Topological band insulators without translational symmetry

TL;DR

This paper introduces a systematic method to construct and analyze topological insulators in amorphous systems lacking translational symmetry, revealing new topological phases and phenomena beyond traditional crystalline frameworks.

Contribution

It develops an isospectral reduction approach to identify topological phases in amorphous systems, expanding the understanding of topological insulators beyond crystalline symmetry constraints.

Findings

Identification of topological phases with edge states in amorphous systems

Discovery of flat bands and bound states in continuum in disordered systems

Validation of topological invariants in non-crystalline structures

Abstract

In the research of the topological band phases, the conventional wisdom is to start from the crystalline translational symmetry systems. Nevertheless, the translational symmetry is not always a necessary condition for the energy bands. Here we propose a systematic method of constructing the topological band insulators without translational symmetry in the amorphous systems. By way of the isospectral reduction approach from spectral graph theory, we reduce the structural-disordered systems formed by different multi-atomic cells into the isospectral effective periodic systems with the energy-dependent hoppings and potentials. We identify the topological band insulating phases with extended bulk states and topological in-gap edge states by the topological invariants of the reduced systems, density of states, and the commutation of the transfer matrix. In addition, when the building blocks of the two multi-atomic cells have different number of the lattice sites, our numerical calculations demonstrate that the existences of the flat band and the macroscopic bound states in the continuum in the amorphous systems. Our findings uncover an arena for the exploration of the topological band states beyond translational symmetry systems paradigm.