Quasiperiodic Mosaic Trimer Lattice

• The paper demonstrates that the quasiperiodic-potential-modulated trimer lattice enables energy-dependent metal-insulator transitions via exact mobility edges.
• It utilizes transfer matrices, Lyapunov exponents, and fractal analyses to reveal localization, resonant extended states, and the emergence of robust Majorana zero modes.
• Experimental realizations in optical and nanophotonic platforms validate the model, highlighting disorder-enhanced topological phases and reentrant transitions.

A quasiperiodic-potential-modulated mosaic trimer lattice is a one-dimensional tight-binding system in which an on-site potential of quasiperiodic character is applied at regular intervals—specifically, every third site—thus forming a "trimer" mosaic superlattice. This construction breaks the standard self-duality of the Aubry–André model, enables energy-dependent metal-insulator transitions via exact mobility edges, and admits highly nontrivial interplay with topological features and disorder, including the emergence of robust Majorana zero modes, topological Anderson insulators, and sharp resonance states. The model is exactly solvable for localization properties and has been realized experimentally in synthetic optical and nanophotonic platforms (Wang et al., 2020, Zeng et al., 2020, Gao et al., 2023, Longhi, 2024, Wang et al., 20 Jan 2026).

1. Lattice Structure, Hamiltonian, and Quasiperiodic Modulation

The defining Hamiltonian of the mosaic trimer lattice involves three-site periodicity with a quasiperiodic potential acting only at positions congruent to zero modulo three: $$H = t \sum_{n} (c_{n+1}^\dagger c_{n} + \mathrm{h.c.}) + \lambda \sum_{n} V_n c_{n}^\dagger c_{n}$$ where $t$ is the nearest-neighbor hopping amplitude, $\lambda$ is the disorder (potential) amplitude, and $V_n$ is the trimer mosaic modulation: $$V_n = \begin{cases} \cos\!\bigl[\,2\pi(\omega n+\theta)\bigr] & n \equiv 0 \pmod 3 \ 0 & n \equiv 1,2 \pmod 3 \end{cases}$$ Here, $\omega\in\mathbb{R}\setminus\mathbb{Q}$ is an irrational frequency (e.g., $(\sqrt{5}-1)/2$), and $\theta$ is a phase. Alternatively, $V_n$ can be replaced by other incommensurate functions, e.g., a $\sec^2$–type potential for analysis of topological transitions (Wang et al., 20 Jan 2026). In the superconducting generalization, a p-wave pairing term is present (Zeng et al., 2020): $$H = -t \sum_n (c_n^\dagger c_{n+1} + \mathrm{h.c.}) + \Delta \sum_n (c_n c_{n+1} + \mathrm{h.c.}) + \sum_n U_n c_n^\dagger c_n$$ with $U_n$ a mosaic quasiperiodic modulation.

This structure introduces a 3-site unit cell, yielding three minibands in the clean (potential-free) limit with gaps supporting nontrivial topological phenomena and robust resonance states (Longhi, 2024).

2. Localization, Mobility Edges, and Lyapunov Analysis

Unlike the traditional Aubry–André model, the trimer mosaic breaks Fourier self-duality, resulting in mobility edges at energies $E_c$ distinguishing extended and localized single-particle states. The localization/delocalization transition is governed by the Lyapunov exponent $\gamma(E)$ (Wang et al., 2020, Gao et al., 2023, Longhi, 2024): $$\gamma(E) = \ln \left| \frac{\lambda}{t} (4\cos^2 q - 1) \right|, \qquad E = 2t \cos q$$ States are extended for $\gamma(E) < 0$ and localized for $\gamma(E) > 0$. Mobility edges occur at $\gamma(E_c) = 0$, yielding four critical energies: $E_c = \pm t \sqrt{1 \pm \frac{t}{\lambda}}$ For $\lambda > t$, all four mobility edges are real and partition the spectrum into alternating extended and localized regions. This outcome is analytically exact and confirmed numerically by inverse participation ratio (IPR) and fractal dimension analyses, which exhibit sharp transitions in $D_2(E)$ at the mobility edges (Wang et al., 2020, Gao et al., 2023, Longhi, 2024).

The following table summarizes mobility edge positions and band structure for representative $\lambda$ at $t=1$ (Longhi, 2024):

$\lambda$	$E_c^\pm$	Extended states	Resonances $E=\pm1$
0.25	$\pm\sqrt{5} \approx 2.236$ (out of band)	All $|E| \leq 2$	Within extended band
1.0	$\pm\sqrt{2} \approx 1.414$	$|E|<1.414$	At mobility edge
2.0	$\pm\sqrt{1.5} \approx 1.225$	$|E|<1.225$	In clean gap
5.0	$\pm\sqrt{1.2} \approx 1.095$	$|E|<1.095$	Deep in gap

If $\lambda \ll t$, all states are extended (mobility edges outside spectrum). As $\lambda$ increases, localization sets in for $|E| > E_c$ progressively from band edges inward.

