Non-Abelian Vortices in Condensed Matter Physics

TL;DR

This paper explores non-Abelian topological vortices in condensed matter systems, demonstrating their unique flux quantization, potential experimental observation, and implications for topological knots, expanding understanding beyond traditional Abelian vortices.

Contribution

It introduces the concept of non-Abelian vortices in condensed matter, establishes their flux quantization, and discusses their experimental realization and topological implications.

Findings

Non-Abelian vortices have flux quantized in units of 4π/g.

Existence of non-Abelian magnetic vortices in two-gap superconductors.

Potential observation of non-Abelian vortices in MgB₂ and Rb-87 condensates.

Abstract

We study the non-Abelian topological vortices in condensed matter physics, whose topological flux quantum number is described by $\pi_2(S^2)$, not by $\pi_1(S^1)$. We present two examples, a magnetic vortex in two-gap superconductor and a vorticity vortex in two-component Bose-Einstein condensate. In both cases the condensates exhibit a global SU(2) symmetry which allows the non-Abelian topology. We establish the non-Abelian flux quantization in two-gap superconductor by demonstrating the existence of non-Abelian magnetic vortex whose flux is quantized in the unit $4\pi/g$, not $2\pi/g$. We also discuss a genuine non-Abelian gauge theory of superconductivity which has a local SU(2) gauge symmetry, and establish the non-Abelian Meissner effect in the non-Abelian superconductor. We compare the non-Abelian vortices with the well-known Abelian Abrikosov vortex, and discuss how these non-Abelian vortices could be observed experimentally in two-gap superconductor made of ${\rm MgB_2}$ and spin-1/2 condensate of $^{87}{\rm Rb}$ atoms. Finally, we argue that the existence of the non-Abelian vortices provides a strong evidence for the existence of topological knots in these condensed matters whose topology is fixed by $\pi_3(S^2)$, which one can construct by twisting and connecting the periodic ends of the non-Abelian vortices.