Classes of integrable spin systems

TL;DR

This paper classifies integrable classical and quantum spin systems, providing explicit solutions for certain classes and characterizing spin graphs with uniform couplings, including enumeration up to five spins.

Contribution

It introduces a recursive property-based classification and characterizes spin graphs with uniform couplings, including enumeration for small systems.

Findings

Explicit time evolution for systems with property P

Characterization of spin graphs without chains of length four

Complete enumeration of spin graphs up to five spins

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.