Spin texture with a topological number Skyrmion is supposed to be topologically stable; Experimentally, it is not stable! Critical re-examination of.

Fate of Topology in Spin-1 Spinor Bose-Einstein Con-

densate

Yun-Tak Oh, Panjin Kim, Jin-Hong Park, Jung Hoon Han,

arXiv:1309.5683

Yun-Tak Oh

Sungkyunkwan University

1. Introduction to Skyrmion texture in spin-1 BEC ( Experiments by SNU group (prof. YI Shin) )

2. Failure of the conventional classification of spin-1 BEC

3. New and complete dynamics of spin-1 BEC

CONTENTS

What is a Skyrmion?

Spin texture with a topological number

Skyrmion is supposed to be topologically stable;Experimentally, it is not stable!

Critical re-examination of ex-isting theory of spinor dy-namics

First successful creation of Skyrmion spin texture in spinor BEC

Shin group, PRL 108, 035301 (2012)

Spin-1 BEC classified as

Spin-spin interaction in the spin-1 condensate:

Dynamics of spin-1 BEC: Gross-Pitaevskii(GP) Equation

antiferromagnetic (AFM) for g2 > 0

ferromagnetic (FM) for g2<0

Where

FMAFM

: Initial state

Spin-1 BEC

Implicitly assumed dynamics occur within AFM or FM man-ifold

Strategy: project onto three orthogonal spinors to get three hydrodynamic equations (Refael, PRB 2009)

Mass continuity eq:

Euler eq:

Landau-Lifshitz eq:

And…

No spatio-temporal fluctuation is allowed within FM mani-fold!!

In FM Limit

!

No spatio-temporal fluctuation is allowed within AFM man-ifold

with ONE EXCEPTION (next talk)

Mass continuity eq:Euler eq:

Landau-Lifshitz eq:

In AFM Limit

Again…!

FMAFM

Spin-1 BEC

All dynamics involves evolution into a mixed state (δ ≠ 0)

Relation to Skyrmion dynamics

From homotopy consideration, stability of Skyrmion only guaranteed within AFM manifold.

However, temporal evolution within AFM manifold is in-trinsically forbidden!!

Therefore, there is no meaning to Skyrmion as a topologi-cal object.

Conclusion:

• Initially tried to understand unstable Skyrmion dynamics

• Instead found neither AFM nor FM sub-manifold supports a well-define d dynamics

(FM; t=0) (FM+AFM, t>0) (AFM; t=0) (AFM+FM, t>0)

• Numerical solution of the Gross-Pitaevskii equation proves our claim (next talk)