- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Noncommutative field theory of the Tkachenko mode: Symmetries and decay rate

Yi-Hsien Du1, Sergej Moroz2,3, Dung Xuan Nguyen4, and Dam Thanh Son1

  • 1Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, IL 60637, USA
  • 2Department of Engineering and Physics, Karlstad University, Karlstad 651 88, Sweden
  • 3Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm 114 19, Sweden
  • 4Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea

PDF

Phys. Rev. Research 6, L012040 – Published 26 February, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L012040

Abstract

We construct an effective field theory describing the collective Tkachenko oscillation mode of a vortex lattice in a two-dimensional rotating Bose-Einstein condensate in the long-wavelength regime. The theory has the form of a noncommutative field theory of a Nambu-Goldstone boson, which exhibits a noncommutative version of dipole symmetry. From the effective field theory, we show that, at zero temperature, the decay width Γ of the Tkachenko mode scales with its energy E as Γ∼E3 in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (55)

  1. V. K. Tkachenko, On vortex lattices, Sov. Phys. JETP 22, 1282 (1965).
  2. V. K. Tkachenko, Stability of vortex lattices, Sov. Phys. JETP 23, 1049 (1966).
  3. V. K. Tkachenko, Elasticity of vortex lattices, Sov. Phys. JETP 29, 945 (1969).
  4. E. B. Sonin, Tkachenko waves, JETP Lett. 98, 758 (2014).
  5. E. Sonin, Vortex-lattice vibrations in a rotating helium II, JETP Lett. 43, 1027 (1976).
  6. G. E. Volovik and V. S. Dotsenko, Jr., Poisson brackets and continuous dynamics of the vortex lattice in rotating He II, JETP Lett. 29, 576 (1979).
  7. G. Baym, Tkachenko modes of vortex lattices in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett. 91, 110402 (2003).
  8. H. Watanabe and H. Murayama, Redundancies in Nambu-Goldstone bosons, Phys. Rev. Lett. 110, 181601 (2013).
  9. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L012040 for discussion of symmetry-breaking redundancies, explicit derivation of the Lifshitz effective theory, resolution of the nonlinear constraint, and the explicit derivation of the GMP algebra.
  10. I. Coddington, P. Engels, V. Schweikhard, and E. A. Cornell, Observation of tkachenko oscillations in rapidly rotating bose-einstein condensates, Phys. Rev. Lett. 91, 100402 (2003).
  11. M. Ruderman, Long period oscillations in rotating neutron stars, Nature (London) 225, 619 (1970).
  12. G. Grinstein, Anisotropic sine-Gordon model and infinite-order phase transitions in three dimensions, Phys. Rev. B 23, 4615 (1981).
  13. S. I. Matveenko and G. V. Shlyapnikov, Tkachenko modes and their damping in the vortex lattice regime of rapidly rotating bosons, Phys. Rev. A 83, 033604 (2011).
  14. M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73, 977 (2001).
  15. V. A. Rubakov, Classical Field Theories. Theories with Fermions. Noncommutative Theories (URSS, 2005) (in Russian).
  16. L. Susskind, The quantum Hall fluid and noncommutative Chern-Simons theory, arXiv:hep-th/0101029.
  17. A. P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, J. High Energy Phys. 04 (2001) 011.
  18. S. Hellerman and M. Van Raamsdonk, Quantum Hall physics equals noncommutative field theory, J. High Energy Phys. 10 (2001) 039.
  19. E. Fradkin, V. Jejjala, and R. G. Leigh, Noncommutative Chern-Simons for the quantum Hall system and duality, Nucl. Phys. B 642, 483 (2002).
  20. V. Pasquier, Quantum Hall effect and non-commutative geometry, Quantum Spaces: Poincaré Seminar 2007, edited by B. Duplantier (Birkhäuser Basel, Basel, 2007), pp. 1–17.
  21. Z. Dong and T. Senthil, Non-commutative field theory and composite Fermi liquids in some quantum Hall systems, Phys. Rev. B 102, 205126 (2020).
  22. Y.-H. Du, U. Mehta, and D. T. Son, Noncommutative gauge symmetry in the fractional quantum Hall effect, arXiv:2110.13875.
  23. T.-L. Ho, Bose-Einstein condensates with large number of vortices, Phys. Rev. Lett. 87, 060403 (2001).
  24. J. Sinova, C. B. Hanna, and A. H. MacDonald, Quantum melting and absence of Bose-Einstein condensation in two-dimensional vortex matter, Phys. Rev. Lett. 89, 030403 (2002).
