In this work we present a study of the dynamics of an Ising chain ( in the presence of a
transverse magnetic field. The model involves only short-range interactions and is described by the interaction of nearest neighbors between two bodies in the chain. We calculate the time-dependent autocorrelation function using the exact diagonalization method. The method consists in considering finite spin chains with increasing sizes to determine the time-dependent autocorrelation function of endless chain. The results for the biggest system studied indicate that the asymptotic behavior in thermodynamical systems for large time has been reached. This means that if we consider larger chains, the results will be the same within the numerical accuracy presented here. The autocorrelation function is oscillatory for large values of the transverse field and is monotonically decreasing for small field values. The system undergoes a crossover from a collective mode excitation scheme to a dynamics of central mode type as the magnetic field is reduced.
New exact results are presented on the long-time asymptotic expansions of the
T=0 autocorrelation functions, and on the leading singularities of their frequency-dependent Fourier transforms, for the one-dimensional
S=21 isotropic
XY model and the
S=21 transverse Ising model at the critical field. High-precision numerical calculations of the latter functions are also given, and experiments are proposed to observe the functional behaviors that are found.