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Topological invariants of non-Hermitian topological systems can be captured by local topological markers defined on the biorthogonal basis. However, unlike the scenario of Hermitian systems, the dynamics of non-Hermitian local topological marker has not yet received much attention so far. Here in this work, we study the dynamic features of local topological markers in non-Hermitian topological systems. In particular, we focus on the propagation of non-Hermitian topological markers in quench dynamics. We find that for the dynamics with topologically distinct pre- and post-quench Hamiltonians, a flow of local topological markers emerges in the bulk, with its propagation speed related to the maximum group velocity. Taking three different non-Hermitian topological models for example, we numerically calculate the propagation speed, and demonstrate that a simple universal relation between the propagation speed and group velocity does not exist, which is unlike the scenarios in previously studied Hermitian systems. Our results reveal the complexity of the local-topological-marker dynamics in non-Hermitian settings, and would stimulate further study on the matter. -
Keywords:
- non-Hermitian topology /
- quench dynamics /
- open systems
[1] Ashida Y, Gong Z, Ueda M 2020 Adv. Phys. 69 249Google Scholar
[2] Kawabata K, Shiozaki k, Ueda M, Sato M 2019 Phys. Rev. X 9 041015Google Scholar
[3] Zhou H, Lee J Y 2019 Phys. Rev. B 99 235112Google Scholar
[4] Bender C M 2007 Rep. Prog. Phys. 70 947Google Scholar
[5] Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002Google Scholar
[6] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2017 Nat. Phys. 14 11Google Scholar
[7] Dalibard J, Castin Y, Mölmer K 1992 Phys. Rev. Lett. 68 580Google Scholar
[8] Carmichael H J 1993 Phys. Rev. Lett. 70 2273Google Scholar
[9] Yao S Y, Wang Z 2018 Phy. Rev. Lett. 121 086803Google Scholar
[10] Yao S Y, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802Google Scholar
[11] Lee C H, Thomale R 2019 Phy. Rev. B 99 201103Google Scholar
[12] McDonald A, Pereg-Barnea T, Clerk A A, 2018 Phys. Rev. X 8 041031Google Scholar
[13] Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phy. Rev. Lett. 121 026808Google Scholar
[14] Yokomizo K, Murakami S 2019 Phys. Rev. Lett. 123 066404Google Scholar
[15] Zhang K, Yang Z, Fang C 2020 Phy. Rev. Lett. 125 126402Google Scholar
[16] Okuma N, Kawabata K, Shiozaki K, Sato M 2020 Phys. Rev. Lett. 124 086801Google Scholar
[17] Lee T E 2016 Phy. Rev. Lett. 16 133903Google Scholar
[18] Zhang X Z, Gong J B, 2020 Phy. Rev. B 101 045415Google Scholar
[19] Zeng Q B, Yang Y B, Xu Y 2020 Phy. Rev. B 101 020201Google Scholar
[20] Wang X R, Guo C X, Kou S P 2020 Phy. Rev. B 101 121116Google Scholar
[21] Li T Y, Zhang Y S, Yi W 2021 Chin. Phys. Lett. 38 030301Google Scholar
[22] Helbig T, Hofmann T, Imhof S, Abdelghany M, Kiessling T, Molenkamp L W, Lee C H, Szameit A, Greiter M, Thomale R 2020 Nat. Phys. 16 747Google Scholar
[23] Xiao L, Deng T S, Wang K K, Zhu G, Wang Z, Yi W, Xue P 2020 Nat. Phys. 16 761Google Scholar
[24] Ghatak A, Brandenbourger M, van Wezel J, Coulais C 2020 Proc. Natl. Acad. Sci. U.S.A. 117 29561Google Scholar
