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Quasiperiodic lattices have been widely studied in cold atoms, which make one study extended-Anderson localization transition in one-dimensional (1D) or two-dimensional (2D) systems. In 2008, Inguscio's group prepared one-dimensional quasiperiodic lattice in cold atomic system and observed Anderson localization, which promoted the theoretical and experimental studies of quasiperiodic systems. Later, Bloch's group observed the many body localization in their prepared 1D and 2D quasiperiodic lattices. Recently, they also successfully observed the mobility edge and many body localization in a system with mobility edge in quasiperiodic lattices. These cold atom experiments have promoted the researches of many body localization and mobility edge. Quasiperiodic lattices have become a platform and its effects on many physical phenomena are being extensively studied, which can be expectantly observed in cold atomic experiments. This paper is based on the authors' some related work and briefly review some recent studies on the 1D quasiperiodic lattices, including some important cold atom experiments, some important properties of the quasiperiodic lattices and their effects on some physical phenomena, such as topological states.
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Keywords:
- optical lattice /
- Anderson localization /
- many body localization /
- topological states
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图 1 基态的倒参与率随
$ \varDelta $ 的变化，这里固定$ J=1 $ 和$ L=1000 $ . 左右的插图分别展示了$ \varDelta=1.9 $ 和$ \varDelta=2.1 $ 时系统的基态波函数的分布Figure 1. IPR of ground states as a function of
$ \varDelta $ for this system with$ J=1 $ and$ L=1000 $ . The left and right insets show the distribution of the ground state with$ \varDelta=1.9 $ and$ \varDelta=2.1 $ respectively.
图 2 实验实现准周期晶格的原理示意图. J 描述的是主晶格最近邻格点之间的跃迁,
$ 2\varDelta $ 是由次晶格导致的在位能最大的差别Figure 2. Sketch of the quasiperiodic lattice realized in the experiment. J describes the hopping between the nearest-neighbor sites of the primary lattice and
$ 2\varDelta $ is the maximum shift of the on-site energy induced by the secondary lattice.
图 3 实验原理图. 制备的初始CDW态，以及在局域、中间和扩展相中，经过一段时间演化后，分别对应的系统的末态 (a)初态分布，制备为CDW态(根据定义，有
$ I > 0 $ ,$ \xi=0 $ ); (b)局域态($ I > 0 $ ,$ \xi=0 $ );(c)中间态，对应于不同的能量存在局域态和扩展态($ I > 0 $ ,$ \xi>0 $ ); (d)扩展态($ I = 0 $ ,$ \xi>0 $ )Figure 3. Schematics of the experiment. Schematic illustration of the initial CDW state and the states reached after time evolution in the localized, intermediate, and extended phase, respectively: (a) Initial state: CDW state (
$ I > 0 $ ,$ \xi=0 $ ); (b) localized phase ($ I > 0 $ ,$ \xi=0 $ ); (c) the intermediate phase, extended and localized states coexist at different energies ($ I > 0 $ ,$ \xi>0 $ ); (d) extended phase ($ I = 0 $ ,$ \xi>0 $ ).
图 4 (a)
$ \langle r\rangle $ 随$ h $ 的变化. 当系统尺寸为$ L=12 $ 和$ L=14 $ 时用的样品数是$ 50 $ , 当$ L=16 $ 时用的样品数是$ 30 $ , 当$ L=18 $ 时用的样品数是$ 20 $ ; (b)平均的纠缠熵$ \langle S \rangle $ 和$ {\rm d} \langle S \rangle / {\rm d}h $ 随$ h $ 的变化. 当$ L=8 $ 和$ L=10 $ 时用$ 500 $ 个样品, 当$ L=12 $ 时用$ 100 $ 个样品, 当$ L=14 $ 时用$ 30 $ 个样品. 相互作用强度始终被固定为$ U=0.4 $ . 这里一个样品指的是任选一个初相位$ \theta $ [21]Figure 4. (a)
$ \langle r\rangle $ as a function of$ h $ . Here we use 50 samples for$ L = 12 $ and$ L =14 $ , 30 samples for$ L = 16 $ , and 20 samples for$ L =18 $ ; (b) averaged entanglement entropy$ \langle S \rangle $ and$ {\rm d} \langle S \rangle / {\rm d}h $ versus$ h $ . Here we use 500 samples for$ L = 8 $ and$ L = 10 $ , 100 samples for$ L = 12 $ and 30 samples for$ L = 14 $ . The interaction strength is fixed at$ U= 0.4 $ . Here a sample is specified by choosing an initial phase$ \theta $ [21].
