Stealthy hyperuniform disorder: A new route to controlling electric states and magnetic phase transition in correlated systems

Abstract: We investigate the effects of stealthy hyperuniform bond distributions on the electronic and magnetic properties of the Hubbard model on the honeycomb lattice. Hyperuniform structures, distinct from random and quasiperiodic ones, have recently attracted considerable interest due to their anomalous suppression of density fluctuations. By diagonalizing the noninteracting Hamiltonian, we show that a linear density of states (DOS) robustly emerges, while the stealth property of the bond distribution changes the wave functions in the higher-energy region extended and significantly modifies the DOS near the band edge. To clarify the impact on magnetism, we apply the real-space Hartree approximation to the Hubbard model. We find that, the phase transition always occurs between semimetallic and antiferromagnetically ordered states and its critical interaction strength is sensitive to the stealth property. A comparison with the quasiperiodic honeycomb tiling further highlights the role of structural correlations. These results demonstrate that stealthy hyperuniform disorder provides a novel route to controlling electronic states and magnetic phase transitions in correlated systems.

• The paper demonstrates that SHU disorder in the honeycomb Hubbard model raises the critical interaction strength (Uc) for magnetic ordering compared to random disorder.
• SHU bond distributions, generated via reverse Monte Carlo optimization, significantly modify the density of states and suppress localization near the band edge.
• Vertex analysis reveals that SHU constraints reduce homogeneous local environments, leading to distinct spatial magnetization patterns and rare-region effects.

Stealthy Hyperuniform Disorder as a Mechanism for Tunable Electronic and Magnetic Phases in the Honeycomb Hubbard Model

Introduction

The study offers a comprehensive investigation into the effects of stealthy hyperuniform (SHU) bond disorder on the electronic and magnetic ground states of the Hubbard model on the honeycomb lattice. SHU patterns, characterized by a structure factor $S({\bf k})$ that vanishes for all $|{\bf k}|$ less than some $k_c$, interpolate between random and highly correlated non-periodic structures. While disordered hyperuniformity has been recognized in soft matter, photonics, and other domains, its impact on strongly correlated electron systems and phase transitions has not been systematically examined at the Hamiltonian level on discrete lattices. This paper introduces SHU bond distributions via reverse Monte Carlo optimization and analyzes their consequences for electronic localization, spectral characteristics, and magnetic phase boundaries in comparison with random and quasiperiodic analogues.

Model and Methodology

The authors consider a half-filled Hubbard model on a honeycomb lattice, with the nearest-neighbor hopping integrals $t_\alpha$ and $t_\beta$ distributed according to either truly random, SHU, or quasiperiodic tilings, maintaining the bond ratio $N_{B\alpha}/N_{B\beta}=\tau$ (golden mean). Bond centers are used to define the structure factor for diagnosing SHU characteristics. SHU configurations are constructed by minimizing a cost functional that enforces $S({\bf k})=0$ within a window $|{\bf k}|<k_c$ in reciprocal space, gradually extending $k_c$ to interpolate between randomness and strong hyperuniformity. The resulting vertex statistics and bond inhomogeneity are then analyzed.

Figure 1: Representation of the honeycomb lattice and its unit vectors; the parallelogram denotes the simulation cell.

Figure 2: Bond distributions and corresponding structure factors for different $k_c$, illustrating the tuning from random to stealthy hyperuniform order as $|{\bf k}|$0 increases.

Noninteracting Electronic Structure: Density of States and Localization

Exact diagonalization of the noninteracting Hamiltonian ($|{\bf k}|$1) reveals that the linear density of states (DOS) at the Dirac point is robust against both random and SHU disorder, provided the global bond ratio is kept fixed. However, significant modifications emerge at higher energies near the band edge for SHU configurations with large $|{\bf k}|$2. The band edges shift to lower energies and the DOS becomes enhanced, with the most notable effects for $|{\bf k}|$3. Concomitantly, the inverse participation ratio (IPR) analysis demonstrates that SHU disorder suppresses localization and leads to more extended one-particle eigenstates over a substantial energy window near the band edge.

Figure 4: (a) DOS for several values of $|{\bf k}|$4; (b) Logarithmic IPR spectra for $|{\bf k}|$5 (random) and $|{\bf k}|$6 (SHU), showing reduced localization for the SHU case at high energies.

The effect is not uniform in $|{\bf k}|$7-space: imposing the stealthy constraint around the $|{\bf k}|$8 point significantly alters the high-energy electronic structure, whereas enforcing stealthiness in windows centered at the K/K' points (the Dirac points) predominantly leaves both the low- and high-energy DOS unchanged except for minor features in the intermediate range.

Figure 3: K-pattern bond configuration, associated structure factor, and DOS/IPR spectra, emphasizing the locality of SHU effects in reciprocal space.