  25. E. B. Sonin, Ground state and Tkachenko modes of a rapidly rotating Bose-Einstein condensate in the lowest-Landau-level state, Phys. Rev. A 72, 021606(R) (2005).
  26. M. Pretko, Subdimensional particle structure of higher rank U(1) spin liquids, Phys. Rev. B 95, 115139 (2017).
  27. M. Pretko and L. Radzihovsky, Fracton-elasticity duality, Phys. Rev. Lett. 120, 195301 (2018).
  28. R. M. Nandkishore and M. Hermele, Fractons, Annu. Rev. Condens. Matter Phys. 10, 295 (2019).
  29. A. Gromov, Towards classification of fracton phases: the multipole algebra, Phys. Rev. X 9, 031035 (2019).
  30. J. Wang, K. Xu, and S.-T. Yau, Higher-rank tensor non-Abelian field theory: Higher-moment or subdimensional polynomial global symmetry, algebraic variety, Noether's theorem, and gauging, Phys. Rev. Res. 3, 013185 (2021).
  31. M. Pretko, X. Chen, and Y. You, Fracton phases of matter, Int. J. Mod. Phys. A 35, 2030003 (2020).
  32. A. Gromov, A. Lucas, and R. M. Nandkishore, Fracton hydrodynamics, Phys. Rev. Res. 2, 033124 (2020).
  33. N. Seiberg and S.-H. Shao, Exotic symmetries, duality, and fractons in 2+1-dimensional quantum field theory, SciPost Phys. 10, 027 (2021).
  34. D. X. Nguyen, A. Gromov, and S. Moroz, Fracton-elasticity duality of two-dimensional superfluid vortex crystals: defect interactions and quantum melting, SciPost Phys. 9, 076 (2020).
  35. E. Lake, M. Hermele, and T. Senthil, Dipolar Bose-Hubbard model, Phys. Rev. B 106, 064511 (2022).
  36. P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Global dipole symmetry, compact Lifshitz theory, tensor gauge theory, and fractons, Phys. Rev. B 106, 045112 (2022).
  37. A. Kapustin and L. Spodyneiko, Hohenberg-Mermin-Wagner-type theorems and dipole symmetry, Phys. Rev. B 106, 245125 (2022).
  38. P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, 2+1d compact Lifshitz theory, tensor gauge theory, and fractons, arXiv:2209.10030.
  39. L. Radzihovsky, Lifshitz gauge duality, Phys. Rev. B 106, 224510 (2022).
  40. A. Gromov and L. Radzihovsky, Fracton matter, Rev. Mod. Phys. 96, 011001 (2024).
  41. D. X. Nguyen and S. Moroz, On quantum melting of superfluid vortex crystals: from Lifshitz scalar to dual gravity, arXiv:2310.13741.
  42. Y.-H. Du, H. T. Lam, and L. Radzihovsky, Quantum vortex lattice via Lifshitz duality, arXiv:2310.13794.
  43. N. Cooper, Rapidly rotating atomic gases, Adv. Phys. 57, 539 (2008).
  44. H. Leutwyler, Phonons as Goldstone bosons, Helv. Phys. Acta 70, 275 (1997).
  45. D. E. Soper, Classical Field Theory (Wiley, New York, 1976).
  46. M. E. Peskin, Mandelstam-'t hooft duality in abelian lattice models, Ann. Phys. 113, 122 (1978).
  47. C. Dasgupta and B. I. Halperin, Phase transition in a lattice model of superconductivity, Phys. Rev. Lett. 47, 1556 (1981).
  48. The hat symbol â on the two sides of Eq. (14) have different meanings: on the left-hand side, it denotes an operator of field theory, on the right-hand side, an operator in the sense of a function of noncommutative coordinates.
  49. One way to obtain the structure [Eq. (17b)] is to start from Uab=∂iXa★∂iXb. To leading order in derivatives, Uab is a symmetric matrix with determinant equal to 1, since the transformation from xi to Xa is area preserving, and thus has two independent components. The two components of Dabϕ are then related to Uab by a rotation by 90 degrees: Dabϕ=−14θ(εacUcb+εbcUac)..
  50. M. Greiter, F. Wilczek, and E. Witten, Hydrodynamic relations in superconductivity, Mod. Phys. Lett. B 03, 903 (1989).
  51. S. Moroz, C. Hoyos, C. Benzoni, and D. T. Son, Effective field theory of a vortex lattice in a bosonic superfluid, SciPost Phys. 5, 039 (2018).
  52. S. Moroz and D. T. Son, Bosonic superfluid on the lowest Landau level, Phys. Rev. Lett. 122, 235301 (2019).
  53. S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B 33, 2481 (1986).
  54. L. Pitaevskii and S. Stringari, Uncertainty principle and off-diagonal long-range order in the fractional quantum Hall effect, Phys. Rev. B 47, 10915 (1993).
  55. G. Baym, Vortex lattices in rapidly rotating Bose-Einstein condensates: Modes and correlation functions, Phys. Rev. A 69, 043618 (2004).

Outline

Information

Phys. Rev. Research 6, L012040– Published 26 February, 2024
  • Received 14 January 2023
  • Revised 23 September 2023
  • Accepted 2 February 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