Non-equilibrium thermo-field dynamics for anti-ferromagnetic spin system

The non-equilibrium thermo-field dynamics for an anti-ferromagnetic spin system interacting with a phonon reservoir is proposed for the case of a non-bilinear unperturbed Hamiltonian, which includes not only a bilinear part but also a non-bilinear part, in the spin–wave approximation. The two kinds of quasi-particle operators are introduced, and their forms are derived for the semi-free boson fields. It is shown that the two quasi-particles decay exponentially with the frequencies and life-times which are different from each other. It is also shown that each quasi-particle changes to the other tilde quasi-particle through the spin–phonon interaction. The spin–spin correlation functions and longitudinal magnetization for the anti-ferromagnetic spin system under an external static magnetic field are derived in the forms convenient for the perturbation expansions. The expectation values of the spin–wave energy and longitudinal magnetization and the spin–spin correlation functions are investigated numerically for an anti-ferromagnetic system of one-dimensional infinite spins interacting with a damped phonon-reservoir, in the region valid for the lowest spin–wave approximation and the narrowing-limit approximation in which the relaxation times of the spin system are much larger than the correlation time of the phonon reservoir. The two-point Green’s function of the semi-free spin–wave for the anti-ferromagnetic spin system is derived by introducing the thermal quartet notation, and it is given in a form of 4×4 matrix.