| ATOMIC AND MOLECULAR PHYSICS |
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Probe Knots and Hopf Insulators with Ultracold Atoms |
| Dong-Ling Deng1,2,3**, Sheng-Tao Wang1,4,3, Kai Sun1, L.-M. Duan1,3 |
1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
2Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA
3Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084
4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA |
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| Cite this article: |
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Dong-Ling Deng, Sheng-Tao Wang, Kai Sun et al 2018 Chin. Phys. Lett. 35 013701 |
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Abstract Knots and links are fascinating and intricate topological objects. Their influence spans from DNA and molecular chemistry to vortices in superfluid helium, defects in liquid crystals and cosmic strings in the early universe. Here we find that knotted structures also exist in a peculiar class of three-dimensional topological insulators—the Hopf insulators. In particular, we demonstrate that the momentum-space spin textures of Hopf insulators are twisted in a nontrivial way, which implies the presence of various knot and link structures. We further illustrate that the knots and nontrivial spin textures can be probed via standard time-of-flight images in cold atoms as preimage contours of spin orientations in stereographic coordinates. The extracted Hopf invariants, knots, and links are validated to be robust to typical experimental imperfections. Our work establishes the existence of knotted structures in Hopf insulators, which may have potential applications in spintronics and quantum information processing.
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Received: 23 November 2017
Published: 03 December 2017
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| PACS: |
37.10.Jk
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(Atoms in optical lattices)
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03.65.Vf
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(Phases: geometric; dynamic or topological)
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73.43.Nq
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(Quantum phase transitions)
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| Fund: D.L.D., S.T.W. and L.M.D. are supported by the ARL, the IARPA LogiQ program, and the AFOSR MURI program, and supported by Tsinghua University for their visits. K.S. acknowledges the support from NSF under Grant No. PHY1402971. D.L.D. is also supported by JQI-NSF-PFC and LPS-MPO-CMTC at the final stage of this paper. |
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| [1] | Kelvin L 1867 Proc. R. Soc. Edinburgh 6 94 | | [2] | Bates A D, Maxwell A and Press O 2005 DNA Topology (Oxford: Oxford University Press) p 2 | | [3] | Meluzzi D, Smith D E and Arya G 2010 Annu. Rev. Biophys. 39 349 | | [4] | Chichak K S, Cantrill S J, Pease A R, Chiu S H, Cave G W, 5 Atwood J L and Stoddart J F 2004 Science 304 1308 | | [5] | Han D, Pal S, Liu Y and Yan H 2010 Nat. Nanotechnol. 