The small-world effect is a universal feature used to explain many different phenomena like percolation, diffusion, and consensus. Starting from any regular lattice of $N$ sites, the small-world effect can be attained by rewiring randomly an $O\left(N\right)$ number of links or by superimposing an equivalent number of new links onto the system. In a classical system this procedure is known to change radically its critical point and behavior, the new system being always effectively mean-field. Here, we prove that at the quantum level the above scenario does not apply: when an $O\left(N\right)$ number of new couplings are randomly superimposed onto a quantum Ising chain, its quantum critical point and behavior both remain unchanged. In other words, at zero temperature quantum fluctuations destroy any small-world effect. This exact result sheds new light on the significance of the quantum critical point as a thermodynamically stable feature of nature that has no analogy at the classical level and essentially prevents a naive application of network theory to quantum systems. The derivation is obtained by combining the quantum-classical mapping with a simple topological argument.

中文翻译：

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在量子水平上没有小世界效应和量子临界点的稳定性

小世界效应是一种通用功能，用于解释许多不同的现象，例如渗透，扩散和共识。从任何规则格开始$ñ$ 站点，则可以通过随机重新布线来实现小世界效应 $Ø（ñ）$链接数或通过将相等数量的新链接叠加到系统上。在经典系统中，已知此过程会从根本上改变其临界点和行为，而新系统始终是有效的均值场。在这里，我们证明在量子水平上上述情况不适用：$Ø（ñ）$许多新的耦合随机地叠加在量子伊辛链上，其量子临界点和行为均保持不变。换句话说，在零温度下，量子涨落会破坏任何小世界效应。这一精确的结果为量子临界点作为自然界的热力学稳定特征的重要性提供了新的线索，在经典水平上没有类似之处，并且本质上阻止了将网络理论简单地应用于量子系统。通过将量子古典映射与一个简单的拓扑参数相结合来获得该推导。