# Hidden Charge Orders in Low-Dimensional Mott Insulators
## 2 Non-Local Orders in Interacting Low Dimensional Insulators  
### Sine Gordon Hamiltonian in One Dimension  
The low-energy physics of a wide class of 1D interacting Fermionic systems at half band filling is characterized by spin-charge separation. This is well captured by an effective Hamiltonian consisting of two decoupled sine-Gordon models describing the charge and spin channels, respectively:  
\[\mathcal{H}_{SG}=\sum_{\
u=c,s}\int dxH_{\
u}(x) \tag{1}\]  
with  
\[H_{\
u}=H_{0\
u}+\frac{m_{\
u}v_{\
u}}{2\pi a^{2}}\cos(\sqrt{8\pi}\phi_{\
u}) \tag{2}\]  
and  
\[H_{0\
u}=\frac{v_{\
u}}{2}\Big{[}K_{\
u}(\partial_{x}\theta_{\
u})^{2}+\frac{ 1}{K_{\
u}}(\partial_{x}\phi_{\
u})^{2}\Big{]} \tag{3}\]  
Here, \(\Pi_{\
u}=-\partial_{x}\theta_{\
u}\) is the conjugate momentum of the field \(\phi_{\
u}\). The mass coefficients \(m_{\
u}\), the Luttinger parameters \(K_{\
u}\) and the velocities \(v_{\
u}\) depend on the coupling constants and the filling of the corresponding microscopic lattice Hamiltonian [9; 11]. Each channel \(\
u=c,s\) of the model in Equation (2) is characterized by a competition between the \(H_{0\
u}\) term in Equation (3), which describes an harmonic oscillator with fluctuating field \(\phi_{\
u}\), and the cosine term, which would favor the pinning of the field around a locked value. Such competition leads to the appearance of different physical phases, as described below.  
The microscopic lattice Hamiltonian corresponding to Equation (1) is formulated instead in terms of Fermionic operators \(c_{j\sigma}\), with \(j\) denoting the lattice site and \(\sigma=\uparrow,\downarrow\) the spin. The connection between the continuum model in Equations (1) and (2) and the underlying lattice models relies on the idea that in most relevant situations the physically interesting energy range of a lattice model is small as compared to the whole bandwidth. It is then customary to introduce an effective model that is equivalent to the lattice model in the low-energy sector. To this purpose, one first rewrites the starting lattice Hamiltonian in terms of four Fermionic fields \(\psi_{a\sigma}(x)\) on the continuum, with \(\alpha=\pm 1\) referring to the two Fermi points \(\pm k_{F}\), related to the lattice operators via \(\psi_{a\sigma}(x)=\sum_{k\simeq ak_{F}}\varepsilon^{ikx}c_{kv}\), where \(c_{kv}\) are the Fourier components of the original lattice operators \(c_{j\sigma}\). Then, by adopting the bosonization formalism [9; 11], one re-expresses the four Fermionic fields as vertex operators \(\psi_{a\sigma}=\eta_{a\sigma}\exp\left[i\sqrt{\frac{\pi}{2}}(\alpha\phi_{c}+ \alpha\sigma\phi_{s}+\theta_{c}+\sigma\theta_{s})\right]/\sqrt{2\pi a}\), where \(\phi_{v},\theta_{v}\) (with \(v=c,s\)) is a pair of bosonic fields related to charge (c) or spin (s) degrees of freedom, \(a\) is a short-distance cut-off, and \(\eta_{a\sigma}\) are anti-commuting Klein factors. By applying this method, e.g. to the Hubbard model, which describes the competition of a band \"hopping\" term and an \"on-site\" two-body Coulomb repulsion with coefficient \(U\), the model in Equations (1) and (2) is obtained, with \(m_{v}\propto U\). The same model is found for a wide class of extended Hubbard-like models that include further interaction beyond the on-site approximation, such as nearest neighbor diagonal interaction, correlated hopping, pair hopping and so on [12; 13; 14; 15], as well as three- and four-body interactions [6] that may be relevant in ultracold atoms or molecules trapped in optical lattices.