Fixed points and dynamic topological phenomena in quench dynamics

Deng Tian-Shu; Yi Wei

doi:10.7498/aps.68.20181928

Abstract

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.

Keywords:

Authors and contacts

Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, China

Corresponding author: Yi Wei, wyiz@ustc.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 15522545).

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References

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Cited By

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 $.

DownLoad: Full-Size Img PowerPoint

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.

DownLoad: Full-Size Img PowerPoint

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.

DownLoad: Full-Size Img PowerPoint

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) $.

DownLoad: Full-Size Img PowerPoint

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[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
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[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
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[50]	Kohei K, Yuto A, Hosho K, Masahito U 2018 Phys. Rev. B 98 085116 Google Scholar

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Vol. 68, Issue 4 - 2019-02-20

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