图 5 AA模型中取不同的
$ \varDelta $ 时$ \sqrt{\langle(\text{δ} x)^2\rangle} $ 随时间t的变化的对数-对数图, 这里固定$ \alpha=\frac{\sqrt{5}-1}{2} $ , 跃迁强度$ J=1 $ , 以及系统尺寸$ L=3000 $ Figure 5. Log-log plot of the width
$ \sqrt{\langle(\text{δ} x)^2\rangle} $ vs time t for several values of$ \varDelta $ in the AA model with$ \alpha=\frac{\sqrt{5}-1}{2} $ ,$ J=1 $ and$ L=3000 $ .
图 6 固定
$ \lambda=1 $ 和$ L=1500 $ , 平均信息熵随周期$ T $ 的变化. 左上角的插图展示了平均纠缠熵的导数随周期$ T $ 的变化, 这里固定$ \lambda=0.8 $ (蓝色),$ \lambda=1.2 $ (红色), 和$ \lambda=1.6 $ (绿色). 右下角的插图展示了平均纠缠熵随$ \lambda $ 的变化, 这里分别固定$ T=0.05 $ (蓝色),$ T=0.3 $ (红色),$ T=0.5 $ (绿色)[55]Figure 6. The mean information entropy as a function of
$ T $ for this system with$ \lambda=1 $ and$ L=1500 $ . The left up inset shows the derivative of the mean information entropy as a function of$ T $ with fixed$ \lambda=0.8 $ (blue),$ \lambda=1.2 $ (red), and$ \lambda=1.6 $ (green). The right down inset shows the derivative of the mean information entropy as a function of$ \lambda $ with$ T\!=\!0.05 $ (blue),$ T\!=\!0.3 $ (red), and$ T\!=\!0.5 $ (green)[55].
图 8
$ \varDelta_{\rm f} $ 取不同值时Loschmidt echo的演化. 初态选准周期势强度为$ \varDelta_{\rm i}\!=\!0.5 $ ((a), (b))和$ \varDelta_{\rm i}\!=\!4 $ ((c), (d))的哈密顿量的基态[66]Figure 8. Evolution of Loschmidt echo in a long time with different
$ \varDelta_{\rm f} $ s. The initial state is chosen to be the ground state of the Hamiltonian with$ \varDelta_{\rm i}=0.5 $ ((a), (b))and$ \varDelta_{\rm i}=4 $ ((c), (d))[66].
图 9 固定系统参数
$ L=1000 $ ,$ T=6\times10^{5} $ 和$ \varDelta_{\rm i}=0.5 $ 时$ m (\varepsilon ) $ 随$ \varDelta_{\rm f} $ 的变化: (a)不同的颜色对应不同的$ \varepsilon $ 值, 这里的初态是初始哈密顿量的基态; (b) 选取不同的初态,$ n $ 表示初始哈密顿量的第n个本征态. 在$ \varDelta_{\rm f}=2 $ 处, 可以清晰地看到一个相边界. 这里固定$ \varepsilon =0.01 $ [66]Figure 9. The behavior of
$ m $ versus$ \varDelta_{\rm f} $ for the system with$ L=1000 $ ,$ T=6\times10^{5} $ and$ \varDelta_{\rm i}=0.5 $ : (a) Different colors correspond to different$ \varepsilon $ s and the initial state is chosen to be the ground state of the initial Hamiltonian; (b) different choice of initial state with$ n $ standing for the$ n{\rm th} $ eigenstates of the initial Hamiltonian$ H(\varDelta_{\rm i}) $ . A clear boundary can be seen at$ \varDelta_{\rm f}=2 $ . Here we choose$ \varepsilon =0.01 $ [66].
