一维准周期晶格的性质及应用

王玉成; 刘雄军; 陈澍

doi:10.7498/aps.68.20181927

摘要

准周期晶格在冷原子领域被广泛研究, 它使得人们可以在一维或者二维系统里研究扩展到安德森局域的转变. 2008年, Inguscio研究组在冷原子系统里制备了一维准周期晶格, 并观测到了安德森局域化现象, 这极大地推动了准周期系统的理论和实验研究. 后来, Bloch研究组在制备的一维和二维准周期晶格中都观测到了多体局域的现象. 最近, 他们还在准周期晶格中成功观测到迁移率边以及存在迁移率边的系统的多体局域现象. 这些冷原子实验推动了多体局域以及迁移率边等方向的研究. 准周期晶格已经成为一个平台, 它对很多物理现象的影响正在被广泛研究, 并可以尝试在冷原子实验中观测到这种影响. 本文结合作者的一些相关工作, 对一维准周期晶格一些近期的研究进行了简要综述, 介绍了一些相关的重要的冷原子实验, 讨论了准周期晶格的一些重要性质, 以及它对一些物理现象(比如拓扑态)的影响.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
南方科技大学物理系, 深圳量子科学与工程研究院, 深圳　518055

2.
北京大学物理学院, 国际量子材料中心, 北京　100871

3.
中国科学技术大学, 量子信息与量子物理协同创新中心, 合肥　230026

4.
中国科学院物理研究所, 北京凝聚态物理国家研究中心, 北京　100190

5.
中国科学院大学物理科学学院, 北京　100049

6.
量子物质科学协同创新中心, 北京　100084

通信作者: 王玉成, wangyc3@sustc.edu.cn ; 陈澍, schen@iphy.ac.cn

基金项目: 国家重点研发计划(批准号: 2016YFA0301604, 2016YFA0300600, 2016YFA0302104)、国家自然科学基金(批准号: 11674301, 11574008, 11761161003, 11425419)、中国青年千人计划和中国科学院战略优先研究计划(批准号: XDB07020000)资助的课题.

Authors and contacts

1.
Shenzhen Institute for Quantum Science and Engineering, Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China

2.
International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China

3.
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China

4.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

5.
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

6.
Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

Corresponding author: Wang Yu-Cheng, wangyc3@sustc.edu.cn ; Chen Shu, schen@iphy.ac.cn

Funds: Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFA0301604, 2016YFA0300600, 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 11674301, 11574008, 11761161003, 11425419), the Thousand-Young-Talent Program of China, and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB07020000).

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施引文献

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

Fig. 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.

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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

Fig. 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.

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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 $)

Fig. 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 $).

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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]

Fig. 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].

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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 $

Fig. 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 $.

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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]

Fig. 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].

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图 7 固定系统尺寸$ L=900 $, 平均信息熵随$ \lambda $和$ T $的变化^[55]

Fig. 7. The mean information entropy versus both $ \lambda $ and $ T $ for the system with $ L=900 $^[55].

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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]

Fig. 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].

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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]

Fig. 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].

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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 $

Fig. 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 $.

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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]

Fig. 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].

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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]

Fig. 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].

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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]

Fig. 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].

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