Topological Classification of Insulators: II. Quasi-Two-Dimensional Locality

TL;DR

This paper introduces a new way to characterize two-dimensional locality in insulators using spectral projections of the Laughlin flux operator, leading to the concept of quasi-2D locality and revealing complex topological behaviors.

Contribution

It provides an alternative spectral characterization of 2D locality and generalizes it to quasi-2D, analyzing the topological classification of such systems.

Findings

Quasi-2D locality exhibits infinitely many -valued indices in the unitary chiral case.

The behavior of quasi-2D systems differs significantly from traditional 2D classifications.

The spectral approach aids in defining Hall conductivity for Fermi projections.

Abstract

We provide an alternative characterization of two-dimensional locality (necessary e.g. to define the Hall conductivity of a Fermi projection) using the spectral projections of the Laughlin flux operator. Using this abstract characterization, we define generalizations of this locality, which we term quasi-2D. We go on to calculate the path-connected components of spaces of unitaries or orthogonal projections which are quasi-2D-local and find a starkly different behavior compared with the actual 2D column of the Kitaev table, exhibiting e.g., in the unitary chiral case, infinitely many $\mathbb{Z}$-valued indices.