Exact mobility edges in quasiperiodic network models with slowly varying potentials

TL;DR

This work addresses the challenge of obtaining exact mobility edges in 1D quasiperiodic systems by introducing mosaic and network models with slowly varying potentials that lack self-duality. The authors develop an effective-Hamiltonian framework obtained by integrating out periodic sites, yielding an energy-dependent potential $g(E)V$ and effective eigenenergy $f(E)$, with mobility edges determined by $f(E)=\pm(2 t^\kappa \pm g(E)V)$ (and analogous forms for network models). They validate the approach analytically for minimal ($\kappa=2$) and ($\kappa=3$) cases and extend it to multipath and non-Hermitian (PT-symmetric) networks, providing exact ME conditions and resonant states, along with numerical confirmations via IPR, NPR, and Lyapunov exponents. Finally, they propose an optical-waveguide experimental platform to realize these models and demonstrate that Anderson transitions can be observed in relatively small systems, highlighting the potential for quantum simulation of exact mobility edges in complex quasiperiodic networks.

Abstract

Quasiperiodic models are important physical platforms to explore Anderson transitions in low dimensional systems, yet the exact mobility edges (MEs) are generally hard to be determined analytically. To date, the MEs in only a few models can be determined exactly. In this manuscript, we propose a new class of network models characterized by quasiperiodic slowly varying potentials and the absence of hidden self-duality, and exactly determine their MEs. We take the mosaic models with slowly varying potentials as examples to illustrate this result and derive its MEs from the effective Hamiltonian. In this method, we can integrate out the periodic sites to obtain an effective Hamiltonian with energy-dependent potentials $g(E)V$ and effective eigenenergy $f(E)$, which directly yields the MEs at $f(E) = \pm(2t^κ\pm g(E)V)$, where $κ\in \mathbb{Z}^+$. With this idea in hand, we then generalize our method to more quasiperiodic network models, including those with much more complicated geometries and non-Hermitian features. Finally, we propose the realization of these models using optical waveguides and show that the Anderson transition can be observed even in small physical systems (with lattice sites about $L = 50 - 100$). Our results provide some key insights into the understanding and realization of exact MEs in experiments.

Figures (10)

• Figure 1: (a) The 1D mosaic model with a slowly varying potential for $\kappa = 2$. (b) The network model when $N = 4$, which can be interpreted as a multipath mosaic model, where $N$ denotes the number of paths. More network models can be found in Ref. hu2025hidden. (c) The effective quasiperiodic chain after integrating out the periodic sites. Here, $V_i$ represents the quasiperiodic slowly varying potential, for example $V \cos(\pi \alpha i^\nu + \phi)$, and $gV_j$ is the corresponding effective potential.
• Figure 2: Fractal dimension $D_2$ of different eigenstates as a function of energy $E$ and potential strength $V$ for mosaic models with slowly varying potentials when (a) $\kappa = 2$ and (b) $\kappa = 3$. The solid lines denote the MEs given by Eqs. (\ref{['eq-ME1']}), (\ref{['eq-ME2']}), and (\ref{['eq-ME3']}), respectively. (c)-(e) Spatial distributions of three representative eigenstates with $E \approx -0.67t$ (extended), $E \approx -0.38t$ (localized), and $E = 0$ (resonant) as marked in Fig. \ref{['fig-fig2']}(a). The other parameters are $\alpha = (\sqrt{5} - 1) / 2$, $\nu = 0.6$, $\phi = 0$ and $L = 987$.
• Figure 3: Lyapunov exponent $\gamma$ as a function of chain length $L$ for the slowly varying models for various $V$ when $E = 0$. The Lyapunov exponent converges to a constant (in the localized phase) with the increasing of chain length $L$. The other parameters are the same as those in Fig. \ref{['fig-fig2']}.
• Figure 4: Lyapunov exponent $\gamma$ as a function of energy $E$ for mosaic models with slowly varying potentials at various $V$ when (a) $\kappa = 2$ and (b) $\kappa = 3$. When the energy satisfies the conditions of Eqs. (\ref{['eq-ME1']}), (\ref{['eq-ME2']}), and (\ref{['eq-ME3']}), $\gamma$ transitions from zero to a finite value, indicating the existence of MEs. The system size is $L = 832040$, and the other parameters are the same as those in Fig. \ref{['fig-fig2']}.
• Figure 5: Finite-size scaling of the IPR (blue asterisk) and NPR (red round) when $\kappa = 2$ and $V = 0.5t$ for (a) localized state $E \approx -2.2t$ and (b) extended state $E \approx -t$. The inset in (b) provides an enlarged view of the scaling of IPR for the extended state. The other parameters are the same as those in Fig. \ref{['fig-fig2']}.