Augmentation and Bulk Edge Correspondence for one dimensional aperiodic tight binding operators

TL;DR

This work develops a comprehensive C*-algebraic framework for 1D aperiodic tight-binding models, introducing augmentation methods (mapping torus and arc-based) to realize bulk-edge correspondences beyond totally disconnected hulls. It connects integrated density of states, spectral flows, and gap-labelling via Connes pairing with Chern cocycles in K-theory, and demonstrates both primary and secondary gap behavior through analytical constructions and Kohmoto-model simulations. By relating edge forces and phason-like motions to spectral flows, and by interpreting Grothendieck-style stacked invariants, the paper unifies previously separate gap-labelling pictures (Bellissard vs Johnson–Moser) in a tangible topological framework. Numerical results illustrate how augmentation reveals observable spectral flows and IDS changes in both 1-cut and 2-cut Kohmoto models, highlighting the physical significance of augmentation and stacking in topological insulators.

Abstract

We consider a particular class of 1D aperiodic models with the aim to understand how their internal degrees of freedom contribute to their topological invariants and the possible relations (correspondences) among them. In order to handle models with finite local complexity we introduce the principle of augmentation. This allows us to relate the values of the Integrated Density of States at gap energies for the bulk system to spectral flows. We consider two different augmentations. The first is based on the mapping torus construction. It leads to an alternative proof of the result that the gap labelling group of Bellissard coincides with that of Johnson-Moser. It furthermore allows for an interpretation of the spectral flow via boundary forces. The second augmentation applies to models obtained by the cut and project method where we find for 2-cut models two different spectral flows, one attached to the edge modes and related to the phason motion whereas the other is an augmented bulk invariant. Our approach is based on the well-established $C^*$-algebraic approach to solid state physics and the description of topological invariants by $K$-theory and cyclic cocycles. We also present numerical simulations to illustrate our theorems.

Figures (13)

• Figure 1: Values of the potential for $\phi=0$, $\theta=0.2$ and $\kappa=0.7$, $t=0.5$.
• Figure 2: Values of the potential for $\phi=0$, $\theta=\kappa=0.4$, $t=0.3$.
• Figure 3: The cut & project scheme for the model $\Xi_{\theta, \kappa}$.
• Figure 4: Graphical representation of $\Xi_{\theta,\kappa}$ (left) and its augmented version $\tilde{\Xi}_{\theta,\kappa}$ (right). The blue (resp. red) numbers correspond to the points of the orbit of $\theta$ (resp. $\kappa$). In the right figure the set $X_0$ is in blue and $X_\kappa$ is in red. Both figures represent only an approximation to the topological spaces in that they show only finitely many cuts or added arcs, respectively. For irrational $\theta$, $\Xi_{\theta,\kappa}$ is actually a Cantor set.
• Figure 5: Spectrum of $H^A_{\xi,t}$ for $t\in [\xi,1]$. The $x$-axis corresponds to the mapping torus parameter $t\in [0,1]$. The vertical scale on the left $y$-axis corresponds to the energies and on the right to the IDS.

Theorems & Definitions (33)

• Definition 2.1
• Theorem 2.1
• proof
• Remark
• Theorem 2.2: Six-Term exact sequence
• Theorem 2.3: Pimsner-Voiculescu Exact Sequence
• Proposition 3.1
• proof
• Lemma 4.1
• proof