搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

平带光子微结构中的新颖现象:从模式局域到实空间拓扑

夏世强 唐莉勤 夏士齐 马继娜 燕文超 宋道红 胡毅 许京军 陈志刚

引用本文:
Citation:

平带光子微结构中的新颖现象:从模式局域到实空间拓扑

夏世强, 唐莉勤, 夏士齐, 马继娜, 燕文超, 宋道红, 胡毅, 许京军, 陈志刚

Novel phenomena in flatband photonic structures: from localized states to real-space topology

Xia Shi-Qiang, Tang Li-Qin, Xia Shi-Qi, Ma Ji-Na, Yan Wen-Chao, Song Dao-Hong, Hu Yi, Xu Jing-Jun, Chen Zhi-Gang
PDF
HTML
导出引用
  • 近年来, 凝聚态物理中平带局域与拓扑等概念与光学体系的有机结合, 使得平带光子学系统的研究迎来了极为快速的发展, 催生了一系列新颖的光物理现象与潜在的应用前景. 目前, 平带结构在光子晶体、光学超构材料以及光子晶格(倏逝波耦合的光学波导阵列)等多种人工光子微结构中得到了实现, 并在其中观察到了很多凝聚态系统中难以直接实现的物理现象. 本文简要综述光子微结构中关于平带物理的最新研究进展. 以光诱导和激光直写光子晶格系统为例, 包括Lieb, Kagome 和超级蜂窝晶格等, 特别介绍平带模式局域与实空间拓扑效应等新颖物理现象. 光子微结构为研究平带物理和拓扑效应提供了一个可调控的平台, 同时其研究结果也对探究电子、声子、等离激元、腔极化子与超冷原子等系统中相关的基本物理问题和应用具有借鉴作用.
    In recent years, flatband systems have aroused considerable interest in different branches of physics, from condensed-matter physics to engineered flatband structures such as in ultracold atoms, various metamaterials, electronic materials, and photonic waveguide arrays. Flatband localization, as an important phenomenon in solid state physics, is of broad interest in the exploration of many fundamental physics of many-body systems. We briefly review the recent experimental advances in light localization in engineered flatband lattices, with the emphasis on the optical induction technique of various photonic lattices and unconventional flatband states. The photonic lattices, established by various optical induction techniques, include quasi-one-dimensional diamond lattices and two-dimensional super-honeycomb, Lieb and Kagome lattices. Nontrivial flatband line states, independent of linear superpositions of conventional compact localized states, are demonstrated in photonic Lieb and super-honeycomb lattices, and they can be considered as an indirect illustration of the non-contractible loop states. Furthermore, we discuss alternative approaches to directly observing the non-contractible loop states in photonic Kagome lattices. These robust loop states are direct manifestation of real-space topology in such flatband systems. In this paper we do not intend to comprehensively account the vast flatband literature, but we briefly review the relevant work on photonic lattices mainly from our group. We hope that the mentioned concepts and techniques can be further explored and developed for subsequent applications in other structured photonic media such as photonic crystals, metamaterials, and other synthetic nanophotonic materials.
      通信作者: 宋道红, songdaohong@nankai.edu.cn ; 许京军, jjxu@nankai.edu.cn ; 陈志刚, zgchen@nankai.edu.cn
    • 基金项目: 国家级-国家自然科学基金( 批准号: 2017YFA0303800)
      Corresponding author: Song Dao-Hong, songdaohong@nankai.edu.cn ; Xu Jing-Jun, jjxu@nankai.edu.cn ; Chen Zhi-Gang, zgchen@nankai.edu.cn
    [1]

    Ramirez A P 1994 Annu. Rev. Mater. Sci. 24 453Google Scholar

    [2]

    Huse D A, Rutenberg A D 1992 Phys. Rev. B 45 7536Google Scholar

    [3]

    Chalker J T, Holdsworth P C, Shender E F 1992 Phys. Rev. Lett. 68 855Google Scholar

    [4]

    Moessner R, Chalker J T 1998 Phys. Rev. B 58 12049Google Scholar

    [5]

    Moessner R, Ramirez A P 2006 Phys. Today 59 24

    [6]

    Bergman D L, Wu C, Balents L 2008 Phys. Rev. B 78 125104Google Scholar

    [7]

    Huber S D, Altman E 2010 Phys. Rev. B 82 184502Google Scholar

    [8]

    Nixon M, Ronen E, Friesem A A, Davidson N 2013 Phys. Rev. Lett. 110 184102Google Scholar

    [9]

    Möller G, Cooper N R 2012 Phys. Rev. Lett. 108 045306Google Scholar

    [10]

    Yin J X, Zhang S S, Li H, Jiang K, Chang G, Zhang B, Lian B, Xiang C, Belopolski I, Zheng H, Cochran T A, Xu S Y, Bian G, Liu K, Chang T R, Lin H, Lu Z Y, Wang Z, Jia S, Wang W, Hasan M Z 2018 Nature 562 91Google Scholar

    [11]

    Yin J X, Zhang S S, Chang G, Wang Q, Tsirkin S S, Guguchia Z, Lian B, Zhou H, Jiang K, Belopolski I, Shumiya N, Multer D, Litskevich M, Cochran T A, Lin H, Wang Z, Neupert T, Jia S, Lei H, Hasan M Z 2019 Nat. Phys. 15 443Google Scholar

    [12]

    Xie Y, Lian B, Jäck B, Liu X, Chiu C L, Watanabe K, Taniguchi T, Bernevig B A, Yazdani A 2019 Nature 572 101Google Scholar

    [13]

    Tang E, Mei J W, Wen X G 2011 Phys. Rev. Lett. 106 236802Google Scholar

    [14]

    Sun K, Gu Z, Katsura H, Sarma S D 2011 Phys. Rev. Lett. 106 236803Google Scholar

    [15]

    Neupert T, Santos L, Chamon C, Mudry C 2011 Phys. Rev. Lett. 106 236804Google Scholar

    [16]

    Sheng D N, Gu Z C, Sun K, Sheng L 2011 Nat. Commun. 2 389Google Scholar

    [17]

    Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C, Jarillo-Herrero P 2018 Nature 556 80Google Scholar

    [18]

    Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43Google Scholar

    [19]

    Kopnin N B, Heikkilä T T, Volovik G E 2011 Phys. Rev. B 83 220503Google Scholar

    [20]

    Tang E, Fu L 2014 Nat. Phys. 10 964Google Scholar

    [21]

    Iglovikov V I, Hébert F, Grémaud B, Batrouni G G, Scalettar R T 2014 Phys. Rev. B 90 094506Google Scholar

    [22]

    Fleischer J W, Segev M, Efremidis N K, Christodoulides D N 2003 Nature 422 147Google Scholar

