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In recent years, a large number of novel phenomena such as the breakdown of conventional bulk-boundary correspondence and non-Hermitian skin effect, have emerged in non-Hermitian systems. In this work, we investigate the localization of the eigenstates and the non-Hermitian skin effect of the disordered non-Hermitian Su-Schrieffer-Heeger (SSH) model by inverse participation rate (IPR) and average inverse participation rate (MIPR). We also investigate the bulk-boundary correspondence ratio of the system. Based on the above, we further investigate the effect of disorder on the non-Hermitian skin effect and the topological properties of the NH system. We find that the disorder does not destroy the localization of the topological edge state due to the protection from the topology of the system. But the eigenstates of bulk are greatly affected by the disorder. In the presence of disorder, the eigenstates of the bulk will rapidly extend into the bulk. Thus, the non-Hermitian skin effect is vulnerable to the disorder. When the disorder is enhanced, the non-Hermitian skin effect will be greatly suppressed. We also show that the disorder will reduce the energy gap and imaginary energy of the system. Our study contributes to the further understanding of the non-Hermitian skin effect.
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Keywords:
- non-Hermitian skin effect /
- disorder /
- inverse participation ratio
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图 1 (a)—(c)拓扑非平凡态, 临界和拓扑平凡态的能谱. 颜色表示能量对应的波函数的IPR. 参数为 (a)
$ {t_1} = 0.4 $ ; (b)${t_1} = $ $ 1.58$ ; (c)$ {t_1} = 2.5 $ . (d)—(f) 分别对应图(a)—(c)中所有本征态在空间的分布. 其余的参数为:$\gamma = 0.4$ , 系统的尺寸L为80Figure 1. (a)–(c) The eigenenergies of topological nontrivial, critical and topological trivial phases with different parameters: (a)
$ {t_1} = 0.4 $ ; (b)$ {t_1} = 1.58 $ ; (c)$ {t_1} = 2.5 $ . The color denotes the IPR of the eigenstates corresponding to the eigenenergies. (d)–(f) The corresponding eigenstates in real space for Figure (a)–(c), respectively. The remaining parameter is$\gamma = 0.4$ . The length of the lattice is 80.图 2 (a) 平均逆参与率(MIPR)随参数
${t_1}$ 变化. 参数为$\gamma = 0.4$ ,$L = 80$ . (b) 平均逆参与率与晶格尺寸L的标度. 绿色线对应${t_1} = 0.4$ ,$\gamma = 0$ . 其他颜色的线分别对应不同的${t_1}$ ,$\gamma = 0.4$ Figure 2. (a) MIPR varies with
$ {t_1} $ , other parameters are$\gamma = 0.4$ and$L = 80$ ; (b) finite-size scaling of MIPR for different$ {t_1} $ . The green line corresponds to${t_1} = 0.4$ ,$\gamma = 0$ , and the other lines correspond to different${t_1}$ with$\gamma = 0.4$ .图 3 (a)
${t_1} \text- \gamma$ 平面的相图. 红色为拓扑相区域, 体边对应率是0.5, 此时存在DES. (b) 体边对应率随微扰大小变化. 当微扰为零时,${R_{\rm BBC}}$ 为0.5, 当微扰增大时,${R_{\rm BBC}}$ 迅速增加到1. 其余参数为$ {t_1} = 0.4 $ ,$\gamma = 0.4$ 以及$L = 80$ Figure 3. (a) Phase diagram in
${t_1} \text- \gamma$ plane. The phase is topological nontrivial in the red regions, where the value of BBC ratio is 0.5. And there exists DES. (b)${R_{BBC}}$ varies with the disturbance$\Delta $ . When the disturbance is zero, the value of${R_{\rm BBC}}$ is 0.5, and when the disturbance increases, the value of${R_{\rm BBC}}$ jumps to 1. The remaining parameters are$ {t_1} = 0.4 $ ,$\gamma = 0.4$ and$L = 80$ .图 4 (a), (b) 开边界条件下的能谱. 红色点表示拓扑零能, 蓝色表示体态本征能量. (c)—(e) 不同无序强度d下的波函数. 黑色线表示体态的波函数, 红色线表示拓扑零能对应的拓扑边缘态的波函数. 无序强度分别为 (c)
