Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries

Bogoliubov theory approach

Daisuke A. Takahashi, Muneto Nitta

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

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
Volume354
DOIs
Publication statusPublished - 2015 Mar 1

Fingerprint

Bogoliubov theory
counting
symmetry
Bose-Einstein condensates
orthogonality
coefficients
mathematics
broken symmetry
solitary waves
vortices
formulations

Keywords

  • 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)

Cite this

Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries : Bogoliubov theory approach. / Takahashi, Daisuke A.; Nitta, Muneto.

In: Annals of Physics, Vol. 354, 01.03.2015, p. 101-156.

Research output: Contribution to journalArticle

@article{821541b5c58141e8964620eb6e8beade,
title = "Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach",
abstract = "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.",
keywords = "Bogoliubov theory, Gross-Pitaevskii equation, Indefinite inner product space, Nambu-Goldstone modes, Spinor Bose-Einstein condensates, Spontaneous symmetry breaking",
author = "Takahashi, {Daisuke A.} and Muneto Nitta",
year = "2015",
month = "3",
day = "1",
doi = "10.1016/j.aop.2014.12.009",
language = "English",
volume = "354",
pages = "101--156",
journal = "Annals of Physics",
issn = "0003-4916",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries

T2 - Bogoliubov theory approach

AU - Takahashi, Daisuke A.

AU - Nitta, Muneto

PY - 2015/3/1

Y1 - 2015/3/1

N2 - 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.

AB - 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.

KW - Bogoliubov theory

KW - Gross-Pitaevskii equation

KW - Indefinite inner product space

KW - Nambu-Goldstone modes

KW - Spinor Bose-Einstein condensates

KW - Spontaneous symmetry breaking

UR - http://www.scopus.com/inward/record.url?scp=84920923376&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920923376&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2014.12.009

DO - 10.1016/j.aop.2014.12.009

M3 - Article

VL - 354

SP - 101

EP - 156

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

ER -