PT phase transition and non-Hermitian skin effect in vacancy-induced subbands of the Haldane model with gain and loss

doi:10.1088/1674-1056/ae44f4

Abstract Defects can profoundly alter the properties of quantum materials, inducing emergent phenomena beyond the scope of their parent counterparts. Here, we construct effective models to investigate the properties of vacancy-induced subbands in the Haldane model with gain and loss, revealing that their behaviors are critically dependent on the spatial configuration of the vacancy array. Specifically, a sequence of parity-time ($\mathcal{PT}$) phase transitions occurs for the one-dimensional vacancy array across different sublattices, whereas the non-Hermitian skin effect (NHSE) — with its nontrivial spectral winding — manifests in a chain confined to a single sublattice. Notably, the NHSE arises in this subband even though the parent system lacks it. Our results demonstrate that vacancy defect engineering serves as a powerful approach to generate subsystems whose properties are decoupled from the parent Hamiltonian.

Keywords: non-Hermitian physics defect parity-time symmetry non-Hermitian skin effect

Received: 13 November 2025
Revised: 11 February 2026
Accepted manuscript online: 12 February 2026

PACS:	71.55.-i	(Impurity and defect levels)
	03.65.Yz	(Decoherence; open systems; quantum statistical methods)
	11.30.Er	(Charge conjugation, parity, time reversal, and other discrete symmetries)
	73.20.-r	(Electron states at surfaces and interfaces)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 12405030), the National Key Research and Development Program of China (Grant No. 2023YFA1406704), the Open Fund of Key Laboratory of Multiscale Spin Physics (Ministry of Education) Beijing Normal University (Grant No. SPIN2024K01), Beijing National Laboratory for Condensed Matter Physics (Grant No. 2025BNLCMPKF023), the Science Foundation of China University of Petroleum, Beijing (Grant No. 2462024SZBH003), Young Elite Scientists Sponsorship Program of the Beijing High Innovation Plan (Grant No. 20250934), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023A1515110081), and Fundamental Research Funds for the Central Universities, China (Grant Nos. FRF-TP-22-098A1 and FRF-IDRY-24-28).

Corresponding Authors: Su-Peng Kou
E-mail: spkou@bnu.edu.cn

Cite this article:

Cui-Xian Guo(郭翠仙), Xiao-Ming Zhao(赵小明), and Su-Peng Kou(寇谡鹏) PT phase transition and non-Hermitian skin effect in vacancy-induced subbands of the Haldane model with gain and loss 2026 Chin. Phys. B 35 047103

