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Chinese Physics Letters

Non-Abelian Topological Excitations in Spinor Condensates

Yunbo Zhang, H. Mäkelä, Kalle‐Antti Suominen

Published Feb 24, 2005• 4 citations
Publisher

Abstract

We investigate the topological defects in atomic spin-1 and spin-2 Bose–Einstein condensates by applying the homotopy group theory. With this rigorous approach we clarify the previously controversial identification of symmetry groups and order parameter spaces for the spin-1 case, and show that the spin-2 case provides a rare example of a physical system with non-Abelian line defects, and the possibility to have winding numbers of 1/3 and its multiples.

DOI
10.1088/0256-307x/22/3/005

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We investigate the topological defects in atomic spin-1 and spin-2 Bose–Einstein condensates by applying the homotopy group theory. With this rigorous approach we clarify the previously controversial identification of symmetry groups and order parameter spaces for the spin-1 case, and show that the spin-2 case provides a rare example of a physical system with non-Abelian line defects, and the possibility to have winding numbers of 1/3 and its multiples.
We apply the homotopy group theory in classifying the topological defects in atomic spin-1 and spin-2 Bose-Einstein condensates. The nature of the defects depends crucially on the spin-spin interaction between the atoms. We find the topologically stable defects both for spin-1 ferromagnetic and anti-ferromagnetic states, and for spin-2 ferromagnetic and cyclic states. With this rigorous approach we clarify the previously controversial identification of symmetry groups and order parameter spaces for the spin-1 anti-ferromagnetic state, and show that the spin-2 cyclic case provides a rare example of a physical system with non-Abelian line defects, like those observed in biaxial nematics. We also show the possibility to produce vortices with fractional winding numbers of 1/2, 1/3 and their multiples in spinor condensates.
We investigate the structure of topological defects in the ground states of spinor Bose–Einstein condensates with spin F = 1 or F = 2. The type and number of defects are determined by calculating the first and second homotopy groups of the order-parameter space. The order-parameter space is identified with a set of degenerate ground state spinors. Because the structure of the ground state depends on whether or not there is an external magnetic field applied to the system, defects are sensitive to the magnetic field. We study both cases and find that the defects in zero and non-zero field are different.
A rich variety of order parameter manifolds of multicomponent Bose-Einstein condensates (BECs) admit various kinds of topological excitations, such as fractional vortices, monopoles, skyrmions, and knots. In this paper, we discuss two topological excitations in spinor BECs: non-Abelian vortices and knots. Unlike conventional vortices, non-Abelian vortices neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. We discuss the collision dynamics of non-Abelian vortices in the cyclic phase of a spin-2 BEC. In the latter part, we show that a knot, which is a unique topological object characterized by a linking number or a Hopf invariant [π3 (S2) = Z], can be created using a conventional quadrupole magnetic field in a cold atomic system.