Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory

TL;DR

This paper develops a new mathematical framework using groupoid algebras and KK-theory to describe topological lattice defects and their bulk-boundary correspondences, providing a general and rigorous approach with concrete examples.

Contribution

It introduces a novel method employing groupoid $C^*$-algebras and Kasparov's KK-theory to analyze lattice defects and establish a general bulk-defect correspondence principle.

Findings

Established a correspondence between 2D lattice defects and KK-theory.

Derived exact sequences of groupoid $C^*$-algebras for defect analysis.

Provided numerical examples demonstrating non-trivial bulk-defect relations.

Abstract

The bulk-boundary and a new bulk-defect correspondence principles are formulated using groupoid algebras. The new strategy relies on the observation that the groupoids of lattices with boundaries or defects display spaces of units with invariant accumulation manifolds, hence they can be naturally split into disjoint unions of open and closed invariant sub-sets. This leads to standard exact sequences of groupoid $C^\ast$-algebras that can be used to associate a Kasparov element to a lattice defect and to formulate an extremely general bulk-defect correspondence principle. As an application, we establish a correspondence between topological defects of a 2-dimensional square lattice and Kasparov's group $KK^1 (C^\ast(\mathbb Z^3),\mathbb C)$. Numerical examples of non-trivial bulk-defect correspondences are supplied.