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Reentrant Localization in Quasiperiodic Thue-Morse Chain

Taylan Yildiz, B. Tanatar

Abstract

We investigate localization and reentrance in a dimerized Su-Schrieffer-Heeger (SSH) tight-binding chain whose on-site energies are given by a quasiperiodic cosine masked by a deterministic Thue-Morse sequence. Working with non-interacting, spinless fermions, we solve the model via exact diagonalization on large Fibonacci sizes and diagnose phases using inverse/normalized participation ratios and the correlation fractal dimension. We identify boundaries separating extended, multifractal (mixed), and localized regimes by constructing a phase diagram in the plane of modulation strength and dimerization ratio. As the quasiperiodic amplitude is increased, the system exhibits reentrant behavior, localizing, partially re-delocalizing into a multifractal regime, and re-localizing, verified via two-size crossings of band-averaged observables and finite-size scaling. We demonstrate that tuning the modulation strength, the SSH dimerization, or the incommensurability parameter provides control over the critical thresholds. Our results suggest a versatile, randomness-free platform for the deterministic control of transport, enabling switching between conducting, multifractal, and insulating states.

Reentrant Localization in Quasiperiodic Thue-Morse Chain

Abstract

We investigate localization and reentrance in a dimerized Su-Schrieffer-Heeger (SSH) tight-binding chain whose on-site energies are given by a quasiperiodic cosine masked by a deterministic Thue-Morse sequence. Working with non-interacting, spinless fermions, we solve the model via exact diagonalization on large Fibonacci sizes and diagnose phases using inverse/normalized participation ratios and the correlation fractal dimension. We identify boundaries separating extended, multifractal (mixed), and localized regimes by constructing a phase diagram in the plane of modulation strength and dimerization ratio. As the quasiperiodic amplitude is increased, the system exhibits reentrant behavior, localizing, partially re-delocalizing into a multifractal regime, and re-localizing, verified via two-size crossings of band-averaged observables and finite-size scaling. We demonstrate that tuning the modulation strength, the SSH dimerization, or the incommensurability parameter provides control over the critical thresholds. Our results suggest a versatile, randomness-free platform for the deterministic control of transport, enabling switching between conducting, multifractal, and insulating states.
Paper Structure (8 sections, 19 equations, 12 figures)

This paper contains 8 sections, 19 equations, 12 figures.

Table of Contents

  1. INTRODUCTION
  2. MODEL and Method
  3. RESULTS
  4. Fractal Dimension Analysis
  5. IPR and NPR ANALYSIS
  6. Critical State Analysis
  7. CONCLUSION
  8. Checking Reentrance with Random Binary Sequences

Figures (12)

  • Figure 1: Illustration of the model described with $3$ unit cells and with length $6$ Thue-Morse sequence
  • Figure 2: Real-space profiles of the middle eigenstate $\psi_i$ (eigen-index $m/L=0.5$) for four potential strengths $\delta=0,\,0.6,\,1.1,$ and $3.0$. Parameters: system size $N=2584$, $J_2=0.7$.
  • Figure 3: Density plot of fractal dimension $D_2$ with respect to eigenstate index with the system size 1597 and with 20 different box sizes
  • Figure 4: Averaged fractal dimension $D_2$ over the middle of the spectrum with respect to $\delta$ for $N=1597,2584,4181,6765$ and extrapolated result for $N\rightarrow\infty$.
  • Figure 5: Average IPR and NPR with respect to potential strength $\delta$ for states in the middle of the spectra and for system size $L=13530$, shaded regions correspond to critical phases. (a) for $J_2=0.7$ (b) for $J_2=1.2$
  • ...and 7 more figures