本論文研究利用拓樸學的物理特性，探討二維三階非厄米特拓樸絕緣體 （third-order Non-Hermitian topological insulator）。 首先說明厄米特與非厄米特拓 樸矩陣的差異，進一步比較一維與二維系統的特性，並從開邊界條件觀察邊界 態，藉由週期邊界條件找出對應的晶格結構。接著，我們在方形結構中加入電路 元件以實現拓樸絕緣體，將原本每兩個節點間的電容與電阻並聯改為串聯。利用 克希荷夫電路定律（Kirchhoff's circuit laws）推導出哈密頓矩陣（Hamiltonian matrix）， 得到三階非厄米特矩陣，進而解出其本徵值（eigenvalue）與本徵向量 （eigenvector）。 最後觀察此電路系統模態（modes）的變化，並分析其邊緣態與 角態是否存在。

This thesis explores the use of topological physics to study a two-dimensional third order non-Hermitian topological insulator. We begin by explaining the differences between Hermitian and non-Hermitian topological matrices, then compare the characteristics of one-dimensional and two-dimensional systems. Boundary states are examined under open boundary conditions, while the corresponding lattice structure is identified using periodic boundary conditions. Next, a square structure is constructed with circuit elements to realize the topological insulator, where the originally parallel connection of capacitors and resistors between each pair of nodes is changed to a series connection. Kirchhoff's circuit laws are applied to derive the Hamiltonian matrix, resulting in a third-order non-Hermitian matrix. The eigenvalues and eigenvectors are then obtained. Finally, we observe changes in the system’s modes and analyze the existence of edge and corner states.

[1] 蔡雅雯、吳杰倫、欒丕綱，〈 從量子霍爾效應到拓樸光子學與拓樸聲子學〉，《科儀新知》， 211期，68–79（2017）。
[2] DAI, Xi, "Topological phases and transitions in condensed matter systems," PHYSICS 45, 757–768 (2016).
[3] Kosterlitz, J. M. & Thouless, D. J., "Ordering, metastability and phase transitions in
two-dimensional systems," J. Phys. C: Solid State Phys 6, 1181–1203 (1973).
[4] Hall, E. H., "On a New Action of the Magnet on Electric Currents," Am. J. Math. 2, 287–292 (1879).
[5] Klitzing, K. v., G. Dorda & M. Pepper, "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance," Phys. Rev. Lett. 45, 494–497 (1980).
[6] Di Xiao, Ming-Che Chang, Qian Niu, "Berry phase effects on electronic properties," Rev. Mod. Phys. 82, 1959–2007 (2010).
[7] D. J. Thouless et al., "Quantized Hall conductance in a two-dimensional periodic potential," Phys. Rev. Lett. 49, 405–408 (1982).
[8] Michael Berry, "Anticipations of the geometric phase," Phys. Today 43, 34–40 (1990).
[9] F. D. M. Haldane, "Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ‘parity anomaly’," Phys. Rev. Lett. 61, 2015–2018 (1988).
[10] F. D. M. Haldane, S. Raghu, "Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry," Phys. Rev. Lett. 100, 013904 (2008).
[11] Y. Yang, H. Jiang, Z. H. Hang, "Topological valley transport in two-dimensional honeycomb photonic crystals," Sci. Rep. 8, 1588 (2018).
[12] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, A. A. Firsov, "Electric field effect in atomically thin carbon films," Science 306, 666669 (2004).
[13] C. L. Kane and E. J. Mele, “Z₂ Topological Order and the Quantum Spin Hall Effect,” Phys. Rev. Lett. 95, 146802 (2005).
[14] T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[15] Y. Xiong, “Why does bulk–boundary correspondence fail in some non-Hermitian topological models,” J. Phys. Commun. 2, 035043 (2018).
[16] M. Z. Hasan & C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[17] C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley (2004).
[18] E. J. Bergholtz, J. C. Budich & F. K. Kunst,“Exceptional topology of non-Hermitian systems,” Rev. Mod. Phys. 93, 015005 (2021).
[19] I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[20] S. Yao & Z. Wang, “Edge States and Topological Invariants of Non-Hermitian Systems,” Phys. Rev. Lett. 121, 086803 (2018).
[21] S. Liu, R. Shao, S. Ma, L. Zhang, O. You, H. Wu, Y. J. Xiang, T. J. Cui & S. Zhang, “Non-Hermitian Skin Effect in a Non-Hermitian Electrical Circuit,” Research, 5608038 (2021).
[22] Victor V. Albert, Leonid I. Glazman, and Liang Jiang, “Topological Properties of Linear Circuit Lattices, ” Phys. Rev. Lett. 114, 173902 (2015).
[23] Motohiko Ezawa, “Higher-order topological electric circuits and topological corner resonance on the breathing kagome and pyrochlore lattices,” Phys. Rev. B 98, 201402(2018).
[24] Ching Hua Lee, Stefan Imhof, Christian Berger, Florian Bayer, Johannes Brehm, Laurens W. Molenkamp, Tobias Kiessling, Ronny Thomale, “Topolectrical circuits,” Commun. Phys. 1,39(2018).
[25] J. K. Asbóth, L. Oroszlány and A. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions, Springer International Publishing Switzerland, Lecture Notes in Physics, 919 (2016).
[26] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder and S. Ryu, “Classification of topological quantum matter with symmetries,” Rev. Mod. Phys. 88, 035005 (2016).
[27] H.-X. Wang, C. Liang, Y. Poo, P.-G. Luan and G.-Y. Guo, “The topological edge modes and Tamm modes in Su–Schrieffer–Heeger LC-resonator circuits,” J. Phys. D: Appl. Phys. 54, 085301 (2021).
[28] W.-P. Su, J. R. Schrieffer and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099 (1980).
[29] D. Obana, F. Liu, & K. Wakabayashi, “Topological edge states in the Su-Schrieffer Heeger model,” Phys. Rev. B 100, 075437 (2019).
[30] F. Liu& K. Wakabayashi, “Novel Topological Phase with a Zero Berry Curvature,” Phys. Rev. Lett . 118, 076803 (2017).
[31] H. C. Wu, L. Jin, & Z. Song, “Topology of an anti-parity-time symmetric nonHermitian Su-Schrieffer-Heeger model,” Phys. Rev. B 103, 235110 (2021).
[32] S. Liu, W. Gao, Q. Zhang, S. Ma, L. Zhang, C. Liu, Y. J. Xiang, T. J. Cui and S.Zhang, “Topologically Protected Edge State in Two-Dimensional Su–Schrieffer–Heeger Circuit,” Research, 8609875 (2019).
[33] M. Ezawa, “Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization,” Phys. Rev. B 99, 201411(R)
(2019).
[34] T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T.Kiessling, D. Wolf, A. Vollhardt, “Reciprocal skin effect and its realization in a topolectrical circuit,” Phys. Rev. Research 2, 023265 (2020).
[35] J. Wu, X. Huang, Y. Yang, W. Deng, J. Lu, W. Deng, & Z. Liu, “Non-Hermitian second-order topology induced by resistances in electric circuits,” Phys. Rev. B 105,
195127 (2022).
[36] J. Wu, X. Huang, J. Lu, Y. Wu, W. Deng, F. Li, & Z. Liu, “Observation of corner states in second-order topological electric circuits,” Phys. Rev. B 102, 104109 (2020).
[37] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, & T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101(R) (2011).