    [23]

    Efremidis N K, Sears S, Christodoulides D N, Fleischer J W, Segev M 2002 Phys. Rev. E 66 046602Google Scholar

    [24]

    Chen Z, Segev M, Christodoulides D N 2012 Rep. Prog. Phys. 75 086401Google Scholar

    [25]

    Longhi S, Marangoni M, Lobino M, Ramponi R, Laporta P, Cianci E, Foglietti V 2006 Phys. Rev. Lett. 96 243901Google Scholar

    [26]

    Szameit A, Garanovich I L, Heinrich M, Sukhorukov A A, Dreisow F, Pertsch T, Nolte S, Tünnermann A, Kivshar Y S 2009 Nat. Phys. 5 271Google Scholar

    [27]

    Schwartz T, Bartal G, Fishman S, Segev M 2007 Nature 446 52Google Scholar

    [28]

    Lahini Y, Avidan A, Pozzi F, Sorel M, Morandotti R, Christodoulides D N, Silberberg Y 2008 Phys. Rev. Lett. 100 013906Google Scholar

    [29]

    Sutherland B 1986 Phys. Rev. B 34 5208Google Scholar

    [30]

    Arai M, Tokihiro T, Fujiwara T, Kohmoto M 1988 Phys. Rev. B 38 1621

    [31]

    Lieb E H 1989 Phys. Rev. Lett. 62 1201Google Scholar

    [32]

    Mielke A 1991 J. Phys. A: Math. Gen. 24 L73Google Scholar

    [33]

    Mielke A 1991 J. Phys. A: Math. Gen. 24 3311Google Scholar

    [34]

    Mielke A 1999 Phys. Rev. Lett. 82 4312Google Scholar

    [35]

    Tasaki H 1992 Phys. Rev. Lett. 69 1608Google Scholar

    [36]

    Tasaki H 1998 Prog. Theor. Phys. 99 489Google Scholar

    [37]

    Aoki H, Ando M, Matsumura H 1996 Phys. Rev. B 54 R17296Google Scholar

    [38]

    Slot M R, Gardenier T S, Jacobse P H, van Miert G C, Kempkes S N, Zevenhuizen S J, Smith C M, Vanmaekelbergh D, Swart I 2017 Nat. Phys. 13 672Google Scholar

    [39]

    Jo G B, Guzman J, Thomas C K, Hosur P, Vishwanath A, Stamper-Kurn D M 2012 Phys. Rev. Lett. 108 045305Google Scholar

    [40]

    Taie S, Ozawa H, Ichinose T, Nishio T, Nakajima S, Takahashi Y 2015 Sci. Adv. 1 e1500854Google Scholar

    [41]

    Guzmán-Silva D, Mejía-Cortés C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A, Vicencio R A 2014 New J. Phys. 16 063061Google Scholar

    [42]

    Makasyuk I, Chen Z, Yang J 2006 Phys. Rev. Lett. 96 223903Google Scholar

    [43]

    Flach S, Leykam D, Bodyfelt J D, Matthies P, Desyatnikov A S 2014 Europhys. Lett. 105 30001Google Scholar

    [44]

    Ramachandran A, Andreanov A, Flach S 2017 Phys. Rev. B 96 161104Google Scholar

    [45]

    Li J, White T P, O’Faolain L, Gomez-Iglesias A, Krauss T F 2008 Opt. Express 16 6227Google Scholar

    [46]

    Schulz S A, Upham J, O’Faolain L, Boyd R W 2017 Opt. Lett. 42 3243Google Scholar

    [47]

    Myoung N, Park H C, Ramachandran A, Lidorikis E, Ryu J W 2019 Sci. Rep. 9 2862Google Scholar

    [48]

    Nakata Y, Okada T, Nakanishi T, Kitano M 2012 Phys. Rev. B 85 205128Google Scholar

    [49]

    Kajiwara S, Urade Y, Nakata Y, Nakanishi T, Kitano M 2016 Phys. Rev. B 93 075126Google Scholar

    [50]

    Huang X, Lai Y, Hang Z H, Zheng H, Chan C T 2011 Nat. Mater. 10 582Google Scholar

    [51]

    Baboux F, Ge L, Jacqmin T, Biondi M, Galopin E, Lemaître A, Gratiet L L, Sagnes I, Schmidt S, Türeci H E, Amo A, Bloch J 2016 Phys. Rev. Lett. 116 066402Google Scholar

    [52]

    Whittaker C E, Cancellieri E, Walker P M, Gulevich D R, Schomerus H, Vaitiekus D, Royall B, Whittaker D M, Clarke E, Iorsh I V, Shelykh I A, Skolnick M S, Krizhanovskii D N 2018 Phys. Rev. Lett. 120 097401Google Scholar

    [53]

    Harder T H, Egorov O A, Beierlein J, Gagel P, Michl J, Emmerling M, Schneider C, Peschel U, Höfling S, Klembt S 2020 arXiv 2002.05405

    [54]

    Milićević M, Montambaux G, Ozawa T, Jamadi O, Real B, Sagnes I, Lemaître A, Gratiet L L, Harouri A, Bloch J, Amo A 2019 Phys. Rev. X 9 031010

    [55]

    Leykam D, Andreanov A, Flach S 2018 Adv. Phys. X 3 1473052

    [56]

    Leykam D, Flach S 2018 APL Photonics 3 070901Google Scholar

    [57]

    Pertsch T, Peschel U, Lederer F, Burghoff J, Will M, Nolte S, Tuennermann A 2004 Opt. Lett. 29 468Google Scholar

    [58]

    Szameit A, Blömer D, Burghoff J, Schreiber T, Pertsch T, Nolte S, Tünnermann A, Lederer F 2005 Opt. Express 13 10552Google Scholar

    [59]

    Plotnik Y, Rechtsman M C, Song D, Heinrich M, Zeuner J M, Nolte S, Lumer Y, Malkova N, Xu J, Szameit A, Chen Z, Segev M 2014 Nat. Mater. 13 57Google Scholar

    [60]

    Rechtsman M C, Zeuner J M, Tünnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photonics 7 153Google Scholar

    [61]

    Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196Google Scholar

    [62]

    Rechtsman M C, Plotnik Y, Zeuner J M, Song D, Chen Z, Szameit A, Segev M 2013 Phys. Rev. Lett. 111 103901Google Scholar

    [63]

    Ivanov S K, Zhang Y, Kartashov Y V, Skryabin D V 2019 APL Photonics 4 126101Google Scholar

    [64]

    Mukherjee S, Spracklen A, Choudhury D, Goldman N, Öhberg P, Andersson E, Thomson R R 2015 Phys. Rev. Lett. 114 245504Google Scholar

    [65]

