Advertisement

Nonlinearity and Topology

  • Avadh SaxenaEmail author
  • Panayotis G. Kevrekidis
  • Jesús Cuevas-Maraver
Chapter
  • 186 Downloads
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 32)

Abstract

The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose–Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons, hopfions, monopoles to name just a few. The interaction among and collision of these nontrivial defects itself constitute topics of significant interest. Curvature and underlying geometry also affect the shape, interaction and behavior of these defects. Such properties can be studied using techniques such as, e.g. the Bogomolnyi decomposition. Some applications of this interplay, e.g. in nonreciprocal photonics as well as topological materials such as Dirac and Weyl semimetals, are also elucidated.

Notes

Acknowledgements

A.S. and P.G.K. acknowledge the support of the U.S. Department of Energy. Specifically, LANL is operated by Triad National Security, L.L.C. for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 892333218NCA000001. J.C.M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1809074 (P.G.K.). Finally, P.G.K. gratefully acknowledges the support of the Leverhulme Trust towards a visiting fellowship at the University of Oxford and the kind hospitality of the Mathematical Institute of the University of Oxford.

References

  1. 1.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    M. Sato, Y. Ando, Rep. Prog. Phys. 80, 076501 (2017)ADSCrossRefGoogle Scholar
  3. 3.
    N.P. Armitage, E.J. Mele, A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018)ADSCrossRefGoogle Scholar
  4. 4.
    J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Kumar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B.A. Bernevig, C. Felser, Nature 575, 315 (2019)ADSCrossRefGoogle Scholar
  5. 5.
    D.J.E. Marsh, K.C. Fong, E.W. Lentz, L. Smejkal, M.N. Ali, Phys. Rev. Lett. 123, 121601 (2019)ADSCrossRefGoogle Scholar
  6. 6.
    S. Gupta, A. Saxena (eds.), The Role of Topology in Materials. Springer Series in Solid-State Sciences, vol. 189 (2018)zbMATHGoogle Scholar
  7. 7.
    W. Chen, D. Leykam, Y.D. Chong, L. Yang, MRS Bull. 43, 443 (2018)CrossRefGoogle Scholar
  8. 8.
    E.B. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976)Google Scholar
  9. 9.
    N. Manton, P. Sutcliffe, Topological Solitons (Cambridge University Press, Cambridge, 2004)zbMATHCrossRefGoogle Scholar
  10. 10.
    Y.M. Shnir, Topological and Non-Topological Solitons in Scalar Field Theories (Cambridge University Press, Cambridge, 2018)zbMATHCrossRefGoogle Scholar
  11. 11.
    Y.X. Zhao, Z.D. Wang, Phys. Rev. Lett. 114, 206602 (2015)ADSCrossRefGoogle Scholar
  12. 12.
    K. Koniaris, M. Muthukumar, Phys. Rev. Lett. 66, 2211 (1991)ADSCrossRefGoogle Scholar
  13. 13.
    S. Das, Y.L. Tang, Z. Hong, M.A.P. Gonçalves, M.R. McCarter, C. Klewe, K.X. Nguyen, F. Gómez-Ortiz, P. Shafer, E. Arenholz, V.A. Stoica, S.-L. Hsu, B. Wang, C. Ophus, J.F. Liu, C.T. Nelson, S. Saremi, B. Prasad, A.B. Mei, D.G. Schlom, J. Íñiguez, P. García-Fernández, D.A. Muller, L.Q. Chen, J. Junquera, L.W. Martin, R. Ramesh, Nature 568, 368 (2019)ADSCrossRefGoogle Scholar
  14. 14.
    Y. Nahas, S. Prokhorenko, L. Louis, Z. Gui, I. Kornev, L. Bellaiche, Nat. Commun. 6, 8542 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    N. Nagaosa, Y. Tokura, Nat. Nanotechnol. 8, 899 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    I. Kézsmŕki, S. Bordács, P. Milde, E. Neuber, L.M. Eng, J.S. White, H.M. Rønnow, C.D. Dewhurst, M. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, A. Loidl, Nat. Mater. 14, 1116 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    A.K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U.K. Rößler, C. Felser, S.S.P. Parkinet, Nature 548, 561 (2017)ADSCrossRefGoogle Scholar
  18. 18.
    T. Van Mechelen, Z. Jacob, Opt. Mater. Express 9, 95 (2019)ADSCrossRefGoogle Scholar
  19. 19.
