Exploring Quantum Phenomena in 2D Materials: From Valley Topology and Exciton Selection Rules to Tuning Magnetic Correlations

Abstract

This thesis presents two independent parts under the common theme of quantum phenomena in 2D materials: Part I introduces a gauge-invariant, quantized interband index for multiband 2D systems and uses it in two applications: A) to analyze valley topology, and B) to derive gauge-invariant exciton selection rules. Part II investigates monolayer Nb₃Cl₈, quantifies magnetic anisotropy, and shows that biaxial strain tunes antiferromagnetic, paramagnetic, and ferromagnetic behavior. Part I, Application A: Interband index and valley topology. We introduce a novel gauge-invariant, quantized interband index in two-dimensional (2D) multiband systems. It provides a bulk topological classification of a submanifold of parameter space (e.g., an electron valley in a Brillouin zone), and therefore overcomes difficulties in characterizing topology of submanifolds. We confirm its topological nature by numerically demonstrating a one-to-one correspondence to the valley Chern number in k·p models (e.g., gapped Dirac fermion model), and the first Chern number in lattice models (e.g., Haldane model). Furthermore, we derive a band-resolved topological charge and demonstrate that it can be used to investigate the nature of edge states due to band inversion in valley systems like multilayer graphene. Part I, Application B: Gauge-invariant optical selection rules for excitons. Excitons are central to the photophysics of 2D semiconductors and photonic devices. Prior circular selection rules for excitons in 2D are successful but gauge-dependent due to assumptions that exclude singular gauge behavior at band edges. By developing a chiral form of the interband index from Application A above, we obtain selection rules that are manifestly gauge-invariant. This framework is directly compatible with numerical workflows used in device modeling, and strengthens the theory of quantum materials, especially two-dimensional semiconductor photophysics. Part II. Strain-tunable magnetism in Nb₃Cl₈. Recent research suggests the possibility of the two-dimensional breathing-Kagome magnet Nb₃Cl₈ hosting a quantum spin liquid state, warranting further study into its magnetic properties. Using ab initio calculations, we show that monolayer Nb₃Cl₈ has short-range antiferromagnetic correlations among Nb₃ trimers with S = 1/2, and becomes magnetically frustrated due to the underlying effective triangular lattice geometry, and is evidenced by a frustration index of f > 1. The high-temperature susceptibility shows a negative Weiss temperature from Monte Carlo calculations. Considering spin-orbit coupling, we investigate the magnetic anisotropy, including anisotropic exchange, single-ion anisotropy and the Dzyaloshinskii–Moriya interaction using the four-state energy mapping formalism. Although the elements have relatively small atomic numbers, the Dzyaloshinskii–Moriya interaction is comparable in magnitude to the anisotropic exchange. Additionally, we show that biaxial strain tunes the short-range correlations between antiferromagnetic, paramagnetic and ferromagnetic. These findings strengthen our understanding of Nb₃Cl₈ and advance its applications in current condensed matter physics and materials science research, including nanoscale mechanical and spintronics applications. Summary. Part I supplies a valley-focused topological index and a gauge-invariant theory of exciton selection rules. Part II elucidates the magnetic anisotropy and tunability of magnetism in monolayer Nb₃Cl₈. Collectively, these findings fulfill the thesis aim of exploring quantum phenomena in 2D materials.