- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Letter
  • Open Access

Gravitational wave analogs in spin nematics and cold atoms

Leilee Chojnacki1,*, Rico Pohle2,3, Han Yan4,5,6, Yutaka Akagi7, and Nic Shannon1

  • *leilee.chojnacki@oist.jp
PDF

Phys. Rev. B 109, L220407 – Published 14 June, 2024

DOI: https://doi.org/10.1103/PhysRevB.109.L220407

Abstract

Large-scale gravitational phenomena are famously difficult to observe, making parallels in condensed matter physics a valuable resource. Here we show how spin nematic phases, found in magnets and cold atoms, can provide an analog to linearized gravity. In particular, we show that the Goldstone modes of these systems are massless spin-2 bosons, in one-to-one correspondence with quantized gravitational waves in flat spacetime. We identify a spin-1 model supporting these excitations and, using simulation, outline a procedure for their observation in a 23Na spinor condensate.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (85)

  1. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  2. G. E. Volovik, Simulation of quantum field theory and gravity in superfluid He3, Low Temp. Phys. 24, 127 (1998).
  3. G. Volovik, Superfluid He3-B and gravity, Phys. B: Condens. Matter 162, 222 (1990).
  4. G. E. Volovik, The Universe in a Helium Droplet, repr ed., International series of monographs on physics No. 117 (Oxford University Press, Oxford, 2009).
  5. W. G. Unruh, Sonic analogue of black holes and the effects of high frequencies on black hole evaporation, Phys. Rev. D 51, 2827 (1995).
  6. S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Measurement of stimulated Hawking emission in an analogue system, Phys. Rev. Lett. 106, 021302 (2011).
  7. G. E. Volovik, Black hole and hawking radiation by type-II Weyl fermions, JETP Lett. 104, 645 (2016).
  8. H. Liu, J.-T. Sun, C. Song, H. Huang, F. Liu, and S. Meng, Fermionic analogue of high temperature Hawking radiation in black phosphorus, Chin. Phys. Lett. 37, 067101 (2020).
  9. M. Hotta, Y. Nambu, Y. Sugiyama, K. Yamamoto, and G. Yusa, Expanding edges of quantum Hall systems in a cosmology language: Hawking radiation from de Sitter horizon in edge modes, Phys. Rev. D 105, 105009 (2022).
  10. Y. Rosenberg, Optical analogues of black-hole horizons, Philos. Trans. R. Soc. London A 378, 20190232 (2020).
  11. J. Petty and F. König, Optical analogue gravity physics: resonant radiation, Philos. Trans. R. Soc. London A 378, 20190231 (2020).
  12. T. Bravo, C. Sabín, and I. Fuentes, Analog quantum simulation of gravitational waves in a Bose-Einstein condensate, EPJ Quantum Technology 2, 3 (2015).
  13. C. Sabín, J. Kohlrus, D. E. Bruschi, and I. Fuentes, Thermal noise in BEC-phononic gravitational wave detectors, EPJ Quantum Technology 3, 8 (2016).
  14. C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, Á. Parra-López, M. Tolosa-Simeón, N. Sánchez-Kuntz, T. Haas, H. Strobel, S. Floerchinger, and M. K. Oberthaler, Quantum field simulator for dynamics in curved spacetime, Nature (London) 611, 260 (2022).
  15. M. Tolosa-Simeón, Á. Parra-López, N. Sánchez-Kuntz, T. Haas, C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, H. Strobel, M. K. Oberthaler, and S. Floerchinger, Curved and expanding spacetime geometries in Bose-Einstein condensates, Phys. Rev. A 106, 033313 (2022).
  16. S. Datta and U. R. Fischer, Inherent nonlinearity of fluid motion and acoustic gravitational wave memory, Phys. Rev. D 105, 022003 (2022).
