TY - JOUR

T1 - Quasi-Nambu-Goldstone modes in nonrelativistic systems

AU - Nitta, Muneto

AU - Takahashi, Daisuke A.

N1 - Publisher Copyright:
© 2015 American Physical Society.

PY - 2015/1/21

Y1 - 2015/1/21

N2 - When a continuous symmetry is spontaneously broken in nonrelativistic systems, there appear either type-I or type-II Nambu-Goldstone modes (NGMs) with linear or quadratic dispersion relations, respectively. When the equation of motion or the potential term has an enhanced symmetry larger than that of Lagrangian or Hamiltonian, there can appear quasi-NGMs if it is spontaneously broken. We construct a theory to count the numbers of type-I and type-II quasi-NGMs and NGMs, when the potential term has a symmetry of a noncompact group. We show that the counting rule based on the Watanabe-Brauner matrix is valid only in the absence of quasi-NGMs because of non-Hermitian generators, while that based on the Gram matrix [D.A. Takahashi and M. Nitta, Ann. Phys. (Amsterdam) 354, 101 (2015)] is still valid in the presence of quasi-NGMs. We show that there exist two types of type-II gapless modes, a genuine NGM generated by two conventional zero modes (ZMs) originated from the Lagrangian symmetry, and quasi-NGM generated by a coupling of one conventional ZM and one quasi-ZM, which is originated from the enhanced symmetry, or two quasi-ZMs. We find that, depending on the moduli, some NGMs can change to quasi-NGMs and vice versa with preserving the total number of gapless modes. The dispersion relations are systematically calculated by a perturbation theory. The general result is illustrated by the complex linear O(N) model, containing the two types of type-II gapless modes and exhibiting the change between NGMs and quasi-NGMs.

AB - When a continuous symmetry is spontaneously broken in nonrelativistic systems, there appear either type-I or type-II Nambu-Goldstone modes (NGMs) with linear or quadratic dispersion relations, respectively. When the equation of motion or the potential term has an enhanced symmetry larger than that of Lagrangian or Hamiltonian, there can appear quasi-NGMs if it is spontaneously broken. We construct a theory to count the numbers of type-I and type-II quasi-NGMs and NGMs, when the potential term has a symmetry of a noncompact group. We show that the counting rule based on the Watanabe-Brauner matrix is valid only in the absence of quasi-NGMs because of non-Hermitian generators, while that based on the Gram matrix [D.A. Takahashi and M. Nitta, Ann. Phys. (Amsterdam) 354, 101 (2015)] is still valid in the presence of quasi-NGMs. We show that there exist two types of type-II gapless modes, a genuine NGM generated by two conventional zero modes (ZMs) originated from the Lagrangian symmetry, and quasi-NGM generated by a coupling of one conventional ZM and one quasi-ZM, which is originated from the enhanced symmetry, or two quasi-ZMs. We find that, depending on the moduli, some NGMs can change to quasi-NGMs and vice versa with preserving the total number of gapless modes. The dispersion relations are systematically calculated by a perturbation theory. The general result is illustrated by the complex linear O(N) model, containing the two types of type-II gapless modes and exhibiting the change between NGMs and quasi-NGMs.

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U2 - 10.1103/PhysRevD.91.025018

DO - 10.1103/PhysRevD.91.025018

M3 - Article

AN - SCOPUS:84921455626

SN - 1550-7998

VL - 91

JO - Physical Review D - Particles, Fields, Gravitation and Cosmology

JF - Physical Review D - Particles, Fields, Gravitation and Cosmology

IS - 2

M1 - 025018

ER -