$ d = 0 $ , (d)$ d = 0.5 $ , (e)$ d = 1 $ . 其余参数为$ {t_1} = 0.4 $ ,$\gamma = 0.4$ 以及$L = 80$ .Figure 4. (a), (b) Energy spectra under the open boundary condition. The red dots are topological zero-mode. (c)–(e) The eigenstate wave functions with different disorders d. The black curves are the bulk-state wave functions and the red curves are the zero-mode wave functions. The disorder strength is: (c) d = 0, (d) d = 0.5, (e) d = 1. The remaining parameters are
$ {t_1} $ = 0.4,$\gamma = 0.4$ and L = 80.图 5 (a) 平均逆参与率与晶格尺寸L的标度. 所有线对应
$ {t_1} = 0.4 $ . 其中绿色线对应$d = 0$ ,$\gamma = 0$ . 红色线对应$d = 0.5$ ,$\gamma = 0$ . 其他颜色的线分别对应不同的$d$ ,$\gamma = 0.4$ . (b) 平均逆参与率(MIPR)随无序强度$d$ 变化. 参数为$ {t_1} = 0.4 $ ,$\gamma = 0.4$ ,$L = 100$ Figure 5. (a) Finite-size scaling of MIPR for different
$d$ . All the lines correspond to${t_1} = 0.4$ .The green line corresponds to$d = 0$ ,$\gamma = 0$ and the red line corresponds to$d = 0.5$ ,$\gamma = 0$ . The other lines correspond to different$d$ with$\gamma = 0.4$ . (b) MIPR varies with$d$ , other parameters are$ {t_1} = 0.4 $ ,$\gamma = 0.4$ and$L = 100$ . -
[1] Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004
Google Scholar
[2] Moore J E, Balents L 2007 Phys. Rev. B 75 121306
Google Scholar
[3] Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803
Google Scholar
[4] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[5] 陈增军, 宁西京 2003 物理学报 52 2683
Google Scholar
Chen Z J, Ning X J 2003 Acta Phys. Sin. 52 2683
Google Scholar
[6] Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2016 Rev. Mod. Phys. 88 035005
Google Scholar
[7] 王洪飞, 解碧野, 詹鹏, 卢明辉, 陈延锋 2019 物理学报 68 224206
Google Scholar
Wang H F, Xie B Y, Zhan P, Lu M H, Chen Y F 2019 Acta Phys. Sin. 68 224206
Google Scholar
[8] 孙孔浩, 易为 2021 物理学报 70 230309
Google Scholar
Sun K H, Yi W 2021 Acta Phys. Sin. 70 230309
Google Scholar
[9] 沈瑞昌, 张国强, 王逸璞, 游建强 2019 物理学报 68 230305
Google Scholar
Shen R C, Zhang G Q, Wang Y P, You J Q 2019 Acta Phys. Sin. 68 230305
Google Scholar
[10] 王学友, 王宇飞, 郑婉华 2020 物理学报 69 024202
Google Scholar
Wang X Y, Wang Y F, Zheng W H 2020 Acta Phys. Sin 69 024202
Google Scholar
[11] Zhou L W, Han W Q 2021 Chin. Phys. B 30 100308
Google Scholar
[12] Zhang S M, Jin L, Song Z 2022 Chin. Phys. B 31 010312
Google Scholar
[13] Wang J H, Tao Y L, Xu Y 2022 Chin. Phys. Lett. 39 010301
Google Scholar
[14] Cheng Z, Yu Z H 2021 Chin. Phys. Lett. 38 070302
Google Scholar
[15] Li L, Lee C H, Mu S, Gong J 2020 Nat. Commun. 11 5491
Google Scholar
[16] Lee C H, Li L, Thomale R, Gong J 2020 Phys. Rev. B 102 085151
Google Scholar
[17] Liu J S, Han Y Z, Liu C S 2020 Chin. Phys. B 29 010302
Google Scholar
[18] Okuma N, Kawabata K, Shiozaki K, Sato M 2020 Phys. Rev. Lett. 124 086801
Google Scholar
[19] Longhi S 2019 Phys. Rev. Res. 1 023013
Google Scholar
[20] Wang H, Ruan J, Zhang H 2019 Phys. Rev. B 99 075130
Google Scholar
[21] Jiang H, Lang L J, Yang C, Zhu S L, Chen S 2019 Phys. Rev. B 100 54301
Google Scholar
[22] Zeng Q B, Xu Y 2020 Phys. Rev. Res. 2 033052
Google Scholar
[23] Liu T, Zhang Y R, Ai Q, Gong Z, Kawabata K, Ueda M, Nori F 2019 Phys. Rev. Lett. 122 076801
Google Scholar
[24] Lee C H, Longhi S 2020 Commun. Phys. 3 147
Google Scholar
[25] Lee C H, Thomale R 2019 Phys. Rev. B 99 201103
Google Scholar
[26] Deng T S, Yi W 2019 Phys. Rev. B 100 035102
Google Scholar
[27] Song F, Yao S, Wang Z 2019 Phys. Rev. Lett. 123 246801
Google Scholar
[28] Kawabata K, Shiozaki K, Ueda M, Sato M 2019 Phys. Rev. X 9 041015
[29] Yang Z, Zhang K, Fang C, Hu J 2020 Phys. Rev. Lett. 125 226402
Google Scholar
[30] Longhi S 2020 Phys. Rev. Lett. 124 066602
Google Scholar
