Keskiner, Mehmet Akif2025-08-292025-08-292025-082025-08-27B149237https://hdl.handle.net/11693/117460Cataloged from PDF version of article.Includes bibliographical references (leaves 89-102).This thesis explores how quasiperiodic geometry, moiré superlattice, and fractionalized spin excitations generate novel electronic and magnetic behavior. Through four theoretical studies, it examines: (i) strictly localized states in the Socolar dodecagonal quasicrystal; (ii) magnetic textures in a twisted moiré superlattice; (iii) Kitaev-type spin liquids on the dual Ammann–Beenker quasicrystal; and (iv) magnetic order induced by Kondo coupling to quantum spin liquids. Localized states: In the Socolar dodecagonal lattice (SDL), a quasicrystal with twelvefold symmetry, we identify 18 distinct types of strictly localized states (LS), accounting for approximately 7.58% of the Hilbert space, closely matching numerical estimates of 7.61%. Through perpendicular space analysis, we demonstrate that at least 3.9% of sites are forbidden from hosting LS due to local connectivity constraints—revealing behavior intermediate between Penrose and Ammann–Beenker quasicrystals. Moiré magnetism: In twisted heterostructures composed of a Mott insulator and a semimetal, we study the emergence of spatially modulated magnetic order arising from nonuniform RKKY interactions. Our Monte Carlo simulations reveal a rich phase structure: AA-stacked regions exhibit antiferromagnetic (AFM) order, AB-stacked regions favor ferromagnetic (FM) alignment, and the intervening regions host ferromagnetic chains coupled antiferromagnetically (FMC). The spatial extent and coexistence of these domains are governed by the inverse decay length, α, of the Kondo interaction—where small α favors extended FMC regions, while larger α leads to the coexistence of FM, AFM, and FMC textures across the moiré unit cell. Quasicrystalline spin liquid: We formulate an exactly solvable Kitaev-type model on the dual Ammann–Beenker lattice (dABL), exploiting its fourfold coordination and partite bond structure. Our comprehensive study uncovers a rich variety of phases, including both gapless and gapped quantum spin liquids with chiral and abelian characteristics, analyzed via Monte Carlo methods and variational techniques. Additionally, incorporating an onsite perturbation refines the ground state selection to 21 unique vison configurations while preserving integrability. This work highlights the complex relationship between quasiperiodicity and emergent quantum magnetic phases. Spin-liquid-mediated magnetism: We investigate how magnetic order emerges among localized spins that interact exclusively via their coupling to a Kitaev-type spin liquid. Studying Kitaev, Yao-Lee, and square-lattice generalization models, we derive effective spin interactions mediated by fractionalized Majorana fermions. Short-range couplings stabilize the spin liquid in the Kitaev model, while the Yao-Lee model exhibits long-range RKKY-like antiferromagnetic order and partial Majorana gap formation. The square-lattice model shows competing anisotropic interactions, leading to dimerized quantum paramagnetism or Ising antiferromagnetism depending on parameters. These results reveal the rich magnetic phases enabled by Kitaev-type spin liquids.xxi, 118 leaves : illustrations, charts ; 30 cm.EnglishQuasicrystalsSpin liquidsKitaev modelMoiré superlatticesKondo latticeLocalized statesEmergent magnetismThesis