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Magnetic Skyrmions N ∈ ℤ = π2(S2)
N = − 1 4π ∫ m ⋅ ( ∂m ∂x × ∂m ∂y ) dxdy
Magnetic Hopfions Paul Sutcliffe Department of Mathematical - - PowerPoint PPT Presentation
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 = 2 ( S 2 ) 4 m ( y ) dxdy N = 1 m x
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Magnetic Skyrmion size ~ 100nm
Experiments on Fe-Co-Si alloy, imaged using transmission electron microscopy (TEM)
Yu et al, Nature 465, 901 (2010)
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Hopfion T
- pology
Q ∈ ℤ = π3(S2)
m(∞) = (0,0,1) −(0,0,1) (1,0,0)
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Hopfion T
- pology
m(∞) = (0,0,1)
Q ∈ ℤ = π3(S2)
(1,0,0) −(0,0,1) −(1,0,0) (0,1,0) −(0,1,0)
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Hopfion as a T wisted Skyrmion T ube Q ∈ ℤ = π3(S2) Q twists
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An example with Q=1
m = (
8xz − 4y(1 − r2) (1 + r2)2
,
8yz + 4x(1 − r2) (1 + r2)2
,
1 + 8z2 − 6r2 + r4 (1 + r2)2
)
m = − (0,0,1) on the circle x2 + y2 = 1 in the plane z = 0 m → (0,0,1) as r2 = x2 + y2 + z2 → ∞
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Energy Minimization
E = ∫ ∫ ∫ {A( ∂m ∂x
2
+ ∂m ∂y
2
+ ∂m ∂z
2
) +˜ A ∂2m ∂x2 + ∂2m ∂y2 + ∂2m ∂z2
2
+H(1 − mz)} dxdydz
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Q=1 Ring
m = (0, 0, −1)
m = (0, − 1 √ 2, − 1 √ 2)
m3
y = 0 x z
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Q=3 Ring
(p, q) = (Z3
1, Z2)
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Q=6 Ring
(p, q) = (Z6
1, Z2)
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Q=6 Ring : circular —> bent
(p, q) = (Z6
1, Z2)
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Q=2+2+2 Link —> Q=6 Ring
(p, q) = (Z3
1, Z2 1 + Z2 2)
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Q=3+2+2 Link
(p, q) = (Z4
1 + Z3 1Z2 + Z3 1 − Z2 1Z2, Z2 1 − Z2 2)
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Q=4+3 Knot —> Q=3+2+2 Link
(p, q) = (Z2
1Z2, Z3 1 + Z2 2)
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Q=7+3 Knot
(p, q) = (Z2
1Z2 2, Z3 1 + Z2 2)
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Energies
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Hopfion Initial Conditions From Polynomials
p(Z1, Z2) & q(Z1, Z2) are polynomials p(0,1) = 0 m = 1 |p|2 + |q|2 (¯ pq + p¯ q, i¯ pq − ip¯ q, |q|2 − |p|2 )
(p, q) = (Zn
1, Z2)
Q = n
(p, q) = (Zα
1 Zβ 2, Za 1 + Zb 2)
Q = aβ + bα
axial rings
(a, b) torus knots/links
(Z1, Z2) = ( 2(x + iy) 1 + r2 , r2 − 1 + 2iz 1 + r2 )
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Hopfions in the Skyrme-Faddeev model
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