Random, uncorrelated displacements of particles on a lattice preserve the hyperuniformity of the original lattice, that is, normalized density fluctuations vanish in the limit of infinite wavelengths. In addition to a diffuse contribution, the scattering intensity from the the resulting point pattern typically inherits the Bragg peaks (long-range order) of the original lattice. Here we demonstrate how these Bragg peaks can be hidden in the effective diffraction pattern of independent and identically distributed perturbations. All Bragg peaks vanish if and only if the sum of all probability densities of the positions of the shifted lattice points is a constant at all positions. The underlying long-range order is then “cloaked” in the sense that it cannot be reconstructed from the pair correlation function alone. On the one hand, density fluctuations increase monotonically with the strength of perturbations $a$, as measured by the hyperuniformity order metric $\overline{\mathrm{\Lambda }}$. On the other hand, the disappearance and reemergence of long-range order, depending on whether the system is cloaked as the perturbation strength increases, is manifestly captured by the $\tau$ order metric. Therefore, while the perturbation strength $a$ may seem to be a natural choice for an order metric of perturbed lattices, the $\tau$ order metric is a superior choice. It is noteworthy that cloaked perturbed lattices allow one to easily simulate very large samples (with at least ${10}^6$ particles) of disordered hyperuniform point patterns without Bragg peaks.

中文翻译：

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隐藏潜在的随机扰动晶格的远程顺序

粒子在晶格上的随机，不相关的位移保留了原始晶格的高度均匀性，也就是说，归一化的密度波动在无限波长的范围内消失了。除了弥散作用外，所得点图案的散射强度通常会继承原始晶格的布拉格峰（远距离顺序）。在这里，我们演示了如何将这些布拉格峰隐藏在独立且均布的扰动的有效衍射图中。当且仅当移动的晶格点的位置的所有概率密度的总和在所有位置都是常数时，所有布拉格峰才消失。然后，就无法掩盖潜在的远程顺序，因为它不能单独从对相关函数中重建。一方面，$\mathrm{一种}$，由超均匀度指标衡量 $\overline{\mathrm{\Lambda }}$。另一方面，远程系统的消失和重新出现，显然取决于系统是否随着扰动强度的增加而被掩盖。$\tau$订单指标。因此，同时扰动强度$\mathrm{一种}$ 对于扰动晶格的阶次度量，似乎是自然的选择， $\tau$订单指标是上乘的选择。值得注意的是，隐蔽的扰动晶格允许人们轻松地模拟非常大的样本（至少${10}^6$ 粒子）无布拉格点的无序超均匀点模式。