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)= H₀( t)+ H‧( t), where H₀( 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 H₀( t). A direct application of this observation is a straightforward rederivation of the results of Yan et al. (Phys. Lett. A 251 (1999) 289, 259 (1999) 207) on the Heisenberg spin system in a changing magnetic field.