3. Resonances, Topological Features, and Robust Extended States

The trimer mosaic lattice supports resonance energies that correspond to robust, potential-insensitive extended states, even deep in the localized phase. For $M=3$, these "dark-site" resonances occur at $E = \pm t$; they are analytic roots of the trimer block kernel and map onto eigenstates supported exclusively on the two unmodulated sublattice sites. Their presence ensures that complete Anderson localization cannot occur across the entire spectrum for any $\lambda$ (Longhi, 2024).

The clean lattice supports topological edge states associated with the trimer structure. In the mosaic Su-Schrieffer-Heeger context, nontrivial Zak phase and polarization, as well as protected edge modes, appear at $1/3$ or $2/3$ filling depending on the ratio of intra- and intercell hopping (Wang et al., 20 Jan 2026). The addition of the quasiperiodic mosaic potential induces reentrant and disorder-enhanced topological phases (e.g., topological Anderson insulator) in regimes of strong modulation.

In the superconducting extension, odd mosaic periods (here, trimer) allow robust Majorana zero modes when the potential strength $|V|$ is below a critical threshold $V_c$, beyond which a topological–trivial transition occurs (Zeng et al., 2020): $|V_c| = 2 (t + \Delta)^2 (t - \Delta)$ For even periods, the topological phase persists for any $V$. For odd periods, Majorana zero modes are topologically protected below $V_c$ and destroyed above.

4. Experimental Realizations and Probing Mobility Edges

Exact mobility edge physics in trimer mosaic lattices has been realized in cold-atom and nanophotonic platforms. In particular, a three-hyperfine-state Raman lattice scheme enables independent control of the trimer modulation and is compatible with optical microscopy of eigenmodes and their localization properties (Wang et al., 2020).

Photonically, integrated Si$_3$N$_4$ waveguide arrays have been fabricated with site-dependent widths to encode the quasiperiodic mosaic modulation. Single-site injection and site-resolved detection allow direct measurement of transport properties, localization, and mobility edge mapping via the spread or localization of optical intensity and the extracted IPR (Gao et al., 2023). Extended regimes exhibit rapid delocalization, while localized regimes exhibit persistent site-confined intensity, allowing identification of mobility edge energies by comparing simulation and experiment.

5. Topological Phase Diagrams and Disorder Effects

In trimer mosaic lattices with quasiperiodic potential, the interplay between topology and disorder gives rise to rich phenomena. For the three-band trimer lattice, the clean limit hosts nontrivial topological phases at both $1/3$ and $2/3$ filling, characterized by quantized Zak phase and polarization $P=1/2$ in real space (Wang et al., 20 Jan 2026). Quasiperiodic disorder, introduced through a mosaic modulation, distinctly affects these phases:

• At $2/3$ filling: The topological phase is enhanced by disorder, resulting in a Topological Anderson Insulator. The phase boundary in the $(\lambda,J)$ plane is analytically and numerically established, with the critical intercell hopping $J_c$ reducing from 2 in the clean limit to 1 for strong disorder.
• At $1/3$ filling: The same disorder compresses and fragments the topological region, causing reentrant transitions between trivial and nontrivial phases as $J$ is varied. This behavior is tracked by the bulk gap, polarization, and confirmed by many-body fidelity susceptibility.

All such transitions are consistent with edge spectra, bulk invariants, and ground state stability (Wang et al., 20 Jan 2026). This demonstrates the fundamentally distinct roles of quasiperiodic disorder for different topological fillings in the trimer mosaic lattice.

6. Analytical Methods and Diagnostic Observables

Key analytical methods for the trimer mosaic lattice include:

• Transfer matrix and Lyapunov exponent: Used to obtain energy-resolved localization properties and exact mobility edge conditions (Wang et al., 2020, Gao et al., 2023, Longhi, 2024, Zeng et al., 2020).
• Zak phase and polarization: Evaluate bulk topological invariants, even in disordered regimes, by using Resta's formula for polarization (Wang et al., 20 Jan 2026).
• Fractal dimension ($D_2$) and inverse participation ratio (IPR): Quantify the localization-delocalization transition, identifying sharp changes at mobility edges (Wang et al., 2020, Gao et al., 2023).
• Many-body fidelity susceptibility: Pinpoints phase transitions between topological and trivial phases (Wang et al., 20 Jan 2026).

These diagnostics are directly accessible numerically and, in engineered quantum platforms, experimentally by single-site addressing and site-resolved detection.

7. Significance and Outlook

The quasiperiodic-potential-modulated mosaic trimer lattice provides an analytically tractable and experimentally accessible platform for investigating energy-dependent localization transitions, robust extended resonances, and topological effects in low-dimensional disordered systems. It challenges conventional wisdom regarding single-particle mobility edges in one dimension and demonstrates disorder-protected topological phases, reentrant transitions, and Majorana robustness specific to odd mosaic periodicities. These features position the model as a canonical framework for exploring the interplay of incommensurability, topology, and localization, with direct relevance to synthetic quantum matter and photonic architectures (Longhi, 2024, Wang et al., 20 Jan 2026, Zeng et al., 2020).