Impact of Stealthy Hyperuniform Magnetic Impurity Configurations on Bulk Magnetism in a Two-dimensional Heisenberg Model

Abstract: We investigate an antiferromagnetic quantum Heisenberg model on a square lattice with high-spin magnetic impurities to clarify how random and stealthy hyperuniform impurity configurations influence the bulk magnetic properties. Stealthy hyperuniform configurations are generated using generalized cost functions that interpolate between square-lattice-like and triangular-lattice-like arrangements. Using linear spin-wave theory for the mixed-spin model, we demonstrate that triangular-lattice-like arrangements yield a larger average staggered magnetization than both random and square-lattice-like cases. This enhancement originates from sublattice effects: while the square-lattice-like configuration enforces nearest-neighbor impurities to occupy opposite sublattices due to its bipartite structure, the triangular-lattice-like arrangement allows same-sublattice nearest-neighbor pairs, thereby strengthening cooperative magnetic enhancement.

• The paper demonstrates that stealthy hyperuniform impurity configurations control bulk staggered magnetization in a mixed-spin square-lattice Heisenberg model using linear spin-wave theory.
• Triangular-lattice-like arrangements produced the strongest enhancement, reaching mean M_AF = 0.3759 versus 0.3729 for random configurations, while reducing sample variability by roughly an order of magnitude.
• A Gaussian-weighted sublattice-pair descriptor F predicts magnetization across random, hyperuniform, and clustered samples, offering a practical design rule for engineering magnetic order.

The study of disordered magnetic systems has long been dominated by either single-impurity perturbation theory or fully periodic superlattice constructions, leaving the intermediate question of how systematically controlled, non-random spatial correlations among impurities affect bulk magnetism largely unaddressed. The work by Asakura, Yamamoto, and Koga addresses this gap by combining the theory of stealthy hyperuniform (SHU) point patterns with linear spin-wave theory for a mixed-spin antiferromagnetic Heisenberg model on a square lattice (2602.22484). The central finding is that the geometry of impurity placement—specifically, whether nearest-neighbor impurity pairs occupy the same or opposite sublattices—controls the bulk staggered magnetization, with triangular-lattice-like SHU arrangements outperforming both random and square-lattice-like configurations.

Model and theoretical framework

The authors consider the nearest-neighbor antiferromagnetic Heisenberg Hamiltonian $H = J\sum_{\langle ij\rangle}\bm{S}_i\cdot\bm{S}_j$ on a square lattice of $L\times L$ sites with periodic supercells, in which host spins carry $S=1/2$ and magnetic impurities carry $S=2$. Impurities are introduced in equal numbers on the two sublattices so that the antiferromagnetic ground state carries no net uniform magnetization. The magnetic properties are computed via linear spin-wave theory: a Holstein-Primakoff expansion yields a quadratic bosonic Hamiltonian with a diagonal on-site block $h$ and an off-diagonal anomalous block $\Delta_{\bm{k}}$, which is diagonalized by a Bogoliubov transformation. The appendix provides a careful derivation of the diagonalization using the generalized (indefinite) inner product defined with the metric $P=\mathrm{diag}(1,\dots,1,-1,\dots,-1)$, ensuring that the bosonic commutation relations are preserved and that eigenvalues come in $\pm$ pairs. The local magnetization follows from the Bogoliubov coefficients, and the spatially averaged staggered magnetization $M_{\mathrm{AF}}$ is obtained by summing with the sublattice sign factor.

Generation of stealthy hyperuniform impurity configurations

The impurity configurations are generated by minimizing a cost function that penalizes deviations of the impurity structure factor $S(\bm{k})$ from a target within a window function. For SHU patterns, the target is $L\times L$0 for $L\times L$1. The key methodological innovation is a generalized cost function $L\times L$2, where the second term enforces stealthiness of the impurity distribution within each sublattice separately. Two representative configurations result:

Configuration	Parameters $L\times L$3	Geometry
(I)	$L\times L$4	Triangular-lattice-like
(II)	$L\times L$5	Square-lattice-like

Increasing $L\times L$6 drives impurities apart, and for large $L\times L$7 the pattern approaches triangular ordering within the supercell. The sublattice-resolved stealthiness constraint in configuration (II) instead forces impurities on each sublattice to separate, producing a square-lattice-like arrangement.