    Vicencio R A, Cantillano C, Morales-Inostroza L, Real B, Mejía-Cortés C, Weimann S, Szameit A, Molina M I 2015 Phys. Rev. Lett. 114 245503Google Scholar

    [66]

    EI Hassan A, Kunst F K, Moritz A, Andler G, Emil J. Bergholtz E J, Bourennane M 2019 Nat. Photonics 13 697Google Scholar

    [67]

    Mukherjee S, Di Liberto M, Öhberg P, Thomson R R, Goldman N 2018 Phys. Rev. Lett. 121 075502Google Scholar

    [68]

    Kremer M, Petrides I, Meyer E, Heinrich M, Zilberberg O, Szameit A2018 arXiv 1805.05209

    [69]

    Zyuzin A A, Zyuzin A Y 2018 Phys. Rev. B 97 041203Google Scholar

    [70]

    Ramezani H 2017 Phys. Rev. A 96 011802Google Scholar

    [71]

    Leykam D, Flach S, Chong Y D 2017 Phys. Rev. B 96 064305Google Scholar

    [72]

    Qi B, Zhang L, Ge L 2018 Phys. Rev. Lett. 120 093901Google Scholar

    [73]

    Ge L 2018 Photonics Res 6 A10Google Scholar

    [74]

    Ge L 2015 Phys. Rev. A 92 052103Google Scholar

    [75]

    Molina M I 2015 Phys. Rev. A 92 063813Google Scholar

    [76]

    Biesenthal T, Kremer M, Heinrich M, Szameit A 2019 Phys. Rev. Lett. 123 183601Google Scholar

    [77]

    Chen Z, Klinger J, Christodoulides D N 2002 Phys. Rev. E 66 066601Google Scholar

    [78]

    Martin H, Eugenieva E D, Chen Z, Christodoulides D N 2004 Phys. Rev. Lett. 92 123902Google Scholar

    [79]

    Song D, Leykam D, Su J, Liu X, Tang L, Liu S, Zhao J, Efremidis N K, Xu J, Chen Z 2019 Phys. Rev. Lett. 122 123903Google Scholar

    [80]

    Song D, Paltoglou V, Liu S, Zhu Y, Gallardo D, Tang L, Xu J, Ablowitz M, Efremidis N K, Chen Z 2015 Nat. Commun. 6 6272Google Scholar

    [81]

    Song D, Liu S, Paltoglou V, Gallardo D, Tang L, Zhao J, Xu J, Efremidis N K Chen Z 2015 2D Mater. 2 034007

    [82]

    Boguslawski M, Rose P, Denz C 2011 Phys. Rev. A 84 013832Google Scholar

    [83]

    Gao Y, Song D, Chu S, Chen Z 2014 IEEE Photonics J. 6 2201806

    [84]

    Shen R, Shao L B, Wang B, Xing D Y 2010 Phys. Rev. B 81 041410Google Scholar

    [85]

    Xia S, Hu Y, Song D, Zong Y, Tang L, Chen Z 2016 Opt. Lett. 41 1435Google Scholar

    [86]

    Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Opt. Lett. 34 1633Google Scholar

    [87]

    Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Phys. Rev. A 80 043806Google Scholar

    [88]

    Boguslawski M, Kelberer A, Rose P, Denz C 2012 Opt. Express 20 27331Google Scholar

    [89]

    Xia S, Ramachandran A, Xia S, Li D, Liu X, Tang L, Hu Y, Song D, Xu J, Leykam D, Flach S, Chen Z 2018 Phys. Rev. Lett. 121 263902Google Scholar

    [90]

    Ma J, Rhim J W, Tang L, Xia S, Wang H, Zheng X, Xia S, Song D, Hu Y, Li Y, Yang B J, Leykam D, Chen Z 2020 Phys. Rev. Lett. 124 183901

    [91]

    Yan W, Zhong H, Song D, Zhang Y, Xia S, Tang L, Leykam D, Chen Z 2020 Adv. Optical Mater. 8 1902174

    [92]

    Xia S, Danieli C, Yan W, Li D, Xia S, Ma J, Lu H, Song D, Tang L, Flach S, Chen Z 2020 APL Photonics 5 016107Google Scholar

    [93]

    Travkin E, Diebel F, Denz C 2017 Appl. Phys. Lett. 111 011104Google Scholar

    [94]

    Diebel F, Leykam D, Kroesen S, Denz C, Desyatnikov A S 2016 Phys. Rev. Lett. 116 183902Google Scholar

    [95]

    Leykam D, Bahat-Treidel O, Desyatnikov A S 2012 Phys. Rev. A 86 031805Google Scholar

    [96]

    Bandres M A, Rechtsman M C, Szameit A, Segev M 2014 Conference on Lasers and Electro-Optics (CLEO)-Laser Science to Photonic Applications San Jose, CA, USA, June 8−13, 2014 p. FF2D.3.

    [97]

    Goda M, Nishino S, Matsuda H 2006 Phys. Rev. Lett. 96 126401Google Scholar

    [98]

    Leykam D, Flach S, Bahat-Treidel O, Desyatnikov A S 2013 Phys. Rev. B 88 224203Google Scholar

    [99]

    Bodyfelt J D, Leykam D, Danieli C, Yu X, Flach S 2014 Phys. Rev. Lett. 113 236403Google Scholar

    [100]

    Gligorić G, Maluckov A, Hadžievski L, Flach S, Malomed B A 2016 Phys. Rev. B 94 144302Google Scholar

    [101]

    Leykam D, Bodyfelt J D, Desyatnikov A S, Flach S 2017 Eur. Phys. J. B 90 1Google Scholar

    [102]

    Vicencio R A, Johansson M 2013 Phys. Rev. A 87 061803Google Scholar

    [103]

    Vicencio R A, Mejía-Cortés C 2014 J. Opt. 16 015706

    [104]

    Zong Y, Xia S, Tang L, Song D, Hu Y, Pei Y, Su J, Li Y, Chen Z 2016 Opt. Express 24 8877Google Scholar

    [105]

    Guo H, Shen S Q, Feng S 2012 Phys. Rev. B 86 085124Google Scholar

    [106]

    Weeks C, Franz M 2012 Phys. Rev. B 85 041104

    [107]

    Liu Z, Bergholtz E J, Fan H, Läuchli A M 2012 Phys. Rev. Lett. 109 186805Google Scholar

    [108]

    Bergholtz E J, Liu Z 2013 Int. J. Mod. Phys. B 27 1330017Google Scholar

    [109]

    Kalesaki E, Delerue C, Smith C M, Beugeling W, Allan G, Vanmaekelbergh D 2014 Phys. Rev. X 4 011010

    [110]

    Pal B 2018 Phys. Rev. B 98 245116Google Scholar

    [111]