    A. Duzgun, J.V. Selinger, A. Saxena, Phys. Rev. E 97, 062706 (2018)ADSCrossRefGoogle Scholar
  20. 20.
    S. Wintz, C. Bunce, A. Neudert, M. Körner, T. Strache, M. Buhl, A. Erbe, S. Gemming, J. Raabe, C. Quitmann, J. Fassbender, Phys. Rev. Lett. 110, 177201 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    S.-Z. Lin, A. Saxena, C.D. Batista, Phys. Rev. B 91, 224407 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    X.Z. Yu, W. Koshibae, Y. Tokunaga, K. Shibata, Y. Taguchi, N. Nagaosa, Y. Tokura, Nature 564, 95 (2018)ADSCrossRefGoogle Scholar
  23. 23.
    D. Foster, C. Kind, P.J. Ackerman, J.-S.B. Tai, M.R. Dennis, I.I. Smalyukh, Nat. Phys. 15, 655 (2019)CrossRefGoogle Scholar
  24. 24.
    B.G.-G. Chen, P.J. Ackerman, G.P. Alexander, R.D. Kamien, I.I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    P.J. Ackerman, I.I. Smalyukh, Nat. Mater. 16, 426 (2017)ADSCrossRefGoogle Scholar
  26. 26.
    I.I. Smalyukh, Y. Lansac, N.A. Clark, R.P. Trivedi, Nat. Mater. 9, 139 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    H.R.O. Sohn, C.D. Liu, Y. Wang, I.I. Smalyukh, Opt. Express 27, 29055 (2019)ADSCrossRefGoogle Scholar
  28. 28.
    P. Sutcliffe, J. Phys. A: Math. Theor. 51, 375401 (2018)CrossRefGoogle Scholar
  29. 29.
    M. Kobayashi, M. Nitta, Phys. Lett. B 728, 314 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    C. Phatak, A.K. Petford-Long, O. Heinonen, M. Tanase, M. De Graef, Phys. Rev. B 83, 174431 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    C. Nisoli, R. Moessner, P. Schiffer, Rev. Mod. Phys. 85, 1473 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    P. Milde, D. Köhler, J. Seidel, L.M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. Mühlbauer, C. Pfleiderer, S. Buhrandt, C. Schütte, A. Rosch, Science 340, 1076 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    M.W. Ray, E. Ruokokoski, K. Tiurev, M. Möttönen, D.S. Hall, Science 348, 544 (2015)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    T. Ollikainen, K. Tiurev, A. Blinova, W. Lee, D.S. Hall, M. Möttönen, Phys. Rev. X 7, 021023 (2017)Google Scholar
  35. 35.
    M. Charilaou, H.B. Braun, J.F. Löffler, Phys. Rev. Lett. 121, 097202 (2018)ADSCrossRefGoogle Scholar
  36. 36.
    F. Kagawa, H. Oike, W. Koshibae, A. Kikkawa, Y. Okamura, Y. Taguchi, N. Nagaosa, Y. Tokura, Nat. Commun. 8, 1332 (2017)ADSCrossRefGoogle Scholar
  37. 37.
    E.H. Fyhn, J. Linder, Phys. Rev. B 100, 214503 (2019)ADSCrossRefGoogle Scholar
  38. 38.
    D. Kleckner, W.T.M. Irvine, Nat. Phys. 9, 253 (2013)Google Scholar
  39. 39.
    L.M. Pismen, Vortices in Nonlinear Fields (Clarendon Press, Oxford, 1999)zbMATHGoogle Scholar
  40. 40.
    Yu.S. Kivshar, B. Luther-Davies, Phys. Rep. 298, 81 (1998)Google Scholar
  41. 41.
    Yu.S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, L. Pismen, Opt. Commun. 152, 198 (1998)Google Scholar
  42. 42.
    A. López Ariste, R. Centeno, E. Khomenko, Astron. Astrophys. 591, A63 (2016)Google Scholar
  43. 43.
    H.J. Lugt, Vortex Flow in Nature and Technology (John Wiley and Sons, New York, 1983)Google Scholar
  44. 44.
    D.L. Whitaker, J. Edwards, Science 329, 406 (2010)ADSCrossRefGoogle Scholar
  45. 45.
    H. Aref, J. Math. Phys. 48, 065401 (2007)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    P.K. Newton, The N-Vortex Problem: Analytical Techniques. Applied Mathematical Sciences, vol. 145 (Springer, Berlin, 2001)zbMATHCrossRefGoogle Scholar
  47. 47.