Single-impurity physics and the sublattice mechanism

The physical origin of the configuration dependence is established through the single-impurity problem. An isolated $L\times L$8 impurity embedded in the $L\times L$9 background has its moment renormalized down to approximately $S=1/2$0, substantially below the value $S=1/2$1 found in the uniform-$S=1/2$2 Heisenberg antiferromagnet, reflecting the enhanced quantum fluctuations from the surrounding smaller spins. Crucially, the response is sublattice-dependent: nearby host spins on the same sublattice as the impurity show enhanced moments, while nearest-neighbor spins on the opposite sublattice remain nearly unchanged. This asymmetry extends over several lattice spacings and implies that the relative sublattice positions of multiple impurities should govern the collective magnetic response at finite concentration—a prediction the finite-density calculations confirm.

Configuration dependence of the bulk magnetization

At an impurity concentration of $S=1/2$3 on a $S=1/2$4 lattice, the magnetization profiles reveal distinct cooperative behaviors. In the random configuration, both impurity separations and nearest-neighbor sublattice relationships vary locally, so magnetic correlations are enhanced in some regions and suppressed in others. In configuration (I), the nearly uniform impurity spacing combined with the possibility of same-sublattice nearest-neighbor pairs produces a cooperative enhancement of the staggered magnetization. Configuration (II), by contrast, enforces opposite-sublattice nearest-neighbor pairs through its bipartite structure, and no cooperative enhancement occurs.

The statistical analysis over independent impurity realizations makes the contrast quantitative. Random configurations yield a broad distribution of $S=1/2$5 with mean $S=1/2$6 and standard deviation $S=1/2$7 (100 samples). Configuration (I) gives a larger mean of $S=1/2$8 with a much narrower spread ($S=1/2$9, 40 samples), while configuration (II) gives $S=2$0 with $S=2$1. Two conclusions follow directly. First, SHU constraints suppress realization-to-realization fluctuations in the magnetic response by an order of magnitude relative to random doping, indicating a more uniform and predictable bulk magnetism. Second, the mean enhancement of configuration (I) over both random and configuration (II) samples is attributable to same-sublattice impurity pairs, whose enhancing and suppressing contributions statistically compensate in the random case.

The authors also test clustered configurations at the same concentration. A mixed-sublattice cluster screens internally—the moments of adjacent $S=2$2 spins are reduced to $S=2$3—and yields a small bulk magnetization, whereas a cluster composed of impurities from a single sublattice enhances correlations inside and around the cluster and produces a significantly larger staggered magnetization of $S=2$4. This demonstrates that clustering per se is insufficient; the sublattice-resolved arrangement is the essential ingredient.

Finally, the authors introduce a scalar descriptor $S=2$5, a Gaussian-weighted sum over impurity pairs of the product of sublattice signs, which is large when same-sublattice impurities are close and small when opposite-sublattice impurities are close. All samples—random and both SHU types—collapse onto a single monotonic curve of $S=2$6 versus $S=2$7, indicating that this quantity captures the essential magnetic response of the system and can serve as a structural predictor of the bulk staggered magnetization. This collapse is arguably the most practically useful result of the paper, as it provides a computable design criterion for impurity arrangements with enhanced magnetic order.

Limitations and open questions

The authors state explicitly that the study is restricted to relatively small system sizes ($S=2$8) and a single impurity density, so finite-size effects and the density dependence of the $S=2$9–$h$0 collapse remain unverified. The analysis is also confined to linear spin-wave theory, which neglects magnon-magnon interactions and may be quantitatively inaccurate for the strongly fluctuating $h$1 background; whether the sublattice-based enhancement mechanism survives beyond the $h$2 expansion is an open question. Additionally, the SHU configurations are constructed within finite supercells, and the thermodynamic-limit behavior of the triangular-lattice-like enhancement is not established. Whether the same-sublattice mechanism generalizes to three-dimensional lattices or to other host spin magnitudes is left unexamined.

Conclusion

This paper demonstrates that systematically controlled spatial correlations—realized through stealthy hyperuniform impurity patterns—can tune the bulk staggered magnetization of a mixed-spin antiferromagnet in ways that random doping cannot. The enhancement in triangular-lattice-like SHU configurations arises from same-sublattice nearest-neighbor impurity pairs, while square-lattice-like bipartite arrangements preclude it, and the single descriptor $h$3 unifies all observed configurations onto a single predictive curve. The work establishes hyperuniform point-process engineering as a concrete framework for controlling magnetic order in compositionally disordered quantum magnets, with high-entropy oxides providing a plausible experimental target for these predictions.