  17. H. Kleinert, Gravity as a theory of defects in a crystal with only second gradient elasticity, Annalen der Physik 499, 117 (1987).
  18. H. Kleinert and J. Zaanen, Nematic world crystal model of gravity explaining absence of torsion in spacetime, Phys. Lett. A 324, 361 (2004).
  19. Y. Zhu and Y. Jiang, World crystal: extending to four dimensional, Gen. Relativ. Gravit. 43, 2935 (2011).
  20. J. Zaanen and A. Beekman, The emergence of gauge invariance: The stay-at-home gauge versus local–global duality, Ann. Phys. 327, 1146 (2012).
  21. A. J. Beekman, J. Nissinen, K. Wu, K. Liu, R.-J. Slager, Z. Nussinov, V. Cvetkovic, and J. Zaanen, Dual gauge field theory of quantum liquid crystals in two dimensions, Phys. Rep. 683, 1 (2017).
  22. A. J. Beekman, J. Nissinen, K. Wu, and J. Zaanen, Dual gauge field theory of quantum liquid crystals in three dimensions, Phys. Rev. B 96, 165115 (2017).
  23. M. Pretko and L. Radzihovsky, Fracton-elasticity duality, Phys. Rev. Lett. 120, 195301 (2018).
  24. M. Pretko, Emergent gravity of fractons: Mach's principle revisited, Phys. Rev. D 96, 024051 (2017).
  25. H. Yan, Hyperbolic fracton model, subsystem symmetry, and holography, Phys. Rev. B 99, 155126 (2019).
  26. H. Yan, Hyperbolic fracton model, subsystem symmetry, and holography. II. The dual eight-vertex model, Phys. Rev. B 100, 245138 (2019).
  27. H. Yan, Geodesic string condensation from symmetric tensor gauge theory: A unifying framework of holographic toy models, Phys. Rev. B 102, 161119(R) (2020).
  28. H. Yan, C. B. Jepsen, and Y. Oz, p-adic holography from the hyperbolic fracton model, arXiv:2306.07203 [hep-th].
  29. D. Hill, V. Slastikov, and O. Tchernyshyov, Chiral magnetism: a geometric perspective, SciPost Phys. 10, 078 (2021).
  30. M. M. Roberts and T. Wiseman, Analog gravity and continuum effective theory of the graphene tight-binding lattice model, Phys. Rev. B 109, 045425 (2024).
  31. A. Andreev and I. Grishchuk, Spin nematics, Zh. Eksp. Teor. Fiz. 87, 467 (1984) Spin nematics [Sov. Phys. JETP 60, 267 (1984)].
  32. A. V. Chubukov, Chiral, nematic, and dimer states in quantum spin chains, Phys. Rev. B 44, 4693 (1991).
  33. N. Papanicolaou, Unusual phases in quantum spin-1 systems, Nucl. Phys. B 305, 367 (1988).
  34. B. A. Ivanov and A. K. Kolezhuk, Effective field theory for the S = 1 quantum nematic, Phys. Rev. B 68, 052401 (2003).
  35. S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai, S. Jonas, C. Broholm, H. Tsunetsugu, Y. Qiu, and Y. Maeno, Spin disorder on a triangular lattice, Science 309, 1697 (2005).
  36. N. Shannon, T. Momoi, and P. Sindzingre, Nematic order in square lattice frustrated ferromagnets, Phys. Rev. Lett. 96, 027213 (2006).
  37. A. Läuchli, F. Mila, and K. Penc, Quadrupolar phases of the S=1 bilinear-biquadratic Heisenberg model on the triangular lattice, Phys. Rev. Lett. 97, 087205 (2006).
  38. H. Tsunetsugu and M. Arikawa, The spin nematic state in triangular antiferromagnets, J. Phys.: Condens. Matter 19, 145248 (2007).
  39. A. Orlova, E. L. Green, J. M. Law, D. I. Gorbunov, G. Chanda, S. Krämer, M. Horvatić, R. K. Kremer, J. Wosnitza, and G. L. J. A. Rikken, Nuclear magnetic resonance signature of the spin-nematic phase in LiCuVO4 at high magnetic fields, Phys. Rev. Lett. 118, 247201 (2017).
  40. Y. Kohama, H. Ishikawa, A. Matsuo, K. Kindo, N. Shannon, and Z. Hiroi, Possible observation of quantum spin-nematic phase in a frustrated magnet, Proc. Natl. Acad. Sci. USA 116, 10686 (2019).
