Isospin of topological defects in Dirac systems

TL;DR

This paper explores the internal isospin degree of freedom in topological defects within Dirac systems, revealing a universal emergent mass order parameter and its implications for quantum numbers across various dimensions.

Contribution

It demonstrates the universal emergence of an internal isospin-1/2 in topological defects, independent of system details, and characterizes the associated local order parameters in Dirac materials.

Findings

Topological defects carry an emergent isospin-1/2 quantum number.

Six local order parameters form two mutually commuting Clifford algebras.

Particle-hole symmetry constrains defects to a single isospin-1/2 quantum number.

Abstract

We study the Dirac quasiparticles in $d$-dimensional lattice systems of electrons in the presence of domain walls ($d=1$), vortices ($d=2$), or hedgehogs ($d=3$) of superconducting and/or insulating, order parameters, which appear as mass terms in the Dirac equation. Such topological defects have been known to carry non-trivial quantum numbers such as charge and spin. Here we discuss their additional internal degree of freedom: irrespectively of the dimensionality of space and the nature of orders that support the defect, an extra mass-order-parameter is found to emerge in their core. Six linearly independent local orders, which close two mutually commuting three-dimensional Clifford algebras are proven to be in general possible. We show how the particle-hole symmetry restricts the defects to always carry the quantum numbers of a single effective isospin-1/2, quite independently of the values of their electric charge or true spin. Examples of this new degree of freedom in graphene and on surfaces of topological insulators are discussed.