Reentrant localized bulk and localized-extended edge in quasiperiodic non-Hermitian systems

GUO Gangfeng; Bao Xixi; TAN Lei; LIU Wuming

doi:10.7498/aps.74.20240933

Abstract

The localization is one of the active and fundamental research areas in topology physics. In this field, a comprehensive understanding of how wave functions distribute within a system is crucial. This work delves into this topic by proposing a novel systematic method based on a generalized Su-Schrieffer-Heeger (SSH) model. This model incorporates a quasiperiodic non-Hermitian term that appears at an off-diagonal position, adding a layer of complexity to the traditional SSH framework.By utilizing this model, we analyze the localization behaviors of both bulk state and edge state. For the bulk states, the analysis reveals a fascinating transition sequence. Specifically, the bulk states can undergo an extended-coexisting-localized-coexisting-localized transition, which is induced by the introduction of quasidisorder. This transition is not arbitrary but is rather conformed by the inverse participation ratio (IPR), a metric that quantifies the degree of localization of a wave function. As quasidisorder increases, the bulk states initially remain extended, but gradually, some states begin to be localized. A coexistence region appears where both extended and localized states are present. Further increase in quasidisorder leads to a complete localization of all bulk states. However, remarkably, within a certain range of quasidisorder strengths, the localized states can once again transition back to an extended state, creating another coexistence region. This complex behavior demonstrates the rich and diverse localization properties of the bulk states in non-Hermitian quasiperiodic systems.In addition to the IPR, other metrics such as the normalized participation ratio (NPR) and the fractal dimension of the eigenstates also play important roles in characterizing the localization behavior. These metrics provide a more in-depth understanding of the transition process and help to confirm the existence of the coexistence regions.Overall, we comprehensively analyze the localization behaviors of bulk and edge states in non-Hermitian quasiperiodic systems based on a generalized SSH model. The proposed systematic method present new insights into the complex interplay between quasidisorder, non-Hermiticity, and localization properties in topological physics.

Keywords:

Authors and contacts

1.
School of Mathematics and Physics, Suqian University, Suqian 223800, China
2.
Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China
3.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author: Bao Xixi, 17172@squ.edu.cn ; TAN Lei, tanlei@lzu.edu.cn ; LIU Wuming, wliu@iphy.ac.cn

Funds: Project supported by the Suqian Sci & Tech Program, China (Grant No. K202330), the Startup Foundation for Newly Recruited Employees of Suqian University, China (Grant No. 2024XRC006), and the National Natural Science Foundation of China (Grant Nos. 11874190, 12174461).

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图 1 (a) ${\rm{IPR}} ^{{\mathrm{B}}} $(红)和${\rm{NPR}} ^{{\mathrm{B}}} $(蓝)随$ W_{1} $的变化, 灰色区域代表共存状态; (b), (c)体能谱的实部和虚部, 附着色代表本征态的${\rm{IPR}} ^{|\varPsi^{n}\rangle} $值; (d)拓扑零能$ |E|^{0} $与对应的$ {\text{IPR}^{0}}/{5} $和${\rm{NPR}} ^{0} $; (e)拓扑不变量. 参数为$ t_{1} = 1 $, $ t_{2} = 2.5 $, $ \gamma = 0.2 $, $ L = 2000 $ 和$ W_{2} = -2\cos(3 W_{1})+2 $

Figure 1. (a) ${\rm{IPR}} ^{{\mathrm{B}}} $ (red) and ${\rm{NPR}} ^{{\mathrm{B}}} $ (blue) versus $ W_{1} $, the shaded regions stand for coexisting regimes; (b), (c) real and imaginary parts of bulk energy spectrum, where the dressed colors stand for different values of ${\rm{IPR}} ^{|\varPsi^{n}\rangle} $ for each bulk eigenstates; (d) topological edge modes $ |E|^{0} $, accompanied by the corresponding $ {\text{IPR}^{0}}/{5} $ and ${\rm{NPR}} ^{0} $; (e) topological invariant as a function of $ W_{1} $. Common parameters are $ t_{1} = 1 $, $ t_{2} = 2.5 $, $ \gamma = 0.2 $, $ L = 2000 $ and $ W_{2} = -2\cos(3 W_{1})+2 $.

