Coherent potential approximation for charge ordering in the extended Hubbard model

We study charge ordering (CO) in the extended Hubbard model with both on-site and nearest-neighbour Coulomb repulsion (U and V , respectively) within the coherent potential approximation. The phase boundary between the homogeneous and charge-ordered phases for the square lattice is obtained for different values of U . It is shown that at quarter-filling for all values of U the CO exists only if the inter-site Coulomb repulsion V exceeds certain a critical value which is of the order of the kinetic energy t . At finite temperature a re-entrant transition is found in some region of V .

The problem of charge ordering (CO) was already attracting the attention of physicists at the end of the 1930s. In the low-density limit, as was first proposed by Wigner [1], the electrons crystallize in form of a lattice in order to keep the Coulomb repulsion as small as possible. Such a Wigner lattice is experimentally realized in a GaAs/AlGaAs heterostructure [2]. The CO may also occur at higher electron concentration if the interaction of electrons with spin degrees of freedom or with phonons drastically reduces the kinetic energy [3]. Recently CO has been extensively observed in real materials at high densities: hole ordering in rare-earth pnictides such as Yb₄ As₃ [4] and CO in the unconventional spinPeierls material α^′ -NaV₂O₅ [5] and in colossal-magnetoresistance compounds, for example R_2−2x A_1+2x Mn₂ O₇ (R = La, Pr; A = Ca, Sr; x ≥ 0.5) [6].

One of the simplest models of interacting electrons that allows for CO is the extended Hubbard model (EHM). This model has been intensively studied both in low dimensions and in the limit of infinite dimension, usually at half-or at quarter-filling. A variety of techniques, such as Hartree-Fock approximation [7], perturbation theory [8], the dynamical mean-field theory (DMFT) [9] and the slave boson approach [10], as well as numerical methods such as quantum Monte Carlo simulation [11] and the Lanczos technique [12], have been employed.

Obviously, each of these approaches is able to describe properly only one of the relevant limits of the model and, despite the many publications devoted to the CO in solids, the physical picture of the phenomenon is far from being clear.

Recently, a melting of the charge-ordered state on decreasing the temperature has been found in Pr_0.65 (Ca_0.7 Sr_0.3 )_0.35 MnO₃ [13] and in La_2−2x Sr_1+2x Mn₂ O₇ (0.47 ≤ x ≤ 0.62) [14, 15]. A re-entrant transition at quarter-filling has been obtained theoretically using the EHM both with electron-phonon interaction [16] and without electron-phonon interaction [9, 12]. The present paper is devoted to a study of the boundary between the charge-ordered and disordered phases for different regimes of the temperature T , the Coulomb interactions U, V and the band filling n. A simple but physically meaningful approximation allowing us to solve this problem is the coherent potential approximation (CPA). This self-consistent approximation is recognized as the best single-site approximation for the spectral properties of disordered systems. Originally, the alloy-analogue approximation was formulated as an approximation scheme for the Hubbard model [17]. To solve the alloy problem the CPA is used as a second step. The CPA was also applied to intermediate-valence and heavy-fermion systems [18]. In the present work this approximation is used for the first time to treat the CO in the EHM.

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