On the Dynamical Invariants and the Geometric Phases for a General Spin System in a Changing Magnetic Field

We consider a class of general spin Hamiltonians of the form $H_s(t)=H_0(t)+H'(t)$ where $H_0(t)$ and $H'(t)$ describe the dipole interaction of the spins with an arbitrary time-dependent magnetic field and the internal interaction of the spins, respectively. We show that if $H'(t)$ is rotationally invariant, then $H_s(t)$ admits the same dynamical invariant as $H_0(t)$. A direct application of this observation is a straightforward rederivation of the results of Yan et al [Phys. Lett. A, Vol: 251 (1999) 289 and Vol: 259 (1999) 207] on the Heisenberg spin system in a changing magnetic field.