Ward-Takahashi Identities in Itinerant Electron Ferromagnetism. I: Zero Temperature Case

Abstract

A systematic calculation of the Ward-Takahashi identities in 1tmerant electron ferro­ magnetism is presented by making use of the functional integral method. The 1/h-singularity of the magnon Green's function for vanishing energy and momentum plays the significant role. Here h denotes a ficticious infinitesimal magnetic field applied to the system. The identities for the multi point vertex functions corresponding to the processes in which the electron, magnon and longitudinal spin fluctuation interact are given. The electron and magnon pole term contributions to these identities are discussed. § I. Introduction So far a number of theoretical works have been made to clarify the effects of the electron-magnon interaction in itinerant electron ferromagnetism. ll. 2) Any theo­ ry in this field is required to keep the rotational invariance when we make quan­ titative discussion. It is well-known that such invariance is conveniently described in the frame of the Ward-Takahashi identities (hereafter referred to as \VTI). In a previous paper,') we have proposed a systematic approach to calculating WTI in ferromagnetically ordered state making use of the functional integral method. In our treatment, a ficticious infinitesimal magnetic field h is introduced to create the ordered state. h is taken as zero after the end of calculation. The ordered state does not have the spin rotational invariance, and as a result of such breakdown of symmetry, the magnon appears as a Goldstone boson in order to recover the original symmetry. In deriving WTI, the 1/h-singularity of the magnon Green's function for vanishing energy and momentum plays the significant role. The pur­ pose of this paper is to present WTI with the help of the above method for multi point vertex functions in which the electron, magnon (transversal spin fluctuation), and longitudinal spin fluctuation interact with each other and themselves. In order to analyze the electron and magnon propagators, we need at least three point vertex functions which correspond to the magnon emission or absorption