    D. Durkin, J. Fajans, Phys. Fluids 12, 289 (2000)ADSCrossRefGoogle Scholar
  48. 48.
    B. Grzybowski, H. Stone, G. Whitesides, Nature 405, 1033 (2000)ADSCrossRefGoogle Scholar
  49. 49.
    B. Grzybowski, H. Stone, G. Whitesides, Proc. Natl, Acad. Sci. USA 99, 4147 (2002)Google Scholar
  50. 50.
    F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999)ADSCrossRefGoogle Scholar
  51. 51.
    C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)Google Scholar
  52. 52.
    L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003)zbMATHGoogle Scholar
  53. 53.
    A. Griffin, D.W. Snoke, S. Stringari (eds.), Bose-Einstein Condensation (Cambridge University Press, Cambridge, 1995)Google Scholar
  54. 54.
    M.H.J. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, Science 269, 198 (1995)ADSCrossRefGoogle Scholar
  55. 55.
    K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)ADSCrossRefGoogle Scholar
  56. 56.
    C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995)ADSCrossRefGoogle Scholar
  57. 57.
    E.A. Cornell, C.E. Wieman, Rev. Mod. Phys. 74, 875 (2002)ADSCrossRefGoogle Scholar
  58. 58.
    W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002)ADSCrossRefGoogle Scholar
  59. 59.
    A.A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957)Google Scholar
  60. 60.
    J.M. Kosterlitz, D.J. Thouless, J. Phys. C: Solid State Phys. 6, 1181 (1973)ADSCrossRefGoogle Scholar
  61. 61.
    Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, J. Dalibard, Nature 441, 1118 (2006)ADSCrossRefGoogle Scholar
  62. 62.
    A. Negretti, C. Henkel, J. Phys. B: At. Mol. Opt. Phys. 37, L385 (2004)ADSCrossRefGoogle Scholar
  63. 63.
    M. Kasevich, Atom systems and Bose Einstein Condensates for metrology and navigation. Presented at the NASA Quantum Future Technologies Conference, 17–21 Jan 2012. https://web.archive.org/web/20161118024357/http://quantum.nasa.gov/materials/2012-01-18-B1-Kasevich.pdf. Accessed 27 Jan 2020Google Scholar
  64. 64.
    T.R. Slatyer, C.M. Savage, Classical Quant. Grav. 22, 3833 (2005)ADSCrossRefGoogle Scholar
  65. 65.
    P. Das Gupta, E. Thareja, Classical Quant. Grav. 34, 035006 (2017)Google Scholar
  66. 66.
    S. Eckel, A. Kumar, T. Jacobson, I.B. Spielman, G.K. Campbell, Phys. Rev. X 8, 021021 (2018)Google Scholar
  67. 67.
    M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman, E.A. Cornell, Phys. Rev. Lett. 83, 2498 (1999)ADSCrossRefGoogle Scholar
  68. 68.
    K.W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Phys. Rev. Lett. 84, 806 (2000)ADSCrossRefGoogle Scholar
  69. 69.
    J.E. Williams, M.J. Holland, Nature 401, 568 (1999)ADSCrossRefGoogle Scholar
  70. 70.
    A. Recati, F. Zambelli, S. Stringari, Phys. Rev. Lett. 86, 377 (2001)ADSCrossRefGoogle Scholar
  71. 71.
    S. Sinha, Y. Castin, Phys. Rev. Lett. 87, 190402 (2001)ADSCrossRefGoogle Scholar
  72. 72.
    K.W. Madison, F. Chevy, V. Bretin, J. Dalibard, Phys. Rev. Lett. 86, 4443 (2001)ADSCrossRefGoogle Scholar
  73. 73.
    C. Raman, J.R. Abo-Shaeer, J.M. Vogels, K. Xu, W. Ketterle, Phys. Rev. Lett. 87, 210402 (2001)ADSCrossRefGoogle Scholar
  74. 74.
    J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Science 292, 476 (2001)ADSCrossRefGoogle Scholar
  75. 75.
    R. Onofrio, C. Raman, J.M. Vogels, J.R. Abo-Shaeer, A.P. Chikkatur, W. Ketterle, Phys. Rev. Lett. 85, 2228 (2000)ADSCrossRefGoogle Scholar
  76. 76.
    D.R. Scherer, C.N. Weiler, T.W. Neely, B.P. Anderson, Phys. Rev. Lett. 98, 110402 (2007)ADSCrossRefGoogle Scholar
  77. 77.