[1] Ashida Y, Gong Z and Ueda M 2020 Adv. Phys. 69 249
[2] Bergholtz E J, Budich J C and Kunst F K 2021 Rev. Mod. Phys. 93 015005
[3] Ding K, Fang C and Ma G 2022 Nat. Rev. Phys. 4 745
[4] Moiseyev N 2011 Non-Hermitian Quantum Mechanics (New York: Cambridge University Press)
[5] Bender C M 2007 Rep. Prog. Phys. 70 947
[6] Yang K, Li Z, König J L K, Rødland L, Stålhammar M and Bergholtz E J 2024 Rep. Prog. Phys. 87 078002
[7] Guo A, Salamo G J, Duchesne D, Morandotti R, VolatierRavat M, Aimez V, Siviloglou G A and Christodoulides D N 2009 Phys. Rev. Lett. 103 093902
[8] Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D 2010 Nat. Phys. 6 192
[9] Chong Y D, Ge L and Stone A D 2011 Phys. Rev. Lett. 106 093902
[10] Regensburger A, Bersch C, MiriMA, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167
[11] Feng L, Xu Y L, Fegadolli W S, Lu M H, Oliveira J E B, Almeida V R, Chen Y F and Scherer A 2013 Nat. Mater. 12 108
[12] Wimmer M, Miri M A, Christodoulides D N and Peschel U 2015 Sci. Rep. 5 17760
[13] Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C and Szameit A 2017 Nat. Mater. 16 433
[14] Zhu X, Ramezani H, Shi C, Zhu J and Zhang X 2014 Phys. Rev. X 4 031042
[15] Popa B I and Cummer S A 2014 Nat. Commun. 5 3398
[16] Fleury R, Sounas D and Alu A 2015 Nat. Commun. 6 5905
[17] Ding K, Ma G, Xiao M, Zhang Z Q and Chan C T 2016 Phys. Rev. X 6 021007
[18] Xiao L, Deng T, Wang K, Wang Z, Yi W and Xue P 2021 Phys. Rev. Lett. 126 230402
[19] Wang K, Xiao L, Budich J C, Yi W and Xue P 2021 Phys. Rev. Lett. 127 026404
[20] Wu Y, LiuW, Geng J, Song X, Ye X, Duan C K, Rong X and Du J 2019 Science 364 878
[21] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243
[22] Hang C, Huang G and Konotop V V 2013 Phys. Rev. Lett. 110 083604
[23] Feng L, Xu Y L, Fegadolli W S, Lu M H, Oliveira J E B, Almeida V R, Chen Y F and A Scherer 2013 Nat. Mater. 12 108
[24] Peng B, Ö zdemir Ş K, Lei F, Monifi F, Gianfreda M, Long G L, Fan S, Nori F, Bender C M and Yang L 2014 Nat. Phys. 10 394
[25] Feng L, Wong Z J, Ma R M, Wang Y and Zhang X 2014 Science 346 972
[26] Hodaei H, Miri M A, Heinrich M, Christodoulides D N and Khajavikhan M 2014 Science 346 975
[27] Weidemann S, Kremer M, Helbig T, Hofmann T, Stegmaier A, Greiter M, Thomale R and Szameit A 2020 Science 368 311
[28] Yao S and Wang Z 2018 Phys. Rev. Lett. 121 086803
[29] Yokomizo K and Murakami S 2019 Phys. Rev. Lett. 123 066404
[30] Kunst F K, Edvardsson E, Budich J C and Bergholtz E J 2018 Phys. Rev. Lett. 121 026808
[31] Zhang K, Yang Z and Fang C 2020 Phys. Rev. Lett. 125 126402
[32] Okuma N, Kawabata K, Shiozaki K and Sato M 2020 Phys. Rev. Lett. 124 086801
[33] Yang Z, Zhang K, Fang C and Hu J 2020 Phys. Rev. Lett. 125 226402
[34] Alvarez V M M, Vargas J E B and Torres L E F F 2018 Phys. Rev. B 97 121401
[35] Borgnia D S, Kruchkov A J and Slager R J 2020 Phys. Rev. Lett. 124 056802
[36] Guo C X, Liu C H, Zhao X M, Liu Y and Chen S 2021 Phys. Rev. Lett. 127 116801
[37] Lee C H and Thomale R 2019 Phys. Rev. B 99 201103
[38] Wang H Y, Song F and Wang Z 2024 Phys. Rev. X 14 021011
[39] Hu H 2025 Sci. Bull. 70 51
[40] Zhang K, Yang Z and Fang C 2022 Nat. Commun. 13 2496
[41] Fu B, Jia Z, Mu W, Yin Y, Zhang J and Tao X 2019 J. Semicond. 40 011804
[42] Klimov P V and Awschalom D D 2025 Appl. Sci. 15 5606
[43] Pearton S J, Norton D P, Ip K, Heo YWand Steiner T 2004 J. Vac. Sci. Technol. B 22 932
[44] Ugeda M M, Brihuega I, Guinea F and Gómez-Rodríguez J M 2010 Phys. Rev. Lett. 104 096804
[45] Schlenk T, Bair M, Wiebe J and Wiesendanger R 2012 Phys. Rev. Lett. 108 066801
[46] Haldane F D M 1988 Phys. Rev. Lett. 61 2015