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图 2 (a1)—(e1)复平面能谱及其${\rm{IPR}} ^{|\varPsi^{n}\rangle} $; (a2)—(e2)蓝色和红色曲线代表一些体态的局域性质, 绿色条代表边缘态的局域性质. 为了显示清楚, 这些物理量已按比例缩放. (a1), (a2) $ W_{1} = 0.15 $, 所有体态均为扩展态; (b1), (b2) $ W_{1} = 0.35 $, 一些态仍为扩展态, 但一些态转为局域态; (c1), (c2) $ W_{1} = 1 $, 所有的体态均转变为局域态; (d1), (d2) $ W_{1} = 2.02 $, 部分已经局域的本征态会再转变为扩展态; (e1), (e2) $ W_{1} = 3.8 $, 所有的体态再次全部转变为局域态. 其他参数和图1取值一致

Figure 2. (a1)–(e1) Energy spectrum on the complex plane with corresponding ${\rm{IPR}} ^{|\varPsi^{n}\rangle} $; (a2)–(e2) both the blue and red lines stand for the localized properties of some bulk eigenstates, and the zero-energy state is shown as the green bar. For a better visibility, the values have been scaled proportionally. (a1), (a2) $ W_{1} = 0.15 $, all bulk eigenstates are extended; (b1), (b2) $ W_{1} = 0.35 $, some eigenstates are still extended while some are localized; (c1), (c2) $ W_{1} = 1 $, all bulk eigenstates are transformed into localized; (d1), (d2) $ W_{1} = 2.02 $, some already localized states will change into extended; (e1)–(e2) $ W_{1} = 3.8 $, all bulk eigenstates are transformed into localized again. Other parameters are the same as the ones in Fig. 1.

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图 3 (a) $ L = 12000 $, 随着γ变化的${\rm{IPR}} ^{{\mathrm{B}}} $(红)和${\rm{NPR}} ^{{\mathrm{B}}} $(蓝), 两个灰色区域代表共存状态; (b) $ \gamma = 4.66946 $, ${\rm{NPR}} ^{{\mathrm{B}}} $的有限尺寸效应分析, 系统尺寸分别为$ L = 20000 $, $ 30000 $和$ 40000 $; (c)体能谱实部, 不同附着色代表$ \eta^{|\varPsi^{n}\rangle} $的不同取值, $ L = 6000 $; (d) $ |E|^{0} $与对应的${\rm{IPR}} ^{0} $和${\rm{NPR}} ^{0} $随着γ的变化趋势, $ L = 2000 $; (e)系统尺寸$ L = 12000 $时${\rm{NPR}} ^{0} $对γ的导数; (f)拓扑不变量随参数γ的变化趋势, $ L = 2000 $. 其他参数为$ t_{1} = 9 $, $ t_{2} = 1 $, $ W_{1} = 0.0039 $和$ W_{2} = 1.563 $

Figure 3. (a) $ L = 12000 $, ${\rm{IPR}} ^{{\mathrm{B}}} $ (red) and ${\rm{NPR}} ^{{\mathrm{B}}} $ (blue) evolving with γ, the two shaded regions stand for intermediate regimes; (b) finite size scaling analysis of ${\rm{NPR}} ^{{\mathrm{B}}} $ at $ \gamma = 4.66946 $ for $ L = 20000 $, $ 30000 $ and $ 40000 $, respectively; (c) real parts of bulk energy spectrum for $ L = 6000 $, where different colors stand for different values of $ \eta^{|\varPsi^{n}\rangle} $; (d) evolution of $ |E|^{0} $ versus γ associated with ${\rm{IPR}} ^{0} $ and ${\rm{NPR}} ^{0} $, $ L = 2000 $; (e) derivative of ${\rm{NPR}} ^{0} $ with respect to γ for $ L = 12000 $; (f) topological invariant versus parameter γ for $ L = 2000 $. Common parameters are $ t_{1} = 9 $, $ t_{2} = 1 $, $ W_{1} = 0.0039 $ and $ W_{2} = 1.563 $.

