A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators

TL;DR

This paper introduces a constructive method to compute topological invariants like Fredholm index and essential spectrum for one-dimensional strictly local operators, especially in quantum walks, providing explicit classifications.

Contribution

It presents an elementary constructive approach to topological invariants for 1D local operators, with explicit calculations for quantum walk models.

Findings

Explicit computation of topological invariants for a quantum walk model

Full classification of these invariants in the studied context

Elementary approach applicable to similar operators

Abstract

In this paper we shall focus on one-dimensional strictly local operators, the notion of which naturally arises in the context of discrete-time quantum walks on the one-dimensional integer lattice. In particular, we give an elementary constructive approach to the following two topological invariants associated with such operators: Fredholm index and essential spectrum. As a direct application, we shall explicitly compute and fully classify these topological invariants for a well-known quantum walk model.