Unified topological characterization of electronic states in spin textures from noncommutative K-theory

TL;DR

This paper introduces a unified algebraic framework using noncommutative K-theory to classify topological electronic states in complex spin textures, enabling the design of topological phases in disordered and non-smooth systems.

Contribution

It provides a novel algebraic approach to categorize topological electronic states in spin textures without relying on smooth-space assumptions.

Findings

Unified classification of topological states via noncommutative K-theory

Applicable to aperiodic, disordered, and non-smooth spin textures

Predicts emergence of topological states in various noncollinear setups

Abstract

The nontrivial topology of spin systems such as skyrmions in real space can promote complex electronic states. Here, we provide a general viewpoint at the emergence of topological electronic states in spin systems based on the methods of noncommutative K-theory. By realizing that the structure of the observable algebra of spin textures is determined by the algebraic properties of the noncommutative hypertorus, we arrive at a unified understanding of topological electronic states which we predict to arise in various noncollinear setups. The power of our approach lies in an ability to categorize emergent topological states algebraically without referring to smooth real- or reciprocal-space quantities. This opens a way towards an educated design of topological phases in aperiodic, disordered, or non-smooth textures of spins and charges containing topological defects.