  41. H. Kim, J.-K. Kim, J. Kwon, J. Kim, H.-W. J. Kim, S. Ha, K. Kim, W. Lee, J. Kim, G. Y. Cho, H. Heo, J. Jang, C. J. Sahle, A. Longo, J. Strempfer, G. Fabbris, Y. Choi, D. Haskel, J. Kim, J. W. Kim et al., Quantum spin nematic phase in a square-lattice iridate, Nature (London) 625, 264 (2024).
  42. T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, J. Phys. Soc. Jpn. 67, 1822 (1998).
  43. E. Demler and F. Zhou, Spinor bosonic atoms in optical lattices: Symmetry breaking and fractionalization, Phys. Rev. Lett. 88, 163001 (2002).
  44. A. Imambekov, M. Lukin, and E. Demler, Spin-exchange interactions of spin-one bosons in optical lattices: Singlet, nematic, and dimerized phases, Phys. Rev. A 68, 063602 (2003).
  45. L. de Forges de Parny, H. Yang, and F. Mila, Anderson tower of states and nematic order of spin-1 bosonic atoms on a 2D lattice, Phys. Rev. Lett. 113, 200402 (2014).
  46. T. Zibold, V. Corre, C. Frapolli, A. Invernizzi, J. Dalibard, and F. Gerbier, Spin-nematic order in antiferromagnetic spinor condensates, Phys. Rev. A 93, 023614 (2016).
  47. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).
  48. S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Cambridge University Press, Cambridge UK, 2019).
  49. M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments (Oxford University Press, Oxford, 2007).
  50. B. S. DeWitt, Quantum theory of gravity. II. The manifestly covariant theory, Phys. Rev. 162, 1195 (1967).
  51. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.109.L220407 for details of the coordinate transformation, details of numerical simulations and animated versions of Figs.  1 and 3.
  52. F. C. Frank, I. Liquid crystals. On the theory of liquid crystals, Discuss. Faraday Soc. 25, 19 (1958).
  53. N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591 (1979).
  54. H. Watanabe and H. Murayama, Unified description of Nambu-Goldstone bosons without Lorentz invariance, Phys. Rev. Lett. 108, 251602 (2012).
  55. K. Remund, R. Pohle, Y. Akagi, J. Romhányi, and N. Shannon, Semi-classical simulation of spin-1 magnets, Phys. Rev. Res. 4, 033106 (2022).
  56. A. Smerald and N. Shannon, Theory of spin excitations in a quantum spin-nematic state, Phys. Rev. B 88, 184430 (2013).
  57. A. Smerald, Theory of the Nuclear Magnetic 1/T1 Relaxation Rate in Conventional and Unconventional Magnets, Springer Theses (Springer International Publishing, Cham, 2013).
  58. T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett. 81, 742 (1998).
  59. J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, Spin domains in ground-state Bose–Einstein condensates, Nature (London) 396, 345 (1998).
  60. D. Jacob, L. Shao, V. Corre, T. Zibold, L. De Sarlo, E. Mimoun, J. Dalibard, and F. Gerbier, Phase diagram of spin-1 antiferromagnetic Bose-Einstein condensates, Phys. Rev. A 86, 061601(R) (2012).
  61. C. Becker, P. Soltan-Panahi, J. Kronjäger, S. Dörscher, K. Bongs, and K. Sengstock, Ultracold quantum gases in triangular optical lattices, New J. Phys. 12, 065025 (2010).
  62. K. Xu, Y. Liu, J. R. Abo-Shaeer, T. Mukaiyama, J. K. Chin, D. E. Miller, W. Ketterle, K. M. Jones, and E. Tiesinga, Sodium Bose-Einstein condensates in an optical lattice, Phys. Rev. A 72, 043604 (2005).
  63. E. Fava, T. Bienaimé, C. Mordini, G. Colzi, C. Qu, S. Stringari, G. Lamporesi, and G. Ferrari, Observation of spin superfluidity in a Bose gas mixture, Phys. Rev. Lett. 120, 170401 (2018).
  64. K. Harada and N. Kawashima, Quadrupolar order in isotropic Heisenberg models with biquadratic interaction, Phys. Rev. B 65, 052403 (2002).
