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Topological edge states with skin effect in a trimerized non-Hermitian lattice

## Topological edge states with skin effect in a trimerized non-Hermitian lattice

Xu Nan, Zhang Yan
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• #### Abstract

In recent years, exploring new topological quantum model structures and in depth analyzing the novel physical properties in various multimerized topological lattices have become a hot topic in the field of quantum optics. Among the different model structures, the multimerized non-Hermitian lattice controlled by different parameters in the future research of topological quantum materials, we believe, can exhibit more meaningful novel topological properties. As one of the most classic topological models, the one-dimensional Aubry-André-Harper (AAH) model has received more and more attention in the study of multimerized lattices. In this paper, we focus on the novel topological properties of a trimerized non-Hermitian lattice, and extend the trimer model structure from a one-dimensional chain to a quasi-one-dimensional zigzag structure. The results show that firstly, if the nearest-neighbor forward coupling coefficient in the unit cell is not equal to the backward coupling coefficient, the chiral inversion symmetry of the system is destroyed. It can be observed that the bulk states and the edge states in the trimerization model will be localized on the same edge of the lattice, and the skin effect will appear in the system. With the increase of the nearest-neighbor coupling coefficient, the width of the edge state changes in which the lower edge state of the imaginary part of the spectrum is narrowed until it disappears. The degree of degeneracy of the system changes, and the number of edge states is reduced from four to two. Remarkably, the generalized bulk-boundary correspondence is shown in certain non-Hermitian topological systems. Secondly, when the trimerization model considers the influence of the next-nearest-neighbor coupling, the numerical results show that the upper and lower energy gaps in the energy spectrum and the edge states in the energy spectrum are asymmetrical as the next-nearest-neighbor coupling coefficient is modulated in an appropriate range. The upper energy gaps and the edge states are narrowed, and the edge states of the lower energy gaps are widened. At the same time, the novel topology features of the system can also be used to achieve the quantitative control of the energy spectrum edge states, and other interesting directions are worth exploring.

#### References

 [1] Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 [2] Klitzing K 1986 Rev. Mod. Phys. 58 519 [3] Haldane F D M 1988 Phys. Rev. Lett. 61 2015 [4] Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904 [5] Raghu S, Haldane F D M 2008 Phys. Rev. A 78 033834 [6] Chang C Z, Zhang J S, Feng X, Xue Q K 2013 Sci. Rep. 340 6129 [7] Goldman N, Budich J C, Zoller P 2016 Nat. Phys. 12 639 [8] 孙晓晨, 何程, 卢明辉, 陈延峰 2017 物理学报 22 224203 Sun X C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 22 224203 [9] 张卫锋, 李春艳, 陈险锋, 黄长明, 叶芳伟 2017 物理学报 22 220201 Zhang W F, Li C Y, Chen X F, Huang C M, Ye F W 2017 Acta Phys. Sin. 22 220201 [10] 陈西浩, 王秀娟 2018 物理学报 19 190301 Chen X H, Wang X J 2018 Acta Phys. Sin. 19 190301 [11] Aubry S, Andreé G, Isr A 1980 Phys. Soc. 322 235 [12] harper P G 1955 Proc. Phys. Soc. London Sect. A 68 874 [13] Lahini Y, Pugatch R, Pozzi F, Sorel M, Morandotti R, Davidson N, Silberberg Y 2009 Phys. Rev. Lett. 103 013901 [14] Biddle J, Wang B, Priour D J, Sarma S D 2009 Phys. Rev. A 80 021603 (R) [15] Ganeshan S, Sun K, Sarma S D 2013 Phys. Rev. Lett. 110 180403 [16] Hatano N, Nelson D R 1997 Phys. Rev. B 56 8651 [17] Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384 [18] Yurkevich I V, Lerner I V 1999 Phys. Rev. Lett. 82 5080 [19] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 [20] Dorey P, Dunning C, Tateo R 2001 J. Phys. A 34 5679 [21] Mostafazadeh A 2002 J. Math. Phys. 43 205 [22] Jones H F 2005 J. Phys. A 38 1741 [23] Klaiman S, Günther U, Moiseyev N 2008 Phys. Rev. Lett. 101 080402 [24] Znojil M 2008 Phys. Rev. D 78 025026 [25] Jin L, Song Z 2009 Phys. Rev. A 80 052107 [26] Rotter I 2009 J. Phys. A: Math. Theor. 42 124206 [27] Moiseyev N 2011 Non-Hermitian Quantum Mechanics (Cambridge: Cambridge University) pp 211—247 [28] Joqlekar Y N, Barnett J L 2011 Phys. Rev. A 84 024103 [29] Longhi S, Valle G D 2012 Phys. Rev. A 85 012112 [30] Longhi S 2013 Phys. Rev. A 88 052102 [31] Longhi S 2016 Phys. Rev. A 94 022102 [32] Jin L, Xin F 2017 Phys. Rev. A 97 012121 [33] Zhu B G, Lü R, Chen S 2014 Phys. Rev. A 89 062102 [34] Xing Y, Qi L, Cao J, Wang D Y, Bai C H, Wang H F, Zhu A D, Zhang S 2017 Phys. Rev. A 96 043810 [35] Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819 [36] Schomerus H, Wiersig J 2014 Phys. Rev. A 90 053819 [37] Cheng Q, Pan Y, Wang Q, Li T, Zhu S 2015 Laser Photonics Rev. 9 392 [38] Longhi S, Gatti D, Della Valle G 2015 Phys. Rev. B 92 094204 [39] Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Phys. Rev. B 97 121401(R) [40] Yao S Y, Wang Z 2018 Phys. Rev. Lett. 121 086803 [41] Jin L 2017 Phys. Rev. A 96 032103

