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One of the most fundamental problems in quantum many-body systems is theidentification of a mean field in spontaneous symmetry breaking which isusually made in a heuristic manner. We propose a systematic method of finding amean field based on the Lie algebra and the dynamical symmetry by introducing aclass of symmetry broken phases which we call $\mu$-symmetry breaking. We showthat for $\mu$-symmetry breaking the quadratic part of an effective Lagrangianof Nambu-Goldstone modes can be block-diagonalized and that homotopy groups oftopological excitations can be calculated systematically.

$\mu$-Symmetry breaking: an algebraic approach to finding mean fields of quantum many-body systems