Renormalisation procedures in the frame of the 1/z expansion of the temperature Green function for localised spins: I. Excitations in the singlet-singlet model

Using the singlet-singlet (transverse Ising) model as an example, shortcomings of the existing diagrammatic method for the temperature Green function of localised spins are discussed in the paramagnetic phase. It is shown that the absence of propagator and vertex renormalisation is the root of these shortcomings. The leading order in the 1/z renormalisation of propagators and vertices (effectively-population factors) is introduced in the frame of the 1/z expansion for the polarisation part. The excitations spectrum and the dynamic susceptibility width are calculated. As a result of the authors’ renormalisation procedure they are essentially modified both now giving the reduction of the phase transition temperature and the proper high-temperature behaviour of the frequency shift and the width. The higher-order renormalisation procedure is indicated in connection with the long-standing problem of removing the erroneous zero response on the frequency which is equal to the bare gap of the system. The problem of self-consistency in the method is also discussed.