#### Cited By

• 图 1  扩展的三聚化非厄密晶格示意图

Figure 1.  Schematic of the generalized trimerized non-Hermitian lattice.

图 2  三聚化非厄密晶格的本征能谱　(a), (c), (e)和(g)分别为$\mu=0, 0.1, 0.15, 0.2$时本征能谱的实部; (b), (d), (f)和(h)分别为$\mu=0, 0.1, 0.15, 0.2$时本征能谱的虚部, 边缘态能带用绿色实线和棕色虚线表示

Figure 2.  The eigen-energy spectrum of the trimerized non-Hermitian lattice. (a), (c), (e) and (g) show the real parts of the energy spectrum for $\mu=0, 0.1, 0.15, 0.2$, respectively; (b), (d), (f) and (h) show the imaginary parts of the energy spectrum for $\mu=0, 0.1, 0.15, 0.2$. Green solid lines and brown dash lines represent the energy band of the edge states.

图 3  系统本征态的光子分布　(a) $\mu=0$; (b) $\mu=0.1$; (c) $\mu=0.2$; (d) $\mu=0.3$

Figure 3.  Photon distributions for eigenstates of the system: (a) $\mu=0$; (b) $\mu=0.1$; (c) $\mu=0.2$; (d) $\mu=0.3$.

图 4  次近邻耦合影响下系统的本征能谱和边缘态的光子分布　(a), (d)和(g)分别为$\nu=0.1, 0.15, 0.2$时本征能谱的实部; (b), (e)和(h)分别为$\nu=0.1, 0.15, 0.2$时上能隙中的边缘态; (c), (f)和(i)分别为$\nu=0.1, 0.15, 0.2$时下能隙的边缘态

Figure 4.  The eigen-energy spectrum and photon distributions for edge states of the system under the influence of the next-nearest-neighbor coupling: (a), (d) and (g) show the real parts of the energy spectrum for $\nu=0.1, 0.15, 0.2$; (b), (e) and (h) show the edge states whose energy band are in the upper gaps for $\nu=0.1, 0.15, 0.2$; (c), (f) and (i) show the edge states whose energy bands are in the lower gaps for $\nu=0.1, 0.15, 0.2$.

