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Topological states and flat bands induced by bound states in the continuum in a ladder-shaped one-dimensional photonic crystal

Sofía Pinto, P. A. Orellana, Sergio Bravo

Abstract

One-dimensional crystals serve as a versatile platform for engineering nontrivial states, which can be easily explored in transport configurations. In this work, we analyze the properties of a periodic structure composed of an H-shaped unit cell, which forms a periodic ladder-shaped system. Using tight-binding models, group-theoretical considerations, and standard band topology, we uncover the influence of bound states in the continuum (BICs) and quasi-BICs formed in the original finite geometry on the creation of nontrivial band states. By designing various textures for the onsite energies, we discovered a topological band inversion between quasi-BIC-induced bands, leading to the emergence of topologically protected edge states that are characterized by a quantized Zak phase. Additionally, we found an on-site configuration that exhibits robust flat bands, induced by a symmetry-protected BIC and linked to special one-sided localized edge states. We present a detailed analysis of the mechanisms driving both effects and discuss the crucial role of symmetry in characterizing the topological phases of these systems.

Topological states and flat bands induced by bound states in the continuum in a ladder-shaped one-dimensional photonic crystal

Abstract

One-dimensional crystals serve as a versatile platform for engineering nontrivial states, which can be easily explored in transport configurations. In this work, we analyze the properties of a periodic structure composed of an H-shaped unit cell, which forms a periodic ladder-shaped system. Using tight-binding models, group-theoretical considerations, and standard band topology, we uncover the influence of bound states in the continuum (BICs) and quasi-BICs formed in the original finite geometry on the creation of nontrivial band states. By designing various textures for the onsite energies, we discovered a topological band inversion between quasi-BIC-induced bands, leading to the emergence of topologically protected edge states that are characterized by a quantized Zak phase. Additionally, we found an on-site configuration that exhibits robust flat bands, induced by a symmetry-protected BIC and linked to special one-sided localized edge states. We present a detailed analysis of the mechanisms driving both effects and discuss the crucial role of symmetry in characterizing the topological phases of these systems.
Paper Structure (8 sections, 4 equations, 3 figures)

This paper contains 8 sections, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model: Structure, parameters and symmetries
  3. Topological bands induced from bound states
  4. Band inversion mechanism
  5. Zak phase characterization
  6. Flat bands induced from mirror anti-symmetric bound states
  7. Bulk-boundary correspondence in finite systems
  8. Discussion

Figures (3)

  • Figure 1: (a) Unit cell configuration for the formation of a generic one-dimensional crystal with ladder shape. (b) Configurations of the onsite energies corresponding to each of the symmetry groups presented in this work. i) $P2'm'm$, ii) $P2m'm'$ and iii) $P2'mm'$. (c) Sample of the periodic crystal showing the ladder structures and the graphical definition of the hopping amplitudes. (d) Graphical representation of the dependence of the critical value of the intercell hopping ($t_i$) that produces the topological band inversion versus the fixed value of the onsite detuning magnitude $\vert \epsilon \vert$ for which this inversion happens.
  • Figure 2: Left: Band structure for a representative case of the $P2'm'm$ group, using as parameters $t_a=t_c=1$, $\vert\epsilon|=0.01$. Right: Zoom in the central pair of bands induced from bound states, showing the evolution with respect to $t_i$ to spot the band inversion associated to the topological transition.
  • Figure 3: (a) Top panel: Eigenvalue spectrum for a finite system with six unit cells in the trivial phase of $P2'm'm$ group. Bottom panel: LDOS for the finite system in the trivial phase of $P2'm'm$ group. The value of intercell parameter is $t_i=0.0015$. (b) Top panel: Eigenvalue spectrum for a finite system with six unit cells in the topological phase of $P2'm'm$ group. Bottom panel: LDOS for the finite system in the trivial phase of $P2'm'm$ group. The value of intercell parameter is $t_i=0.15$. (c) Top panel: Eigenvalue spectrum for a finite system with six unit cells with flat bands for the $P2'mm'$ group. Bottom panel: LDOS for the finite system with flat bands for $P2'mm'$ group. The value of intercell parameter is $t_i=0.09$. For all the above calculations, the other parameters where set to: $t_a=t_c=1$ and $\vert \epsilon \vert = 0.1$. For all top panels: red circle indicates the states for which the LDOS is computed. For all bottom panels: The intensity of the sites indicates the magnitude of the modulus of the wavefunction.