# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 2 Non-Local Orders in Interacting Low Dimensional Insulators
### Non-Local Orders  
The standard way of characterizing the physical features of different ordered phases is to analyze the asymptotic behavior of correlation functions of appropriate local operators. In 1D, they asymptotically decay to zero with a power or exponential law: the \"order\" of the phase is described by the correlation function which decays slower, except when the ground state spontaneously breaks some discrete symmetry of the Hamiltonian. Only in the latter case, one of the correlation functions remains finite, thus capturing the long range order of the phase through a local operator, whose expectation value can be regarded to as an order parameter. Recently it has been realized [2; 5; 17] that, even when the Hamiltonian has only continuous internal symmetries, non-local operators may order in some phases, meaning that their average value is non-zero in these phases and vanishes at the phase transition. Importantly, such operators turn out to be non-local in the lattice representation, which is consistent with the absence of spontaneous symmetry breaking. They can be built by means of the on site charge and spin density operators \(s_{vj}\), here defined as \(s_{cj}=(n_{j\uparrow}+n_{j\downarrow}-1)/2\), and \(s_{sj}=(n_{j\uparrow}-n_{j\downarrow})/2\), \(n_{j\sigma}\doteq c^{\dagger}_{j\sigma}c_{j\sigma}\) being the Fermion number operator at site \(j\) with spin \(\sigma\). Upon introducing the on-site parity \(P^{v}_{j}\doteq\mathrm{e}^{2\pi iS_{vj}}\) (\(P_{v}=\pm 1\)), one can finally define two NL operators as  
\[\mathcal{O}^{v}_{\mathcal{P}}(r)=\prod_{j=1}^{r}P^{v}_{j}\quad,\quad\mathcal{O }^{v}_{\mathcal{S}}(r)=2s_{v\tau}\prod_{j=1}^{r-1}P^{v}_{j} \tag{4}\]  
with asymptotic correlation functions  
\[\mathcal{C}^{v}_{\mathcal{A}}=\lim_{r\to\infty}\langle\frac{\mathcal{O}^{v \dagger}_{\mathcal{A}}(0)\mathcal{O}^{v}_{\mathcal{A}}(r)}{c^{v}_{\mathcal{A }}(r)}\rangle \tag{5}\]  
with \(\mathcal{A}=\mathcal{P},\mathcal{S}\). In each channel, the operators \(\mathcal{O}^{v}_{\mathcal{A}}(r)\) are better known as parity (\(\mathcal{A}=\mathcal{P}\)) and string (\(\mathcal{A}=\mathcal{S}\)) operators, respectively. It turns out that the non-vanishing of the average of one of the two operators in the charge channel, together with the behavior of their companions in the spin channel are able to capture all the possible insulating phases that may be induced by interaction in these systems. This is shown in column NLO of Table 1, where for each phase it is reported which NL operators are non-vanishing. The above results may be understood in the continuum limit by evaluating the NL operators in Equation (4) in terms of the bosonic fields. They read [2; 17]:  
\[\mathcal{O}^{v}_{\mathcal{P}}\sim\cos(\sqrt{2\pi}\phi_{v})\quad,\quad\mathcal{ O}^{v}_{\mathcal{S}}\sim\sin(\sqrt{2\pi}\phi_{v})\quad. \tag{6}\]Unlike the case of local operators, each of the above NL operators is associated to a single bosonic field. Thus, while their averages, as well as the correlation functions in Equation (5), are all vanishing when both fields are unpinned (LL phase), if some field is locked to a fixed value, at least one NL operator has non-zero average. More precisely, in each channel, when the field \(\phi_{\
u}\) is pinned to one of the two possible values in Table 1, the average of one of the two corresponding NL operators becomes finite, whereas the other remains identically zero. It is straightforward to verify that in both cases just one of the correlation functions \(\mathcal{C}_{\mathcal{P}}^{\
u}\propto\langle\cos\sqrt{2\pi}\phi_{\
u}\rangle^ {2}\), \(\mathcal{C}_{\mathcal{S}}^{\
u}\propto\langle\sin\sqrt{2\pi}\phi_{\
u}\rangle^ {2}\) remains finite [2], whereas the other one still vanishes. The average values of \(\mathcal{O}_{\mathcal{A}}^{\
u}\) (\(\mathcal{A}=\mathcal{P},\mathcal{S}\)) are thus identified as order parameters in the \(\
u\) channel for the two possible gapped phases.