}_{j\sigma}c_{j+1\sigma}+h.c. \right)+U\sum_{j}n_{j\uparrow}n_{j\downarrow} \tag{13}\] \[+ V\sum_{i,j}^{\prime}\frac{\mathrm{e}^{-\lambda|i-j|}}{|i-j|^{ \alpha}}n_{i}n_{j}\quad,\]  
where \(\sigma=\uparrow,\downarrow\) is the spin index; \(c^{\dagger}_{j\sigma}\) and \(c_{j\sigma}\) are the Fermionic creation and annihilation operators, respectively; \(n_{j\sigma}=c^{\dagger}_{j\sigma}c_{j\sigma}\) counts the number of particles with spin \(\sigma\); and \(n_{j}=\sum_{\sigma}n_{j\sigma}\). The coupling coefficients \(t,U\) describe the hopping process, and the on-site interaction, respectively, whereas \(V\) is the overall scale of the density-density interaction between further sites. Moreover, \(\lambda\) is its screening coefficient, and \(\alpha\) is the exponent of its power law decay. Finally, the \({}^{\prime}\) in the summation is reminiscent of the fact that in the numerical simulations it is limited to sites up to a given truncation distance \(R_{T}\) (in units of lattice constants): in other words, the summation is over pairs of sites \((i,j)\) obeying the condition \(1\leq|i-j|\leq R_{T}\in\mathbf{N}\). For instance, the standard extended Hubbard model discussed at the beginning of this section amounts to the choice \(R_{T}=1\), \(\lambda=0\).