    Khomeriki R, Flach S 2016 Phys. Rev. Lett. 116 245301Google Scholar

    [112]

    Long Y, Ren J 2017 arXiv 1706.01107

    [113]

    Mukherjee S, Thomson R R 2017 Opt. Lett. 42 2243Google Scholar

    [114]

    Kolovsky A R, Ramachandran A, Flach S 2018 Phys. Rev. B 97 045120Google Scholar

    [115]

    Yulin A V, Konotop V V 2013 Opt. Lett. 38 4880Google Scholar

    [116]

    Longhi S 2014 Opt. Lett. 39 5892Google Scholar

    [117]

    Mukherjee S, Thomson R R 2015 Opt. Lett. 40 5443Google Scholar

    [118]

    Rhim J W, Yang B J 2019 Phys. Rev. B 99 045107Google Scholar

    [119]

    Shima N, Aoki H 1993 Phys. Rev. Lett. 71 4389Google Scholar

    [120]

    Zhong H, Zhang Y, Zhu Y, Zhang D, Li C, Zhang Y, Li F, Belić M R, Xiao M 2017 Ann. Phys. 529 1600258Google Scholar

    [121]

    Longhi S 2019 Opt. Lett. 44 287Google Scholar

    [122]

    Lim L K, Fuchs J N, Piéchon F, Montambaux G 2020 Phys. Rev. B 101 045131Google Scholar

    [123]

    Jiang W, Kang M, Huang H, Xu H, Low T, Liu F 2019 Phys. Rev. B 99 125131Google Scholar

    [124]

    Wang P, Zheng Y, Chen X, Huang C, Kartashov Y V, Torner L, Konotop V V, Ye F 2020 Nature 577 42Google Scholar

    [125]

    Abouelkomsan A, Liu Z, Bergholtz E J 2020 Phys. Rev. Lett. 124 106803Google Scholar

  • 图 1  几种典型的平带晶格结构 (a) 菱形晶格; (b) Lieb晶格; (c) Kagome晶格; (d) 超级蜂窝型晶格; 彩色点为常规紧凑排列的局域本征模式示意图, 灰色点振幅为0, 红色点代表振幅为1, 蓝色点为–1, 即两者振幅大小相同而相位差为π

    Fig. 1.  Typical examples of flatband lattices: (a) Quasi-one-dimensional (1D) rhombic lattice; (b) Lieb lattice; (c) Kagome lattice; (d) super-honeycomb lattice (sHCL). Compact localized states (CLSs) are depicted as colored sites, where zero amplitudes are denoted by gray color, and those with non-zero amplitudes of opposite phase are denoted by red and blue colors.

    图 2  光子系统中平带研究的典型实例 (a) 左图为表面等离激元平带的金属Kagome超构材料, 右图为对应的太赫兹波透射图, 存在一个全角度的无辐射平带(水平虚线所示)[48]; (b) 左图为四方结构的全介质零折射率超构材料实现隐身效应实验装置, 右图为适当介质参数时得到的四方结构色散关系, 含有一个狄拉克锥与一个无色散平带[50]; (c) 一维“枕”型结构的微柱光学腔及其色散关系, 中间为平带[51]

    Fig. 2.  Examples of photonic flatband structures: (a) Kagome lattice for terahertz spoof plasmons, displaying an omnidirectional minimum in the transmission at the flatband frequency (dashed line) in the right panel[48]; (b) experimental setup exposing the zero-refractive-index all-dielectric metamaterials with a square lattice to realize cloaking inside a channel with the Dirac point, and the right panel shows corresponding three-dimensional dispersion diagrams consisting of a Dirac cone and a flatband[50]; (c) structured microcavity forming a 1D stub lattice and its photoluminescence spectrum, revealing a flatband in the middle[51].

    图 3  利用飞秒激光直写技术制备的平带光子晶格 (a) 用于研究平带局域模式的Lieb光子晶格[64,65]; (b) 用于研究拓扑角态的Kagome光子晶格[66]; (c) 用于实现Aharonov-Bohm效应的纵向弯曲菱形光子晶格[67,68]

    Fig. 3.  Examples of femtosecond laser-writing flatband photonic lattices: (a) A photonic Lieb lattice for demonstration of flatband compact localized states[64,65]; (b) a photonic Kagome lattice established for demonstration of topological corner states[66]; (c) a driven photonic rhombic lattice for experimental observation of Aharonov-Bohm cages[67,68].

    图 4  多光束干涉光诱导技术制备光子晶格[80,83], 其中第一排为实验光路图(PBS, 偏振分光棱镜; Mask, 掩膜板; SBN, 铌酸锶钡晶体); 第二排为实验制作的典型光子晶格 (a) 蜂窝型光子晶格; (b) 对称破缺的蜂窝型光子晶格; (c) 涡旋晶格; (d) Kagome晶格

    Fig. 4.  Examples of photonic lattices created by multiple-beam optical induction method[80,83]. Top panel shows schematic of experimental setup. PBS, polarized beam splitter; SBN, strontium barium niobite. Bottom panel shows typical examples of photonic lattices realized in experiment: (a) A “perfect” honeycomb lattice; (b) an inversion-symmetry-breaking honeycomb lattice; (c) a vortex lattice; (d) a Kagome lattice.

    图 5  连续激光直写技术制备不同平带光子晶格, 其中, 第一排为实验光路图(SLM, 空间光调制器; BS, 分光棱镜; FM, 傅里叶掩膜板); 第二排为典型的直写光子晶格 (a)“胡须”边界(“bearded” edges)的Lieb光子晶格[89]; (b) 平滑边界的Kagome光子晶格[90]; (c) 超级蜂窝晶格[91]; (d) 横向折射率梯度变化的菱形光子晶格[92]

    Fig. 5.  Examples of photonic lattices created by direct cw-laser-writing technique in a nonlinear bulk crystal. Top panel shows illustration of experimental setup. SLM, spatial light modulator; BS, beam splitter; FM, Fourier mask. Bottom panel shows typical examples of photonic lattices created by direct cw-laser-writing method: (a) A photonic Lieb lattice with “bearded” edges[89]; (b) a photonic Kagome lattice with flat boundary[90]; (c) a photonic sHCL[91]; (d) a driven photonic rhombic lattice with refractive index gradient parallel to the ribbon[92].