[1]	High-precision calculations of highly excited and autoionizing states of the nickel atom Sheng-Bo Niu(牛生波), Jun-Yao Zhang(张钧尧), Rui Jin(金锐), and Yi-Zhi Qu(屈一至). Chin. Phys. B, 2026, 35(6): 063101.
[2]	Observation of exceptional points in anisotropic ReS₂ microcavities Huihua Chen(陈辉华), Yuquan Zhou(周玉全), Xin Li(李昕), Jinyang Lou(娄金阳), Song Luo(罗松), Hang Zhou(周航), Xinyue Zhang(张新悦), Zhao Xu(徐钊), Yan Liu(刘艳), Zheng Lv(吕峥), Yuxin Duan(段雨欣), Haodong Cheng(成浩东), Hongming Zhang(张鸿铭), Yaofeng Zhu(朱耀峰), Anpeng Li(李安鹏), Jian Ren(任坚), Xiao Wang(王潇), Lixin Zhang(章立心), Long Zhang(张龙), and Zhanghai Chen(陈张海). Chin. Phys. B, 2026, 35(5): 057105.
[3]	Site occupation of Al doping in Lu₂SiO₅: The role of ionic radius versus chemical valence Xuejiao Sun(孙雪娇), Yu Cui(崔宇), Feng Gao(高峰), Zhongjun Xue(薛中军), Shuwen Zhao(赵书文), Dongzhou Ding(丁栋舟), Fan Yang(杨帆), and Yi-Yang Sun(孙宜阳). Chin. Phys. B, 2025, 34(9): 096101.
[4]	Improved ferroelectricity in Mn-doped HfO₂ (111) epitaxial thin films through controlled doping and substrate orientation Jiayi Gu(顾嘉仪), Haiyi Zhang(张海义), Weijin Pan(潘炜进), Haifeng Bu(卜海峰), Zhijian Shen(沈志健), Shengchun Shen(沈胜春), Yuewei Yin(殷月伟), and Xiaoguang Li(李晓光). Chin. Phys. B, 2025, 34(8): 087701.
[5]	Non-quantized Zak phases, PT/APT symmetry transitions, and doubly degenerate exceptional points in a non-Hermitian spin-orbit coupled SSH model Jun-Xing Huo(霍俊行), Jian Li(李健), Qing-Xu Li(李清旭), and Jia-Ji Zhu(朱家骥). Chin. Phys. B, 2025, 34(7): 070301.
[6]	Characterization of antisite defects and in-gap states in antiferromagnetic MnSb₂Te₄ Junming Zhang(张峻铭), Ming Xi(席明), Yuchong Zhang(张羽翀), Hang Li(李航), Jiali Zhao(赵佳丽), Hechang Lei(雷和畅), Zhongxu Wei(魏忠旭), and Tian Qian(钱天). Chin. Phys. B, 2025, 34(7): 076801.
[7]	Simulation of the non-Hermitian Kitaev chain by electrical circuits Jiali Xu(徐佳莉), Hao Geng(耿昊), Abdul Wahab, Xiaosen Yang(杨孝森), Yuee Xie(谢月娥), and Yuanping Chen(陈元平). Chin. Phys. B, 2025, 34(6): 068401.
[8]	Effects of helium ion irradiation and thermal annealing on the optical and structural properties of hexagonal boron nitride Guan-Lin Liu(刘冠麟), Ji-Lian Xu(徐辑廉), Peng-Tao Jing(景鹏涛), Jing-Jing Shao(邵京京), Xu Guo(郭旭), Yun-Tao Wu(吴韵涛), Feng Qin(覃凤), Zhen Cheng(程祯), Deming Liu(刘德明), Yang Bao(鲍洋), Hai Xu(徐海), Li-Gong Zhang(张立功), Da Zhan(詹达), Jia-Xu Yan(闫家旭), Lei Liu(刘雷), and De-Zhen Shen(申德振). Chin. Phys. B, 2025, 34(5): 057801.
[9]	Three-body physics under dissipative spin-orbit coupling Xi Zhao(赵茜). Chin. Phys. B, 2025, 34(3): 033101.
[10]	Comparison of high-order harmonic generation in defect-free and defective solids with different time delays Shujie Zhao(赵书杰), Yuanzuo Li(李源作), Jun Zhang(张军), and Xuefei Pan(潘雪飞). Chin. Phys. B, 2025, 34(12): 123201.
[11]	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.
[12]	Degradation mechanisms of Schottky p-GaN gate AlGaN/GaN HEMTs under high-temperature reverse bias stress Fei Hu(胡飞), Chengbing Pan(潘成兵), Xinyuan Zheng(郑鑫源), Yibo Ning(宁一博), Xueyan Li(李雪燕), and Lixia Zhao(赵丽霞). Chin. Phys. B, 2025, 34(11): 118501.
[13]	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.
[14]	Lamb wave TDTE super-resolution imaging assisted by deep learning Liu-Jia Sun(孙刘家), Qing-Bang Han(韩庆邦), and Qi-Lin Jin(靳琪琳). Chin. Phys. B, 2025, 34(1): 014301.
[15]	Visualizing extended defects at the atomic level in a Bi₂Sr₂CaCu₂O_8+δ superconducting wire Kejun Hu(胡柯钧), Shuai Wang(王帅), Boyu Li(李泊玉), Ying Liu(刘影), Binghui Ge(葛炳辉), and Dongsheng Song(宋东升). Chin. Phys. B, 2024, 33(9): 096101.

