Near-local density approximation approach to one-dimensional lattice systems

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.Early on in our education, we were taught that everything around us exists in one of three states: solid, liquid and gas. This may sound like a natural observation to make or a mere consequence of our reality, but how important is this distinction and how does it affect our lives? To reformulate this question in a clearer way, let us take the following example of water in a glass. A normal glass of tap water would have water in the liquid state. If we put the same glass in the freezer overnight, the water would turn into ice which is the solid state of water. If we instead boil the water in the glass, it would evaporate and turn into water vapor, which is the gas state of water. Would it help us in any way to know the ins and outs of the different states the water in the glass displayed? It turns out that it is indeed important to study the different states of matter and especially the solid state. By reaching the end of this page, you will know the answer to why. Matter around us appears predominantly in the solid state, for example spoons, light bulbs, windows, helmets,...etc. Most solids, however, can be found naturally in crystalline form. Crystalline form, or simply a crystal, is an arrangement of a repeated pattern of constituents. If we for example take ten identical tennis balls and arrange them on a straight line with equal spacing between them, then we have a crystal of tennis balls as constituents. Solids are extremely analogous to our arrangement of tennis balls with various constituents. Our understanding of different natural phenomena, such as electrical conduction and light reflection, is highly dependent on studying matter in the solid state, i.e. solid state physics. Our smart-phones, TVs, computers and many other devices we use in our everyday life are also a product of our advancement in said field. Indeed, investigating and understanding those crystals has been of high applicability and has led to an easier everyday life and a more prosperous human society. However, is there more to it? The work of this thesis is to further investigate a certain type of crystals; to broaden our understanding of matter in the solid state and to serve as a step forward in harnessing the full extent of our creativity. The study revolves around electrons living on a crystal and interacting with each other. Imagine the same crystal of tennis balls but instead, the tennis balls are replaced by small electron houses called sites. We can imagine an electron as an individual and its house as a small single studio flat. As one may imagine, two humans can not live in a single studio flat. If they did, there is likely going to be interaction in the form of a quarrel for space until one would go for a walk to get some air. The situation is analogous in the case of electrons on a crystal. This form of interaction can have many implications and can (in some cases) be responsible for different electrical properties in solids. While quarrels may not be desirable for us humans, electron quarrels can yield interesting effects. Finally, crystals with a lot of quarrel in their vicinity are categorized under the term Quantum Materials. Quantum Materials are believed to be the next generation of application in various fields of science and everyday life. A new generation of solar cells, ultra-fast spectroscopy, different types of superconductors,...etc. They even extend to intelligent windows which control the amount of heat in a room automatically using quarreling electrons! All in all, these are few examples to show the extent in which studying different types of matter can reach