    图 6  平带光子晶格常规紧凑局域模式的实验结果(从左至右: 实验制备的光诱导光子晶格结构、紧束缚近似下能带结构、入射面探测光、同相位输出以及反相位输出) (a) Lieb光子晶格中基于平带模式的图像传输[85]; (b) Kagome光子晶格中的平带局域模式[104]; (c) 折射率梯度场下菱形光子晶格中U = 2的局域本征模式[92]

    Fig. 6.  Experimental results of CLSs in flatband lattices: (a) Linear image (formed by CLSs) propagation through an optically induced Lieb photonic lattice[85]; (b) a bound-state transmission in a Kagome photonic lattice[104]; (c) observation of a quincunx-shaped (U = 2) compact localized state which spans over two-unit cells in a photonic rhombic lattice[92]. From left to right: shown are the lattices, calculated band structures in the tight-binding approximation, probe beam inputs, and their in-phase and out-of-phase outputs.

    图 7  Lieb与超级蜂窝光子晶格中非常规线形平带模式的实验结果[89,91] (a), (d)非常规线形平带局域模式的示意图, 其中图(d)右下插图为能带结构; (b1), (e1) 反相位入射的探测光; (b2), (e2) 无晶格时的输出; (b3), (e3) 有晶格时的输出; (b4), (e4) 长距离传输的数值模拟结果; 图(c1)−(c4)完全对应(b1)−(b4), 但为探测光同相位入射时的结果; 图(f1)−(f4)对应(e1)−(e4), 但为探测光同相位入射时的结果; (b5), (c5)为对应(b3), (c3)的k空间能谱, 虚线框为第一布里渊区边界; (e5) 为对应于图(e3)的k空间能量分布, 其中虚线框表示第一与第二布里渊区边界

    Fig. 7.  Demonstration of unconventional line state in photonic Lieb and super-honeycomb lattices[89,91]: (a), (d) Schematic of flatband line states in infinite lattices, and insect in (d) shows the band structure of sHCL; (b1), (e1) out-of-phase input line beam; (b2), (e2) out-of-phase output without the lattice; (b3), (e3) out-of-phase output through the lattice; (b4), (e4) simulation results showing the out-of-phase line beam remains intact but the in-phase line deteriorates after propagating a long distance through the lattice; (b5) measured k-space spectrum of (b3) with a dashed square marking the first Brillouin zone (BZ); (c1)−(c5) the same as in (b1)−(b5), and (f1)−(f4) the same as in (e1)−(e4) except that the line beam is in in-phase excitation condition; (e5) momentum space spectrum of (e3), where the white dashed lines outline the first and second BZs.

    图 8  (a) Kagome晶格中两种不可压缩环形态示意图; (b) 模拟无限Kagome晶格两个不可压缩环形态的圆环面; (c) 在平滑边界的Kagome晶格中两种稳定的平带边界局域模, 橘黄色线代表模式一, 绿色虚线代表模式二; (d) Corbino-Kagome结构示意图, 橘黄色圆环代表不可压缩的局域环形态; (e) 从左至右, (e1), (e2) 边界模式一的反相位输出与对应数值模拟结果; (e3), (e4) 边界模式二的反相位输出与对应数值模拟结果; (f1) 实验制备的Corbino-Kagome光子晶格; (f2), (f3)反相位激发的不可压缩环形态及其对应数值模拟结果; (f4) 40 mm传输时数值模拟结果; 图中所有插图均代表输入探测光; 同相位激发破坏所有平带局域模式的形成[90]

    Fig. 8.  (a) Illustration of the noncontractible loop states (NLSs) in an infinitely extended Kagome lattice; (b) a torus showing two NLSs mimicking the 2D infinite lattice; (c) two robust boundary modes (RBMs) in a Kagome lattice with flat cutting edges, where black sites are of zero-amplitude, while blue and red ones distinguish non-zero sites with opposite phase; (d) schematic diagram of the Corbino-shaped Kagome lattice, where the NLS is illustrated by the orange circle; (e) from left to right (e1), (e3) experimental results of RBM1 and RBM2 under out-of-phase condition; (e2), (e4) simulation result corresponding to (e1) and (e3); (f1) experimentally established finite-sized Kagome lattice in a Corbino-geometry; (f2)-(f4) the NLS observed in (f2) experiment and (f3) simulations after propagation to 10 mm and (f4) 40 mm under out-of-phase condition. All insets are from input ring necklace of the probe beam. In-phase excitation destroys all localized states[90].

  • [1]

    Ramirez A P 1994 Annu. Rev. Mater. Sci. 24 453Google Scholar

    [2]

    Huse D A, Rutenberg A D 1992 Phys. Rev. B 45 7536Google Scholar

    [3]

    Chalker J T, Holdsworth P C, Shender E F 1992 Phys. Rev. Lett. 68 855Google Scholar

    [4]

    Moessner R, Chalker J T 1998 Phys. Rev. B 58 12049Google Scholar

    [5]

    Moessner R, Ramirez A P 2006 Phys. Today 59 24

    [6]

    Bergman D L, Wu C, Balents L 2008 Phys. Rev. B 78 125104Google Scholar

    [7]

    Huber S D, Altman E 2010 Phys. Rev. B 82 184502Google Scholar

    [8]

    Nixon M, Ronen E, Friesem A A, Davidson N 2013 Phys. Rev. Lett. 110 184102Google Scholar

    [9]

    Möller G, Cooper N R 2012 Phys. Rev. Lett. 108 045306Google Scholar

    [10]

    Yin J X, Zhang S S, Li H, Jiang K, Chang G, Zhang B, Lian B, Xiang C, Belopolski I, Zheng H, Cochran T A, Xu S Y, Bian G, Liu K, Chang T R, Lin H, Lu Z Y, Wang Z, Jia S, Wang W, Hasan M Z 2018 Nature 562 91Google Scholar

    [11]

    Yin J X, Zhang S S, Chang G, Wang Q, Tsirkin S S, Guguchia Z, Lian B, Zhou H, Jiang K, Belopolski I, Shumiya N, Multer D, Litskevich M, Cochran T A, Lin H, Wang Z, Neupert T, Jia S, Lei H, Hasan M Z 2019 Nat. Phys. 15 443Google Scholar

    [12]

    Xie Y, Lian B, Jäck B, Liu X, Chiu C L, Watanabe K, Taniguchi T, Bernevig B A, Yazdani A 2019 Nature 572 101Google Scholar

    [13]

    Tang E, Mei J W, Wen X G 2011 Phys. Rev. Lett. 106 236802Google Scholar

    [14]

    Sun K, Gu Z, Katsura H, Sarma S D 2011 Phys. Rev. Lett. 106 236803Google Scholar

    [15]

    Neupert T, Santos L, Chamon C, Mudry C 2011 Phys. Rev. Lett. 106 236804Google Scholar

    [16]

    Sheng D N, Gu Z C, Sun K, Sheng L 2011 Nat. Commun. 2 389Google Scholar

    [17]

    Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C, Jarillo-Herrero P 2018 Nature 556 80Google Scholar

    [18]

    Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43Google Scholar

    [19]

