Some exact calculations on a chain of spins 1/2 II

The chain of spins 1/2 with anisotropic nearest neighbour interaction, known as the X–Y model, which was introduced by Lieb, Schultz and Mattis1) and studied in ref. 2*) in the presence of a constant field along the z axis, is now studied with a small oscillating field superimposed on the constant field. Since the autocorrelation function of the magnetization is known from I, the susceptibility, as given by the Kubo formalism, can now easily be calculated. The implicit assumption of Kubo and Tomita3) that the aftereffects of the adiabatic switching on of the oscillating field vanish, can now be rigorously justified for this simple case by giving an exact procedure for solving the time-dependent Schrödinger equation in successive orders of the amplitude of the oscillating field. The imaginary part of the susceptibility is different from zero in a certain frequency range with the effect that energy will be absorbed by the chain if the frequency of the oscillating field lies within this range. If one assumes for the sake of simplicity that the chain is thermally completely isolated and that it always is in thermal equilibrium, the temperature will obviously increase in time since energy is being absorbed. An exact equation for the temperature as a function of time is derived in terms of the parameters characterizing the system in the limit of high temperatures and the temperature is seen to increase exponentially in time