Hidden self-duality and exact mobility edges in quasiperiodic network models

TL;DR

This work reveals hidden self-duality in a class of quasiperiodic network models by deriving an energy-dependent effective Hamiltonian after integrating out periodic sites. The resulting framework, characterized by two functions $f(E)$ and $g(E)$, predicts exact mobility edges through a generalized self-duality condition and a resonant-state mechanism defined by $g(E)=0$. The authors construct explicit MEs for several network geometries, including side-coupled and multipath chains, and for multiple quasiperiodic potentials, with extensions to long-range hopping and non-Hermitian cases. Their approach provides a practical, designable route to engineer Anderson transitions and MEs in low-dimensional systems, with potential experimental realizations in optical and acoustic waveguide arrays, and offers new insights into the structure of mobility edges in quasiperiodic lattices.

Abstract

In one-dimensional quasiperiodic systems, only a few models with exact mobility edges (MEs) have been constructed using generalized self-duality theory, Avila's global theory, or the renormalization group method. This raises an intriguing question that whether we can realize more physical models with exact solvable MEs. In this work, we uncover the hidden self-duality within a class of quasiperiodic network models constituted by periodic and quasiperiodic sites. Although the original Hamiltonians appear to lack self-duality, their effective Hamiltonians obtained by integrating out the periodic sites exhibit self-duality, which yield MEs. The well-studied mosaic model, which is the simplest case of quasiperiodic network models, was previously thought to exhibit MEs due to the absence of self-duality, but we show that they actually arise from the hidden self-duality. Using the effective Hamiltonian, we further introduce the concept of resonant states to understand the shape of MEs. Finally, we present in detail how to determine the MEs in various network models, including some non-Hermitian models, based on the hidden self-duality. These predictions can be experimentally realized using optical and acoustic waveguide arrays. Our work can greatly advance our understanding of MEs in Anderson transition.

Figures (11)

• Figure 1: (a) The generalized one-dimensional quasiperiodic network model constituted by periodic network sites between quasiperiodic sites and its reduced effective quasiperiodic model. Here, $V_{i+1}$ and $g(E) V_{i+1}$ denote the potentials before and after integrating out the periodic network sites. (b) The quasiperiodic mosaic model and its reduced effective quasiperiodic model. In both figures, the blue, red, and green solid points denote the lattice sites with quasiperiodic, periodic, and effective energy-dependent potentials, respectively.
• Figure 2: Energy-dependent $g(E)$ and their corresponding MEs. (a1)-(d1) illustrate linear and quadratic $g(E)$ and (a2)-(d2) show their possible MEs. The shape of the MEs depends on the root structure of $g(E) = 0$. Realizations of MEs in the quasiperiodic mosaic model are presented in (a3) with $\kappa = 2$, $U_1 = t$, (b3) with $\kappa = 3$, $U_1 = -U_2 = t$, (c3) with $U_1 = -U_2 = it$, and (d3) with $U_1 = -U_2 = 1.5it$. Non-Hermitian models in (c3) and (d3) exhibit parity-time (PT) symmetry.
• Figure 3: Fractal dimension $D_2$ of different eigenstates versus $E$ and potential strength $V$ for (a) $\bar{U}_1 = -\bar{U}_2 = t$, (b) $\bar{U}_1 = \bar{U}_2 = 0$, (c) $\bar{U}_1 = -\bar{U}_2 = it$, and (d) $\bar{U}_1 = -\bar{U}_2 = 1.2it$ using the quasiperiodic potential $V_i = V \cos (2 \pi \alpha i)/[1-b \cos (2 \pi \alpha i)]$. The periodic network is given by the first structure of Table \ref{['table']}. The solid lines denote the MEs $b f(E) = \pm 2t - V g(E)$ with $f(E)$ and $g(E)$ given in Table \ref{['table']}. The other parameters are $\alpha = (\sqrt{5} - 1) / 2$, $b = 0.5$ and $L = 600$.
• Figure S1: Two kinds of quasiperiodic network models, which can be mapped to an effective nearest-neighbor chain.
• Figure S2: Verification of the generalized self-dual points with quasiperiodic mosaic models (a) $\kappa = 2$, $U_1 = t$, (b) $\kappa = 3$, $U_1 = -U_2 = t$, (c) $U_1 = -U_2 = it$, and (d) $U_1 = -U_2 = 1.5it$. The models in (c) and (d) have PT symmetry. The black lines in each figure denote the MEs given by Eq. (\ref{['eq-S6']}). The other parameters are $\alpha = (\sqrt{5} - 1)/2$, and $L = 987$.