    Kopnin N B, Heikkilä T T, Volovik G E 2011 Phys. Rev. B 83 220503Google Scholar

    [20]

    Tang E, Fu L 2014 Nat. Phys. 10 964Google Scholar

    [21]

    Iglovikov V I, Hébert F, Grémaud B, Batrouni G G, Scalettar R T 2014 Phys. Rev. B 90 094506Google Scholar

    [22]

    Fleischer J W, Segev M, Efremidis N K, Christodoulides D N 2003 Nature 422 147Google Scholar

    [23]

    Efremidis N K, Sears S, Christodoulides D N, Fleischer J W, Segev M 2002 Phys. Rev. E 66 046602Google Scholar

    [24]

    Chen Z, Segev M, Christodoulides D N 2012 Rep. Prog. Phys. 75 086401Google Scholar

    [25]

    Longhi S, Marangoni M, Lobino M, Ramponi R, Laporta P, Cianci E, Foglietti V 2006 Phys. Rev. Lett. 96 243901Google Scholar

    [26]

    Szameit A, Garanovich I L, Heinrich M, Sukhorukov A A, Dreisow F, Pertsch T, Nolte S, Tünnermann A, Kivshar Y S 2009 Nat. Phys. 5 271Google Scholar

    [27]

    Schwartz T, Bartal G, Fishman S, Segev M 2007 Nature 446 52Google Scholar

    [28]

    Lahini Y, Avidan A, Pozzi F, Sorel M, Morandotti R, Christodoulides D N, Silberberg Y 2008 Phys. Rev. Lett. 100 013906Google Scholar

    [29]

    Sutherland B 1986 Phys. Rev. B 34 5208Google Scholar

    [30]

    Arai M, Tokihiro T, Fujiwara T, Kohmoto M 1988 Phys. Rev. B 38 1621

    [31]

    Lieb E H 1989 Phys. Rev. Lett. 62 1201Google Scholar

    [32]

    Mielke A 1991 J. Phys. A: Math. Gen. 24 L73Google Scholar

    [33]

    Mielke A 1991 J. Phys. A: Math. Gen. 24 3311Google Scholar

    [34]

    Mielke A 1999 Phys. Rev. Lett. 82 4312Google Scholar

    [35]

    Tasaki H 1992 Phys. Rev. Lett. 69 1608Google Scholar

    [36]

    Tasaki H 1998 Prog. Theor. Phys. 99 489Google Scholar

    [37]

    Aoki H, Ando M, Matsumura H 1996 Phys. Rev. B 54 R17296Google Scholar

    [38]

    Slot M R, Gardenier T S, Jacobse P H, van Miert G C, Kempkes S N, Zevenhuizen S J, Smith C M, Vanmaekelbergh D, Swart I 2017 Nat. Phys. 13 672Google Scholar

    [39]

    Jo G B, Guzman J, Thomas C K, Hosur P, Vishwanath A, Stamper-Kurn D M 2012 Phys. Rev. Lett. 108 045305Google Scholar

    [40]

    Taie S, Ozawa H, Ichinose T, Nishio T, Nakajima S, Takahashi Y 2015 Sci. Adv. 1 e1500854Google Scholar

    [41]

    Guzmán-Silva D, Mejía-Cortés C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A, Vicencio R A 2014 New J. Phys. 16 063061Google Scholar

    [42]

    Makasyuk I, Chen Z, Yang J 2006 Phys. Rev. Lett. 96 223903Google Scholar

    [43]

    Flach S, Leykam D, Bodyfelt J D, Matthies P, Desyatnikov A S 2014 Europhys. Lett. 105 30001Google Scholar

    [44]

    Ramachandran A, Andreanov A, Flach S 2017 Phys. Rev. B 96 161104Google Scholar

    [45]

    Li J, White T P, O’Faolain L, Gomez-Iglesias A, Krauss T F 2008 Opt. Express 16 6227Google Scholar

    [46]

    Schulz S A, Upham J, O’Faolain L, Boyd R W 2017 Opt. Lett. 42 3243Google Scholar

    [47]

    Myoung N, Park H C, Ramachandran A, Lidorikis E, Ryu J W 2019 Sci. Rep. 9 2862Google Scholar

    [48]

    Nakata Y, Okada T, Nakanishi T, Kitano M 2012 Phys. Rev. B 85 205128Google Scholar

    [49]

    Kajiwara S, Urade Y, Nakata Y, Nakanishi T, Kitano M 2016 Phys. Rev. B 93 075126Google Scholar

    [50]

    Huang X, Lai Y, Hang Z H, Zheng H, Chan C T 2011 Nat. Mater. 10 582Google Scholar

    [51]

    Baboux F, Ge L, Jacqmin T, Biondi M, Galopin E, Lemaître A, Gratiet L L, Sagnes I, Schmidt S, Türeci H E, Amo A, Bloch J 2016 Phys. Rev. Lett. 116 066402Google Scholar

    [52]

    Whittaker C E, Cancellieri E, Walker P M, Gulevich D R, Schomerus H, Vaitiekus D, Royall B, Whittaker D M, Clarke E, Iorsh I V, Shelykh I A, Skolnick M S, Krizhanovskii D N 2018 Phys. Rev. Lett. 120 097401Google Scholar

    [53]

    Harder T H, Egorov O A, Beierlein J, Gagel P, Michl J, Emmerling M, Schneider C, Peschel U, Höfling S, Klembt S 2020 arXiv 2002.05405

    [54]

    Milićević M, Montambaux G, Ozawa T, Jamadi O, Real B, Sagnes I, Lemaître A, Gratiet L L, Harouri A, Bloch J, Amo A 2019 Phys. Rev. X 9 031010

    [55]

    Leykam D, Andreanov A, Flach S 2018 Adv. Phys. X 3 1473052

    [56]

    Leykam D, Flach S 2018 APL Photonics 3 070901Google Scholar

    [57]

    Pertsch T, Peschel U, Lederer F, Burghoff J, Will M, Nolte S, Tuennermann A 2004 Opt. Lett. 29 468Google Scholar

    [58]

    Szameit A, Blömer D, Burghoff J, Schreiber T, Pertsch T, Nolte S, Tünnermann A, Lederer F 2005 Opt. Express 13 10552Google Scholar

    [59]

    Plotnik Y, Rechtsman M C, Song D, Heinrich M, Zeuner J M, Nolte S, Lumer Y, Malkova N, Xu J, Szameit A, Chen Z, Segev M 2014 Nat. Mater. 13 57Google Scholar

    [60]

    Rechtsman M C, Zeuner J M, Tünnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photonics 7 153Google Scholar

    [61]

    Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196Google Scholar

    [62]