图 10 (a) 固定两个p波配对强度
$ \varDelta=0.5 $ 和$ \varDelta=0.8 $ 时，MIPR随准周期势强度$ V $ 的变化，这里用的系统尺寸是$ L=1000 $ ; (b) 系统随p波配对强度$ \varDelta $ 和准周期势强度$ V $ 变化的相图，I:扩展相，II:临界相，III:局域相. 这里固定$ J=1 $ Figure 10. (a) MIPR as a function of the incommensurate potential strength
$ V $ at two p-wave pairing strength$ \varDelta=0.5 $ and$ \Delta=0.8 $ . Here use$ L=1000 $ ; (b) phase diagram of this system with a p-wave pairing strength$ \varDelta $ and incommensurate potential strength$ V $ . I: extended phase, II: critical phase and III: localized phase. Here fix$ J=1 $ .
图 11 (a) 在开边界条件下, 固定
$ \varDelta=0.5 $ 和$ L=500 $ 时系统的能谱; (b), (c)不同的$ V $ 值时最低激发模的$ \phi_i $ ((b))和$ \psi_i $ ((c))的分布[16]Figure 11. (a) Energy spectra of this system with
$ \varDelta=0.5 $ and$ L=500 $ under OBC. The distributions of$ \phi_i $ (b) and$ \psi_i $ (c) for the lowest excitation with different$ V $ [16].
图 12 第N个本征态的IPR((a))和MIPR((b))随
$ k_x $ 和$ k_y $ 的变化, 这里固定$ V=1.9 $ ; 第N个本征态的IPR((c))和MIPR((d))随$ k_x $ 和$ V $ 的变化, 这里固定$ k_y=\frac{\text{π}}{2} $ ; 第N个本征态的IPR((e))和MIPR((f))作为$ k_y $ 和$ V $ 的函数, 这里固定$ k_x=0 $ . 其他参数是$ L=300 $ 和$ t_x=t_y=t_z=1 $ [91]Figure 12. IPR((a))and MIPR((b)) as a function of
$ k_x $ and$ k_y $ with fixed$ V=1.9 $ ; IPR((c)) and MIPR((d)) as a function of$ k_x $ and$ V $ with fixed$ k_y=\frac{\text{π}}{2} $ ; IPR((e)) and MIPR((f)) as a function of$ k_y $ and$ V $ with fixed$ k_x=0 $ . The lattice size is$ L=300 $ and$ t_x=t_y=t_z=1 $ [91].
