Exceptional point-induced knot structure transformations in non-Abelian braids

doi:10.1088/1674-1056/ae2265

Abstract The strong connection between braids and knots provides valuable insights into studying the topological state and phase classification of various physical systems. The phenomenon of non-Hermitian (NH) two- and three-band braiding has received widespread attention. However, a systematic exploration and visualization of non-Abelian braiding and the associated knot transformations in four-band systems remains unexplored. Here, we propose a theoretical model of NH four-band braiding, provide its phase diagram, and establish its trivial, Abelian, and non-Abelian braiding rules. Additionally, we report on special knots, such as the Hopf and Solomon links in braided knots, and reveal that their transformations are accompanied by and mediated through exceptional points. Our work provides a detailed case for studying NH multiband braiding and knot structures in four-band systems, which could offer insights for topological photonics and analog information processing applications.

Keywords: exceptional point knot structures phase transitions non-Abelian braids

Received: 16 October 2025
Revised: 19 November 2025
Accepted manuscript online: 21 November 2025

PACS:	02.20.-a	(Group theory)
	03.65.Vf	(Phases: geometric; dynamic or topological)
	68.35.Rh	(Phase transitions and critical phenomena)

Fund: This work was supported by the National Natural Science Foundation of China (Grant Nos. 62575099, 62075059, and 61405058), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2024A1515011353), Open Project of the State Key Laboratory of Advanced Optical Communication Systems and Networks of China (Grant No. 2024GZKF20), the Natural Science Foundation of Hunan Province (Grant Nos. 2020JJ4161 and 2017JJ2048), and Scientific Research Foundation of Hunan Provincial Education Department (Grant No. 21A0013).

Corresponding Authors: Jian-Jun Liu
E-mail: jianjun.liu@hnu.edu.cn

Cite this article:

Lin-Sheng Bao(包淋升), Jia-Yun Ning(宁佳运), Ao-Qian Shi(史奥芊), Peng Peng(彭鹏), Zhen-Nan Wang(王瑱男), Chao Peng(彭超), Shuang-Chun Wen(文双春), and Jian-Jun Liu(刘建军) Exceptional point-induced knot structure transformations in non-Abelian braids 2026 Chin. Phys. B 35 010203