[1]	TANG GUI-DE (唐贵德), NIE XIANG-FU (聂向富), SUN HUI-YUAN (孙会元), MENG GUANG-QING (孟广庆), HAN BAO-SHAN (韩宝善). ANNIHILATION OF VERTICAL-BLOCH-LINE CHAINS IN THE WALLS OF THE SECOND KIND OF DUMBBELL DOMAINS SUBJECTED TO AN IN-PLANE FIELD[J]. Acta Physica Sinica (Overseas Edition), 1993, 2(11): 863 -869 .
[2]	LIN ZHEN-QUAN (林振权), YANG ZHAN-RU (杨展如), QIN YONG (秦勇). RENORMALIZATION GROUP APPROACH TO THE BOND PERCOLATION ON SIERPINSKI CARPETS[J]. Acta Physica Sinica (Overseas Edition), 1997, 6(4): 257 -265 .
[3]	LIU GUO-ZHI (刘国治), SONG XIAO-XIN (宋晓欣). SELF-MAGNETIC-FIELD-LIMITING CURRENT OF INTENSE RELATIVISTIC ELECTRON BEAM UNDER EXTERNALLY APPLIED MAGNETIC FIELD[J]. Acta Physica Sinica (Overseas Edition), 1998, 7(4): 288 -291 .
[4]	Feng Mang (冯芒), Hai Wen-hua (海文华), Zhu Xi-wen (朱熙文), Gao Ke-lin (高克林), Shi Lei (施磊). RESONANT INTERACTION BETWEEN A PAUL-TRAPPED ION AND A STANDING WAVE LASER[J]. Chinese Physics, 2000, 9(4): 250 -258 .
[5]	Zheng Shao-kuan (郑绍宽), Chen Zhong (陈忠), Chen Zhi-wei (陈志伟), Zhong Jian-hui (钟健晖). DIRECT MEASUREMENT OF TRANSVERSE RELAXATION TIME OF INTERMOLECULAR MULTIPLE QUANTUM COHERENCES IN NMR[J]. Chinese Physics, 2001, 10(6): 558 -563 .
[6]	Zhao Jian-lin (赵建林), Wang Bin (王彬), Wu Jian-jun (吴建军), Yang De-xing (杨德兴), S. Kapphan, R. Pankrath. INVESTIGATION OF PHOTOREFRACTIVE TWO-WAVE COUPLING IN Cr-DOPED STRONTIUM BARIUM NIOBATE CRYSTAL[J]. Chinese Physics, 2001, 10(8): 739 -742 .
[7]	Zhang Sheng-Hai (张胜海), Shen Ke (沈柯). Generalized synchronization of chaos in erbium-doped dual-ring lasers[J]. Chinese Physics, 2002, 11(9): 894 -899 .
[8]	Li Shao-Hui (李邵辉), Li Ru-Xin (李儒新), Ni Guo-Quan (倪国权), Xu Zhi-Zhan (徐至展). Electron impact ionization of large krypton clusters[J]. Chinese Physics, 2004, 13(10): 1684 -1688 .
[9]	Sun Jian-Cheng (孙建成), Zhang Tai-Yi (张太镒), Liu Feng (刘枫). Modelling of chaotic systems based on modified weighted recurrent least squares support vector machines[J]. Chinese Physics, 2004, 13(12): 2045 -2052 .
[10]	Chen Xiao-Fan (陈小凡), Yang Xue-Dong (杨学栋), Han Ling (韩玲). Two-pion interferometry at small relative momentum for pion sources with transverse and longitudinal expansions in relativistic heavy ion collisions[J]. Chinese Physics, 2004, 13(3): 341 -343 .

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

PT phase transition and non-Hermitian skin effect in vacancy-induced subbands of the Haldane model with gain and loss

Cite this article:

Online attention