Exact Time-Dependent Analysis for the One-DimensionalXYModel

An exact solution of the Liouville equation for the $\mathrm{XY}$ model with time-dependent external magnetic fields is obtained. Standard transformations allow a decomposition of the space of states into a direct product of four-dimensional subspaces. The Liouville equation is reduced to a set of coupled differential equations in each subspace. These equations have been explicitly solved for several cases.