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In this review, we discuss the recent progress on the study of dynamic topological phenomena in quench dynamics. In particular, we focus on dynamic quantum phase transition and dynamic topological invariant, both of which are hinged upon the existence of fixed points in the dynamics. Further, the existence of these fixed points are topologically protected, in the sense that their existence are closely related to static topological invariants of pre- and post-quench Hamiltonians. We also discuss under what condition these dynamic topological phenomena are robust in non-unitary quench dynamics governed by non-Hermitian Hamiltonians. So far, dynamic topological phenomena have been experimentally observed in synthetic systems such as cold atomic gases, superconducting qubits, and linear optics. These studies extend our understanding of topological matter to the non-equilibrium regime.
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Keywords:
- topological phase /
- quench dynamics /
- dynamic topological invariant /
- ultracold atoms
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图 1 Bloch球上的动力学演化 (a) 态矢量在Bloch球上绕
$ {h}^f $ 运动; (b) 动力学不动点对应于$ {h}^i\cdot {h}^f=\pm 1 $ ; (c) 临界点对应于$ {h} ^i\cdot {h}^f=0 $ . 实线代表$ {h}^i $ (绿色)与$ {h}^f $ (红色), 虚线代表态矢量; 假设初态处于$ H_k $ 基态上, 即$ t=0 $ 时态矢量与$ {h}^i $ 方向相反Figure 1. Visualizing dynamics on the Bloch sphere: (a) State vector revolving around the
$ {h}^f $ axis; (b) illustration of fixed points when$ {h}^i\cdot {h}^f=\pm 1 $ ; (c) illustration of critical points with$ {h} ^i\cdot {h}^f=0 $ .
图 2 淬火前后哈密顿量具有不同拓扑数时的典型斯格明子结构. 黑色箭头为自旋在平面内方向, 背景颜色对应自旋在与平面垂直方向上的分量大小, 蓝色对应向内, 黄色对应向外. 竖直虚线为不动点位置, 红色实线表示不同动量
$ k $ 点的周期Figure 2. Momentum-time skyrmions when pre- and post-quench Hamiltonians possess different winding numbers.
图 3 非厄米SSH模型及其拓扑相图 (a) 非厄米SSH模型. 在厄米SSH模型的基础上, 每个格点上均有增益或损耗; (b) 体系拓扑相图. 实线为拓扑边界, 虚线为宇称-时间对称与对称破缺区域的边界.
$ v $ ,$ w $ 为SSH模型的跃迁系数,$ u $ 为增益损耗系数,$ \nu $ 为绕数Figure 3. Non-Hermitian SSH model and its topological phase diagram: (a) Non-Hermitian SSH model; (b) topological phase diagram.
图 4 非厄密SSH模型淬火中的典型动力学自由能
$ g(t) $ 与动力学拓扑序参量$ \nu^D(t) $ (a) 动力学自由能$ g(t) $ ; (b) 动力学拓扑序参量$ \nu^D(t) $ . 在非厄米淬火过程中存在两个临界时间尺度及两个动力学拓扑序参量Figure 4. Dynamic free energy
$ g(t) $ and dynamic topological order parameter$ \nu^D(t) $ in the quench dynamics of non-Hermitian SSH model: (a) Dynamic free energy$ g(t) $ ; (b) dynamic topological order parameter$ \nu^D(t) $ . -
[1] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[2] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[3] Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T 2014 Nature 515 237
Google Scholar
[4] Fläschner N, Rem B S, Tarnowski M, Vogel D, Lühmann D S, Sengstock K, Weitenberg C 2016 Science 352 1091
Google Scholar
[5] Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y, Chen S, Liu X J, Pan J W 2016 Science 354 83
Google Scholar
[6] Song B, Zhang L, He C, Poon T F J, Haiiyev E, Zhang S, Liu X J, Jo G B 2018 Sci. Adv. 4 4748
Google Scholar
[7] Poli C, Bellec M, Kuhl U, Mortessagne F, Schomerus H 2015 Nat. Commun. 6 6710
Google Scholar
[8] Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C, Szameit A 2017 Nat. Mater. 16 433
Google Scholar
[9] Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N, Yi W, Obuse H, Sanders B C, Xue P 2017 Nat. Phys. 13 1117
Google Scholar
