Magnon decay in gapped quantum spin systems

In the O(3) $\sigma$-model description of gapped spin systems, $S=1$ magnons can only decay into {\em three\/} lower energy magnons. We argue that the symmetry of the quantum spin Hamiltonian often allows decay into {\em two\/} magnons, and compute this decay rate in model systems. For a realistic model describing two-dimensional spin dimer material $\rm (C_{4}H_{12}N_{2})Cu_{2}Cl_{6}$ (known as PHCC), we compare our results for the momentum-dependent magnon linewidth with recent measurements by Stone {\em et al.} (e-print cond-mat/0511266) and extract new information on the exchange coupling pattern in this material. For $S=1$ Haldane chains, we show that two-magnon decay is allowed in the full lattice description, even though it cannot be induced by any allowed term written in powers and gradients of the $\sigma$-model field. We present estimates for the behavior of the magnon linewidth in Haldane gap chains and discuss relation to the recent experimental work.