[25] Weidemann S, Kremer M, Helbig T, Hofmann T, Stegmaier A, Greiter M, Thomale R, Szameit A 2020 Science 368 311Google Scholar
[26] Longhi S 2019 Phys. Rev. Res. 1 023013Google Scholar
[27] Longhi S 2019 Opt. Lett. 44 5804Google Scholar
[28] Xiao L, Deng T S, Wang K K, Wang Z, Yi W, Xue P 2021 Phys. Rev. Lett. 126 230402Google Scholar
[29] Li T Y, Sun J Z, Zhang Y S, Yi W 2021 Phys. Rev. Res. 3 023022Google Scholar
[30] Wang K, Li T, Xiao L, Han Y, Yi W, Xue P 2021 arXiv: 2107.14741
[31] Luo X W, Zhang C W 2019 arXiv: 1912.10652
[32] Song F, Yao S, Wang Z 2019 Phys. Rev. Lett. 123 246801Google Scholar
[33] Bianco R, Resta R 2011 Phys. Rev. B 84 241106Google Scholar
[34] Caio M D, Mller G, Cooper N R, Bhaseen M J 2019 Nat. Phys. 15 257Google Scholar
[35] Privitera L and Santoro G E 2016 Phy. Rev. B 93 241406Google Scholar
[36] Pozo O, Repellin C, Grushin A G 2019 Phys. Rev. Lett. 123 247401Google Scholar
[37] Qi X L, Wu Y S, Zhang S C 2006 Phys. Rev. B 74 085308Google Scholar
[38] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar
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图 1 非厄米局域拓扑指标的标度行为. 系统中心的非厄米局域拓扑指标随着
$m$ 的变化遵循$(m-m_{\rm c})L^{\tfrac{1}{\nu}}$ 的标度行为. 在$L$ 分别取$15, 17, 19, 21, 23, 25, 27$ 等值时(对应于图中不同颜色的曲线), 所有曲线经变换后遵循同一函数形式, 经拟合得到$\nu\approx 1.1259$ . 在本文的数值计算中, 取$t_1=1$ ,$\gamma_x=\gamma_y=0.3$ Figure 1. Scaling behavior of the non-Hermitian local topological marker near the topological phase transition. The numerically calculated local topological markers for different system sizes (
$L=15, 17, 19, 21, 23, 25, 27$ ) collapse to the same curve, with the functional form$(m-m_{\rm c})L^{\tfrac{1}{\nu}}$ and$\nu\approx 1.12(6)$ . For all calculations, we take$t_1=1$ ,$\gamma_x= $ $ \gamma_y=0.3$ 图 2 非厄米局域拓扑指标的传播速度 (a) 淬火过程中系统中心格点处的非厄米局域拓扑指标的时间演化. 系统尺寸为
$L=23$ , 取开边界. 初始时${\boldsymbol{H}}\left(t_1=1,\; m=1\right)$ 处于拓扑非平庸相, 淬火后${\boldsymbol{H}}\left(t^{\prime}_1=1,\; m^{\prime}=6\right)$ 处于拓扑平庸相, 淬火前后$\gamma_x=\gamma_y=0.3$ . 这里只考虑$x$ 方向的动力学演化,$y$ 方向的演化有类似性质. (b) 非厄米局域拓扑指标的传播速度随末态哈密顿量参数$t^{\prime}_1$ 的变化. 淬火参数与图(a)相同. 分别数值模拟了体系大小为$L=11, 15, 19, 23, 27$ 时的淬火过程, 并通过线性拟合$t^\ast$ 与$L$ 的关系, 数值得到了局域拓扑指标的传播速度Figure 2. Dynamics of the non-Hermitian local topological marker. (a) Spatial distribution (along the
$x$ -direction) of the non-Hermitian local topological marker at different times of the quench dynamics. The parameters are:$L=23$ ,${\boldsymbol{H}}_{\rm i}(t_1=1, \;m=1)$ ,${\boldsymbol{H}}_{\rm f}(t'_1=1, \;m'=6)$ ,$\gamma_x=\gamma_y=0.3$ . (b) Propagation speed of the local topological marker as a function of$t'_1$ . Other parameters are the same as those in panel (a).图 3 模型(5)的非厄米局域拓扑指标的传播速度随
$t^{\prime}_1$ 的变化. 初始时,$t_x=t_y=1, m=1$ ; 淬火后,$t^{\prime}_x=t^{\prime}_y=t^{\prime}_1, $ $ m^{\prime}=6$ . 淬火前后$\gamma_x= \gamma_y=\gamma=0.35$ Figure 3. Propagation speed of the non-Hermitian local topological marker for Hamiltonian (5). We fix
$\gamma_x=\gamma_y=\gamma=0.35$ , and the pre- and post-quench parameters are$t_x=t_y=1, $ $ \; m=1$ and$t^{\prime}_x=t^{\prime}_y=t^{\prime}_1, \; m^{\prime}=6$ , respectively.图 4 模型(8)的非厄米局域拓扑指标的传播速度随
$t^{\prime}_1$ 的变化. 初始时,$t_x=t_y=1, m=1$ ; 淬火后,$t^{\prime}_x=t^{\prime}_y=t^{\prime}_1, $ $ m^{\prime}=6$ . 淬火前后$\gamma=0.3$ Figure 4. Propagation speed of the non-Hermitian local topological marker for Hamiltonian (8). We fix
