Quasiperiodicity-Engineered Re-entrant Localization-Delocalization aspects in a Diamond Lattice

Abstract: We investigate localization in a quasiperiodically engineered diamond lattice with strand-dependent Aubry-André-Harper onsite modulations, highlighting the decisive roles of the modulation ratio $s$ and the averaged potential on the middle strand. The upper strand hosts the primary potential $λ$, the lower strand carries a weaker modulation $λ/s$, and the middle strand follows their average, generating a correlated quasiperiodic landscape across each plaquette. By tuning $λ$ for selected values of $s$, we probe spectral and eigenstate properties via the inverse participation ratio (IPR), normalized participation ratio (NPR), and fractal dimension $D_2$. We uncover a pronounced re-entrant localization behavior, where eigenstates repeatedly switch between extended and localized regimes, which persists only within a finite range of $s$ and crucially relies on the averaged potential construction. This unconventional sequence arises from the interplay of $s$, the correlated potential, and the intrinsic diamond geometry, producing a highly nontrivial interference landscape. Our results reveal localization physics beyond the standard Aubry-André paradigm, further supported by the evolution of extended states, system-size scaling of $\langle \mathrm{NPR} \rangle$ and $\langle D_2 \rangle$, and dynamical signatures from the time-dependent root-mean-square displacement, confirming the robustness of the re-entrant transitions.

• The paper demonstrates that a diamond lattice with engineered strand-averaged AAH modulations exhibits nonmonotonic re-entrant localization–delocalization transitions.
• It employs spectral (IPR, NPR, fractal dimensions) and dynamical analyses (root-mean-square displacement) to characterize eigenstate behaviors.
• The study identifies the modulation ratio s as a key parameter for tuning mobility edges, offering promising applications in photonic and cold-atom systems.

Quasiperiodicity-Induced Re-entrant Localization–Delocalization in Diamond Lattices

Introduction

This work presents a systematic study of localization phenomena in a quasiperiodically modulated diamond lattice with engineered Aubry–André–Harper (AAH) onsite potentials distributed in a strand-dependent fashion. The research delineates how the interplay between quasiperiodic modulation strength, the modulation ratio $s$, and the spatial averaging of onsite potentials on the middle strand drives a robust sequence of re-entrant localization–delocalization transitions, constituting an atypical localization scenario beyond the conventional AAH paradigm. The findings are substantiated through extensive spectral and eigenstate analyses exploiting both static and dynamical measures.

Model and Methodology

The lattice under study is a diamond chain featuring three strands (I: upper, II: middle, III: lower) with nearest-neighbor hopping. The modulations are configured such that the upper and lower strands host AAH onsite potentials of strengths $\lambda$ and $\lambda/s$, respectively, with the middle strand receiving their arithmetic average. The incommensurate modulation employs an irrational $\beta = (1+\sqrt{5})/2$.

Localization properties are evaluated via standard diagnostics:

• Inverse participation ratio (IPR): Quantifies the spatial concentration of eigenstates, distinguishing extended ($\sim 1/L$) from localized ($O(1)$) character.
• Normalized participation ratio (NPR): Complementary to IPR; approaches unity for extended states and zero for localized.
• Fractal dimension ($D_2$): Assesses the multifractality of eigenstates, identifying critical behavior for intermediate values.
• Spectral and eigenstate averaging: Used for global characterization.
• Dynamical probe: Time-dependent root-mean-square displacement, capturing wavepacket evolution.

Finite-size scaling and systematic parameter sweeps ensure robustness against finite-size effects and confirm the generality of the observed phenomena.

Results and Analysis

Re-entrant Localization–Delocalization Behavior

A prominent hallmark observed is the nonmonotonic, re-entrant sequence of localization–delocalization transitions as the modulation strength $\lambda$ increases. For $2 \lesssim s \lesssim 5$, eigenstates (particularly near the band center) transition from extended to localized and then back to extended states, before ultimately localizing at larger $\lambda$. This oscillatory alternation is absent for both $\lambda$0 and large $\lambda$1, establishing $\lambda$2 as a crucial control parameter. The effect is critically dependent on the use of the averaged onsite potential for the middle strand. When this is replaced with an uncoupled AAH modulation, the re-entrant phenomena are suppressed.