•  [1] Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 [2] Klitzing K 1986 Rev. Mod. Phys. 58 519 [3] Haldane F D M 1988 Phys. Rev. Lett. 61 2015 [4] Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904 [5] Raghu S, Haldane F D M 2008 Phys. Rev. A 78 033834 [6] Chang C Z, Zhang J S, Feng X, Xue Q K 2013 Sci. Rep. 340 6129 [7] Goldman N, Budich J C, Zoller P 2016 Nat. Phys. 12 639 [8] 孙晓晨, 何程, 卢明辉, 陈延峰 2017 物理学报 22 224203 Sun X C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 22 224203 [9] 张卫锋, 李春艳, 陈险锋, 黄长明, 叶芳伟 2017 物理学报 22 220201 Zhang W F, Li C Y, Chen X F, Huang C M, Ye F W 2017 Acta Phys. Sin. 22 220201 [10] 陈西浩, 王秀娟 2018 物理学报 19 190301 Chen X H, Wang X J 2018 Acta Phys. Sin. 19 190301 [11] Aubry S, Andreé G, Isr A 1980 Phys. Soc. 322 235 [12] harper P G 1955 Proc. Phys. Soc. London Sect. A 68 874 [13] Lahini Y, Pugatch R, Pozzi F, Sorel M, Morandotti R, Davidson N, Silberberg Y 2009 Phys. Rev. Lett. 103 013901 [14] Biddle J, Wang B, Priour D J, Sarma S D 2009 Phys. Rev. A 80 021603 (R) [15] Ganeshan S, Sun K, Sarma S D 2013 Phys. Rev. Lett. 110 180403 [16] Hatano N, Nelson D R 1997 Phys. Rev. B 56 8651 [17] Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384 [18] Yurkevich I V, Lerner I V 1999 Phys. Rev. Lett. 82 5080 [19] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 [20] Dorey P, Dunning C, Tateo R 2001 J. Phys. A 34 5679 [21] Mostafazadeh A 2002 J. Math. Phys. 43 205 [22] Jones H F 2005 J. Phys. A 38 1741 [23] Klaiman S, Günther U, Moiseyev N 2008 Phys. Rev. Lett. 101 080402 [24] Znojil M 2008 Phys. Rev. D 78 025026 [25] Jin L, Song Z 2009 Phys. Rev. A 80 052107 [26] Rotter I 2009 J. Phys. A: Math. Theor. 42 124206 [27] Moiseyev N 2011 Non-Hermitian Quantum Mechanics (Cambridge: Cambridge University) pp 211—247 [28] Joqlekar Y N, Barnett J L 2011 Phys. Rev. A 84 024103 [29] Longhi S, Valle G D 2012 Phys. Rev. A 85 012112 [30] Longhi S 2013 Phys. Rev. A 88 052102 [31] Longhi S 2016 Phys. Rev. A 94 022102 [32] Jin L, Xin F 2017 Phys. Rev. A 97 012121 [33] Zhu B G, Lü R, Chen S 2014 Phys. Rev. A 89 062102 [34] Xing Y, Qi L, Cao J, Wang D Y, Bai C H, Wang H F, Zhu A D, Zhang S 2017 Phys. Rev. A 96 043810 [35] Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819 [36] Schomerus H, Wiersig J 2014 Phys. Rev. A 90 053819 [37] Cheng Q, Pan Y, Wang Q, Li T, Zhu S 2015 Laser Photonics Rev. 9 392 [38] Longhi S, Gatti D, Della Valle G 2015 Phys. Rev. B 92 094204 [39] Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Phys. Rev. B 97 121401(R) [40] Yao S Y, Wang Z 2018 Phys. Rev. Lett. 121 086803 [41] Jin L 2017 Phys. Rev. A 96 032103
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•  Citation:
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• Abstract views:  506
• Cited By: 0
##### Publishing process
• Received Date:  21 January 2019
• Accepted Date:  11 March 2019
• Available Online:  01 May 2019
• Published Online:  20 May 2019

## Topological edge states with skin effect in a trimerized non-Hermitian lattice

###### Corresponding author: Zhang Yan, zhangy345@nenu.edu.cn
• School of Physics, Northeast Normal University, Changchun 130024, China

Abstract: In recent years, exploring new topological quantum model structures and in depth analyzing the novel physical properties in various multimerized topological lattices have become a hot topic in the field of quantum optics. Among the different model structures, the multimerized non-Hermitian lattice controlled by different parameters in the future research of topological quantum materials, we believe, can exhibit more meaningful novel topological properties. As one of the most classic topological models, the one-dimensional Aubry-André-Harper (AAH) model has received more and more attention in the study of multimerized lattices. In this paper, we focus on the novel topological properties of a trimerized non-Hermitian lattice, and extend the trimer model structure from a one-dimensional chain to a quasi-one-dimensional zigzag structure. The results show that firstly, if the nearest-neighbor forward coupling coefficient in the unit cell is not equal to the backward coupling coefficient, the chiral inversion symmetry of the system is destroyed. It can be observed that the bulk states and the edge states in the trimerization model will be localized on the same edge of the lattice, and the skin effect will appear in the system. With the increase of the nearest-neighbor coupling coefficient, the width of the edge state changes in which the lower edge state of the imaginary part of the spectrum is narrowed until it disappears. The degree of degeneracy of the system changes, and the number of edge states is reduced from four to two. Remarkably, the generalized bulk-boundary correspondence is shown in certain non-Hermitian topological systems. Secondly, when the trimerization model considers the influence of the next-nearest-neighbor coupling, the numerical results show that the upper and lower energy gaps in the energy spectrum and the edge states in the energy spectrum are asymmetrical as the next-nearest-neighbor coupling coefficient is modulated in an appropriate range. The upper energy gaps and the edge states are narrowed, and the edge states of the lower energy gaps are widened. At the same time, the novel topology features of the system can also be used to achieve the quantitative control of the energy spectrum edge states, and other interesting directions are worth exploring.

Reference (41)

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