In recent times, there has been a significant interest in low-dimensional materials due totheir unique electronic, optical, magnetic, and topological properties that differ from 3Dbulk materials. This dissertation focuses on a specific class of 1D carbon structures known asgraphene nanoribbons (GNRs), which can be synthesized atomically with precision througha bottom-up method. The theoretical tools employed in this study are primarily topologicaltheory and quantum many-body first-principles calculations.Chapter 1 introduces some basics about density functional theory, GW many-body perturbationtheory and the Belthe-Salpeter equation. Chapter 2 of this dissertation delves intothe topology of GNRs when chiral symmetry is approximately maintained. Building onthis theory and in collaboration with experimentalists, Chapter 3 explores a metallic 1Dnanowire known as saw-tooth GNRs, while Chapter 4 investigates various quantum dot systemswith unique bonding and anti-bonding characters. In Chapter 5, a different type ofmetallic GNRs is studied using zero-mode (topologically protected in-gap electronic states)engineering. Chapter 6 takes the study beyond the Hermitian Hamiltonian and introducesthe non-Hermitian skin effect. When 1D or 0D structures are interconnected, nanoporousgraphene is formed. Its electronic properties are studied in Chapter 8. Furthermore, Chapter9 examines a carbon kagome lattice’s excitonic properties. The content of each Chapter iselaborated as the following:• Chapter 1 provides a foundational understanding of density functional theory (DFT)for ground state properties by introducing the Kohn-Sham equation and different functionals.We also discuss the GW perturbation theory, which allows us to incorporatemany-body effects into our calculations of excited-state properties. Specifically, weexplore how the GW method can be utilized to calculate quasi-particle excitations.To study the two-particle excitation problem for optical properties, we introduce theBethe-Salpeter equation (BSE) method. This equation provides a framework for calcu2lating the interaction between an excited electron and the hole it leaves behind, whichis crucial for understanding optical properties such as absorption and emission spectra.• In Chapter 2, we examine GNR structures under the first nearest neighbor tightbindingmodel, assuming chiral symmetry holds. In this scenario, we utilize the firstChern number to obtain a Z index for general 1D materials. From the general Zindex formula, we derive the Chiral phase index in vector form, which enables us toobtain the analytic Z index formula for all types of unit cells in GNRs. Finally, weexplore a spin-chain formed by topological junction states that exhibit strong spin-spininteractions.[1]• Chapter 3 builds on the chiral classification theory introduced in Chapter 2 by utilizingthe topological junction states as building blocks and connecting them in a symmetricmanner to form a 1D metallic nanowire. We use first-principles DFT calculations tostudy the electronic bandstructure, local density of states (LDOS), and mapping ofwavefunctions. Our results are then compared with experimental STM measurements,and we achieve good agreement. In addition, we also investigate the topological propertiesof asymmetrically connected structures, and the predicted junction/end statematches well the corresponding experimental evidence.[2]• In Chapter 4, we employed the topological junction states that arise from the connectionbetween 7-armchair graphene nanoribbons (7AGNR) and 9-armchair graphenenanoribbons (9AGNR) to construct topological quantum dots. We investigated twodistinct types of quantum dots by means of DFT calculations, with the aim of studyingtheir electronic properties, such as the bonding and anti-bonding traits of their valenceand conduction states. In addition, we devised a tight-binding theory to elucidate theunderlying factors contributing to the characteristics of the wavefunctions.[3]• In Chapter 5, we focus on a different variety of metallic graphene nanoribbon (GNR)called Olympicene GNRs that does not exhibit the Stoner instability, which was observedin the sawtooth GNRs presented in Chapter 3. This new GNR features coveshapededges, and its low-energy behavior is governed by zero modes. The most notabledistinction between this GNR and the sawtooth GNR is that the nearest zero modeslocalize on different sublattices, leading to a significant increase in electron hoppingand precluding any magnetic instability. To verify this, we conduct DFT calculationsand compare our findings with experimental observations.• In Chapter 6, we explore the topology of 1D non-Hermitian systems, extending ouranalysis beyond Hermitian topological classification. Specifically, we investigate a 1Dnon-Hermitian system with no symmetry constraints, and use a Z index that canbe employed to classify such systems. We examine the well-known skin effect fornon-trivial non-Hermitian topological models and identify a promising GNR material,Co-4AGNR, which could potentially be realized in experiments. By conducting firstprinciplesDFT and full-frequency GW calculations, we establish that the material3exhibits non-trivial topology. Lastly, we present evidence of the asymmetric transportproperties in this material by calculating the Green’s function for a finite segment ofthis system.• In Chapter 7, we examine the 2D carbon structure that results from linking 1D metallicGNRs. To accomplish this, we created a theoretical model with low energy statesusing modes that are found in the pentagons located at the edge of GNRs as thebases. This effective tight-binding model provides a description of a unique, distortedsuper-graphene. We also conducted DFT calculations and compared our findings withexperimental results provided by our colleagues.[4]• Chapter 8 focuses on the examination of a kagome lattice that is formed by linkingtriangulene building blocks. This unique structure was predicted to exhibit excitonicinsulator (EI) behavior. In partnership with experimentalists, we conducted an investigationof the electronic properties of this structure using multiple levels of theory,such as DFT, GW-BSE, and Bardeen-Cooper-Schrieffer (BCS) theory. Our researchrevealed that DFT based single-particle theory was insufficient for accurately capturingthe features of the LDOS map observed in STM measurements. By incorporating aBCS-like theory for condensation of excitons, we were able to provide an explanationfor the experimental observations.[5]In addition to the projects above, I was also involved in 3 other projects, including onestudying the color center in twisted BN [6], one studying the kondo effect in magnetic NdopedchevronGNR [7], one studying the pseodo-atomic orbitals in graphene nanoribbons [8].These research projects are also very interesting, but beyond the scope of this dissertation.