图 13 (a) 不同晶格尺寸
$ N $ 时,$ \rho(0) $ 随$ V $ 的变化, 这里固定$ \sigma=0.02 $ ; (b)固定$ N=300 $ , 取不同的准周期势强度$ V $ 时系统的态密度随能量的变化[91]Figure 13. (a)
$ \rho(0) $ versus$ V $ for different lattice size$ N $ with fixed$ \sigma=0.02 $ ; (b) DOS with$ N=300 $ as a function of energy for various values of incommensurate potential strength$ V $ [91]. -
[1] 王义遒 2007 原子的激光冷却与陷俘(北京: 北京大学出版社)
Wang Y Q 2007 Atomic Laser Cooling And Trapping (Beijing: Peking University Press) (in Chinese)
[2] Stwalley W C 1976 Phys. Rev. Lett. 37 1628
Google Scholar
[3] Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885
Google Scholar
[4] Lin Y J, Jimenez-Garcia K, Spielman I B 2011 Nature 471 83
Google Scholar
[5] Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y J, Chen S, Liu X J, Pan J W 2016 Science 354 83
Google Scholar
[6] Zhang L, Liu X J 2018 arXiv:1806.05628
[7] Cooper N R, Dalibard J, Spielman I B 2018 arXiv:1803.00249
[8] Jian C M, Zhai H 2011 Phys. Rev. B 84 060508
Google Scholar
[9] Hu J, Feng L, Zhang Z, Chin C, 2018 arXiv:1807.07504
[10] Aubry S, André G 1980 Ann. Israel Phys. Soc 3 18
[11] Das Sarma S, He S, Xie X C 1988 Phys. Rev. Lett. 61 2144
Google Scholar
[12] Das Sarma S, He S, Xie X C 1990 Phys. Rev. B 41 5544
Google Scholar
[13] Biddle J, Das Sarma S 2010 Phys. Rev. Lett. 104 070601
Google Scholar
[14] Biddle J, Priour Jr D J, Wang B, Das Sarma S 2011 Phys. Rev. B 83 075105
Google Scholar
[15] Roati G, D’Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, Inguscio M 2008 Nature 453 895
Google Scholar
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Google Scholar
[17] DeGottardi W, Sen D, Vishveshwara S 2013 Phys. Rev. Lett. 110 146404
Google Scholar
[18] Wang J, Liu X J, Gao X, Hu H 2016 Phys. Rev. B 93 104504
Google Scholar
[19] Wang Y, Wang Y, Chen S 2016 Eur. Phys. J. B 89 254
Google Scholar
[20] Iyer S, Oganesyan V, Refael G, Huse D A 2013 Phys. Rev. B 87 134202
Google Scholar
[21] Wang Y, Hu H, Chen S 2016 Eur. Phys. J. B 89 77
Google Scholar
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Google Scholar
[23] Schreiber M, Hodgman S S, Bordia P, Lüschen H P, Fischer M H, Vosk R, Altman E, Schneider U, Bloch I 2015 Science 349 842
Google Scholar
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Google Scholar
[25] Thouless D J 1974 Phys. Rep. 13 93
Google Scholar
[26] Schreiber M 1985 J. Phys. C 18 2493
Google Scholar
[27] Hashimoto Y, Niizeki K, Okabe Y 1992 J. Phys. A 25 5211
Google Scholar
[28] Modugno M 2009 New J. Phys. 11 033023
Google Scholar
[29] 郎利君 2014 博士学位论文(北京: 中国科学院大学)
Lang L J 2014 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese)
[30] Li X, Li X, Das Sarma S 2017 Phys. Rev. B 96 085119
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[34] Pal A, Huse D A 2010 Phys. Rev. B 82 174411
Google Scholar
[35] Vosk R, Huse D A, Altman E 2015 Phys. Rev. X 5 031032
[36] Agarwal K, Gopalakrishnan S, Knap M, Müller, Demler E 2015 Phys. Rev. Lett. 114 160401
Google Scholar
[37] Fan R, Zhang P, Shen H, Zhai H 2017 Sci. Bull. 62 707
Google Scholar
[38] Else D V, Bauer B, Nayak C 2016 Phys. Rev. Lett. 117 090402
Google Scholar
[39] Oganesyan V, Huse D A 2007 Phys. Rev. B 75 155111
Google Scholar
[40] Vosk R, Altman E 2013 Phys. Rev. Lett. 110 067204
Google Scholar
[41] Bauer B, Nayak C 2013 J. Stat. Mech. 2013 P09005
Google Scholar
[42] 王玉成 2018 博士学位论文(北京: 中国科学院大学)
Wang Y 2018 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese)
[43] Serbyn M, Papić, Abanin D A 2013 Phys. Rev. Lett. 111 127201
Google Scholar
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