    A.E. Leanhardt, A. Görlitz, A.P. Chikkatur, D. Kielpinski, Y. Shin, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 89, 190403 (2002)ADSCrossRefGoogle Scholar
  78. 78.
    Y. Shin, M. Saba, M. Vengalattore, T.A. Pasquini, C. Sanner, A.E. Leanhardt, M. Prentiss, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 93, 160406 (2004)ADSCrossRefGoogle Scholar
  79. 79.
    C.N. Weiler, T.W. Neely, D.R. Scherer, A.S. Bradley, M.J. Davis, B.P. Anderson, Nature 455, 948 (2008)ADSCrossRefGoogle Scholar
  80. 80.
    D.V. Freilich, D.M. Bianchi, A.M. Kaufman, T.K. Langin, D.S. Hall, Science 329, 1182 (2010)ADSCrossRefGoogle Scholar
  81. 81.
    S. Middelkamp, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, P. Schmelcher, D.V. Freilich, D.S. Hall, Phys. Rev. A 84, 011605(R) (2011)ADSCrossRefGoogle Scholar
  82. 82.
    P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, P. Schmelcher, D.S. Hall, Phys. Lett. A 375, 3044 (2011)ADSCrossRefGoogle Scholar
  83. 83.
    T.W. Neely, E.C. Samson, A.S. Bradley, M.J. Davis, B.P. Anderson, Phys. Rev. Lett. 104, 160401 (2010)ADSCrossRefGoogle Scholar
  84. 84.
    R. Navarro, R. Carretero-González, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, M.W. Ray, E. Altuntaş, D.S. Hall, Phys. Rev. Lett. 110, 225301 (2013)ADSCrossRefGoogle Scholar
  85. 85.
    A.V. Zampetaki, R. Carretero-González, P.G. Kevrekidis, F.K. Diakonos, D.J. Frantzeskakis, Phys. Rev. E 88, 042914 (2014)ADSCrossRefGoogle Scholar
  86. 86.
    J.A. Seman, E.A.L. Henn, M. Haque, R.F. Shiozaki, E.R.F. Ramos, M. Caracanhas, P. Castilho, C. Castelo Branco, P.E.S. Tavares, F.J. Poveda-Cuevas, G. Roati, K.M.F. Magalhaes, V.S. Bagnato, Phys. Rev. A 82, 033616 (2010)Google Scholar
  87. 87.
    S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, P. Schmelcher, Phys. Rev. A 82, 013646 (2010)ADSCrossRefGoogle Scholar
  88. 88.
    V. Koukouloyannis, G. Voyatzis, P.G. Kevrekidis, Phys. Rev. E 89, 042905 (2014)ADSCrossRefGoogle Scholar
  89. 89.
    N. Kyriakopoulos, V. Koukouloyannis, C. Skokos, P.G. Kevrekidis, Chaos 24, 024410 (2014)ADSMathSciNetCrossRefGoogle Scholar
  90. 90.
    P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, Emergent Nonlinear Phenomena in Bose-Einstein Condensates (Springer, Berlin, 2008)zbMATHCrossRefGoogle Scholar
  91. 91.
    A.L. Fetter, A.A. Svidzinsky J. Phys. Condens. Mat. 13, R135 (2001)Google Scholar
  92. 92.
    A.L. Fetter, Rev. Mod. Phys. 81, 647 (2009)ADSCrossRefGoogle Scholar
  93. 93.
    T. Kolokolnikov, P.G. Kevrekidis, R. Carretero-González, Proc. Roy. Soc. Lond. A 470, 20140048 (2014)ADSCrossRefGoogle Scholar
  94. 94.
    A. Aftalion, I. Danaila, Phys. Rev. A 68, 023603 (2003)ADSCrossRefGoogle Scholar
  95. 95.
    S. Komineas, N.R. Cooper, N. Papanicolaou, Phys. Rev. A 72, 053624 (2005)ADSCrossRefGoogle Scholar
  96. 96.
    T.P. Simula, Phys. Rev. A 84, 021603(R) (2011)ADSCrossRefGoogle Scholar
  97. 97.
    R.J. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991)Google Scholar
  98. 98.
    A.S. Rodrigues, P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, P. Schmelcher, T.J. Alexander, Yu.S. Kivshar, Phys. Rev. A 79, 043603 (2009)Google Scholar
  99. 99.
    R. Carretero-González, N. Whitaker, P.G. Kevrekidis, D.J. Frantzeskakis, Phys. Rev. A 77, 023605 (2008)ADSCrossRefGoogle Scholar
  100. 100.
