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Quasi-Nambu-Goldstone modes in nonrelativistic systems

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 relation, respectively. When 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 non-compact 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 [DT & MN, arXiv:1404.7696, Ann. Phys. 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 ZMand 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.