非厄米特拓樸電路系統之研究

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姓名

古鴻瑋(Hung-Wei Ku) 查詢紙本館藏

畢業系所

光電科學與工程學系

論文名稱

非厄米特拓樸電路系統之研究
(The research of non-Hermitian topological circuit systems)

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摘要(中)

本論文主要探討 Su-Schrieffer-Heeger (SSH) 模型在厄米特 (Hermitian) 與非厄米特 (Non-Hermitian) 情況下的行為，以及比較其不同的拓樸形態。以電路系統為主要模擬的對象，在非厄米特的拓樸能帶圖可以發現在厄米特系統中所沒有的新拓樸相。論文中分析非厄米特趨膚效應 (Non-Hermitian skin effect) 對拓樸邊緣模態 (topological edge modes) 與 Tamm 模態 (Tamm mode) 的影響。
第一章主要介紹拓樸材料的近代發展，先從介紹整數量子霍爾效開始，再
延伸至不同系統的拓樸相 (topological phase)，此外也介紹布洛赫能定理 (Bloch theorem)、貝瑞相(Berry phase)以及厄米特與非厄米特矩陣等與拓樸相有關的知識。第二章介紹一維和二維厄米特和非厄米特量子系統的矩陣表達式的相關知識並應用在電路系統。藉由使用運算放大器 (Operational Amplifier) 建構出非厄米特電路系統，導出其漢米頓量 (Hamiltonian) 並計算出能帶圖。第三章介紹柯希荷夫電流定律 (KCL) 以及運用節點分析法進行電路分析和計算札克相 (Zak phase)來建立區分拓樸性質的拓樸不變量 (topological invariant)。第四章為此論文主體，在其中我們主要比較 Tamm 模態 (Tamm mode) 、體模態 (bulk eigenmodes)，以及拓樸零模 (topological zero mode) 之邊界局域化程度，以及探討三種模態在調整運算放大器電壓增幅下的變化趨勢。第五章對厄米特與非厄米特系統進行了總結比較，並探討了對未來的展望。

摘要(英)

This thesis mainly studiesthe behaviors of the one-dimensional and two-dimensional Su-Schrieffer-Heeger (SSH) models in the Hermitian and Non-Hermitian cases, and
compare their different topological properties. Circuit periodic systems consisting of
capacitors, inductors, and amplifiers (for the non-hermitian cases) are chosen as the model systems. As the main simulation results, new topological phases are revealed in
the non-Hermitian topological band diagrams, which are not found in the corresponding Hermitian systems. We study the influence of the non-Hermitian skin effect on the
topological edge modes and Tamm mode numerically.
The first chapter mainly introduces the development of topological physics in modern times. We start from the introduction of integer quantum Hall effect, and then
extend the concepts extracted from topological insulators to other systems with topological phases. We also introduce the corresponding knowledge about Bloch theorem, Berry phase, and Hermitian and non-Hermitian matrices. In chapter two the
knowledge of matrix expressions of typical one-dimensional and two-dimensional Hermitian and non-Hermitian systems is introduced and applied to the circuit systems.
Operational Amplifiers are used to help construct the non-hermitian circuit systems.
The Hamiltonian is derived, and the energy band diagrams are calculated. Chapter 3 introduces Kochhoff’s Laws (KCL) and the use of nodal analysis method for circuit analysis and the calculation of Zak phase to characterize the topological properties of the topological circuit system. In chapter 4 we simulate the topological circuits numerically. We compare the band diagrams of one-dimensional and two-dimensional
non-Hermitian circuit systems, and find a new edge state: Tamm mode. The localization degrees of these modes were compared by tuning the amplification factor of the Operational Amplifiers. In chapter 5 we give the conclusive comparison of Hermitian and non-Hermitian systems, and discuss possible future works.