    R. Carretero-González, B.P. Anderson, P.G. Kevrekidis, D.J. Frantzeskakis, C.N. Weiler, Phys. Rev. A 77, 033625 (2008)ADSCrossRefGoogle Scholar
  101. 101.
    J. Ruostekoski, Z. Dutton, Phys. Rev. A 72, 063626 (2005)ADSCrossRefGoogle Scholar
  102. 102.
    B.P. Anderson, P.C. Haljan, C.A. Regal, D.L. Feder, L.A. Collins, C.W. Clark, E.A. Cornell, Phys. Rev. Lett. 86, 2926 (2001)ADSCrossRefGoogle Scholar
  103. 103.
    I. Shomroni, E. Lahoud, S. Levy, J. Steinhauer, Nat. Phys. 5, 193 (2009)CrossRefGoogle Scholar
  104. 104.
    K. Sasaki, N. Suzuki, H. Saito, Phys. Rev. A 83, 033602 (2011)ADSCrossRefGoogle Scholar
  105. 105.
    P.H. Roberts, J. Grant, J. Phys. A: Gen. Phys. 4, 55 (1971)ADSCrossRefGoogle Scholar
  106. 106.
    M. Abad, M. Guilleumas, R. Mayol, M. Pi, Laser Phys. 18, 648 (2008)ADSCrossRefGoogle Scholar
  107. 107.
    M. Guilleumas, D.M. Jezek, R. Mayol, M. Pi, M. Barranco, Phys. Rev. A 65, 053609 (2002)ADSCrossRefGoogle Scholar
  108. 108.
    A.L. Fetter, Phys. Rev. A 69, 043617 (2004)ADSCrossRefGoogle Scholar
  109. 109.
    T.P. Simula, T. Mizushima, K. Machida, Phys. Rev. A 78, 053604 (2008)ADSCrossRefGoogle Scholar
  110. 110.
    T.-L. Horng, S.-C. Gou, T.-C. Lin, Phys. Rev. A 74, 041603(R) (2006)ADSCrossRefGoogle Scholar
  111. 111.
    F. Chevy, S. Stringari, Phys. Rev. A 68, 053601 (2003)ADSCrossRefGoogle Scholar
  112. 112.
    J.J. García-Ripoll, V.M. Pérez-García, Phys. Rev. A 64, 053611 (2001)ADSCrossRefGoogle Scholar
  113. 113.
    F. Maggioni, S. Alamri, C.F. Barenghi, R.L. Ricca, Phys. Rev. E 82, 026309 (2010)ADSMathSciNetCrossRefGoogle Scholar
  114. 114.
    J.L. Helm, C.F. Barenghi, A.J. Youd, Phys. Rev. A 83, 045601 (2011)ADSCrossRefGoogle Scholar
  115. 115.
    S. Lin, J.-X. Zhu, A. Saxena, Phys. Rev. B 99, 140408(R) (2019)ADSCrossRefGoogle Scholar
  116. 116.
    K. Sasaki, N. Suzuki, H. Saito, Phys. Rev. A 83, 053606 (2011)ADSCrossRefGoogle Scholar
  117. 117.
    P.G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, 1992)zbMATHGoogle Scholar
  118. 118.
    R.N. Bisset, W. Wang, C. Ticknor, R. Carretero-González, D.J. Frantzeskakis, L.A. Collins, P.G. Kevrekidis, Phys. Rev. A 92, 043601 (2015)ADSCrossRefGoogle Scholar
  119. 119.
    C. Ticknor, W. Wang, P.G. Kevrekidis, Phys. Rev. A 98, 033609 (2018)ADSCrossRefGoogle Scholar
  120. 120.
    W. Wang, R.N. Bisset, C. Ticknor, R. Carretero-González, D.J. Frantzeskakis, L.A. Collins, P.G. Kevrekidis, Phys. Rev. A 95, 043638 (2017)ADSCrossRefGoogle Scholar
  121. 121.
    R.M. Caplan, J.D. Talley, R. Carretero-González, P.G. Kevrekidis, Phys. Fluids 26, 097101 (2014)ADSCrossRefGoogle Scholar
  122. 122.
    J. Ruostekoski, J.R. Anglin, Phys. Rev. Lett. 86, 3934 (2001)ADSCrossRefGoogle Scholar
  123. 123.
    C.M. Savage, J. Ruostekoski, Phys. Rev. Lett. 91, 010403 (2003)ADSCrossRefGoogle Scholar
  124. 124.