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图 4 所有归一化体态的空间分布. 为了表示清楚, 系统尺寸取$ L = 1000 $, 并适当缩减能量值范围　(a) $ \gamma = 0.2 $, 扩展态与局域态一起出现; (b) $ \gamma = 1 $, 所有扩展态均转变为局域态; (c) $ \gamma = 4.66946 $, 一些局域态会转回扩展态; (d) $ \gamma = 5.5 $, 所有态再次变为局域态. 其他参数和图3 一致

Figure 4. Spatial distributions of normalized bulk eigenstates with the change of eigenvalues. For enhanced visibility, the system size is $ L = 1000 $, and the values of the energies have been cut down appropriately: (a) $ \gamma = 0.2 $, the extended and localized eigenstates appearing together; (b) $ \gamma = 1 $, all extended eigenstates being transformed into the localized; (c) $ \gamma = 4.66946 $, some localized eigenstates turning to the extended; (d) $ \gamma = 5.5 $, all extended eigenstates being transformed into the localized once again. Other parameters are the same as the ones in Fig. 3.

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图 5 (a)拓扑不变量随$ W_{1} $的变化趋势; (b) $ |E|^{0} $与对应的${\rm{IPR}} ^{0} $和${\rm{NPR}} ^{0} $随$ W_{1} $的演化; (c) 图(b)其中一部分的详细信息; (d) ${\rm{NPR}} ^{0} $的导数, 显然其中存在一些不连续的点; (e)以$ W_{1} $为自变量的${\rm{IPR}} ^{{\mathrm{B}}} $(红)和${\rm{NPR}} ^{{\mathrm{B}}} $(蓝), 阴影区域表示扩展态和局域态共存; (f)体能谱的实部, 附着在其上的不同颜色代表不同的${\rm{IPR}} ^{|\varPsi^{n}\rangle} $值. 图(a), (b), (e) 和(f) $ L = 2000 $, 图(c)和(d) $ L = 10000 $, 其他参数为$ t_{1} = 1 $, $ t_{2} = 1.3 $, $ \gamma = 0.05 $和$ W_{2} = W_{1} $

Figure 5. (a) Topological invariant versus $ W_{1} $; (b) evolution of $ |E|^{0} $ via $ W_{1} $, associated with ${\rm{IPR}} ^{0} $ and ${\rm{NPR}} ^{0} $; (c) detailed information on the localization of the eigenstates with the lowest energy; (d) derivative of ${\rm{NPR}} ^{0} $, where there exist some discontinuity points; (e) ${\rm{IPR}} ^{{\mathrm{B}}} $ (red) and ${\rm{NPR}} ^{{\mathrm{B}}} $ (blue) being plotted as a function of $ W_{1} $. The shaded region indicates the coexisting regime, with extended and localized eigenstates being coexistence; (f) real part of bulk energy spectrum, where different colors stand for different values of ${\rm{IPR}} ^{|\varPsi^{n}\rangle} $. For panels (a), (b), (e) and (f) $ L = 2000 $, and for panels (c) and (d) $ L = 10000 $. Same parameters are $ t_{1} = 1 $, $ t_{2} = 1.3 $, $ \gamma = 0.05 $ and $ W_{2} = W_{1} $.

