準晶體由於缺乏平移對稱性，導致基於布里淵區與布洛赫定理的傳統拓樸能帶理論失效。本論文旨在建立一套完整的理論與數值框架，利用緊束縛模型分析二維 Penrose 鋪排（Penrose tiling）的電子結構。我們系統地研究了系統對三種關鍵微擾的響應：磁通量（Peierls 相位）、子晶格對稱破缺（二分位能）以及連結度調變（拉普拉斯項）。

本研究涵蓋兩大主軸：拓樸與對稱性。首先，在實空間拓樸不變量方面，我們發現局域陳數標記（Local Chern Marker）雖然定義明確，但在準晶體內部並不量子化，而是呈現出強烈的空間漲落，這反映了準晶體波函數的臨界特性。為了解析其參數空間，我們嚴格證明了在規範場的週期邊界條件下，磁通量參數構成了一個環面拓樸結構，這有效地取代了布里淵區，使我們得以重新定義連續的能帶結構。

其次，我們利用準晶體保有精確 $C_5$ 旋轉對稱性的特點，引入了晶格角動量算符。我們揭示了一個基本的幾何約束：任何具備 $C_5$ 對稱性的 Penrose 區塊必然包含一個旋轉不動點。理論分析預測，該不動點會導致零角動量子空間的維度不平衡。最後，我們利用格林函數形式進行同調傳輸模擬，證實了中心位點作為完美的角動量濾波器，僅允許 $m=0$分量通過。本研究成果展示了如何以角動量取代晶格動量、以磁通空間取代倒易空間，為非週期性量子系統的特徵分析提供了穩健的方法論。

The absence of translational symmetry in quasicrystals renders conventional topological band theory, which relies on the Brillouin zone and Bloch's theorem, inapplicable. In this thesis, we develop a comprehensive theoretical and numerical framework to analyze the electronic structure of the two-dimensional Penrose tiling using a tight-binding model. We investigate the system's response to three key perturbations: magnetic flux (Peierls phase), sublattice symmetry breaking (bipartite potential), and connectivity modulation (Laplacian term).

Our study proceeds in two main directions: topology and symmetry. First, we examine real-space topological invariants. We find that the local Chern marker, while well-defined, fails to quantize in the quasicrystalline bulk, exhibiting strong spatial fluctuations characteristic of critical wavefunctions. To resolve the parameter space, we prove that under periodic boundary conditions of the gauge field, the magnetic flux parameters form a toroidal topology, effectively replacing the Brillouin zone and restoring a notion of continuous spectral bands.

Second, we exploit the exact $C_5$ rotational symmetry to introduce a lattice angular momentum operator. We identify a fundamental geometric constraint: the inevitable existence of a rotational fixed point in any $C_5$-symmetric Penrose patch. Theoretical analysis predicts that this fixed point induces a dimension imbalance specifically in the zero-angular-momentum (m=0) sector. Finally, using Green's function formalism, we simulate coherent transport and confirm that the central site acts as a perfect angular momentum filter, selectively transmitting $m=0$ components. Our results establish a robust methodology for characterizing aperiodic quantum systems by replacing crystal momentum with angular momentum and reciprocal space with flux space.

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