The Loring--Schulz-Baldes Spectral Localizer Revisited

TL;DR

This paper revisits the spectral localizer method for computing topological invariants of quantum Hamiltonians, providing a new spectral-theoretic proof and connecting it to higher-dimensional topological insulators.

Contribution

It offers a direct, elementary proof for the spectral localizer in 1D and 2D cases and reinterprets it within the framework of higher-dimensional topological insulators.

Findings

Elementary spectral-theoretic proof for $d=1,2$ cases

Unified treatment of 1D and 2D localizers

Reinterpretation as higher-dimensional topological insulator

Abstract

The spectral localizer, introduced by Loring in 2015 and Loring and Schulz-Baldes in 2017, is a method to compute the (infinite volume) topological invariant of a quantum Hamiltonian on $\ZZ^d$, as the signature of the (finite) localizer matrix. We present a direct and elementary spectral-theoretic proof treating the $d=1$ and $d=2$ cases on an almost equal footing. Moreover, we re-interpret the localizer as a higher-dimensional topological insulator via the bulk-edge correspondence.