    Rechtsman M C, Plotnik Y, Zeuner J M, Song D, Chen Z, Szameit A, Segev M 2013 Phys. Rev. Lett. 111 103901Google Scholar

    [63]

    Ivanov S K, Zhang Y, Kartashov Y V, Skryabin D V 2019 APL Photonics 4 126101Google Scholar

    [64]

    Mukherjee S, Spracklen A, Choudhury D, Goldman N, Öhberg P, Andersson E, Thomson R R 2015 Phys. Rev. Lett. 114 245504Google Scholar

    [65]

    Vicencio R A, Cantillano C, Morales-Inostroza L, Real B, Mejía-Cortés C, Weimann S, Szameit A, Molina M I 2015 Phys. Rev. Lett. 114 245503Google Scholar

    [66]

    EI Hassan A, Kunst F K, Moritz A, Andler G, Emil J. Bergholtz E J, Bourennane M 2019 Nat. Photonics 13 697Google Scholar

    [67]

    Mukherjee S, Di Liberto M, Öhberg P, Thomson R R, Goldman N 2018 Phys. Rev. Lett. 121 075502Google Scholar

    [68]

    Kremer M, Petrides I, Meyer E, Heinrich M, Zilberberg O, Szameit A2018 arXiv 1805.05209

    [69]

    Zyuzin A A, Zyuzin A Y 2018 Phys. Rev. B 97 041203Google Scholar

    [70]

    Ramezani H 2017 Phys. Rev. A 96 011802Google Scholar

    [71]

    Leykam D, Flach S, Chong Y D 2017 Phys. Rev. B 96 064305Google Scholar

    [72]

    Qi B, Zhang L, Ge L 2018 Phys. Rev. Lett. 120 093901Google Scholar

    [73]

    Ge L 2018 Photonics Res 6 A10Google Scholar

    [74]

    Ge L 2015 Phys. Rev. A 92 052103Google Scholar

    [75]

    Molina M I 2015 Phys. Rev. A 92 063813Google Scholar

    [76]

    Biesenthal T, Kremer M, Heinrich M, Szameit A 2019 Phys. Rev. Lett. 123 183601Google Scholar

    [77]

    Chen Z, Klinger J, Christodoulides D N 2002 Phys. Rev. E 66 066601Google Scholar

    [78]

    Martin H, Eugenieva E D, Chen Z, Christodoulides D N 2004 Phys. Rev. Lett. 92 123902Google Scholar

    [79]

    Song D, Leykam D, Su J, Liu X, Tang L, Liu S, Zhao J, Efremidis N K, Xu J, Chen Z 2019 Phys. Rev. Lett. 122 123903Google Scholar

    [80]

    Song D, Paltoglou V, Liu S, Zhu Y, Gallardo D, Tang L, Xu J, Ablowitz M, Efremidis N K, Chen Z 2015 Nat. Commun. 6 6272Google Scholar

    [81]

    Song D, Liu S, Paltoglou V, Gallardo D, Tang L, Zhao J, Xu J, Efremidis N K Chen Z 2015 2D Mater. 2 034007

    [82]

    Boguslawski M, Rose P, Denz C 2011 Phys. Rev. A 84 013832Google Scholar

    [83]

    Gao Y, Song D, Chu S, Chen Z 2014 IEEE Photonics J. 6 2201806

    [84]

    Shen R, Shao L B, Wang B, Xing D Y 2010 Phys. Rev. B 81 041410Google Scholar

    [85]

    Xia S, Hu Y, Song D, Zong Y, Tang L, Chen Z 2016 Opt. Lett. 41 1435Google Scholar

    [86]

    Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Opt. Lett. 34 1633Google Scholar

    [87]

    Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Phys. Rev. A 80 043806Google Scholar

    [88]

    Boguslawski M, Kelberer A, Rose P, Denz C 2012 Opt. Express 20 27331Google Scholar

    [89]

    Xia S, Ramachandran A, Xia S, Li D, Liu X, Tang L, Hu Y, Song D, Xu J, Leykam D, Flach S, Chen Z 2018 Phys. Rev. Lett. 121 263902Google Scholar

    [90]

    Ma J, Rhim J W, Tang L, Xia S, Wang H, Zheng X, Xia S, Song D, Hu Y, Li Y, Yang B J, Leykam D, Chen Z 2020 Phys. Rev. Lett. 124 183901

    [91]

    Yan W, Zhong H, Song D, Zhang Y, Xia S, Tang L, Leykam D, Chen Z 2020 Adv. Optical Mater. 8 1902174

    [92]

    Xia S, Danieli C, Yan W, Li D, Xia S, Ma J, Lu H, Song D, Tang L, Flach S, Chen Z 2020 APL Photonics 5 016107Google Scholar

    [93]

    Travkin E, Diebel F, Denz C 2017 Appl. Phys. Lett. 111 011104Google Scholar

    [94]

    Diebel F, Leykam D, Kroesen S, Denz C, Desyatnikov A S 2016 Phys. Rev. Lett. 116 183902Google Scholar

    [95]

    Leykam D, Bahat-Treidel O, Desyatnikov A S 2012 Phys. Rev. A 86 031805Google Scholar

    [96]

    Bandres M A, Rechtsman M C, Szameit A, Segev M 2014 Conference on Lasers and Electro-Optics (CLEO)-Laser Science to Photonic Applications San Jose, CA, USA, June 8−13, 2014 p. FF2D.3.

    [97]

    Goda M, Nishino S, Matsuda H 2006 Phys. Rev. Lett. 96 126401Google Scholar

    [98]

    Leykam D, Flach S, Bahat-Treidel O, Desyatnikov A S 2013 Phys. Rev. B 88 224203Google Scholar

    [99]

    Bodyfelt J D, Leykam D, Danieli C, Yu X, Flach S 2014 Phys. Rev. Lett. 113 236403Google Scholar

    [100]

    Gligorić G, Maluckov A, Hadžievski L, Flach S, Malomed B A 2016 Phys. Rev. B 94 144302Google Scholar

    [101]

    Leykam D, Bodyfelt J D, Desyatnikov A S, Flach S 2017 Eur. Phys. J. B 90 1Google Scholar

    [102]

    Vicencio R A, Johansson M 2013 Phys. Rev. A 87 061803Google Scholar

    [103]

    Vicencio R A, Mejía-Cortés C 2014 J. Opt. 16 015706

    [104]

    Zong Y, Xia S, Tang L, Song D, Hu Y, Pei Y, Su J, Li Y, Chen Z 2016 Opt. Express 24 8877Google Scholar

    [105]

    Guo H, Shen S Q, Feng S 2012 Phys. Rev. B 86 085124Google Scholar

    [106]

    Weeks C, Franz M 2012 Phys. Rev. B 85 041104

    [107]

