Topological Insulators in Random Lattices

TL;DR

This paper demonstrates the existence of topological insulators in random lattices, expanding the understanding of topological phases beyond crystalline structures and suggesting new avenues for experimental realization.

Contribution

It introduces models of topological insulators on random lattices, showing phase transitions and topological properties in non-crystalline systems across multiple symmetry classes.

Findings

Topological phases can exist in random lattice systems.

Transitions from trivial to topological phases are tunable by site density.

Explicit example of a 3D $Z_2$ topological insulator on a random lattice.

Abstract

Our understanding of topological insulators is based on an underlying crystalline lattice where the local electronic degrees of freedom at different sites hybridize with each other in ways that produce nontrivial band topology, and the search for material systems to realize such phases have been strongly influenced by this. Here we theoretically demonstrate topological insulators in systems with a random distribution of sites in space, i. e., a random lattice. This is achieved by constructing hopping models on random lattices whose ground states possess nontrivial topological nature (characterized e. g., by Bott indices) that manifests as quantized conductances in systems with a boundary. By tuning parameters such as the density of sites (for a given range of fermion hopping), we can achieve transitions from trivial to topological phases. We discuss interesting features of these transitions. In two spatial dimensions, we show this for all five symmetry classes (A, AII, D, DIII and C) that are known to host nontrivial topology in crystalline systems. We expect similar physics to be realizable in any dimension and provide an explicit example of a $Z_2$ topological insulator on a random lattice in three spatial dimensions. Our study not only provides a deeper understanding of the topological phases of non-interacting fermions, but also suggests new directions in the pursuit of the laboratory realization of topological quantum matter.