[10] 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
[11] 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
[12] Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402
Google Scholar
[13] Chen Y, Zhai H 2018 Phys. Rev. B 98 245130
[14] Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phys. Rev. Lett. 121 026808
Google Scholar
[15] Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803
Google Scholar
[16] Yao S, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802
Google Scholar
[17] Caio M D, Cooper N R, Bhaseen M J 2015 Phys. Rev. Lett. 115 236403
Google Scholar
[18] D’Alessio L, Rigol M 2015 Nat. Commun. 6 8336
Google Scholar
[19] Wang C, Zhang P, Chen X, Yu J, Zhai H 2017 Phys. Rev. Lett. 118 185701
Google Scholar
[20] Yang C, Li L, Chen S 2018 Phys. Rev. B 97 060304
Google Scholar
[21] Gong Z, Ueda M 2018 Phys. Rev. Lett. 121 250601
[22] Zhang L, Zhang L, Niu S, Liu X J 2018 Science Bulletin 63 1385
Google Scholar
[23] Zhang L, Zhang L, Liu X J 2018 arXiv: 1807.10782 [cond-mat.quant-gas]
[24] Fläschner N, Vogel D, Tarnowski M, Rem B S, Lühmann D S, Heyl M, Budich J C, Mathey L, Sengstock K, Weitenberg C 2018 Nat. Phys. 14 265
Google Scholar
[25] Tarnowski M, Nur-Unal F, Flaschner N, Rem B S, Eckard A, Sengstock K, Weitenberg C 2017 arXiv:1709.01046 [cond-mat.quant-gas]
[26] Sun W, Yi C R, Wang B Z, Zhang W W, Sanders B C, Xu X T, Wang Z Y, Schmiedmayer J, Deng Y J, Liu X J, Chen S, Pan J W 2018 Phys. Rev. Lett. 121 250403
[27] Guo X Y, Yang C, Zeng Y, Peng Y, Li H K, Deng H, Jin Y R, Chen S, Zheng D N, Fan H 2018 arXiv:1806.09269 [cond-mat.stat-mech]
[28] Wang K, Qiu X, Xiao L, Zhan X, Bina Z, Yi W, Xue P 2019 Phys. Rev. Lett. 122 020501
[29] Tian T, Ke K, Zhang L, Lin L, Shi Z, Huang P, Lee C, Du J 2018 arXiv:1807.04483 [quant-ph]
[30] Xu X Y, Wang Q Q, Heyl M, Budich J C, Pan W W, Chen Z, Jan M, Sun K, Xu J S, Han Y J, Li C F, Guo G C 2018 arXiv:1808.03930 [quant-ph]
[31] Wang K, Qiu X, Xiao L, Zhan X, Bian Z, Yi W, Xue P 2018 arXiv:1808.06446 [quant-ph]
[32] Heyl M, Polkovnikov A, Kehrein S 2013 Phys. Rev. Lett. 110 135704
Google Scholar
[33] Heyl M 2015 Phys. Rev. Lett. 115 140602
Google Scholar
[34] Heyl M 2018 Rep. Prog. Phys. 81 054001
Google Scholar
[35] Budich J C, Heyl M 2016 Phys. Rev. B 93 085416
Google Scholar
[36] Huang Z, Balatsky A V 2016 Phys. Rev. Lett. 117 086802
Google Scholar
[37] Vajna S, Dora B 2015 Phys. Rev. B 91 155127
Google Scholar
[38] Zhou L W, Wang Q H, Wang H L 2018 Phys. Rev. A 98 022129
Google Scholar
[39] Gu J, Sun K 2016 Phys. Rev. B 94 12511
Google Scholar
[40] Qiu X, Deng T S, Guo G C, Yi W 2018 Phys. Rev. A 98 021601
Google Scholar
[41] Qiu X, Deng T S, Hu Y, Xue P, Yi W 2018 arXiv:1806.10268[cond-mat.quant-gas]
[42] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[43] Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401
Google Scholar
[44] Bender C M 2007 Rep. Prog. Phys. 70 947
Google Scholar
[45] Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698
Google Scholar
[46] Zhu B, Lu R, Chen S 2014 Phys. Rev. A 89 062102
Google Scholar
[47] Garrison J, Wright E 1988 Phys. Lett. A 128 177
Google Scholar
[48] Brody D C 2014 J. Phys. A: Math. Theor. 47 035305
Google Scholar
[49] Li J, Harter A K, Liu J, de Melo L, Joglekar Y N, Luo L 2016 arXiv:1608.05061[cond-mat.quant-gas]
[50] Kohei K, Yuto A, Hosho K, Masahito U 2018 Phys. Rev. B 98 085116
Google Scholar
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