    J. Choi, W.J. Kwon, Y. Shin, Phys. Rev. Lett. 108, 035301 (2012)ADSCrossRefGoogle Scholar
  125. 125.
    L.S. Leslie, A. Hansen, K.C. Wright, B.M. Deutsch, N.P. Bigelow, Phys. Rev. Lett. 103, 250401 (2009)ADSCrossRefGoogle Scholar
  126. 126.
    J. Lovegrove, M.O. Borgh, J. Ruostekoski, Phys. Rev. Lett. 112, 075301 (2014)ADSCrossRefGoogle Scholar
  127. 127.
    Y.V. Kartashov, B.A. Malomed, Y. Shnir, L. Torner, Phys. Rev. Lett. 113, 264101 (2014)ADSCrossRefGoogle Scholar
  128. 128.
    Y.M. Bidasyuk, A.M. Chumachenko, O.O. Prikhodko, S.I. Vilchinskii, M. Weyrauch, A.I. Yakimenko, Phys. Rev. A 92, 053603 (2015)ADSCrossRefGoogle Scholar
  129. 129.
    R.N. Bisset, W. Wang, C. Ticknor, R. Carretero-González, D.J. Frantzeskakis, L.A. Collins, P.G. Kevrekidis, Phys. Rev. A 92, 063611 (2015)MathSciNetCrossRefGoogle Scholar
  130. 130.
    D. Proment, M. Onorato, C.F. Barenghi, Phys. Rev. E 85, 036306 (2012)ADSCrossRefGoogle Scholar
  131. 131.
    D. Proment, M. Onorato, C.F. Barenghi, J. Phys.: Conf. Ser. 544, 012022 (2014)Google Scholar
  132. 132.
    D. Kleckner, L.H. Kauffman, W.T.M. Irvine, Nat. Phys. 12, 650 (2016)CrossRefGoogle Scholar
  133. 133.
    The Knot Atlas, http://katlas.org. Accessed 27 Jan 2020
  134. 134.
    V.P. Ruban, JETP Lett. 107, 307 (2018)ADSCrossRefGoogle Scholar
  135. 135.
    V.P. Ruban, JETP Lett. 108, 605 (2018)ADSCrossRefGoogle Scholar
  136. 136.
    C. Ticknor, V.P. Ruban, P.G. Kevrekidis, Phys. Rev. A 99, 063604 (2019)ADSCrossRefGoogle Scholar
  137. 137.
    R. Dandoloff, S. Villain-Guillot, A. Saxena, A.R. Bishop, Phys. Rev. Lett. 74, 813 (1995)ADSCrossRefGoogle Scholar
  138. 138.
    A.A. Belavin, A.M. Polyakov, JETP Lett. 22, 245 (1975)ADSGoogle Scholar
  139. 139.
    S. Villain-Guillot, R. Dandoloff, A. Saxena, Phys. Lett. A 188, 343 (1994)Google Scholar
  140. 140.
    A. Saxena, R. Dandoloff, Phys. Rev. B 58, R563 (1998)ADSCrossRefGoogle Scholar
  141. 141.
    A. Saxena, R. Dandoloff, Phys. Rev. B 66, 104414 (2002)ADSCrossRefGoogle Scholar
  142. 142.
    J. Benoit, A. Saxena, T. Lookman, J. Phys. A: Math. Gen. 34, 9417 (2001)ADSCrossRefGoogle Scholar
  143. 143.
    F.C. Marques, A. Neves, Annals Math. 179, 683 (2014)MathSciNetCrossRefGoogle Scholar
  144. 144.
    T.O. Wehling, A.M. Black-Schaffer, A.V. Balatsky, Adv. Phys. 63, 1 (2014)ADSCrossRefGoogle Scholar
  145. 145.
    X.L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)ADSCrossRefGoogle Scholar
  146. 146.
    Y. Ando, L. Fu, Annu. Rev. Condens. Matter Phys. 6, 361 (2015)ADSCrossRefGoogle Scholar
  147. 147.
    O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014)ADSCrossRefGoogle Scholar
  148. 148.
    N. Sirica, R.I. Tobey, L.X. Zhao, G.F. Chen, B. Xu, R. Yang, B. Shen, D.A. Yarotski, P. Bowlan, S.A. Trugman, J.-X. Zhu, Y.M. Dai, A.K. Azad, N. Ni, X.G. Qiu, A.J. Taylor, R.P. Prasankumar, Phys. Rev. Lett. 122, 197401 (2019)ADSCrossRefGoogle Scholar
  149. 149.
