Existence and Uniqueness of Nearest Stealthy Hyperuniform Configurations from Random Initial Conditions

Technical Summary: Existence and Uniqueness of Nearest Stealthy Hyperuniform Configurations

Problem Statement
Disordered hyperuniform (DHU) systems occupy a unique state between crystalline order and complete randomness, characterized by the suppression of long-wavelength density fluctuations (vanishing structure factor $S(k)$ as $k \to 0$). While numerical constructions of DHU states via "stealthy" potentials (constraining a finite set of low-$k$ Fourier modes to vanish) have demonstrated that such states exist and can be disordered, a fundamental geometric question remained unresolved: Given a generic random configuration of $N$ particles, does a well-defined, locally unique "nearest" hyperuniform configuration exist? Furthermore, under what conditions is this nearest state disordered rather than crystalline? Previous work relied on energy minimization from random initial conditions but lacked a rigorous geometric proof of the existence and uniqueness of the projection onto the hyperuniform manifold.

Methodology
The author employs a dual approach combining rigorous differential geometry with computational optimization:

1. Mathematical Formulation:

   • The problem is framed in a $2N$-dimensional configuration space $(T^2)^N$ for $N$ particles in a 2D periodic square domain.
   • A constraint map $F$ is defined to enforce the "stealthy" condition: the real and imaginary parts of $m$ low-$k$ collective density modes $\hat{\rho}(k_\alpha)$ are set to zero. The solution set $M = F^{-1}(0)$ represents the hyperuniform manifold.
   • Assumption: The author assumes a full-rank Jacobian condition ($\text{rank } DF(X) = 2m$), which holds with probability one for uniformly random configurations. This is explicitly verified via a lattice-based witness construction (Lemma 1).
   • Theoretical Tools: The Implicit Function Theorem is used to establish $M$ as a smooth submanifold of codimension $2m$. The author then applies the theory of metric projections to prove the existence and uniqueness of the nearest point on $M$ to a generic random configuration $X_0$.

2. Computational Implementation:

   • A gradient-based generator network is implemented to numerically approximate the theoretical projection.
   • The network minimizes a composite loss functional $\mathcal{L} = \mathcal{L}_{\text{hyper}} + \mathcal{L}_{\text{rep}} + \mathcal{L}_{\text{smooth}}$:
     • $\mathcal{L}_{\text{hyper}}$: Penalizes deviations of $S(k)$ from a small target value for $|k| < k_c$.
     • $\mathcal{L}_{\text{rep}}$: Enforces short-range repulsion to prevent overlaps.
     • $\mathcal{L}_{\text{smooth}}$: Regularizes high-frequency fluctuations to prevent crystallization.
   • Simulations are conducted for $N=10^4$ particles in a 2D box, starting from uniform random initial conditions.

Key Contributions

• Rigorous Existence and Uniqueness: The paper proves that for almost every random initial configuration, there exists a unique nearest stealthy hyperuniform configuration, up to "floppy-mode" displacements (tangent directions that preserve the constraints to first order).
• Geometric Characterization: The author establishes that the set of finite-mode hyperuniform states forms a smooth submanifold. They identify the "hyperuniform inherent structure" as the unique projection of a random configuration onto this manifold, analogous to the inherent structure paradigm in energy landscapes but defined by minimal Euclidean displacement rather than energetic relaxation.
• Quantitative Approximation: The closed-form solution for the linearized displacement ($\delta X^*$) is derived, and the author shows that the Gauss-Newton iteration converges quadratically to the exact nearest point, providing a constructive approximation.
• Disorder Verification: Numerical experiments confirm that within the regime of constrained-mode fraction $\chi = m/(dN)$, the resulting states remain disordered, lacking Bragg peaks and long-range bond-orientational order.

Results

• Theoretical: Theorem 1 establishes that the metric projection $\pi_M(X_0)$ exists and is unique within a local basin of validity. The displacement orthogonal to the manifold is uniquely determined, while residual degrees of freedom correspond to floppy modes.
• Numerical:
  • The generator network successfully drives random configurations toward the hyperuniform manifold.
  • The number-variance exponent $\alpha$ transitions from the Poisson value ($\alpha \approx 2$) to the class-I hyperuniform value ($\alpha \approx 1$), confirming the suppression of long-wavelength fluctuations.
  • Effective Suppression: While only $m=22$ modes were explicitly constrained ($\chi \approx 1.1 \times 10^{-3}$), the structure factor $S(k)$ is suppressed over a much wider reciprocal-space window, effectively constraining $\approx 5.7 \times 10^3$ modes ($\chi_{\text{eff}} \approx 0.28$). This remains below the known threshold for crystallization in 2D ($\chi \approx 0.5$).
  • Stability: Particle positions do not diffuse significantly during optimization (Pearson correlation coefficients remain near 1), and local bond-orientational analysis confirms the absence of crystalline order.

Significance
The paper provides a rigorous geometric foundation for understanding stealthy hyperuniformity. By connecting the concept of hyperuniformity to the theory of inherent structures, it redefines these states not merely as energy minima of specific potentials, but as geometrically determined "inherent structures" of random particle arrangements. This work clarifies the conditions under which a random configuration admits a unique, disordered hyperuniform representative, bridging the gap between abstract geometric constraints and constructive computational generation. The framework offers a principled method for generating DHU states from arbitrary initial conditions, validated by both analytical proofs and numerical convergence.