[1] Yang Y, Yang B, Ma G, Li J, Zhang S and Chan C T 2024 Science 383 eadf9621
[2] Slager R J, Bouhon A and Unal F N 2024 Nat. Commun. 15 1144
[3] Pang Z, Wong B T T, Hu J and Yang Y 2024 Phys. Rev. Lett. 132 043804
[4] Sun X C, Wang J B, He C and Chen Y F 2024 Phys. Rev. Lett. 132 216602
[5] Long Y, Wang Z, Zhang C, Xue H, Zhao Y X and Zhang B 2024 Phys. Rev. Lett. 132 236401
[6] Shan Z L, Sun Y K, Tao R, Chen Q D, Tian Z N and Zhang X L 2024 Phys. Rev. Lett. 133 053802
[7] Nayak C, Simon S H, Stern A, Freedman M and Das Sarma S 2008 Rev. Mod. Phys. 80 1083
[8] Iqbal M, Tantivasadakarn N, Verresen R and Vishwanath A 2023 Phys. Rev. Lett. 131 060405
[9] Campbell S L, Dreiling J M, Figgatt C, Gaebler J P, Johansen J, Mills M, Moses S A, Pino J M, Ransford A, Rowe M, Siegfried P, Stutz R P, Foss-Feig M, Vishwanath A and Dreyer H 2024 Nature 626 501
[10] Tsintzis A, Souto R S, Flensberg K, Danon J and Leijnse M 2024 PRX Quantum 5 010323
[11] Petiziol F 2024 Phys. Rev. Lett. 133 036601
[12] Long Y, Xue H and Zhang B 2024 Nat. Mach. Intell. 6 904
[13] Chen J, Wang Z, Tan Y T, Wang C and Ren J 2024 Commun. Phys. 7 209
[14] Gukov S, Halverson J and Ruehle F 2024 Nat. Rev. Phys. 6 310
[15] Wang K, Dutt A, Wojcik C C and Fan S 2021 Nature 598 59
[16] Patil Y S S, Holler J, Henry P A, Guria C, Zhang Y, Jiang L, Kralj N, Read N and Harris J G E 2022 Nature 607 271
[17] Guo C X, Chen S, Ding K and Hu H 2023 Phys. Rev. Lett. 130 157201
[18] Zhang Q, Li Y, Sun H, Liu X, Zhao L, Feng X, Fan X and Qiu C 2023 Phys. Rev. Lett. 130 017201
[19] Okuma N, Kawabata K, Shiozaki K and Sato M 2020 Phys. Rev. Lett. 124 086801
[20] Zhu W and Gong J 2023 Phys. Rev. B 108 035406
[21] Li Z, Wang L W, Wang X, Lin Z K, Ma G and Jiang J H 2024 Nat. Commun. 15 6544
[22] Chen J, Shi A, Peng Y, Peng P and Liu J 2024 Chin. Phys. Lett. 41 037103
[23] Shi A, Bao L, Peng P, Ning J, Wang Z and Liu J 2025 Phys. Rev. B 111 094109
[24] Peng Y, Shi A, Peng P and Liu J 2025 Phys. Rev. B 111 085148
[25] Wojcik C C, Sun X Q, Bzdusek T and Fan S 2020 Phys. Rev. B 101 205417
[26] Li Z and Mong R S K 2021 Phys. Rev. B 103 155129
[27] Zhang Q, Zhao L, Liu X, Feng X, Xiong L, Wu W and Qiu C 2023 Phys. Rev. Res. 5 L022050
[28] Hu H and Zhao E 2021 Phys. Rev. Lett. 126 010401
[29] Li Z, Ding K and Ma G 2023 Phys. Rev. Res. 5 023038
[30] Rafi-Ul-Islam S M, Siu Z B, Razo M S H, Sahin H and Jalil M B A 2024 Phys. Rev. B 110 045444
[31] Cao W, Zhang W and Zhang X 2024 Phys. Rev. B 109 165128
[32] Zhu B, Wang Q, Wang Y, Wang Q J and Chong Y D 2024 Phys. Rev. B 110 134317
[33] Yang Z, Chiu C K, Fang C and Hu J 2020 Phys. Rev. Lett. 124 186402
[34] Wang K, Dutt A, Yang K Y, Wojcik C C, Vuckovic J and Fan S 2021 Science 371 1240
[35] Ding K, Fang C and Ma G 2022 Nat. Rev. Phys. 4 745
[36] Tang W, Ding K and Ma G 2022 Nat. Sci. Rev. 9 nwac010
[37] Bao L, Shi A, Peng Y, Peng P, Ning J, Wang Z, Peng C and Liu J 2024 Phys. Rev. B 110 L020104
[38] Lee C H, Li L, Thomale R and Gong J 2020 Phys. Rev. B 102 085151
[39] Hu H, Sun S and Chen S 2022 Phys. Rev. Res. 4 L022064
[40] Shen H, Zhen B and Fu L 2018 Phys. Rev. Lett. 120 146402
[41] Li Y, Ji X, Chen Y, Yan X and Yang X 2022 Phys. Rev. B 106 195425
[42] Zhang X, Yu F, Chen Z, Tian Z, Chen Q D, Sun H B and Ma G 2022 Nat. Photon. 16 390
[43] Chen Z, Zhang R, Chan C T and Ma G 2022 Nat. Phys. 18 179
[44] Zhong Q, Khajavikhan M, Christodoulides D N and El-Ganainy R 2018 Nat. Commun. 9 4808
[45] Midya B 2023 Appl. Phys. Lett. 123 123101
[46] Yao S and Wang Z 2018 Phys. Rev. Lett. 121 086803
[47] See supplementary material for the derivation of phase diagrams for the tight-binding model, the definition and characteristics of the braid group B₄, the NH four-band braiding with u = 5 and u = 6 and the topological phase transitions and transition states of braided knots in complex E–k space
[48] Guria C, Zhong Q, Ozdemir S K, Patil Y S S, El-Ganainy R and Harris J G E 2024 Nat. Commun. 15 1369
[49] Tang Z, Chen T, Tang X and Zhang X 2024 Light Sci. Appl. 13 167
[50] Guo Q, Jiang T, Zhang R Y, Zhang L, Zhang Z Q, Yang B, Zhang S and Chan C T 2021 Nature 594 195
[51] Yan Q C, Zhao B H, Zhou R, Ma R, Lyu Q H, Chu S S, Hu X Y and Gong Q H 2023 Nanophotonics 12 2247
[52] Ke S L, Wang B, Long H, Wang K and Lu P X 2017 Opt. Quant. Electron. 49 224
[53] Zhang B, Yan W and Chen F 2025 Adv. Photon. 7 034002
[54] Yu D, Song W, Wang L, Srikanth R, Sridhar S, Chen T, Huang C, Li G, Qiao X, Wu X, Dong Z, He Y, Xiao M, Chen X, Dutt A, Gadway B and Yuan L 2025 Photon. Insights 4 R06
[55] Leykam D, Andreanov A and Flach S 2018 Adv. Phys. X 3 1473052

1. .pdf(7332KB)

