Topological junctions for one-dimensional systems

TL;DR

This paper introduces a systematic framework for understanding symmetry-protected edge modes at junctions of one-dimensional materials, providing a new proof of the topological classification and demonstrating the inevitability of edge modes at interfaces with differing topological indices.

Contribution

It offers a novel, symmetry-based proof of the 1D topological insulator classification and a comprehensive framework for analyzing protected edge modes without relying on periodicity.

Findings

Edge modes arise at junctions with different topological indices.

The framework applies to both continuous and discrete models.

Provides a new proof of the periodic table of 1D topological insulators.

Abstract

We study and classify the emergence of protected edge modes at the junction of one-dimensional materials. Using symmetries of Lagrangian planes in boundary symplectic spaces, we present a novel proof of the periodic table of topological insulators in one dimension. We show that edge modes necessarily arise at the junction of two materials having different topological indices. Our approach provides a systematic framework for understanding symmetry-protected modes in one-dimension. It does not rely on periodic nor ergodicity and covers a wide range of operators which includes both continuous and discrete models.