Interplay of Antiferromagnetism and Quasiperiodicity in a Hubbard Ring: Localization Insights

Abstract: We study localization in a quasiperiodic spinful antiferromagnetic Hubbard ring within a self-consistent Hartree-Fock framework, emphasizing the interplay of quasiperiodicity, staggered Zeeman-field-induced antiferromagnetic order, and electron correlations. Localization properties are characterized through inverse participation ratios, normalized participation ratios, and multifractality, and are consistently supported by a broad class of real-space mean-field observables, including double occupancy, density fluctuations, local entropy, spin-density-wave (SDW) order, and other related correlation measures. We uncover a pronounced nonmonotonic evolution of localization with interaction strength, featuring an intermediate regime marked by enhanced localization, strong spatial inhomogeneity, and magnetic ordering, followed by a re-entrant tendency toward delocalization at stronger interaction regime. Phase diagrams constructed from complementary localization and mean-field indicators reveal extended, localized, and critical regimes governed by the interplay of quasiperiodicity and interactions. Furthermore, real-time wave-packet dynamics of eigenstates provide direct evidence of ballistic spreading, confinement, and re-entrant transport, in agreement with the underlying spectral characteristics. These results establish a unified framework where diverse mean-field observables and dynamical probes consistently capture correlation-driven localization phenomena in quasiperiodic systems.

• The paper demonstrates nonmonotonic localization behavior as Hubbard U tunes transitions from extended to localized states in a quasiperiodic ring.
• The research employs a self-consistent Hartree–Fock approach to analyze spin-resolved eigenstates using IPR, NPR, and fractal dimensions.
• The study shows tunable phase boundaries and re-entrant delocalization, with dynamic simulations linking equilibrium spectral properties to transport.

Interplay of Antiferromagnetism and Quasiperiodicity in a Hubbard Ring: Localization Insights

Model and Methodology

The study rigorously analyzes a one-dimensional spinful Hubbard ring incorporating quasiperiodic modulation in the hopping amplitude and a staggered spin-dependent Zeeman field, within a self-consistent Hartree–Fock mean-field framework. The lattice employs an incommensurate modulation through a cosine function with irrational frequency, forcing deterministic aperiodicity. The antiferromagnetic order arises from the Zeeman field, which alternates in sign, explicitly breaking spin symmetry. The Hubbard $U$ term models on-site interactions for both spin channels.

The electron–electron interactions are decoupled within Hartree–Fock, generating coupled, site- and spin-dependent effective potentials. The resultant spin-resolved effective single-particle Hamiltonians are diagonalized self-consistently to obtain the spin-dependent eigenstates for observable calculations.

The localization properties are quantified using inverse participation ratio (IPR), normalized participation ratio (NPR), and the averaged second-order fractal dimension ($D_2$). Additional equilibrium and real-space mean-field observables include local density variance, spin-density amplitude, double occupancy, local entropy, and single-particle excitation gap, providing complementary measures of spatial inhomogeneity, spin ordering, and electronic correlations. Time-dependent dynamics of initial wave packets are simulated for direct assessment of nonequilibrium transport.

Interaction-Induced Localization, Spectral Reconstruction, and Spin Asymmetry

The primary result is the nonmonotonic dependence of localization on the Hubbard repulsion $U$, manifesting distinct physical regimes as a function of both interaction strength and the magnitudes of the quasiperiodic ($\lambda$) and Zeeman ($h_z$) fields. For weak interactions, the system predominantly remains in a delocalized phase. Upon increasing $U$, an intermediate regime arises, where the IPR reflects the proliferation of localized states, formation of additional sharply localized bands, and pronounced spatial inhomogeneity. This is accompanied by enhanced density variance, double occupancy and local entropy, and the opening of a single-particle gap.

Notably, at larger interaction strengths, the system exhibits a re-entrant delocalization, evidenced by the reduction of IPR and restoration of extended eigenstates. This regime is a result of strong Hartree renormalization overtaking quasiperiodic inhomogeneity.