[1]	Efficient and controlled symmetric and asymmetric Bell-state transfers in a dissipative Jaynes-Cummings model Qi-Cheng Wu(吴奇成), Yu-Liang Fang(方玉亮), Yan-Hui Zhou(周彦辉), Jun-Long Zhao(赵军龙), Yi-Hao Kang(康逸豪), Qi-Ping Su(苏奇平), and Chui-Ping Yang(杨垂平). Chin. Phys. B, 2026, 35(1): 010304.
[2]	Role of symmetry in antiferromagnetic topological insulators Sahar Ghasemi and Morad Ebrahimkhas. Chin. Phys. B, 2025, 34(7): 077302.
[3]	Random flux manipulating topological phase transitions in Chern insulators Jinkun Wang(王锦坤) and Wu-Ming Liu(刘伍明). Chin. Phys. B, 2025, 34(6): 067301.
[4]	Topological transmission and topological corner states combiner in all-dielectric honeycomb valley photonic crystals Ming Sun(孙铭), Xiao-Fang Xu(许孝芳), Yun-Feng Shen(沈云峰), Ya-Qing Chang(常雅箐), and Wen-Ji Zhou(周文佶). Chin. Phys. B, 2025, 34(3): 034206.
[5]	Pattern description of quantum phase transitions in the transverse antiferromagnetic Ising model with a longitudinal field Yun-Tong Yang(杨贇彤), Fu-Zhou Chen(陈富州), and Hong-Gang Luo(罗洪刚). Chin. Phys. B, 2025, 34(12): 127504.
[6]	Observation of exceptional points and realization of high sensitivity sensing in electric circuits Jiaxi Cai(蔡家希), Xinyi Wang(王鑫怡), Xiaomin Zhang(张晓敏), Taoran Yue(乐陶然), and Jijun Wang(王纪俊). Chin. Phys. B, 2025, 34(12): 128402.
[7]	Exceptional rings and non-Abelian topology in non-Hermitian high-spin systems Peng-Zhen Sun(孙鹏震), Zhou-Tao Lei(雷周涛), and Yuan-Gang Deng(邓元刚). Chin. Phys. B, 2025, 34(11): 110303.
[8]	Emergent ferroelectricity in the two-dimensional Janus MoSSe monolayer driven by nondegenerate phonon instability Zhi-Long Cao(曹智龙), Chen Cao(曹琛), Jia-Jun Xu(徐佳俊), Jia-Xu Yan(闫家旭), Lei Liu(刘雷), and De-Zhen Shen(申德振). Chin. Phys. B, 2025, 34(11): 117305.
[9]	Noise-induced phase transition in the Vicsek model through eigen microstate methodology Yongnan Jia(贾永楠), Jiali Han(韩佳丽), and Qing Li(李擎). Chin. Phys. B, 2024, 33(9): 090501.
[10]	First-principles study on stability and superconductivity of ternary hydride LaYH_x (x =2, 3, 6 and 8) Xiao-Zhen Yan(颜小珍), Xing-Zi Zhou(周幸姿), Chao-Fei Liu(刘超飞), Yin-Li Xu(徐寅力), Yi-Bin Huang(黄毅斌), Xiao-Wei Sheng(盛晓伟), and Yang-Mei Chen(陈杨梅). Chin. Phys. B, 2024, 33(8): 086301.
[11]	Design of a high sensitivity and wide range angular rate sensor based on exceptional surface Xinsheng Ding(丁鑫圣), Wenyao Liu(刘文耀), Shixian Wang(王师贤), Yu Tao(陶煜), Yanru Zhou(周彦汝), Yu Bai(白禹), Lai Liu(刘来), Enbo Xing(邢恩博), Jun Tang(唐军), and Jun Liu(刘俊). Chin. Phys. B, 2024, 33(8): 084204.
[12]	Mode coupling with Fabry-Perot modes in photonic crystal slabs Ken Qin(秦恳), Peng Hu(胡鹏), Jie Liu(刘杰), Hong Xiang(向红), and De-Zhuan Han(韩德专). Chin. Phys. B, 2024, 33(8): 084205.
[13]	Exceptional points and quantum dynamics in a non-Hermitian two-qubit system Yi-Xi Zhang(张益玺), Zhen-Tao Zhang(张振涛), Zhen-Shan Yang(杨震山), Xiao-Zhi Wei(魏晓志), and Bao-Long Liang(梁宝龙). Chin. Phys. B, 2024, 33(6): 060308.
[14]	Interacting topological magnons in a checkerboard ferromagnet Heng Zhu(朱恒), Hongchao Shi(施洪潮), Zhengguo Tang(唐政国), and Bing Tang(唐炳). Chin. Phys. B, 2024, 33(3): 037503.
[15]	Valley modulation and topological phase transition in staggered kagome ferromagnets Yuheng Xing(邢玉恒), Wenjuan Qiu(邱文娟), Xinxing Wu(吴新星), and Yue Tan(谭悦). Chin. Phys. B, 2024, 33(12): 127503.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Exceptional point-induced knot structure transformations in non-Abelian braids

Cite this article:

Online attention