關鍵字(中)

★ 非厄米特電路
★ 拓樸絕緣體
★ 非厄米特趨膚效應
★ 廣義布里淵區

關鍵字(英)

論文目次

目錄
摘要............................................................... Ⅰ
Abstract........................................................... Ⅱ
致謝............................................................... Ⅲ
目錄.............................................................. Ⅳ
圖目錄............................................................ Ⅵ
第一章緒論.........................................................1
1-1 前言............................................................ 1
1-2 布洛赫定理與貝瑞相.............................................. 3
1-3 厄米特矩陣與非厄米特矩陣........................................ 5
第二章拓樸模型理論................................................ 7
2-1 一維 Su-Schrieffer-Heeger(SSH)模型................................ 7
2-2 一維非厄米特 Su-Schrieffer-Heeger(SSH)模型........................ 9
2-3 一維厄米特 SSH 模型電路系統.................................... 11
2-4 二維厄米特 SSH 模型電路系統.................................... 13
2-5 運算放大器(Operational Amplifier)................................. 15
2-6 負電阻增益與損耗之非厄米特 SSH 電路系統........................ 18
第三章非厄米特模型與研究方法..................................... 24
3-1 如何建構電路系統............................................... 24
3-1-1 克希荷夫電流定律……………………………………………………….. 24
3-1-2 電路分析………………………………………………………………….. 25
3-2 一維非厄米特 SSH模型電路系統.................................. 25
3-3 二維非厄米特 SSH模型電路系統.................................. 28
3-4 拓樸不變量與非厄米特趨膚效應................................... 30
3-4-1 用札克相解釋非厄米特系統…………………………………………….. 30
第四章.研究模擬與討論..............................................32
4-1 一維 SSH 模型數值分析........................................... 32
4-1-1 一維厄米特 SSH 模型數值分析………………………………………....... 32
4-1-2 一維非厄米特 SSH 模型數值分析………………………………………… 33
4-1-3 一維厄米特 SSH 電路系統數值分析……………………………………… 35
4-1-4 一維非厄米特 SSH 電路系統數值分析…………………………………….36
4-1-5 一維非厄米特 SSH 電路系統邊緣態探討………………………………….37
4-2 二維 SSH 模型數值分析........................................... 43
4-2-1 二維厄米特 SSH 電路系統數值分析……………………………………… 43
4-2-2 二維厄米特 SSH 電路系統數值分析……………………………………… 44
4-2-3 二維非厄米特 SSH 電路系統數值分析…………………………………... 45
第五章結論與未來展望..............................................48
V
5-1 結論........................................................... 48
5-2 未來展望....................................................... 49
參考文獻.......................................................... 50

