Nambu–Goldstone Bosons Characterized by the Order Parameter in Spontaneous Symmetry Breaking
Journal of the Physical Society of Japan
We present explicitly a relation between the Nambu-Goldstone boson and the order parameter in non-relativistic systems with spontaneous symmetry breaking. We show that the Nambu-Goldstone bosons are characterized by transformation property of the order parameter under symmetry transformation of a system. We give an explicit formula for the Nambu-Goldstone boson for a general Lie group G, and then the number of the Nambu-Goldstone boson is derived straightforwardly from the symmetry of the order
... parameter, i.e. the type of symmetry breaking. We show that the Ward-Takahashi identity is modified in the presence of the Nambu-Goldstone boson, where the generalized Ward-Takahashi identity includes the coupling (the vertex function) between fermions and Nambu-Goldstone bosons. The closed equation for the Green's functions of Nambu-Goldstone bosons is derived by introducing the fermion-Nambu-Goldstone boson vertex function. Examples are given for G=SU(2) (ferromagnetic), U(1) (superconductor) and SU(3) symmetry breaking.