$\gamma=0.3$ , and the pre- and post-quench parameters are$t_x=t_y=1, m=1$ and$t^{\prime}_x=t^{\prime}_y=t^{\prime}_1, m^{\prime}=6$ , respectively. -
[1] Ashida Y, Gong Z, Ueda M 2020 Adv. Phys. 69 249Google Scholar
[2] Kawabata K, Shiozaki k, Ueda M, Sato M 2019 Phys. Rev. X 9 041015Google Scholar
[3] Zhou H, Lee J Y 2019 Phys. Rev. B 99 235112Google Scholar
[4] Bender C M 2007 Rep. Prog. Phys. 70 947Google Scholar
[5] Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002Google Scholar
[6] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2017 Nat. Phys. 14 11Google Scholar
[7] Dalibard J, Castin Y, Mölmer K 1992 Phys. Rev. Lett. 68 580Google Scholar
[8] Carmichael H J 1993 Phys. Rev. Lett. 70 2273Google Scholar
[9] Yao S Y, Wang Z 2018 Phy. Rev. Lett. 121 086803Google Scholar
[10] Yao S Y, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802Google Scholar
[11] Lee C H, Thomale R 2019 Phy. Rev. B 99 201103Google Scholar
[12] McDonald A, Pereg-Barnea T, Clerk A A, 2018 Phys. Rev. X 8 041031Google Scholar
[13] Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phy. Rev. Lett. 121 026808Google Scholar
[14] Yokomizo K, Murakami S 2019 Phys. Rev. Lett. 123 066404Google Scholar
[15] Zhang K, Yang Z, Fang C 2020 Phy. Rev. Lett. 125 126402Google Scholar
[16] Okuma N, Kawabata K, Shiozaki K, Sato M 2020 Phys. Rev. Lett. 124 086801Google Scholar
[17] Lee T E 2016 Phy. Rev. Lett. 16 133903Google Scholar
[18] Zhang X Z, Gong J B, 2020 Phy. Rev. B 101 045415Google Scholar
[19] Zeng Q B, Yang Y B, Xu Y 2020 Phy. Rev. B 101 020201Google Scholar
[20] Wang X R, Guo C X, Kou S P 2020 Phy. Rev. B 101 121116Google Scholar
[21] Li T Y, Zhang Y S, Yi W 2021 Chin. Phys. Lett. 38 030301Google Scholar
[22] Helbig T, Hofmann T, Imhof S, Abdelghany M, Kiessling T, Molenkamp L W, Lee C H, Szameit A, Greiter M, Thomale R 2020 Nat. Phys. 16 747Google Scholar
[23] Xiao L, Deng T S, Wang K K, Zhu G, Wang Z, Yi W, Xue P 2020 Nat. Phys. 16 761Google Scholar
[24] Ghatak A, Brandenbourger M, van Wezel J, Coulais C 2020 Proc. Natl. Acad. Sci. U.S.A. 117 29561Google Scholar
[25] Weidemann S, Kremer M, Helbig T, Hofmann T, Stegmaier A, Greiter M, Thomale R, Szameit A 2020 Science 368 311Google Scholar
[26] Longhi S 2019 Phys. Rev. Res. 1 023013Google Scholar
[27] Longhi S 2019 Opt. Lett. 44 5804Google Scholar
[28] Xiao L, Deng T S, Wang K K, Wang Z, Yi W, Xue P 2021 Phys. Rev. Lett. 126 230402Google Scholar
[29] Li T Y, Sun J Z, Zhang Y S, Yi W 2021 Phys. Rev. Res. 3 023022Google Scholar
[30] Wang K, Li T, Xiao L, Han Y, Yi W, Xue P 2021 arXiv: 2107.14741
[31] Luo X W, Zhang C W 2019 arXiv: 1912.10652
[32] Song F, Yao S, Wang Z 2019 Phys. Rev. Lett. 123 246801Google Scholar
[33] Bianco R, Resta R 2011 Phys. Rev. B 84 241106Google Scholar
[34] Caio M D, Mller G, Cooper N R, Bhaseen M J 2019 Nat. Phys. 15 257Google Scholar
[35] Privitera L and Santoro G E 2016 Phy. Rev. B 93 241406Google Scholar
[36] Pozo O, Repellin C, Grushin A G 2019 Phys. Rev. Lett. 123 247401Google Scholar
[37] Qi X L, Wu Y S, Zhang S C 2006 Phys. Rev. B 74 085308Google Scholar
[38] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar
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