Level repulsion in integrable and almost-integrable quantum spin models.

The repartition of the separation between energy levels of various isotropic S=1/2 antiferromagnetic chains is studied numerically with the aim of investigating the transition from integrable to noninte- grable systems. We begin by displaying the level separation distribution of the integrable Bethe chain. Then two nonintegrable systems, two coupled chains, and a next-nearest-neighbor coupled chain, are studied as a function of the coupling. We examine how the level spacing evolves from the Poisson distribution to the Gaussian-orthogonal-ensemble distribution. Finally we consider the Haldane-Shastry 1/${\mathit{r}}^{2}$ model. A number of conclusions regarding the behavior and relevance of the level-spacing distribution in these spin systems is drawn.