Key spectral signatures across the full energy spectrum and restricted band-center regions include:

• Multiple IPR/NPR oscillations with respect to $\lambda$3, corresponding to successive LD transitions.
• The band-center states being most sensitive, displaying highly nontrivial alternation between extended, critical, and localized regimes.
• Fractal dimension ($\lambda$4) analysis reveals a hierarchy of critical and multifractal states, mapping onto the observed re-entrant transitions.

The nontrivial interference landscape stemming from the incommensurate, correlated strand-dependent potentials and the intrinsic geometry of the diamond chain produces persistent mobility edges and self-similar spectral features, distinguishing this model from traditional AAH and its single-stranded generalizations.

Robustness and System-Size Effects

Finite-size analysis demonstrates that the nonmonotonic LD–DL–LD transition is not a finite-size artifact. The sequence persists for system sizes ranging from $\lambda$5 to $\lambda$6 plaquettes, and for both narrow ($\lambda$7) and wide ($\lambda$8) energy windows. The dependence on $\lambda$9 is highly nontrivial: increasing $\lambda/s$0 sharpens and shifts the re-entrant features, further confirming its precise control over the window and amplitude of the re-entrant regime.

Extended State Counting and Spectral Regions

The number of extended eigenstates as a function of $\lambda/s$1 exhibits a clearly nonmonotonic profile, with a re-emergence of extended states at intermediate disorder strengths even after an initial depletion—a notable contradiction to monotonic localization expected in standard AAH or Anderson-like scenarios.

Dynamical Signatures

Time-evolution studies using root-mean-square displacement display equivalent nonmonotonic dependence on $\lambda/s$2, with an initial suppression in spreading followed by revival, substantiating the dynamical manifestation of the re-entrant extended regime.

Theoretical and Practical Implications

Beyond Standard Localization Paradigms

These results position the strand-averaged diamond lattice as an archetype of correlated quasiperiodic systems with localization–delocalization behavior decoupled from universal self-dual symmetry constraints. The findings indicate that nontrivial geometry and engineered multi-strand correlations can produce mobility-edge phenomena and re-entrant transitions inaccessible to 1D AAH, Fibonacci, or even conventional multi-chain models.

Tunable Anomalous Transport

The explicit tunability via $\lambda/s$3 and the rich critical regime open possibilities for engineering custom mobility edges, multifractal states, and dynamical transport regimes in photonic, cold-atom, or superconducting circuit realizations, where precise site-dependent and strand-resolved potentials can be imposed with modern experimental technologies.

Perspectives for Further Study

The investigation motivates several pressing directions:

• Inclusion of interactions: The impact of many-body effects and possible realization of critical many-body phases.
• Extension to higher dimensions: Exploring analogous re-entrant transitions in 2D/3D networks with more complex geometries.
• Driven and non-Hermitian generalizations: Examining how periodic driving and non-Hermitian perturbations (e.g., gain/loss) affect re-entrant phenomenology.
• Disorder vs. quasiperiodicity: Elucidating the interplay when true random potentials compete with engineered quasiperiodicity.

Conclusion

This work establishes that a diamond lattice with engineered, strand-averaged AAH modulations supports robust and tunable re-entrant localization–delocalization transitions. The phase diagram, controlled by the modulation ratio $\lambda/s$4 and the nature of spatial correlations, reveals spectral, multifractal, and dynamical features beyond the canonical AAH and Anderson models. These findings delineate a promising direction for the controlled study of criticality, anomalous transport, and mobility edge phenomena in artificial quantum lattices, facilitating future explorations in both theoretical and experimental condensed matter physics.

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Reference:

"Quasiperiodicity-Engineered Re-entrant Localization-Delocalization aspects in a Diamond Lattice" (2603.29173)