Vertex Structure and Local Correlations

A classification of vertex types (C$|{\bf k}|$9) according to their local bond environments reveals that SHU patterns selectively reduce the fraction of locally homogeneous sites (C$k_c$0, C$k_c$1) as $k_c$2 increases. The vertex statistics for SHU and quasiperiodic tilings differ, indicating the nontrivial role of spatial correlations beyond the random ensemble, which impacts local electronic and magnetic properties.

Figure 5: (a) Four vertex types in the honeycomb lattice and their bond environments; (b) Vertex-type fractions as a function of $k_c$3, with comparisons to the golden-mean quasiperiodic tiling.

Magnetic Order and Phase Transition

Employing the real-space Hartree approximation, the phase diagram is analyzed for the Mott semimetal–antiferromagnet transition. Across all bond distributions, a finite $k_c$4 is required for magnetic order. However, the critical interaction strength $k_c$5 is observed to be sensitive to stealthiness: the SHU-disordered system demonstrates a larger $k_c$6 than its random counterpart for fixed bond ratio and vertex statistics. The averaged staggered magnetization grows more slowly with $k_c$7 for SHU patterns and is consistently smaller than the random case in the vicinity of the transition.

Figure 7: Average local magnetization versus $k_c$8 for random, SHU, and quasiperiodic cases, with the golden-mean tiling result displayed for comparison.

The Hartree-level critical $k_c$9 is underestimated compared to exact Monte Carlo results but the qualitative trends regarding bond distribution survive.

Fine Structure of Magnetization: Inverse Participation Ratio Analysis

The spatial inhomogeneity and rare-region effects in the developing antiferromagnetic order are quantified using first and second moment IPRs of the local magnetization. For large $t_\alpha$0, all systems exhibit nearly uniform magnetization reflecting saturated order. For $t_\alpha$1, however, the SHU system exhibits smaller IPR$t_\alpha$2 but larger IPR$t_\alpha$3 compared to random, indicating more homogeneous but sharper rare-peak distributions—a signature of spatially isolated strongly ordered regions amidst an otherwise weakly ordered background.

Figure 6: IPR$t_\alpha$4 (main) and IPR$t_\alpha$5 (inset) of the magnetization as a function of $t_\alpha$6 for SHU, random, and quasiperiodic bond structures, with error bands from disorder averaging.

Spatial magnetization maps support this: the SHU case produces more regular patterns with isolated strongly ordered sites, while the random case yields more widespread magnetization with smaller contrast.

Figure 8: Real-space maps of local magnetic moments at $t_\alpha$7 for SHU and random bond distributions, normalized to average (upper panels) and maximum (lower panels) values.

Interplay with Quasiperiodic Order

Comparison with quasiperiodic tilings reveals that higher fractions and clustering of highly magnetizable vertex types (C$t_\alpha$8) lead to stabilization of the ordered phase at smaller $t_\alpha$9 than in both SHU and random systems. Thus, the precise nature of structural correlations—periodic, quasiperiodic, or SHU—strongly modifies the balance of rare region effects and collective order.

Vertex-Constrained SHU Patterns

Applying SHU constraints simultaneously to bonds and vertex distributions further suppresses C$t_\beta$0 sites, but induces only minor additional changes in the DOS, IPR, and magnetization profiles compared to bond-only SHU, indicating the robustness of the main findings to the details of constraint implementation.

Figure 9: Top: Joint SHU constraint on bonds and vertices with the corresponding structure factor; the C$t_\beta$1 fraction is strongly reduced.

Figure 10: (a) DOS and (b) IPR for SHU vertex constraint; (c) magnetization versus $t_\beta$2, showing near quantitative agreement with SHU bond-only results.

Implications and Outlook

This study establishes SHU disorder as a distinctive route to patterning the electronic spectrum and tuning magnetic phase boundaries in correlated systems via the suppression and spatial structuring of rare regions. The results highlight the sensitivity of correlated ground states to nontrivial long-range correlations in disorder, going beyond conventional random or patterned lattices. Practically, this enables the prospect of harnessing SHU microstructure in engineered quantum materials for customizable spectral and phase behavior, including possible stabilization of Griffiths-like phases. Theoretically, it motivates further work employing many-body techniques surpassing mean-field, and the extension to multi-orbital or topological correlated models.

Conclusion

Stealthy hyperuniform disorder fundamentally modifies both one-particle and many-body electronic properties in the honeycomb Hubbard model. While the linear low-energy DOS and Dirac physics are robust, SHU patterns induce substantial delocalization at high energies and systematically raise the threshold for magnetic order relative to random and quasiperiodic analogues. The observed modulation of rare region statistics, vertex type distributions, and spatial inhomogeneity of magnetization demonstrates SHU disorder as a flexible and impactful tool for controlling quantum electronic and magnetic states in correlated lattices.

---

Reference: "Stealthy hyperuniform disorder: A new route to controlling electric states and magnetic phase transition in correlated systems" (2604.19041)