Continuum Model of Isospectrally Patterned Lattices

TL;DR

The paper develops a continuum analogue (CIPL) of isospectrally patterned lattices to obtain analytic expressions for the eigenvalue spectrum and eigenstates, enabling a closed-form localization length that scales as the square root of the ratio between inter-cell coupling and phase gradient. Using a linearized, small-a/L expansion, the authors derive a tractable spectrum ${E^{(\pm)}_n = \pm\sqrt{\lambda^2 + 2 g n}}$ with $\lambda = 1+\epsilon$ and $g = \frac{\pi \epsilon a}{2L}$, finding a missing negative-ground state due to broken chiral symmetry. A symmetry analysis clarifies the continuum model’s structure relative to a chirally symmetric Hamiltonian, showing a preserved partner-state pairing for $n>0$ but not for $n=0$. The work provides analytical insight into localization from phase gradients and coupling in IPLs and highlights open questions for going beyond the linear approximation, likely requiring numerical methods. The results have implications for designing nonperiodic, degeneracy-driven lattices with controllable localization properties.

Abstract

Isospectrally patterned lattices (IPL) have recently been shown to exhibit a rich band structure comprising both regimes of localized as well as extended states. The localized states show a single center localization behaviour with a characteristic localization length. We derive a continuum analogue of the IPL which allows us to determine analytically its eigenvalue spectrum and eigenstates thereby obtaining an expression for the localization length which involves the ratio of the coupling among the cells of the lattice and the phase gradient across the lattice. This continuum model breaks chiral symmetry but still shows a pairing of partner states with positive and negative energies except for the ground state. We perform a corresponding symmetry analysis which illuminates the continuum models structure as compared to a corresponding chirally symmetric Hamiltonian.

Figures (2)

• Figure 1: The profiles of the eigenstates with energies $\mathcal{E}_0=1.5$ (a), $\mathcal{E}^{(-)}_{1}=-\sqrt{1.6}$ (b), and $\mathcal{E}^{(-)}_{2}=-\sqrt{1.7}$ (c), for ${\cal{H}}_{cl}^{(1)}$ with $\lambda=1.5$ and $g=0.05$.
• Figure 2: The profiles of the eigenstates with energies $\mathcal{E}^{(+)}_1=\sqrt{1.6}$ (a), and $\mathcal{E}^{(+)}_{2}=\sqrt{1.7}$ (b) for ${\cal{H}}_{cl}^{(1)}$ with $\lambda=1.5$ and $g=0.05$. These are the partner eigenstates to those presented in Fig.\ref{['fig1']}(b) and Fig.\ref{['fig1']}(c), respectively.