# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 2 Non-Local Orders in Interacting Low Dimensional Insulators
### Fractional Parity Orders in Trivial Low Dimensional Mott Insulators  
The possibility to generalize to higher dimension the Haldane insulating phase, with its non-trivial topological nature, strongly relies on the lattice geometry. In particular, in the case of spinful electrons on a square lattice, it is not possible to obtain such topological phase. It is much easier instead to generalize to higher dimension the trivial Mott insulating phase, characterized in the one dimensional case by a non-vanishing value of the charge parity operator. As already mentioned, its physics amounts to a background of singly occupied sites, in which holons and doublons occur in entangled pairs with a finite coherence length. These in principle can occur in arbitrary dimension. A two-dimensional example is depicted in the cartoon of Figure 2. In presence of such physics, the parity operator must be properly generalized in order to remain non-vanishing in the insulating phase, and zero elsewhere. Let us start with the case of a \(M\) legs ladder: rather than considering a string of site parities delimited by two boundary sites, as in the definition in Equations (4) and (5), one now takes _barnes_ of sites delimited by boundary strings, as proposed in [23]. Explicitly, we can introduce:  
\[O_{P}(r,M)=\prod_{0\leq x<r}\prod_{0\leq y<M}\mathcal{P}^{c}_{x,y} \tag{10}\]  
Figure 1: Cartoon of the Haldane phase, showing the presence of two degenerate states with fractional edge modes. The Haldane order consists of the superposition of all possible configurations with a string of alternated holons (empty blue circles) and doublons (patterned red circles) diluted in a background of single electrons with arbitrary spin orientation (black lines). For an open chain, the string starting with the doublon (**top**) and that starting with the holon (**bottom**) are distinct states. Therefore, each edge acquires a fractional charge given by the superposition of either doublons and single occupations or holons and single occupations. where \(x,y\) denote the coordinates of sites. Even with the above generalization of the parity operator, an increase of the length of the boundary turns out to diminish its average value also in the insulating phase. In fact, the longer the boundary, the higher the number of holon-doublon pairs divided by it, which are responsible for the switch of the parity sign. Such effect has been quantified as a \"perimeter law\": the parity decreases to zero in the two-dimensional limit \(M\to\infty\) with \(e^{-\pi bM}\), where \(b\) is a positive constant and \(M\) is the length of the boundary. To overcome such difficulty, a fractional generalization of the brane parity was introduced [4], in which the site parities are multiplied by a fractional angle \(\pi/M^{\alpha}\), instead of \(\pi\) characteristic of the one-dimensional case. Explicitly, one defines a generalized brane parity operator as:  
\[O_{p}^{(\alpha)}(r,M)\equiv[O_{p}(r,M)]^{(\frac{1}{M})^{\alpha}}\,, \tag{11}\]  
where \(\alpha\) possibly depends on the model Hamiltonian. It turns out that the insulating phase of the two-dimensional Hubbard model is characterized by a non-vanishing value of its expectation value. More precisely, by first taking \(r=L/2\) and then the limit \(L\to\infty\), one can prove that:  
\[C_{p}^{(\alpha)}\doteq\lim_{M\to\infty}\lim_{L\to\infty}\langle O_{p}^{( \alpha)}(L/2,M)\rangle \tag{12}\]  
is finite, solely in the MI phase, for \(\alpha\geq\frac{1}{2}\), in the bosonic case. In fact, it is exactly equal to \(1\) in case the inequality is strict, whereas its finite value depends on the Coulomb repulsion \(U\) for \(\alpha=1/2\). A similar result holds also for the Fermionic 2D Hubbard model at half filling [24]. Noticeably, in the latter case, it is found that also the fractional spin parity, defined in full analogy with Equation (10) upon replacing \(\mathcal{P}^{c}\to\mathcal{P}^{s}\), remains finite.