Charge-Ordered States and the Phase Diagram of the Extended Hubbard Model on the Bethe lattice

TL;DR

The paper analyzes the extended Hubbard model on a Bethe lattice using a broken-symmetry Hartree mean-field approach to map ground-state and finite-temperature phase diagrams. It identifies three principal phases—charge-ordered insulator (COI), charge-order metallic (COM), and non-charge-ordered (NO)—and derives analytical expressions for key states at $T=0$, complemented by finite-temperature transitions governed by the interplay of on-site $U$ and nearest-neighbor $V$ interactions. Spectral functions and a conduction-carrier measure $c$ are employed to distinguish insulating versus metallic behavior, revealing how increasing $U$ suppresses charge order and can drive insulator-to-metal transitions, with reentrant effects and phase-separation regions captured by Maxwell constructions. The results offer a clear, analytically tractable view of correlation-induced charge ordering, provide a pedagogical benchmark against DMFT, and illustrate the utility and limitations of MFA for ordering phenomena in strongly correlated lattice models.

Abstract

We study the extended Hubbard model (EHM) with both onsite Hubbard interaction and the intersite density-density interaction between nearest-neighbors using the standard Hartree mean-field approximation (MFA) on the Bethe lattice. We found that, at the ground state, the system can be in a charge-ordered insulating (COI), a charge-order metallic (COM) or a non-charge-ordered (NO) state. Moreover, the finite-temperature phase diagrams are presented. Several observables like a charge-order parameter, a spectral function, and particularly at finite temperatures, a charge carrier concentration (to visualize the degree of metallicity) are analyzed. The results show that increasing onsite repulsion suppresses charge order and change the properties of the system from insulating to metallic. Worth noting, that a number of phenomena can be found within the MFA, where their analysis is much simpler than in more advanced approaches. The method used for the EHM on the Bethe lattice also allows for a series of analytical derivations and simplification to see general geometry-independent features and analytical results, avoiding the numerical inaccuracies and other issues that appear with a purely numerical solution.

Figures (5)

• Figure 1: Spectral functions for the ground state. (a) and (b) are spectral functions of the non-charge-ordered (NO, $n=0.597$, $\Delta=0$) metal and charge-ordered insulator (COI, $n=1$, $\Delta=0.865$) for the same point of the phase diagram where their grand potentials ($\Omega$) are equal (the point on a discontinuous phase transition line). (c) is a spectral function evolution of the COI phase at $\bar{\mu}=0D$ when $zV - \frac{U}{2}$ approaches $0D$ (the continuous transition to the half-filled NO phase).
• Figure 2: Ground-state phase diagrams and properties. (a) $\bar{\mu}$-$V$ phase diagram for $U=0D$; (b) dependencies of COI-phase properties; (c) evolution of the $\bar{\mu}$-$U$ phase diagram when $V$ changes; (d) $\bar{\mu}$-$U$ phase diagram for $V=1.5D$. On the phase diagrams: dashed and solid black lines denote continuous and discontinuous transitions, respectively; color scales are for concentration $n$, charge polarization $\Delta$ and spectral function $A(i\eta)D$ (as labeled). The white region on the $A(i\eta)D$ color-contour plot corresponds to the model parameters, where the value of $A(i\eta)D$ is far out of a range of the presented colorbar (it is unrepresentative to extend the colorbar range), the corresponding phase is the COM phase. Only the phases with the lowest grand potential are shown (no metastable phases).
• Figure 3: Tendencies of the ground-state critical points: a relation between the interaction strengths of the critical points, alongside with a point of the continuous COI-NO transition at $\bar{\mu}=0D$ (a); and dependencies of the interaction strengths of the critical points on the shifted chemical potential (b) and the electron density of the NO phase (c), alongside with a point of the discontinuous COI-NO transition at $U=0D$ ($V_\text{c}$, Fig. \ref{['fig:GS']}a).
• Figure 4: Finite-temperature phase diagrams. (a) the $(zV-\frac{U}{2})$-$T$ phase diagram for $\bar{\mu}=0D$; (b) the evolution of the $\bar{\mu}$-$T$ phase diagram when the interaction strengths change; (c) the $\bar{\mu}$-$T$ phase diagram for $U=0D$ and $V=1.5D$. On the phase diagrams: dashed and solid black lines are for continuous and discontinuous transitions, respectively; color scales are for concentration $n$, charge polarization $\Delta$ and charge carrier concentration $c$ (as labeled). Both short and long dashed lines on panel (b) denote continuous transitions and are different to distinguish the $U=0D$ lines and the $zV-\frac{U}{2}=1D$ lines. Only the phases with the lowest grand potential are shown (no metastable phases).
• Figure 5: Tendencies of the finite-temperature tricritical point for $U=0D$: relation between the intersite interaction strength and temperature of the tricritical point, alongside with a point of the continuous CO-NO transition at $\bar{\mu}=0D$ (eq. (\ref{['eq:finT-halffil']}), Fig. \ref{['fig:finT']}a) (a); and dependencies of the intersite interaction strength and temperature of the tricritical point on the shifted chemical potential (b) and the electron density of the NO phase (c), alongside with a point of the discontinuous COI-NO transition at $T=0D$ ($V_\text{c}$, Fig. \ref{['fig:GS']}a).