Quantum Geometry and Localization in Crystalline and Disordered Solids.

다국어 초록 (Multilingual Abstract)

We study the localization properties of electrons in solid-state systems, both with and without disorder. A key focus is to describe how the real-space extent of electronic eigenstates influences measurable physical quantities, such as the AC conductivity and dielectric properties. An essential link in this analysis, which connects microscopic localization and macroscopic responses, is the quantum metric - a geometric measure that quantifies the strength of dipole fluctuations in the many-electron ground state.In the first part of this thesis (Chapters 2 and 3), we study the quantum metric and the electric susceptibility (or capacitance), both analytically and numerically, across a diverse range of material systems, ranging from the microscale two-dimensional electron gases (2DEG) in magnetic fields to moire systems, and down to nanoscale conventional and topological insulators and semiconductors. This analysis reveals that the dielectric properties of matter can serve as diagnostics for certain correlated states (e.g., fractional quantum Hall phases) and exotic localization phenomena, such as the zero flux localization occurring in twisted bilayer graphene at the magic angle. Moreover, we propose that the relationship between the quantum metric and the dielectric constant can be instrumental in determining the dominant bonding character of the valence electrons (covalent vs. ionic) and even in detecting the non-trivial wavefunction topology.In the second part of the study (Chapters 4 and 5), we focus on Anderson insulators - materials where electrons are localized due to the destructive wavefunction self-interference induced by the presence of impurities. In these materials, hybridized pairs of localized eigenstates, known as Mott resonances, play a crucial role in the transport phenomena. We focus on investigating this mechanism in chiral disordered topological insulators, where we show that the hybridizing pairs of topological zero modes give rise to remarkable transport properties. In particular, in the chains with bond disorder, we identify the existence of an unusual "Anderson metal" phase, in which the electronic eigenstates appear localized yet exhibit finite DC conductivity. We also predict a novel phase, the superdielectric matter, characterized by a finite quantum metric (vanishing DC conductivity) and a divergent dielectric constant.Our work establishes the quantum metric and the hybridization analysis of disorder-localized eigenstates as crucial and unifying frameworks for understanding how the features of microscopic localization influence macroscopic observables, providing insight into how the transport properties of correlated, disordered, and topological systems can be efficiently studied.