发表机构
CNR-Istituto dei Sistemi Complessi; Dipartimento di Fisica, Universitá La Sapienza di Roma（意大利国家研究委员会复杂系统研究所; 罗马大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明二维随机粒子构型在约束低波矢模式消失时存在唯一最近隐身超均匀构型，并通过梯度生成器网络验证了从随机初始条件生成无序超均匀态的方法。
AI 中文摘要
我们证明，在二维周期方形域中，对于几乎每一个由$N$个粒子组成的均匀随机构型，当一组有限的$m$个低波矢傅里叶模式被约束为精确消失时，存在唯一的最近隐身超均匀构型（模去软模位移）。在显式的满秩雅可比假设下，垂直于有限模式超均匀流形切空间方向的位移分量被唯一确定，而剩余的切空间自由度对应于保持受约束低波矢密度涨落的一阶软模。我们将注意力限制在约束模式比例$\chi=m/(dN)$的区域内，在该区域内已知此类构型保持无序而非结晶。为验证这一框架，我们实现了一个基于梯度的生成器网络，通过迭代移动粒子来最小化由超均匀性、短程排斥和平滑性组成的组合损失函数，其动力学预期近似于理论中到超均匀流形的最小正交投影。该网络将通用随机构型驱动至无序超均匀代表构型，逐步抑制长波长密度涨落（通过结构因子和数方差指数测量），其作用范围延伸至远超明确约束模式的倒空间窗口，并且我们证明该窗口受结构因子求和规则约束。局域键取向分析和布拉格峰的缺失证实所得状态保持无序。这种分析与计算相结合的方法为理解隐身超均匀性以及从通用随机构型生成其无序代表构型提供了严格的几何基础。
英文摘要
We prove that, for almost every uniformly random configuration of $N$ particles in a two-dimensional periodic square domain, there exists a unique nearest stealthy hyperuniform configuration, up to floppy-mode displacements, when a finite set of $m$ low-$k$ Fourier modes is constrained to vanish exactly. Under an explicit full-rank Jacobian assumption, the displacement component orthogonal to the tangent space of the finite-mode hyperuniform manifold is uniquely determined, while residual tangent-space degrees of freedom correspond to first-order floppy modes that preserve the constrained low-$k$ density fluctuations. We restrict attention to the regime of constrained-mode fraction $χ=m/(dN)$ in which such configurations are known to remain disordered rather than crystallize. To corroborate this framework, we implement a gradient-based generator network that iteratively displaces particles to minimize a combined loss functional of hyperuniformity, short-range repulsion, and smoothness, and whose dynamics is expected to approximate the theory's minimal orthogonal projection onto the hyperuniform manifold. The network drives generic random configurations toward disordered hyperuniform representatives, progressively suppressing long-wavelength density fluctuations as measured by the structure factor and number-variance exponent\textcolor{black}{, over a reciprocal-space window that extends well beyond the explicitly constrained modes and is bounded, as we show, by the structure-factor sum rule. Local bond-orientational analysis and the absence of Bragg peaks confirm that the resulting states remain disordered}. This integrated analytical and computational approach provides a rigorous geometric foundation for understanding stealthy hyperuniformity and for generating its disordered representatives from generic random configurations.
