Relaxation method and TCLE method of linear response in terms of thermo-field dynamics

The general formulae of the dynamic susceptibility are derived using the relaxation method and the TCLE method for the linear response by introducing the renormalized hat-operator in terms of thermo-field dynamics (TFD). In the former method, the Kubo formula is calculated for systems with no external driving fields, while in the latter method the admittance is directly calculated from time-convolutionless equations with external driving terms. The relation between the two methods is analytically investigated, and also the fluctuation–dissipation theorem is examined for the two methods in terms of TFD. The TCLE method is applied to an interacting spin system, and a formula of the transverse magnetic susceptibility is derived for such a system. The transverse magnetic susceptibility of an interacting spin system with S=1/2 spins is obtained up to the first order in powers of the spin–spin interaction.