    Liu Z, Bergholtz E J, Fan H, Läuchli A M 2012 Phys. Rev. Lett. 109 186805Google Scholar

    [108]

    Bergholtz E J, Liu Z 2013 Int. J. Mod. Phys. B 27 1330017Google Scholar

    [109]

    Kalesaki E, Delerue C, Smith C M, Beugeling W, Allan G, Vanmaekelbergh D 2014 Phys. Rev. X 4 011010

    [110]

    Pal B 2018 Phys. Rev. B 98 245116Google Scholar

    [111]

    Khomeriki R, Flach S 2016 Phys. Rev. Lett. 116 245301Google Scholar

    [112]

    Long Y, Ren J 2017 arXiv 1706.01107

    [113]

    Mukherjee S, Thomson R R 2017 Opt. Lett. 42 2243Google Scholar

    [114]

    Kolovsky A R, Ramachandran A, Flach S 2018 Phys. Rev. B 97 045120Google Scholar

    [115]

    Yulin A V, Konotop V V 2013 Opt. Lett. 38 4880Google Scholar

    [116]

    Longhi S 2014 Opt. Lett. 39 5892Google Scholar

    [117]

    Mukherjee S, Thomson R R 2015 Opt. Lett. 40 5443Google Scholar

    [118]

    Rhim J W, Yang B J 2019 Phys. Rev. B 99 045107Google Scholar

    [119]

    Shima N, Aoki H 1993 Phys. Rev. Lett. 71 4389Google Scholar

    [120]

    Zhong H, Zhang Y, Zhu Y, Zhang D, Li C, Zhang Y, Li F, Belić M R, Xiao M 2017 Ann. Phys. 529 1600258Google Scholar

    [121]

    Longhi S 2019 Opt. Lett. 44 287Google Scholar

    [122]

    Lim L K, Fuchs J N, Piéchon F, Montambaux G 2020 Phys. Rev. B 101 045131Google Scholar

    [123]

    Jiang W, Kang M, Huang H, Xu H, Low T, Liu F 2019 Phys. Rev. B 99 125131Google Scholar

    [124]

    Wang P, Zheng Y, Chen X, Huang C, Kartashov Y V, Torner L, Konotop V V, Ye F 2020 Nature 577 42Google Scholar

    [125]

    Abouelkomsan A, Liu Z, Bergholtz E J 2020 Phys. Rev. Lett. 124 106803Google Scholar

  • [1] 陆展鹏, 徐志浩. 具有平带的一维十字型晶格中重返局域化现象. 物理学报, 2024, 73(3): 037202. doi: 10.7498/aps.73.20231393
    [2] 古杰, 马立国. 莫尔晶格中的激子绝缘体. 物理学报, 2023, 72(6): 067101. doi: 10.7498/aps.72.20230079
    [3] 郑雅欣, 那仁满都拉. 可压缩液体中气泡的声空化特性. 物理学报, 2022, 71(1): 014301. doi: 10.7498/aps.71.20211266
    [4] 孙海明. 一维螺旋型Se原子链中的Rashba效应和平带性质. 物理学报, 2022, 71(14): 147102. doi: 10.7498/aps.71.20220646
    [5] 王仲锐, 姜宇航. 转角二维量子材料中平带相关的新奇电子态物性. 物理学报, 2022, 71(12): 127202. doi: 10.7498/aps.71.20220064
    [6] 张若寒, 任慧莹, 何林. 二维材料平带的实现及其新奇量子物态. 物理学报, 2022, 71(12): 127302. doi: 10.7498/aps.71.20220225
    [7] 郑雅欣, 那仁满都拉. 可压缩液体中气泡的声空化特性. 物理学报, 2021, (): . doi: 10.7498/aps.70.20211266
    [8] 胡嘉懿, 张文欢, 柴振华, 施保昌, 汪一航. 三维不可压缩流的12速多松弛格子Boltzmann模型. 物理学报, 2019, 68(23): 234701. doi: 10.7498/aps.68.20190984
    [9] 邱流潮. 基于不可压缩光滑粒子动力学的黏性液滴变形过程仿真. 物理学报, 2013, 62(12): 124702. doi: 10.7498/aps.62.124702
    [10] 张晓娟, 赵建林. 光子晶体光纤带隙及模场的一种综合模拟分析方法. 物理学报, 2010, 59(9): 6299-6305. doi: 10.7498/aps.59.6299
    [11] 王立锋, 叶文华, 李英骏. 二维不可压缩流体Kelvin-Helmholtz不稳定性的二次谐波产生. 物理学报, 2008, 57(5): 3038-3043. doi: 10.7498/aps.57.3038
    [12] 王新军, 王玲玲, 黄维清, 唐黎明, 邹炳锁, 陈克求. 三元合金缺陷层对有限超晶格中局域界面光学声子模的影响. 物理学报, 2007, 56(1): 429-436. doi: 10.7498/aps.56.429
    [13] 龚志强, 贺梦冬. 两耦合半无限超晶格中的局域界面声子-极化激元. 物理学报, 2007, 56(11): 6600-6607. doi: 10.7498/aps.56.6600
    [14] 张大成, 王鹿霞, 刘德胜, 韩圣浩, 解士杰. 扰动对一维局域模的影响. 物理学报, 2003, 52(12): 3191-3196. doi: 10.7498/aps.52.3191
    [15] 周光辉, 夏庆林, 颜家壬. 一维金刚石结构晶格的非线性局域模. 物理学报, 2000, 49(9): 1741-1746. doi: 10.7498/aps.49.1741
    [16] 俞慧丹, 赵凯华. 模拟可压缩流体的格子Boltzmann模型. 物理学报, 1999, 48(8): 1470-1476. doi: 10.7498/aps.48.1470
    [17] 丁建文, 颜晓红, 方显承, 段祝平. 纳米结构链的hopping电导实空间重正化群方法. 物理学报, 1999, 48(2): 314-319. doi: 10.7498/aps.48.314
    [18] 蒋祺, 陶瑞宝. 有任意带满的紧束缚哈密顿量的实空间重整化群研究. 物理学报, 1989, 38(11): 1778-1784. doi: 10.7498/aps.38.1778
    [19] 熊诗杰, 蔡建华. 非均匀无序系统中Anderson局域化的标度理论——实空间重整化群途径. 物理学报, 1985, 34(12): 1530-1538. doi: 10.7498/aps.34.1530
    [20] 卢鹤绂. 可压缩流体之散逸函数. 物理学报, 1951, 8(2): 143-149. doi: 10.7498/aps.8.143
计量
  • 文章访问数:  10534
  • PDF下载量:  672
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-03-14
  • 修回日期:  2020-04-07
  • 上网日期:  2020-05-09
  • 刊出日期:  2020-08-05

/

返回文章
返回