Classes of integrable spin systems

Abstract

We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property $P$ saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property $P$. For these systems the time evolution can be explicitely calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing $N-1$ independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property $P$ and can be characterized as spin graphs not containing chains of length four. We completely enumerate and characterize all spin graphs up to N=5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed.