# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 2 Non-Local Orders in Interacting Low Dimensional Insulators
### Classification of Insulating Phases  
For the above reasons, the sine-Gordon model in Equations (1)-(2) plays a crucial role in the characterization of 1D Fermion lattice models. The result of the competition of the two terms appearing in Equation (2) depends on the sign and the magnitude of the mass term \(m_{v}\) in each channel \(\
u\), and can be established by a well known RG analysis of the sine-Gordon model [11]. In brief, for \(4(\sqrt{K_{v}}-1)<|m_{\
u}|\), the cosine term becomes relevant, and a gap opens in the related channel \(\
u=c,s\), with the field \(\phi_{v}\) pinned at either \(0\) (for \(m_{v}<0\)) or \(\pm\sqrt{\pi/8}\) (for \(m_{\
u}>0\)). The case where no gap opens in both channels corresponds to the Luttinger liquid (LL) phase, where \(\phi_{c}\) and \(\phi_{s}\) are both unpinned and the effective Hamiltonian is \(H_{LL}=\sum_{\
u}\int H_{0\
u}\). The remaining eight phases, are characterized by a gap in at least one channel. To classify all possible insulating phases, one should focus on the cases in which the charge gap is open, i.e., the charge field is pinned. These are summarized in Table 1, with the pinning value of the fields provided in the first and second columns.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\sqrt{8\pi}\phi_{c}\) & \(\sqrt{8\pi}\phi_{s}\) & NLO & SP \\ \hline Mott & \(0\) & \(u\) & \(\mathcal{O}_{P}^{c}\) & triv \\ HI & \(\pm\pi\) & \(u\) & \(\mathcal{O}_{S}^{c}\) & P, T \\ BOW & \(0\) & \(0\) & \(\mathcal{O}_{P}^{c}\), \(\mathcal{O}_{P}^{c}\) & triv \\ CDW & \(\pm\pi\) & \(0\) & \(\mathcal{O}_{S}^{c}\), \(\mathcal{O}_{P}^{c}\) & P \\ SDW & \(0\) & \(\pm\pi\) & \(\mathcal{O}_{P}^{c}\), \(\mathcal{O}_{S}^{c}\) & P, T \\ BSDW & \(\pm\pi\) & \(\pm\pi\) & \(\mathcal{O}_{S}^{c}\), \(\mathcal{O}_{S}^{c}\) & P \\ \hline \hline \end{tabular}
\end{table}
Table 1: Left-hand side: Classification of the insulating phases for \(\mathcal{H}_{SG}\) at half-filling. The first column denotes the two possible pinning values for the charge (\(\phi_{c}(x)\)) field. The second gives the value of the spin field, with \(u\) standing for “unpinned”. The third column denotes the non-vanishing NL order parameters \(\mathcal{O}_{A}^{\
u}\), \(\
u=c,s\). Right-hand side: The phase without topological features are denoted by “triv”, whereas we report the type of symmetry (T and or P) protecting the phases with degenerate edge modes.  
In the table, the denomination of the different insulating phases is borrowed form the one adopted in the case of some correspondent lattice model. Indeed, it is possible to see [9] that the phase denoted here as Mott is the one characteristic of the repulsive Hubbard model at half-filling, with no local order and dominant spin density wave (SDW) correlations. Similarly, the other partly gapped insulating phase, which we denote as Haldane insulator (HI) for reasons clarified below, corresponds to no local order and dominant CDW correlations. Both are characterized by the presence of a gap in only the charge channel. In contrast, BOW, CDW, spin density wave (SDW), and bond spin density wave (BSDW) are fully gapped phases, also characterized by a gap in the spin channel and by a finite local order parameter. In these latter cases, in the ground state, a spontaneous symmetry breaking can be observed. The explicit expression of the parameters in terms of the coupling constants of the microscopic Hubbard-like models are given in previous works [6; 12; 13; 16] and are not reported here. As an example, in the next section, we discuss in more details the lattice model with diagonal density-density interaction, also known as extended Hubbard model.