參考文獻

參考文獻
[1] K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy
Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,
Phys. Rev. Lett. 45, 494 (1980).
[2] F. Haldane, Model for a Quantum Hall Effect without Landau Levels: CondensedMatter Realization of the Parity Anomaly,Phys. Rev. Lett. 61, 18 (1988).
[3] C. L. Kane and E.J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect,
Phys. Rev. Lett. 95, 146802 (2005).
[4] C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett.
95, 226801 (2005).
[5] F. D. M. Haldane and S. Raghu, Possible Realization of Directional Optical
Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Phys. Rev.
Lett. 100, 013904 (2008).
[6] Jun Mei, Zeguo Chen, and Ying Wu, Pseudo-time-reversal symmetry and
topological edge states in two-dimensional acoustic crystals, Scientific Reports. 6,
32752 (2016).
[7] Yuting Yang, Hua Jiang, and Zhi Hong Hang, Topological Valley Transport in
Two-dimensional Honeycomb Photonic Crystals, Scientific Reports. 8, 1588 (2018).
[8] Sunkai, Topological insulators part III: tight-binding models, Ch5. Phys 620 course
(2013).
[9] Toshikaze Kariyado and Yasuhiro Hatsugai, Manipulation of Dirac Cones in
Mechanical Graphene, Scientific Reports. 5, 18107 (2016).
[10] Ramy EI-Ganainy , Konstantinos G. Makris , Mercedeh Khajavikhan, Ziad H.
Musslimani , Stefan Rotter and Demetrios N. Christodoulides, Non-Hermitian physics
and PT symmetry, Nature Physics.14, 11-19 (2018).
[11] T. Thonhauser and David Vanderbilt, Insulator/Chern-insulator transition in the
Haldane model, Phys. Rev. Lett. 74, 235111 (2006).
[12] Asbóth, János K., Oroszlány, László, Pályi, András, A Short Course on
Topological Insulators (Springer, Lecture Note in Physics, 2016).
[13] Shunyu Yao and Zhong Wang, Edge states and topological invariants of nonHermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
[14] Kazuki Yokomizo and Shuichi Murakami, Non-Bloch Band Theory of NonHermitian Systems, Phys. Rev. Lett. 123, 066404 (2019).
[15] Nobuyuki Okuma, Kohei Kawabata,Ken Shiozaki, and Masatoshi Sato,
Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett.124, 086801 (2020).
[16] Ching Hua Lee, Stefan Imhof, Christian Berger, Florian Bayer, Johannes Brehm,
Laurens W. Molenkamp, Tobias Kiessling and Ronny Thomale, Topolectrical Circuits,
communications physics.1, 39 (2018).
[17]Shuo Liu, Wenlong Gao, Qian Zhang, Shaojie Ma, Lei Zhang,Changxu Liu, Yuan
Jiang Xiang, Tie Jun Cui and Shuang Zhang, Topologically Protected Edge State in
Two-Dimensional Su–Schrieffer–Heeger Circuit, Research, 8609875 (2019).
[18] Shuo Liu, Ruiwen Shao, Shaojie Ma, Lei Zhang, Oubo You, Haotian Wu, Yuan
Jiang Xiang, Tie Jun Cui, and Shuang Zhang, Non-Hermitian Skin Effect in a NonHermitian Electrical Circuit, Research. 5608038 (2021).
[19] X. Z. Zhang and Z. Song, Partial topological Zak phase and dynamical
confinement in non-Hermitian bipartite system, Phys. Rev. Applied. 99, 012113(2019).
[20] You Wang, Li-Jun Lang, Ching Hua Lee, Baile Zhang and Y.D Chong,
Topologically enhanced harmonic generation in a nonlinear transmission line
metamaterial, Nature Communications. 10, 1102 (2019).
[21] Shuo Liu, Shaojie Ma, Cheng Yang, Lei Zhang, Wenlong Gao, Yuan Jiang Xiang,
Tie Jun Cui, and Shuang Zhang, Gain- and Loss-Induced Topological Insulating Phase
in a Non-Hermitian Electrical Circuit, Phys. Rev. Applied. 13, 014047 (2020).
[22] Feng Liu, Novel Topological Phase with a Zero Berry Curvature, Phys. Rev. Lett.
118, 076803 (2017).
[23] Hai-Xiao Wang, Chengpeng Liang, Yin Poo, Pi-Gang Luan and Guang-Yu Guo,
The topological edge modes and Tamm modes in Su–Schrieffer–Heeger LC-resonator
circuits, J. Phys. D: Appl. Phys. 54, 435301 (2021).
[24] Imhof, S. et al, Topolectrical circuit realization of topological corner modes, Nat.
Phys. 14, 925–929 (2018).
[25] Albert, V. V., Glazman, L. I. and Jiang, L, Topological properties of linear circuit