    F. Cooper, A. Khare, B. Mihaila, A. Saxena, Phys. Rev. E 82, 036604 (2010)ADSMathSciNetCrossRefGoogle Scholar
  150. 150.
    J. Cuevas-Maraver, P.G. Kevrekidis, F.G. Mertens, A. Saxena, Phys. Rev. E 100, 022210 (2019)ADSMathSciNetCrossRefGoogle Scholar
  151. 151.
    J. Cuevas-Maraver, P.G. Kevrekidis, A. Saxena, A. Comech, R.M. Lan, Phys. Rev. Lett. 116, 214101 (2016)ADSMathSciNetCrossRefGoogle Scholar
  152. 152.
    J. Cuevas-Maraver, N. Boussaïd, A. Comech, R. Lan, P.G. Kevrekidis, A. Saxena, in Nonlinear Systems, vol. 1, ed. by V. Carmona, J. Cuevas-Maraver, F. Fernández-Sánchez, E. García-Medina (Springer, Cham, 2018), p. 89Google Scholar
  153. 153.
    S. Seki, X.Z. Yu, S. Ishiwata, Y. Tokura, Science 336, 198 (2012)ADSCrossRefGoogle Scholar
  154. 154.
    C. Phatak, O. Heinonen, M. De Graef, A. Petford-Long, Nano Lett. 16, 4141 (2016)ADSCrossRefGoogle Scholar
  155. 155.
    J. Chen, L. Wang, M. Zhang, L. Zhou, L. Jin, X. Wang, H. Qin, Y. Qiu, J. Mei, F. Ye, B. Xi, H. He, B. Li, G. Wang, Nano Lett. 19, 6144 (2019)ADSCrossRefGoogle Scholar
  156. 156.
    X. Wan, A.M. Turner, A. Vishwanath, S.Y. Savrasov, Phys. Rev. B 83, 205101 (2011)ADSCrossRefGoogle Scholar
  157. 157.
    R.-J. Slager, J. Phys. Chem. Solids 128, 24 (2019)ADSCrossRefGoogle Scholar
  158. 158.
    L. Bi, MRS Bull. 43, 408 (2018)CrossRefGoogle Scholar
  159. 159.
    X. Zhou, Y. Wang, D. Leykam, Y.D. Chong, New J. Phys. 19, 095002 (2017)ADSCrossRefGoogle Scholar
  160. 160.
    Z. Wang, Y. Chong, J.D. Joannopoulos, M. Soljacic, Nature 461, 772 (2009)ADSCrossRefGoogle Scholar
  161. 161.
    W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22, 2099 (1980)ADSCrossRefGoogle Scholar
  162. 162.
    Y. Hadad, V. Vitelli, A. Alu, ACS Photonics 4, 1974 (2017)CrossRefGoogle Scholar
  163. 163.
    R.K. Pal, M. Schaeffer, M. Ruzzene, J. Appl. Phys. 199, 084305 (2016)ADSCrossRefGoogle Scholar
  164. 164.
    R. Süsstrunk, S.D. Huber, Science 349, 47 (2015)ADSCrossRefGoogle Scholar
  165. 165.
    L.M. Nash, D. Kleckner, A. Read, V. Vitelli, A.M. Turner, W.T.M. Irvine, Proc. Natl, Acad. Sci. USA 112, 14495 (2015)Google Scholar
  166. 166.
    S.H. Mousavi, A.B. Khanikaev, Z. Wang, Nat. Commun. 6, 8682 (2015)ADSCrossRefGoogle Scholar
  167. 167.
    P. Wang, L. Lu, K. Bertoldi, Phys. Rev. Lett. 115, 104302 (2015)ADSCrossRefGoogle Scholar
  168. 168.
    S.D. Huber, Nat. Phys. 12, 621 (2016)CrossRefGoogle Scholar
  169. 169.
    C.L. Kane, T.C. Lubensky, Nat. Phys. 10, 39 (2013)CrossRefGoogle Scholar
  170. 170.
    R. Süsstrunk, S.D. Huber, Proc. Natl, Acad. Sci. USA 113, E4767 (2016)Google Scholar
  171. 171.
    D.D. Snee, Y.-P. Ma, Extreme Mech. Lett. 30, 100487 (2019)CrossRefGoogle Scholar
  172. 172.
    B.G. Chen, N. Upadhyaya, V. Vitelli, Proc. Natl. Acad. Sci. U. S. A. 111, 13004 (2014)Google Scholar
  173. 173.
