Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

Daisuke A. Takahashi, Muneto Nitta

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31 Citations (Scopus)


When continuous symmetry is spontaneously broken, there appear Nambu-Goldstone modes (NGMs) with linear or quadratic dispersion relation, which is called type-I or type-II, respectively. We propose a framework to count these modes including the coefficients of the dispersion relations by applying the standard Gross-Pitaevskii-Bogoliubov theory. Our method is mainly based on (i) zero-mode solutions of the Bogoliubov equation originated from spontaneous symmetry breaking and (ii) their generalized orthogonal relations, which naturally arise from well-known Bogoliubov transformations and are referred to as "σ-orthogonality" in this paper. Unlike previous works, our framework is applicable without any modification to the cases where there are additional zero modes, which do not have a symmetry origin, such as quasi-NGMs, and/or where spacetime symmetry is spontaneously broken in the presence of a topological soliton or a vortex. As a by-product of the formulation, we also give a compact summary for mathematics of bosonic Bogoliubov equations and Bogoliubov transformations, which becomes a foundation for any problem of Bogoliubov quasiparticles. The general results are illustrated by various examples in spinor Bose-Einstein condensates (BECs). In particular, the result on the spin-3 BECs includes new findings such as a type-I-type-II transition and an increase of the type-II dispersion coefficient caused by the presence of a linearly-independent pair of zero modes.

Original languageEnglish
Pages (from-to)101-156
Number of pages56
JournalAnnals of Physics
Publication statusPublished - 2015 Mar 1


  • Bogoliubov theory
  • Gross-Pitaevskii equation
  • Indefinite inner product space
  • Nambu-Goldstone modes
  • Spinor Bose-Einstein condensates
  • Spontaneous symmetry breaking

ASJC Scopus subject areas

  • Physics and Astronomy(all)


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