  65. H. Tsunetsugu and M. Arikawa, Spin nematic phase in s = 1 triangular antiferromagnets, J. Phys. Soc. Jpn. 75, 083701 (2006).
  66. E. M. Stoudenmire, S. Trebst, and L. Balents, Quadrupolar correlations and spin freezing in s=1 triangular lattice antiferromagnets, Phys. Rev. B 79, 214436 (2009).
  67. R. K. Kaul, Spin nematic ground state of the triangular lattice S=1 biquadratic model, Phys. Rev. B 86, 104411 (2012).
  68. A. Völl and S. Wessel, Spin dynamics of the bilinear-biquadratic s=1 Heisenberg model on the triangular lattice: A quantum Monte Carlo study, Phys. Rev. B 91, 165128 (2015).
  69. D. M. Stamper-Kurn and M. Ueda, Spinor Bose gases: Symmetries, magnetism, and quantum dynamics, Rev. Mod. Phys. 85, 1191 (2013).
  70. V. L. Berezinskiǐ, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems, Zh. Eksp. Teor. Fiz. 61, 1144 (1972) [Sov. Phys. JETP 34, 610 (1972)].
  71. J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973).
  72. H. Kawamura and A. Yamamoto, Vortex-induced topological transition of the bilinear–biquadratic Heisenberg antiferromagnet on the triangular lattice, J. Phys. Soc. Jpn. 76, 073704 (2007).
  73. Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and J. Dalibard, Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas, Nature (London) 441, 1118 (2006).
  74. X. Tang and J. V. Selinger, Orientation of topological defects in 2D nematic liquid crystals, Soft Matter 13, 5481 (2017).
  75. S. Kang, S. W. Seo, H. Takeuchi, and Y. Shin, Observation of wall-vortex composite defects in a spinor Bose-Einstein condensate, Phys. Rev. Lett. 122, 095301 (2019).
  76. P. Kunkel, M. Prüfer, S. Lannig, R. Rosa-Medina, A. Bonnin, M. Gärttner, H. Strobel, and M. K. Oberthaler, Simultaneous readout of noncommuting collective spin observables beyond the standard quantum limit, Phys. Rev. Lett. 123, 063603 (2019).
  77. I. Carusotto and E. J. Mueller, Imaging of spinor gases, J. Phys. B: At., Mol. Opt. Phys. 37, S115 (2004).
  78. F. Michaud, F. Vernay, and F. Mila, Theory of inelastic light scattering in spin-1 systems: Resonant regimes and detection of quadrupolar order, Phys. Rev. B 84, 184424 (2011).
  79. L. Savary and T. Senthil, Probing hidden orders with resonant inelastic x-ray scattering, arXiv:1506.04752 [cond-mat.str-el].
  80. A. Nag, A. Nocera, S. Agrestini, M. Garcia-Fernandez, A. C. Walters, S.-W. Cheong, S. Johnston, and K.-J. Zhou, Quadrupolar magnetic excitations in an isotropic spin-1 antiferromagnet, Nat. Commun. 13, 2327 (2022).
  81. B. A. Ivanov, R. S. Khymyn, and A. K. Kolezhuk, Pairing of solitons in two-dimensional s=1 magnets, Phys. Rev. Lett. 100, 047203 (2008).
  82. H. T. Ueda, Y. Akagi, and N. Shannon, Quantum solitons with emergent interactions in a model of cold atoms on the triangular lattice, Phys. Rev. A 93, 021606(R) (2016).
  83. Y. Akagi, Y. Amari, N. Sawado, and Y. Shnir, Isolated skyrmions in the CP2 nonlinear sigma model with a Dzyaloshinskii-Moriya type interaction, Phys. Rev. D 103, 065008 (2021).
  84. Y. Akagi, Y. Amari, S. B. Gudnason, M. Nitta, and Y. Shnir, Fractional skyrmion molecules in a CPN−1 model, J. High Energy Phys. 11 (2021) 194.
  85. Y. Amari, Y. Akagi, S. B. Gudnason, M. Nitta, and Y. Shnir, CP2 skyrmion crystals in an SU(3) magnet with a generalized Dzyaloshinskii-Moriya interaction, Phys. Rev. B 106, L100406 (2022).

Outline

Information

Phys. Rev. B 109, L220407– Published 14 June, 2024
  • Received 14 December 2023
  • Revised 8 May 2024
  • Accepted 9 May 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation