# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 1 Introduction  
As predicted by sir [PERSON] almost seventy years ago [1], interaction is capable of opening energy gaps in bands of otherwise conducting materials, so that at sufficiently low temperature they become insulating. The microscopic arrangement of the electronic degrees of freedom consistent with such insulating phase, as well as the corresponding macroscopic orders, have been more elusive. In fact, classical local order parameters, such as magnetic or charge order, are often vanishing both in the metallic and in the insulating state of low-dimensional insulators. This is the case for instance for the Mott phase of the paradigmatic Hubbard model. In one dimension, and at zero temperature, it takes place for any non-zero value of the on-site Coulomb repulsion. Only recently [2] it was proved that, despite the absence of a local order parameter, the insulating phase is characterized by the ordering of a non-local (NL) parity operator. The result has been successively extended also to ladders [3], and to the two-dimensional bosonic case [4]. For Fermions in 1D, it was also proved on more general grounds by means of a bosonization analysis and a renormalization-group (RG) study of the continuum limit that, in the case of spin-charge separation and local interaction, in each (spin or charge) channel, the two possible gapped phases are univocally associated to the non-vanishing of the expectation value of one of two hidden NL operators: string or parity [2; 5; 6]. The result provides a clear insight into the basicmicroscopic structure of the possible insulating phases induced by interaction through the opening of a gap in the charge channel. For instance, the non-zero expectation value of the NL parity tells us that the Mott insulating phase of the Hubbard model amounts to a uniform background of singly occupied sites in which empty (holons) and doubly occupied (doublons) sites are charge fluctuation appearing only in correlated pairs of finite coherence length. In addition, a more peculiar \"topological\" Haldane-like ordering of holons and doublons can occur in the insulating phase of other lattice models, and is characterized by the non-vanishing of the NL string order. The above bosonization predictions apply to short-ranged (on-site or local) interactions, and can be verified in ultracold systems of Fermionic atoms trapped onto optical lattices. In particular, NL order parameters can be measured by means of in-situ imaging techniques, as in [7; 8]. For more general applications in both condensed matter and atomic physics, it would be interesting to understand how such orders and phases are modified by the presence of a long-ranged interaction, i.e., an interaction decaying with distance \(r\) as \(r^{-\alpha}\), with \(\alpha>0\). This is the case for instance for both dipolar interaction (\(\alpha=3\)), and unscreened Coulomb repulsion (\(\alpha=1\)). In fact, when the decay is sufficiently fast (\(\alpha>1\)), it is expected that the phase diagram remains qualitatively the same [9], and a similar behavior is derived also for the non-local parameters [10]. However, it is not possible to obtain analogous analytic predictions for lower \(\alpha\)'s values. In this article, we first review the general classification of the possible one-dimensional insulating phases for Fermionic systems described by a double sine Gordon Hamiltonian in terms of hidden orders. We then summarize how insulating phases characterized by a parity order can also be found in two-dimensional systems, upon appropriate generalization to branes of fractional parity. We discuss how instead a finite value of solely the string charge operator is peculiar of the Haldane insulating phase and is accompanied by fractionalized edge modes in the charge channel. Finally, we specialize our discussion to the repulsive extended Hubbard model at half-filling.