5 10 | | [6] | Ponnuswamy N, Cougnon F B, Clough J M, Pantoş G D and Sanders J K 2012 Science 338 783 | | [7] | de Gennes P G 1979 Scaling Concepts in Polymer Physics (New York: Cornell University Press) | | [8] | Faddeev L and Niemi A J 1997 Nature 387 58 | | [9] | Kedia H, Bialynicki-Birula I, Peralta-Salas D and Irvine W T M 2013 Phys. Rev. Lett. 111 150404 | | [10] | Ricca R L, Samuels D C and Barenghi C F 1999 J. Fluid Mech. 391 29 | | [11] | Woltjer L 1958 Proc. Natl. Acad. Sci. USA 44 489 | | [12] | Kawaguchi Y, Nitta M and Ueda M 2008 Phys. Rev. Lett. 100 180403 | | [13] | Machon T and Alexander G P 2013 Proc. Natl. Acad. Sci. USA 110 14174 | | [14] | Machon T and Alexander G P 2014 Phys. Rev. Lett. 113 027801 | | [15] | Alexander G P, Chen B G g, Matsumoto E A and Kamien R D 2012 Rev. Mod. Phys. 84 497 | | [16] | Buniy R V and Kephart T W 2003 Phys. Lett. B 576 1 | | [17] | Witten E 1989 Commun. Math. Phys. 121 351 | | [18] | Senyuk B, Liu Q, He S, Kamien R D, Kusner R B, Lubensky T C and Smalyukh I I 2013 Nature 493 200 | | [19] | Tkalec U, Ravnik M, Čopar S, Žumer S and Muševič I 2011 Science 333 62 | | [20] | Chen B G g, Ackerman P J, Alexander G P, Kamien R D and Smalyukh I I 2013 Phys. Rev. Lett. 110 237801 | | [21] | Martinez A, Ravnik M, Lucero B, Visvanathan R, Žumer S and Smalyukh I I 2014 Nat. Mater. 13 258 | | [22] | Dennis M R, King R P, Jack B, O'Holleran K and PadgettM J 2010 Nat. Phys. 6 118 | | [23] | Kleckner D and Irvine W T 2013 Nat. Phys. 9 253 | | [24] | Irvine W T and Kleckner D 2014 Nat. Mater. 13 229 | | [25] | Babaev E 2002 Phys. Rev. Lett. 88 177002 | | [26] | Babaev E, Faddeev L D and Niemi A J 2002 Phys. Rev. B 65 100512 | | [27] | Hall D S, Ray M W, Tiurev K, Ruokokoski E, Gheorghe A H and Möttönen M 2016 Nat. Phys. 12 478 | | [28] | Yuan X X, He L, Wang S T, Deng D L, Wang F, Lian W Q, Wang X, Zhang C H, Zhang H L, Chang X Y and Duan L M 2017 Chin. Phys. Lett. 34 060302 | | [29] | Berezinskii V 1971 Sov. J. Exp. Theor. Phys. 32 493 | | [30] | Kosterlitz J M and Thouless D J 1973 J. Phys. C 6 1181 | | [31] | CageM E, Klitzing K, Chang A, Duncan F, Haldane M, Laughlin R, Pruisken A, Thouless D, Prange R E and Girvin S M 2012 The Quantum Hall Effect (Springer Science & Business Media) | | [32] | Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057 | | [33] | Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 | | [34] | Moore J E 2010 Nature 464 194 | | [35] | Chen X, Gu Z C, Liu Z X and Wen X G 2012 Science 338 1604 | | [36] | Thouless D J, Kohmoto M, Nightingale M P and den Nijs M 1982 Phys. Rev. Lett. 49 405 | | [37] | Qi X L, Li R, Zang J and Zhang S C 2009 Science 323 1184 | | [38] | Rosenberg G, Guo H M and Franz M 2010 Phys. Rev. B 82 041104 | | [39] | Rosenberg G and Franz M 2010 Phys. Rev. B 82 035105 | | [40] | Faddeev L 1975 Princeton preprint IAS-75-QS70 | | [41] | Lin Y J, Jimenez-Garcia K and Spielman I 2011 Nature 471 83 | | [42] | Galitski V and Spielman I B 2013 Nature 494 49 | | [43] | Beeler M, Williams R, Jimenez-Garcia K, LeBlanc L, Perry A and Spielman I 2013 Nature 498 201 | | [44] | Dalibard J, Gerbier F, Juzeliūnas G and Öhberg P 2011 Rev. Mod. Phys. 83 1523 | | [45] | Bloch I, Dalibard J and Nascimbène S 2012 Nat. Phys. 8 267 | | [46] | Goldman N, Budich J and Zoller P 2016 Nat. Phys. 12 7 | | [47] | Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D and Esslinger