Recently it was suggested that certain perturbations of integrable spin chains lead to a weak breaking of integrability in the sense that integrability is preserved at the first order in the coupling. Here we examine this claim using level spacing distribution. We find that the volume dependent crossover between integrable and chaotic level spacing statistics which marks the onset of quantum chaotic behaviour, is markedly different for weak vs. strong breaking of integrability. In particular, for the gapless case we find that the crossover coupling as a function of the volume $L$ scales with a $1/L^2$ law for weak breaking as opposed to the $1/L^3$ law previously found for the strong case.

中文翻译：

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弱可积性断裂和水平间距分布

最近有人提出，可积自旋链的某些扰动会导致可积性的弱破坏，因为可积性在耦合中的一阶保持不变。在这里，我们使用级别间距分布来检查此声明。我们发现可积和混沌水平间距统计之间的体积相关交叉标志着量子混沌行为的开始，对于可积性的弱破坏与强破坏有显着不同。特别是，对于无间隙情况，我们发现作为交易量 $L$ 的函数的交叉耦合与弱破坏的 $1/L^2$ 定律相反，而不是先前发现的 $1/L^3$ 定律用于强案。