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arXiv:2605.18064 · Physical review. B./Physical review. B

Localization transitions in a half-filled helical Aubry-André model

Anonymous, B. Tanatar, Balázs Hetényi

Published Jul 31, 2026• 0 citations
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Abstract

We investigate localization in a one-dimensional quasiperiodic lattice obtained by extending the Aubry-André model with a structured N th -neighbor hopping term of strength J N . This additional channel connects successive windings of an effective helical chain, introducing both a hopping scale and a geometric length scale into the localization problem. Under periodic boundary conditions, we identify the localization transition using a geometric Binder cumulant U 4 constructed from the polarization amplitudes of the occupied many-body Slater determinant. The finite-size critical potential is extracted from the zero crossing of U 4 . At fixed helical range N , increasing J N shifts the transition toward larger quasiperiodic potential strengths. In contrast, varying N produces a strongly nonmonotonic phase boundary with pronounced spikelike enhancements of the critical potential. Using Fibonacci system sizes together with a Zeckendorf-shift construction for the particle number, we show that the dominant spike structure survives the thermodynamic-limit extrapolation. The largest enhancements are associated with helical ranges for which the phase shift 2 π β N approaches an in-phase or antiphase resonance condition. We supplement the polarization analysis with occupied-state IPR and NPR, energy-resolved IPR maps, and the single-particle Fermi gap. The Fermi gap begins to open at the same crossover onset at which U 4 departs from its conducting-regime plateau, providing a complementary spectral signature of the same insulating transition. The participation ratios show that, in the half-filled sectors studied here, this transition is also accompanied by a gradual and energy-dependent localization of the single-particle eigenstates.

DOI
10.1103/c5wk-3ym5

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We investigate localization in a one-dimensional quasiperiodic lattice obtained by extending the Aubry-André model with a structured N th -neighbor hopping term of strength J N . This additional channel connects successive windings of an effective helical chain, introducing both a hopping scale and a geometric length scale into the localization problem. Under periodic boundary conditions, we identify the localization transition using a geometric Binder cumulant U 4 constructed from the polarization amplitudes of the occupied many-body Slater determinant. The finite-size critical potential is extracted from the zero crossing of U 4 . At fixed helical range N , increasing J N shifts the transition toward larger quasiperiodic potential strengths. In contrast, varying N produces a strongly nonmonotonic phase boundary with pronounced spikelike enhancements of the critical potential. Using Fibonacci system sizes together with a Zeckendorf-shift construction for the particle number, we show that the dominant spike structure survives the thermodynamic-limit extrapolation. The largest enhancements are associated with helical ranges for which the phase shift 2 π β N approaches an in-phase or antiphase resonance condition. We supplement the polarization analysis with occupied-state IPR and NPR, energy-resolved IPR maps, and the single-particle Fermi gap. The Fermi gap begins to open at the same crossover onset at which U 4 departs from its conducting-regime plateau, providing a complementary spectral signature of the same insulating transition. The participation ratios show that, in the half-filled sectors studied here, this transition is also accompanied by a gradual and energy-dependent localization of the single-particle eigenstates.
We study the finite size scaling of the bulk polarization in a quasiperiodic (Aubry-Andr\'e) model using the geometric analog of the Binder cumulant. As a proof of concept, we show that the geometric Binder cumulant method described here can reproduce the known literature values for the flat and raised cosine distributions, which are the two distributions that occur in the delocalized phase. For the Aubry-Andr\'e model at half-filling, the phase transition point is accurately reproduced. Not only is the correct size scaling exponent of the variance obtained in the extended and the localized phases, but the geometric Binder cumulant undergoes a sign change at the phase transition. We also calculate the state resolved Binder cumulant as a function of disorder strength to gain insight into the mechanism of the localization transition.
In this study, we investigate the localization transition and quantum criticality {in the ground state of the} disordered Aubry-André-Harper (AAH) model, where a quasiperiodic potential is hybridized with a disordered potential. In the clean limit, the AAH model undergoes a localization transition from an extended phase to a localized phase via an intermediate critical phase as the strength of the quasiperiodic potential is varied. While the staggered potential merely shifts the critical point to a lower value, Fibonacci and Thue-Morse potentials induce immediate localization. This contrast reveals the sensitivity of localization behavior to the structural complexity of the potential, with the onset of localization correlating with the sequence's complexity. More specifically, the system follows a hierarchy defined by the complexity measures of the applied potentials. In addition, the typical fidelity susceptibility exhibits a power-law scaling behavior at the localization transition, enabling reliable extraction of the critical exponent. We focus on the AAH model with the Fibonacci potential due to its minimal finite-size effects compared to other cases. For the disordered AAH model with the Fibonacci potential, we determine critical exponents that differ from those of the AAH model without disorder and the Anderson model. Moreover, despite differences in localization behavior, we find that the disordered AAH models with the staggered potential and the Fibonacci potential share the same correlation-length critical exponent. These findings provide a unified framework for understanding localization transitions in quasiperiodic systems and are amenable to experimental validation using emerging techniques.
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We investigate localization transitions and spectral topology in a one-dimensional non-Hermitian generalization of the Aubry-André model in which both the nearest-neighbor and the next-nearest-neighbor hopping amplitudes are nonreciprocal. By extending the Fermi-surface point-matching method to nonreciprocal hopping, we derive a closed-form expression for the energy-dependent mobility edge in which the two nonreciprocity parameters are absorbed into exponentially renormalized effective hopping amplitudes. The mobility edge forms a single parabola in the energy--potential plane: nearest-neighbor nonreciprocity rigidly shifts the localization boundary toward stronger potentials, whereas next-nearest-neighbor nonreciprocity reduces the curvature of the boundary and thereby broadens the energy window in which extended and localized states coexist. Exact diagonalization confirms the analytical boundary for purely nearest-neighbor, purely next-nearest-neighbor, and combined nonreciprocity, and recovers the known Hermitian mobility edge in the reciprocal limit. We further analyze the spectral topology under periodic boundary conditions and show that the spectral winding numbers evaluated at base energies near the two band edges directly bracket the mixed phase: the winding number at the lower band edge drops when the mobility edge enters the spectrum and the first localized states appear, while the winding number at the upper band edge drops when the last extended states localize, delineating the full potential-strength window over which extended and localized states coexist. These results provide a compact analytical framework that connects energy-dependent localization, spectral topology, and nonreciprocity in quasiperiodic lattices, and they are directly testable in photonic, atomic, and electrical-circuit platforms.
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