T 2014 Nature 515 237 | | [48] | Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y, Chen S, Liu X J and Pan J W 2016 Science 354 83 | | [49] | Moore J E, Ran Y and Wen X G 2008 Phys. Rev. Lett. 101 186805 | | [50] | Deng D L, Wang S T, Shen C and Duan L M 2013 Phys. Rev. B 88 201105 | | [51] | Fu L, Kane C L and Mele E J 2007 Phys. Rev. Lett. 98 106803 | | [52] | Moore J E and Balents L 2007 Phys. Rev. B 75 121306 | | [53] | Roy R 2009 Phys. Rev. B 79 195322 | | [54] | Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R and Hasan M Z 2008 Nature 452 970 | | [55] | Xia Y, Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor Y, Cava R et al 2009 Nat. Phys. 5 398 | | [56] | Schnyder A P, Ryu S, Furusaki A and Ludwig A W W 2008 Phys. Rev. B 78 195125 | | [57] | Kitaev A 2009 AIP Conf. Proc. 1134 22 | | [58] | Deng D L, Wang S T and Duan L M 2014 Phys. Rev. B 89 075126 | | [59] | Whitehead J H 1947 Proc. Natl. Acad. Sci. USA 33 117 | | [60] | See the supplemental material for details of the stereographic coordinates, a sketch of the spin texture, adding random perturbations, the $\epsilon$-neighborhood method and an experimental protocol to realize Hopf insulators with ultracold atoms in optical lattices | | [61] | Kauffman L H 2013 Knots Physics (Singapore: World Scientific) Vol 53 | | [62] | Hatcher A 2002 Algebraic Topology (Cambridge: Cambridge University Press) | | [63] | Hietarinta J, Palmu J, Jäykkä J and Pakkanen P 2012 New J. Phys. 14 013013 | | [64] | Deng D L, Wang S T and Duan L M 2014 Phys. Rev. A 90 041601 | | [65] | Alba E, Fernandez-Gonzalvo X, Mur-Petit J, Pachos J K and Garcia-Ripoll J J 2011 Phys. Rev. Lett. 107 235301 | | [66] | Skyrme T H R 1962 Nucl. Phys. 31 556 | | [67] | Yu X, Onose Y, Kanazawa N, Park J, Han J, Matsui Y, Nagaosa N and Tokura Y 2010 Nature 465 901 | | [68] | Rößler U, Bogdanov A and Pfleiderer C 2006 Nature 442 797 | | [69] | Jaksch D and Zoller P 2003 New J. Phys. 5 1 | | [70] | Miyake H, Siviloglou G A, Kennedy C J, Burton W C and Ketterle W 2013 Phys. Rev. Lett. 111 185302 | | [71] | Aidelsburger M, Atala M, Lohse M, Barreiro J T, Paredes B and Bloch I 2013 Phys. Rev. Lett. 111 185301 | | [72] | Aidelsburger M, Atala M, Nascimbène S, Trotzky S, Chen Y A and Bloch I 2011 Phys. Rev. Lett. 107 255301 | | [73] | Wang S T, Deng D L and Duan L M 2014 Phys. Rev. Lett. 113 033002 | | [74] | Zheng W and Zhai H 2014 Phys. Rev. A 89 061603 | | [75] | Goldman N and Dalibard J 2014 Phys. Rev. X 4 031027 | | [76] | Hauke P, Tieleman O, Celi A, Ölschläger C, Simonet J, Struck J, Weinberg M, Windpassinger P, Sengstock K, Lewenstein M, 6 and Eckardt A 2012 Phys. Rev. Lett. 109 145301 | | [77] | Struck J, Ölschläger C, Weinberg M, Hauke P, Simonet J, Eckardt A, Lewenstein M, Sengstock K and Windpassinger P 2012 Phys. Rev. Lett. 108 225304 | | [78] | Baur S K, Schleier-Smith M H and Cooper N R 2014 Phys. Rev. A 89 051605 | | [79] | Wang C, Zhang P, Chen X, Yu J and Zhai H 2017 Phys. Rev. Lett. 118 185701 | | [80] | Creffield C E and Sols F 2014 Phys. Rev. A 90 023636 | | [81] | Jungwirth T, Wunderlich J, Novák V, Olejnik K, Gallagher B, Campion R, Edmonds K, Rushforth A, Ferguson A and Němec P 2014 Rev. Mod. Phys. 86 855 |
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