[31] Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803
Google Scholar
[32] Zhang K, Yang Z, Fang C 2020 Phys. Rev. Lett. 125 126402
Google Scholar
[33] Yokomizo K, Murakami S 2019 Phys. Rev. Lett. 123 066404
Google Scholar
[34] Yao S, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802
Google Scholar
[35] Ghatak A, Das T 2019 J. Phys. Condens. Matter 31 263001
Google Scholar
[36] Jin L, Song Z 2019 Phys. Rev. B 99 081103
Google Scholar
[37] Kawabata K, Shiozaki K, Ueda M 2018 Phys. Rev. B 98 165148
Google Scholar
[38] Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402
Google Scholar
[39] Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phys. Rev. Lett. 121 026808
Google Scholar
[40] Cao P C, Peng Y G, Li Y, Zhu X F 2022 Chin. Phys. Lett. 39 057801
Google Scholar
[41] Bergholtz E J, Budich J C, Kunst F K 2021 Rev. Mod. Phys. 93 015005
Google Scholar
[42] Araki H, Yoshida T, Hatsugai Y 2021 J. Phys. Soc. Jpn. 90 053703
Google Scholar
[43] Luo X W, Zhang C 2019 Phys. Rev. Lett. 123 073601
Google Scholar
[44] Yi Y, Yang Z 2020 Phys. Rev. Lett. 125 186802
Google Scholar
[45] Fu Y, Hu J, Wan S 2021 Phys. Rev. B 103 045420
Google Scholar
[46] Cao Y, Li Y, Yang X 2021 Phys. Rev. B 103 075126
Google Scholar
[47] Hu Y M, Song F, Wang Z 2021 Acta Phys. Sin. 70 230307
Google Scholar
[48] Helbig T, Hofmann T, Imhof S, Abdelghany M, Kiessling T, Molenkamp L W, Lee C H, Szameit A, Greiter M, Thomale R 2020 Nature Phys. 16 747
Google Scholar
[49] Weidemann S, Kremer M, Helbig T, Hofmann T, Stegmaier A, Greiter M, Thomale R, Szameit A 2020 Science 368 311
Google Scholar
[50] Gou W, Chen T, Xie D, Xiao T, Deng T S, Gadway B, Yi W, Yan B 2020 Phys. Rev. Lett. 124 070402
Google Scholar
[51] Yoshida T, Mizoguchi T, Hatsugai Y 2020 Phys. Rev. Res. 2 022062
Google Scholar
[52] Mandal S, Banerjee R, Ostrovskaya E A, Liew T C H 2020 Phys. Rev. Lett. 125 123902
Google Scholar
[53] Gao P, Willatzen M, Christensen J 2020 Phys. Rev. Lett. 125 206402
Google Scholar
[54] Zhu X, Wang H, Gupta S K, Zhang H, Xie B, Lu M, Chen Y 2020 Phys. Rev. Res. 2 013280
Google Scholar
[55] Hofmann T, Helbig T, Schindler F, Salgo N, Brzezińska M, Greiter M, Kiessling T, Wolf D, Vollhardt A, Kabaši A, Lee C H, Bilusic A, Thomale R, Neupert T 2020 Phys. Rev. Res. 2 023265
Google Scholar
[56] Brandenbourger M, Locsin X, Lerner E, Coulais C 2019 Nat. Commun. 10 4608
Google Scholar
[57] Rosa M I N, Ruzzene M 2020 New J. Phys. 22 053004
Google Scholar
[58] Zhong J, Wang K, Park Y, Asadchy V, Wojcik C C, Dutt A, Fan S 2021 Phys. Rev. B 104 125416
Google Scholar
[59] Zhang L, Yang Y, Ge Y, Guan Y J, Chen Q, Yan Q, Chen F, Xi R, Li Y, Jia D, Yuan S Q, Sun H X, Chen H, Zhang B 2021 Nat. Commun. 12 6297
Google Scholar
[60] Xiao L, Deng T, Wang K, Zhu G, Wang Z, Yi W, Xue P 2020 Nat. Phys. 16 761
Google Scholar
[61] Wu H, An J H 2020 Phys. Rev. B 102 041119
Google Scholar
[62] Rudner M S, Lindner N H, Berg E, Levin M 2013 Phys. Rev. X 3 031005
[63] Yao S, Yan Z, Wang Z 2017 Phys. Rev. B 96 195303
Google Scholar
[64] Li T Y, Zhang Y S, Yi W 2021 Chin. Phys. Lett. 38 030301
Google Scholar
[65] Zhai L J, Huang G Y, Yin S 2021 Phys. Rev. B 104 014202
[66] Lee T E 2016 Phys. Rev. Lett. 116 133903
Google Scholar
[67] Wang X R, Guo C X, Kou S P 2020 Phys. Rev. B 101 121116
Google Scholar
[68] Wang X R, Guo C X, Du Q, Kou S P 2020 Chin. Phys. Lett. 37 117303
Google Scholar
[69] Anderson P W 1958 Phys. Rev. 109 1492
Google Scholar
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