Magnetic Hopfions Paul Sutcliffe Department of Mathematical - PowerPoint PPT Presentation

Magnetic Hopfions Paul Sutcliffe Department of Mathematical Sciences Durham University Physical Review Letters, 118, 247203 (2017); arXiv:1705.10966

Magnetic Skyrmions N ∈ ℤ = π 2 ( S 2 ) 4 π ∫ m ⋅ ( ∂ y ) dxdy N = − 1 ∂ m ∂ x × ∂ m

Magnetic Skyrmion size ~ 100nm Yu et al, Nature 465, 901 (2010) Experiments on Fe-Co-Si alloy, imaged using transmission electron microscopy (TEM)

Hopfion T opology m ( ∞ ) = (0,0,1) − (0,0,1) (1,0,0) Q ∈ ℤ = π 3 ( S 2 )

Hopfion T opology m ( ∞ ) = (0,0,1) − (0,0,1) (1,0,0) Q ∈ ℤ = π 3 ( S 2 ) − (1,0,0) (0,1,0) − (0,1,0)

Hopfion as a T wisted Skyrmion T ube Q ∈ ℤ = π 3 ( S 2 ) Q twists

An example with Q=1 m = ( ) 1 + 8 z 2 − 6 r 2 + r 4 8 xz − 4 y (1 − r 2 ) 8 yz + 4 x (1 − r 2 ) , , (1 + r 2 ) 2 (1 + r 2 ) 2 (1 + r 2 ) 2 m → (0,0,1) as r 2 = x 2 + y 2 + z 2 → ∞ m = − (0,0,1) on the circle x 2 + y 2 = 1 in the plane z = 0

Energy Minimization E = ∫ ∫ ∫ { A ( ∂ m ∂ m ∂ m 2 2 2 ) + + ∂ x ∂ y ∂ z ∂ 2 m ∂ x 2 + ∂ 2 m ∂ y 2 + ∂ 2 m 2 + ˜ A ∂ z 2 + H (1 − m z ) } dxdydz

Q=1 Ring m 3 y = 0 z m = (0 , 0 , − 1) x m = (0 , − 1 2 , − 1 2) √ √

Q=3 Ring ( p , q ) = ( Z 3 1 , Z 2 )

Q=6 Ring ( p , q ) = ( Z 6 1 , Z 2 )

Q=6 Ring : circular —> bent ( p , q ) = ( Z 6 1 , Z 2 )

Q=2+2+2 Link —> Q=6 Ring ( p , q ) = ( Z 3 1 , Z 2 1 + Z 2 2 )

Q=3+2+2 Link ( p , q ) = ( Z 4 1 + Z 3 1 Z 2 + Z 3 1 − Z 2 1 Z 2 , Z 2 1 − Z 2 2 )

Q=4+3 Knot —> Q=3+2+2 Link ( p , q ) = ( Z 2 1 Z 2 , Z 3 1 + Z 2 2 )

Q=7+3 Knot ( p , q ) = ( Z 2 1 Z 2 2 , Z 3 1 + Z 2 2 )

Energies

Hopfion Initial Conditions From Polynomials ( Z 1 , Z 2 ) = ( ) 1 + r 2 , r 2 − 1 + 2 iz 2( x + iy ) 1 + r 2 p ( Z 1 , Z 2 ) & q ( Z 1 , Z 2 ) are polynomials p (0,1) = 0 1 | p | 2 + | q | 2 ( ¯ ) q , | q | 2 − | p | 2 m = pq + p ¯ q , i ¯ pq − ip ¯ ( p , q ) = ( Z n axial rings 1 , Z 2 ) Q = n ( a , b ) torus knots/links 1 Z β ( p , q ) = ( Z α 2 , Z a 1 + Z b Q = a β + b α 2 )

Hopfions in the Skyrme-Faddeev model

THE END