Near-local density approximation approach to one-dimensional lattice systems
Abstract: Describing many-body quantum systems has been an analytically and computationally challenging task since the advent of quantum mechanics. However, in the past 50 years as a result of our technological advancement and the emergence of methods such as density-functional theory (DFT), we have taken crucial steps forward regarding our ability to study and understand large quantum systems. In this work,we have studied the extended Hubbard model using a mean-field approximation where we have tested an approach to the long-ranged electronic interaction that is not entirely local. We have developed a near-local approximation (NLA) for the model under study, where the exact non-local electronic interaction at site i is approximated using the densities at a pair of sites neighbouring i in addition to the density at i. As preliminary to a discussion of NLA, we showed results for the density of states of systems with a three-site supercell, thus providing a simple characterization of the mean-field treatment (in the case of local interactions only). When it comes to the main results of our work, i.e. a derivation and the testing of the NLA, our findings can be summarised as follows: we have found three NLA variants, namely a left site approximation (LSA), a right site approximation (RSA) and a center site approximation (CSA). Furthermore, we have found that CSA performs better than a local approximation both under the effect of a parabolic external potential and a distorted one. We have also found that the performance of LSA and RSA are scarce in general. Albeit this work (being based on the extended Hubbard model under a mean-field approximation) does not address exchange and correlation effects directly, it provides a first step towards future work where a near-local treatment is carried out on more accurate grounds.
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