    B. Deng, P. Wang, Q. He, V. Tournat, K. Bertoldi, Nat. Commun. 9, 3410 (2018)ADSCrossRefGoogle Scholar
  174. 174.
    R.K. Pal, J. Vila, M. Leamy, M. Ruzzene, Phys. Rev. E 97, 032209 (2018)ADSCrossRefGoogle Scholar
  175. 175.
    R. Chaunsali, E. Kim, A. Thakkar, P.G. Kevrekidis, J. Yang, Phys. Rev. Lett. 119, 024301 (2017)ADSCrossRefGoogle Scholar
  176. 176.
    R. Chaunsali, G. Theocharis, Phys. Rev. B 100, 014302 (2019)ADSCrossRefGoogle Scholar
  177. 177.
    T. Zhao, A.S. Ramos Jr., G.H. Paulino, Int. J. Numer. Meth. Eng. 118, 804 (2019)CrossRefGoogle Scholar
  178. 178.
    M. Abdi, I. Ashcroft, R. Wildman, Eng. Optimiz. 50, 1850 (2018)CrossRefGoogle Scholar
  179. 179.
    B. Senyuk, Q. Liu, S. He, R.D. Kamien, R.B. Kusner, T.C. Lubensky, I.I. Smalyukh, Nature 493, 200 (2013)ADSCrossRefGoogle Scholar
  180. 180.
    A. Biuhon, A.M. Black-Schaffer, R.-J. Slager, Phys. Rev. B 100, 195135 (2019)ADSCrossRefGoogle Scholar
  181. 181.
    J.O. Island, X. Cui, C. Lewandowski, J.Y. Khoo, E.M. Spanton, H. Zhou, D. Rhodes, J.C. Hone, T. Taniguchi, K. Watanabe, L.S. Levitov, M.P. Zaletel, A.F. Young, Nature 571, 85 (2019)ADSCrossRefGoogle Scholar
  182. 182.
    Z. Song, L. Elcoro, N. Regnault, B.A. Bernevig, Fragile phases as affine monoids: full classification and material examples. arXiv:1905.03262
  183. 183.
    D. Castelvecchi, Nature 571, 17 (2019)ADSCrossRefGoogle Scholar
  184. 184.
    M. Karoubi, K-theory. An introduction. Classics in Mathematics (Springer, Berlin Heidelberg, 1978). arXiv:math/0602082zbMATHCrossRefGoogle Scholar
  185. 185.
    S. Das Sarma, M. Freedman, C. Nayak, NPJ Quantum Inform. 1, 15001 (2015)Google Scholar
  186. 186.
    J.-J. Feng, Z. Huang, Z. Wang, Q. Niu, Phys. Rev. B 98, 13415 (2018)CrossRefGoogle Scholar
  187. 187.
    C.E. Bardyn, A. Imamoglu, Phys. Rev. Lett. 109, 253606 (2012)ADSCrossRefGoogle Scholar
  188. 188.
    I.C. Christov, R.J. Decker, A. Demirkaya, V.A. Gani, P.G. Kevrekidis, A. Khare, A. Saxena, Phys. Rev. Lett. 122, 171601 (2019)ADSCrossRefGoogle Scholar
  189. 189.
    F. Wilczek, Phys. Rev. Lett. 109, 160401 (2012)ADSCrossRefGoogle Scholar
  190. 190.
    K. Nakatsugawa, T. Fujii, A. Saxena, S. Tanda, J. Phys. A: Math. Theor. 53, 025301 (2020)ADSCrossRefGoogle Scholar
  191. 191.
    E. Babaev, L.D. Faddeev, A.J. Niemi, Phys. Rev. B 65, 100512(R) (2001)Google Scholar
  192. 192.
    F.N. Rybakov, J. Garaud, E. Babaev, Phys. Rev. B 100, 094515 (2019)Google Scholar
  193. 193.
    K. Sokalski, L.T. Stepien, D. Sokalska, J. Phys. A: Math. Gen. 35, 6157 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  194. 194.
    L.T. Stepien, J. Phys. A: Math. Theor. 49, 175202 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Avadh Saxena
    • 1
    Email author
  • Panayotis G. Kevrekidis
    • 2
  • Jesús Cuevas-Maraver
    • 3
    • 4
  1. 1.Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  2. 2.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  3. 3.Departamento de Fisica Aplicada I, Grupo de Fisica No LinealUniversidad de SevillaSevillaSpain
  4. 4.Instituto de Matemáticas de la Universidad de Sevilla (IMUS)Edificio Celestino MutisSevillaSpain

Personalised recommendations