Spin-resolved analysis uncovers a finite window of enhanced spin-dependent localization, where the small mean-field spin-density imbalance, amplified by $U$, leads to a measurable dichotomy in the localization characteristics for spin-up and spin-down channels, as pinpointed by significant maxima in $\Delta\mathrm{IPR}$ and $\Delta\mathrm{NPR}$. However, across most of the spectrum and outside the intermediate $U$ regime, spin-resolved localization properties overlap, indicating strong restoration of effective spin symmetry at large $D_2$0.

Phase Diagrams, Multifractality, and Magnetic Ordering

Phase diagrams constructed from the combined localization indicator $D_2$1, IPR extrema, and average fractal dimension $D_2$2 reveal the coexistence and competition of extended, localized, and critical multifractal regimes. The regime boundaries, and the thickness of the intermediate (critical) phase, are tunable via $D_2$3 and $D_2$4. For weak $D_2$5, a broad delocalized window is observed, whereas strong $D_2$6 leads to suppression of delocalization and an augmentation of the critical and localized regimes.

Analysis of the spin-density-wave (SDW) amplitude confirms re-entrant nonmonotonic ordering: at fixed $D_2$7, $D_2$8 grows with $D_2$9 at small interaction, is strongly suppressed in the intermediate, localized regime, and increases again at higher $U$0. This provides magnetic corroboration of the interaction-induced localization–delocalization crossovers. Increasing $U$1 further broadens the intermediate, low-SDW region, consistent with enhancement of the critical/multifractal regime.

Real-Time Dynamics and Connection to Static Diagnostics

Time evolution of site- and spin-resolved wave packets under the self-consistent mean-field Hamiltonian yields ballistic expansion in the extended regime (low and high $U$2), strong confinement in the localized and critical windows (intermediate $U$3), and redistribution of localization between edge and bulk states as a function of $U$4. These effects are observed through the root-mean-square displacement $U$5 and long-time survival probability $U$6, which quantitatively track the static localization diagnostics: minima in $U$7 and maxima in $U$8 correlate with enhanced mean-field IPR and multifractal dimension suppression.

Theoretical and Practical Implications

This work illustrates that the interplay of quasiperiodicity, spin-dependent Zeeman fields, and interactions yields a highly tunable platform for accessing, controlling, and interrogating extended, localized, and multifractal regimes:

• Re-entrant localization/delocalization and nontrivial spin-resolved localization: these effects are robust across the spectrum and persist out-of-equilibrium, implying that mean-field-induced spectral reorganization and spin asymmetry are generic features in correlated, quasiperiodic quantum materials.
• Static–dynamic correspondence: the explicit connection between equilibrium spectral properties and nonequilibrium transport dynamics validates both as mutual diagnostics and as tools for probing correlation-induced effects in experiment.
• Phase tunability: the width and nature of localized and critical regimes can be engineered by adjusting $U$9, $\lambda$0, and $\lambda$1, with implications for quantum control, state preparation, and the realization of tunable criticality in quantum simulation experiments based on ultracold atoms or engineered synthetic lattices.

Future Outlook

The mean-field, Hartree–Fock approach captures salient features of interaction-driven reconstruction but omits fluctuation effects and possible spontaneous symmetry breaking. Future research incorporating exact diagonalization, DMRG, or dynamical mean-field theory may further elucidate beyond-mean-field physics, including many-body localization, emergent correlations, and finite-temperature properties. Extensions to higher dimensions, inclusion of disorder, and coupling to driven or dissipative baths are natural follow-ups, directly relevant to both condensed-matter systems and quantum simulation platforms. The demonstrated static–dynamic correspondence suggests that nonequilibrium probes, such as quantum quenches or time-resolved transport, will remain powerful diagnostic tools.

Conclusion

The investigation establishes a comprehensive framework for understanding the complex correlation between antiferromagnetism and quasiperiodicity in Hubbard rings. The work demonstrates that interactions can both enhance and suppress localization in a nonmonotonic fashion, with clear dynamical manifestations and a significant, yet controlled, impact on spin-selective localization. The results are directly relevant for engineered quantum systems and inform both theory and experiment on the conditions for realizing and manipulating nontrivial localization and transport phenomena in correlated quasiperiodic media.

---

Reference: "Interplay of Antiferromagnetism and Quasiperiodicity in a Hubbard Ring: Localization Insights" (2603.29177)