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[1]	Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004 Google Scholar
[2]	Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2016 Rev. Mod. Phys. 88 035005 Google Scholar
[3]	Asbóth J, Oroszlány L, Pályi A 2016 A Short Course on Topological Insulators 919 2 Google Scholar
[4]	Bernevig B A, Hughes T L 2013 Topological Insulators and Topological Superconductors (Princeton: Princeton University Press
[5]	Shen S Q 2012 Topological Insulators Dirac Equation in Condensed Matters (Vol. 174) (Berlin: Springer Google Scholar
[6]	Yokomizo K, Murakami S 2020 Phys. Rev. Research 2 043045 Google Scholar
[7]	Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Phys. Rev. B 97 121401 Google Scholar
[8]	Wu H, Wang B Q, An J H 2021 Phys. Rev. B 103 L041115 Google Scholar
[9]	He P, Huang Z H 2020 Phys. Rev. A 102 062201 Google Scholar
[10]	Torres L E F F 2019 J. Phys.: Mater. 3 014002 Google Scholar
[11]	Esaki K, Sato M, Hasebe K, Kohmoto M 2011 Phys. Rev. B 84 205128 Google Scholar
[12]	Hu Y C, Hughes T L 2011 Phys. Rev. B 84 153101 Google Scholar
[13]	Zhu B, Lü R, Chen S 2014 Phys. Rev. A 89 062102 Google Scholar
[14]	Malzard S, Poli C, Schomerus H 2015 Phys. Rev. Lett. 115 200402 Google Scholar
[15]	Xu Y, Wang S T, Duan L M 2017 Phys. Rev. Lett. 118 045701 Google Scholar
[16]	Zhan X, Xiao L, Bian Z, Wang K, Qiu X, Sanders B C, Yi W, Xue P 2017 Phys. Rev. Lett. 119 130501 Google Scholar
[17]	Zeuner J M, Rechtsman M C, Plotnik Y, Lumer Y, Nolte S, Rudner M S, Segev M, Szameit A 2015 Phys. Rev. Lett. 115 040402 Google Scholar
[18]	Rufangura P, Folland T G, Agrawal A, Caldwell J D, Iacopi F 2020 J. Phys.: Mater. 3 032005 Google Scholar
[19]	Jin L, Song Z 2019 Phys. Rev. B 99 081103 Google Scholar
[20]	Borgnia D S, Kruchkov A J, Slager R J 2020 Phys. Rev. Lett. 124 056802 Google Scholar
[21]	Takata K, Notomi M 2018 Phys. Rev. Lett. 121 213902 Google Scholar
[22]	Chen Y, Zhai H 2018 Phys. Rev. B 98 245130 Google Scholar
[23]	Lang L J, Wang Y, Wang H, Chong Y D 2018 Phys. Rev. B 98 094307 Google Scholar
[24]	Harari G, Bandres M A, Lumer Y, Rechtsman M C, Chong Y D, Khajavikhan M, Christodoulides D N, Segev M 2018 Science 359 eaar4003 Google Scholar
[25]	Xiao L, Deng T, Wang K, Zhu G, Wang Z, Yi W, Xue P 2020 Nat. Phys. 16 761 Google Scholar
[26]	Zhou H, Peng C, Yoon Y, Hsu C W, Nelson K A, Fu L, Joannopoulos J D, Soljačić M, Zhen B 2018 Science 359 1009 Google Scholar
[27]	Deng T S, Yi W 2019 Phys. Rev. B 100 035102 Google Scholar
[28]	Ezawa M 2019 Phys. Rev. B 99 201411 Google Scholar
[29]	Kawabata K, Higashikawa S, Gong Z, Ashida Y, Ueda M 2019 Nat. Commun. 10 297 Google Scholar
[30]	Kawabata K, Ashida Y, Katsura H, Ueda M 2018 Phys. Rev. B 98 085116 Google Scholar
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[32]	Lee C H, Li L, Gong J 2019 Phys. Rev. Lett. 123 016805 Google Scholar
[33]	Yoshida T, Kudo K, Hatsugai Y A 2019 Sci. Rep. 9 16895 Google Scholar
[34]	Luitz D J, Piazza F 2019 Phys. Rev. Research 1 033051 Google Scholar
[35]	Rosenthal E I, Ehrlich N K, Rudner M S, Higginbotham A P, Lehnert K W 2018 Phys. Rev. B 97 220301 Google Scholar
[36]	Lee T E 2016 Phys. Rev. Lett. 116 133903 Google Scholar
[37]	Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phys. Rev. Lett. 121 026808 Google Scholar
[38]	Xiong Y 2018 J. Phys. Commun. 2 035043 Google Scholar
[39]	Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803 Google Scholar
[40]	Yokomizo K, Murakami S 2019 Phys. Rev. Lett. 123 066404 Google Scholar
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[42]	Yang Z, Zhang K, Fang C, Hu J 2020 Phys. Rev. Lett. 125 226402 Google Scholar
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[45]	Bao X X, Guo G F, Tan L 2021 J. Phys. Condens. Matter 33 465403 Google Scholar
[46]	Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M 2018 Phys. Rev. X 8 031079 Google Scholar
[47]	Ghatak A, Das T 2019 J. Phys. Condens. Matter 31 263001 Google Scholar
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