lattices, Phys. Rev. Lett. 114, 173902 (2015).
[26] Serra-Garcia, M., Süsstrunk, R. and Huber, S. D, Observation of quadrupole
transitions and edge mode topology in an LC network, Phys. Rev. B. 99, 020304 (R)
(2019).
[27]Fleury, R., Khanikaev, A. B., and Alù, A, Floquettopological insulators for sound,
Nat. Commun. 7, 11744 (2016).
[28] Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H.
Musslimani, Stefan Rotter, and Demetrios N. Christodoulides, Non-Hermitian physics
and PT symmetry, Nature Physics. 14, 11-19 (2018).
[29] Carl M. Bender and Stefan Boettcher, Real Spectra in Non-Hermitian
Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243 (1998).
[30] Dan S. Borgnia, Alex Jura Kruchkov, and Robert-Jan Slager, Non-Hermitian
Boundary Modes and Topology, Phys. Rev. Lett. 124, 056802 (2020).
[31] Motohiko Ezawa, Electric circuits for non-Hermitian Chern insulators, Phys. Rev.
B. 100, 081401 (2019).
[32] Ya-Jie Wu, and Junpeng Hou, Symmetry-protected localized states at defects in
non-Hermitian systems, Phys. Rev. Applied. 99, 062107 (2019).
[33] Baogang Zhu, Rong Lu, and Shu Chen, PT symmetry in the non-Hermitian SuSchrieffer-Heeger model with complex boundary potentials, Phys. Rev. Applied. 89,
062102 (2014).
[34] L. Jin, and Z. Song, Bulk-boundary correspondence in a non-Hermitian system in
one dimension with chiral inversion symmetry, Phys. Rev. B. 99, 081103 (2019).
[35]Shunyu Yao, Fei Song, and Zhong Wang, Non-Hermitian Chern Bands, Phys. Rev.
Lett. 121,136802 (2018).
[36] Daniel Leykam, Konstantin Y. Bliokh, Chunli Huang, Y. D. Chong, and Franco
Nori,Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems,
Phys. Rev. Lett.118, 040401 (2017).
[37]Simon Lieu, Topological phases in the non-Hermitian Su-Schrieffer-Heeger model,
Phys. Rev. B. 97, 045106 (2018).
[38] Kenta Esaki, Masatoshi Sato, Kazuki Hasebe, and Mahito Kohmoto, Edge states
and topological phases in non-Hermitian systems, Phys. Rev. B. 84, 205128 (2011).
[39] Zongping Gong,1, Yuto Ashida,1, Kohei Kawabata,Kazuaki Takasan, Sho
Higashikawa, and Masahito Ueda, Topological Phases of Non-Hermitian Systems,
Phys. Rev. X. 8, 031079 (2018).
[40] Hui Jiang, Chao Yang, and Shu Chen, Topological invariants and phase diagrams
for one-dimensional two-band non-Hermitian systems without chiral symmetry, Phys.
Rev. Applied. 98, 052116 (2018).
[41]Mingsen Pan, Han Zhao, Pei Miao, Stefano Longhi, and Liang Feng, Photonic zero
mode in a non-Hermitian photonic lattice, Nat. Commun. 9, 1308 (2018).
[42] Julia M. Zeuner, Mikael C. Rechtsman, Yonatan Plotnik, Yaakov Lumer, Stefan
Nolte, Mark S. Rudner, Mordechai Segev, and Alexander Szameit, Observation of a
Topological Transition in the Bulk of a Non-Hermitian System, Phys. Rev. Lett.115,
040402 (2015).
[43]Huitao Shen, Bo Zhen, and Liang Fu, Topological Band Theory for Non-Hermitian
Hamiltonians, Phys. Rev. Lett.120,146402 (2018).
[44]Chuanhao Yin, Hui Jiang, Linhu Li, Rong Lü, and Shu Chen, Geometrical meaning
of winding number and its characterization of topological phases in one-dimensional
chiral non-Hermitian systems, Phys. Rev. Applied. 97, 052115 (2018).
[45]Yu Chen,and Hui Zhai, Hall conductance of a non-Hermitian Chern insulator, Phys.
Rev. B. 98, 245130 (2018).
[46] Timothy M. Philip, Mark R. Hirsbrunner, and Matthew J. Gilbert, Loss of Hall
conductivity quantization in a non-Hermitian quantum anomalous Hall insulator, Phys.
Rev. B. 98, 155430 (2018).
[47] Li-Jun Lang , Yijiao Weng, Yunhui Zhang, Enhong Cheng , and Qixia Liang ,
Dynamical robustness of topological end states in nonreciprocal Su-Schrieffer-Heeger
models with open boundary conditions, Phys. Rev. B. 103, 014302 (2021).
[48] Han-Ting Chen,Chia-Hsun Chang, and Hsien-chung Kao, The Zak phase and
Winding number, arXiv:1908.06700 (2019).

指導教授

欒丕綱

審核日期

2021-10-27

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