本論文以石墨烯的六角晶格出發，運用緊束縛(tight binding)方法後，結合最近鄰原子和次近鄰原子的效應，一步步推導哈密頓量(Hamiltonian)，且加入了破壞對稱性的參數後，最後得到的哈密頓量就是霍爾丹模型(The Haldane model)，將其帶入陳數(Chern number)[1]公式中，就可以發現在|M/t'|=|3√3 sin ϕ|範圍內時會有響應而在範圍之外時為零，可知數值結果與理論相符(圖3.1b,d,f)。

陳數所考慮的性質為第一布理淵區中將其各點做積分加總所得到的全域性質，但如果要考慮各點的響應，也就是量子幾何張量[2]所描述的局部性質，其實部和虛部分別為量子度量(Quantum metric)[3]和貝里曲率(Berry curvature)，當產生拓撲相變時也會產生不同的變化，在具有拓譜相時極值會集中在K或K'點上，也就是能隙最小值的區域，且根據參數不同會具有偶函數和奇函數的性質，和後續討論光電流時具有高度關聯性。

光電流是一個重要的物理性質，利用微擾理論將其拆解成多階的光電流，二階光電流就是我們本次討論的重點，可以分成平移電流和注入電流[4]，根據有無拓撲相會有不同的響應，有拓撲相時平移電流在動量空間中的響應集中在K'或K點上，而注入電流的響應會集中在兩點周圍，而方向上和量子幾何張量有關，可以從其奇偶性去推測光電流是否為非零響應。

This thesis begins with the hexagonal lattice structure of graphene and applies the tight-binding method, incorporating both nearest neighbor and next nearest neighbor interactions to systematically derive the Hamiltonian. By introducing symmetry-breaking parameters, the resulting Hamiltonian corresponds to the Haldane model. Substituting this model into the Chern number[1]formula reveals that a topological response exists within the range |M/t'|=|3√3 sin ϕ|, while it vanishes outside of this region. This numerical result is consistent with the theoretical prediction, as shown in Fig.3.1 (b, d, f).

The Chern number represents a global property obtained by integrating the Berry Curvature over the entire First Brillouin Zone. To explore local properties in momentum space, one must consider the Quantum Geometric Tensor [2], whose real part and imaginary parts correspond to the Quantum Metric [3] and Berry Curvature, respectively. These local quantities vary with topological phase transitions. When the system is in a topological phase, the extrema tend to concentrate around the K or K' points, where the energy gap is minimized. Depending on the system parameters, these quantities can exhibit even or odd functional behavior, which is strongly correlated with the nonlinear photocurrent response discussed later.

Photocurrent is a significant physical property of materials and can be expanded into multiple orders based on the electric field strength. In this work, we focus on the second-order photocurrent,which includes both shift current and injection current [4]. The presence of a topological phase affects their responses: the shift current tends to be localized around the $K$ or $K'$ points, whereas the injection current concentrates near these points. Moreover, the directionality of the photocurrent is closely linked to the parity of the Quantum Geometric Tensor, allowing us to predict whether a nonzero response exists in a specific direction.

[1] Javier Sivianes and Julen Ibañez-Azpiroz. Shift photoconductivity in the haldane model. Phys. Rev. B, 2023.
[2] Tomoki Ozawa and Bruno Mera. Relations between topology and the quantum metric for chern insulators. Phys. Rev. B, 2021.
[3] Takahiro Kashihara, Yoshihiro Michishita, and Robert Peters1. Quantum metric on the brillouin zone in correlated electron systems and its relation to topology for chern insulators. Phys. Rev. B, 2023.
[4] Junyeong Ahn, Guang-Yu Guo, and Naoto Nagaosa. Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals. Phys. Rev. B, 2020.
[5] F. D. M. Haldane. Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”. Phys. Rev. Lett, 1988.
[6] Fengcheng Wu, Timothy Lovorn, Emanuel Tutuc, Ivar Martin, and A.H. MacDonald. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402, 2019.
[7] Swati Chaudhary, Cyprian Lewandowski, and Gil Refael. Shift-current response as a probe of quantum geometry and electron-electron interactions in twisted bilayer graphene. Phys. Rev. Research 4, 013164, 2022.
[8] Hiroki Yoshida and Shuichi Murakami. Diverging shift current responses in the gapless limit of two-dimensional systems. Phys. Rev. B 111, 2025.
[9] Motohiko Ezawa. Bulk photovoltaic effects in altermagnets. Phys. Rev. B 111, 2025.
[10] Yang Zhang, Tobias Holder, Hiroaki Ishizuka, Fernando de Juan, Naoto Nagaosa, Claudia Felser, and Binghai Yan. Switchable magnetic bulk photovoltaic effect in the two-dimensional magnet cri3. Nature Communications volume 10, Article number:3783, 2019.
[11] Hua Wang and Xiaofeng Qian. Electrically and magnetically switchable nonlinear photocurrent in pt-symmetric magnetic topological quantum materials. npj Computational Materials volume 6,Article number: 199, 2020.
[12] Sunje Kim, Yoonah Chung, Yuting Qian, Soobin Park, Chris Jozwiak, Eli Rotenberg, Aaron Bostwick, Keun Su Kim, , and Bohm-Jung Yang. Direct measurement of the quantum metric tensor in solids. Vol 388, Issue 6751 pp. 1050-1054, 2025.
[13] Anton R. Akhmerov and Carlo W. J. Beenakker. Topology in condensed matter: Tying quantum knots. https://topocondmat.org, 2015. Online course, Delft University of Technology.
[14] Mark O. Goerbig. Quantum hall effects. Phys. Rev. X, 2015.
[15] T.-W. Chen, Z.-R. Xiao, D.-W. Chiou, and G.-Y. Guo. High chern number quantum anomalous hall phases in single‑layer graphene with haldane orbital coupling. Physical Review B, 2011.