novel topological phases and surface states in interacting...
TRANSCRIPT
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2469-13
Workshop and Conference on Geometrical Aspects of Quantum States in Condensed Matter
Ashvin Vishwanath
1 - 5 July 2013
University of California at Berkeley
Novel Topological Phases and Surface States in Interacting Systems
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• •
– • Hd+1(G,U(1))
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•
(ρ, jx, jy) =∇× a
2π
• I,J•
L =∑I,J
KIJ
4πaI · ∇ × aJ
KKIJL =∑I,J
KKIJJJJJ
44444πaI · ∇∇∇ aJ
(∂t, ∂x, ∂y) = ∇
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•
• •
L =∑I,J
KIJ
4πaI · ∇ × aJ Ledge =
1
4πKIJ φ̇I∂xφJ
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•
• •
KE8 =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
2 −1 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0
0 −1 2 −1 0 0 0 −1
0 0 −1 2 −1 0 0 0
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 −1 0 0 0 0 2
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
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– –
K =
(0 11 0
)q =
(11
)
L =1
2πa1 · ∇ × a2 +
∇×A
2π(a1 + a2)
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•
Ledge = φ̇1∂xφ2
2π
φ
ρ =∂xφ2
2π
1
2πa1 · ∇ × a2
Δ φ
φ1 → φ1 + ε φ1 → φ1 + ε
φ2 → φ2 + εφ2 → φ2
(ϕ)
(2θ)
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• φ,θ).
θ
φ φ θ
φ θ φ
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νσ Θ=π.
σσ Θ=2π.⎜
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•
• –
•
– σ
–
vortex = (ψ↑, ψ↓)
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• – –
– •
•
vortex = (ψ↑, ψ↓)
∇× a
2π= n
Ledge = |(∂ − ia+ iσ
2A)ψσ|2 +
∇× a
2π·A+ La
ψ↑ a =1
2A Lem =
1
4πA · ∇ ×A
σxy = 1 (Θ = 2π)
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κ
Ε8 κ
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•
• • ••
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• – –
π
π
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•
•
1
D
∑a
d2aθa = ei2πc−/8
•
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•
– – –
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σ, τH = −
∑V
AV −∑P
BP
AV =
(∏i
σxi +
∏i
τxi
)i
BP =∏�
σz +∏�
τzU O BP = [σx]O∏�
σz[σxτx]U + [σxτx]O∏�
τz[τx]U
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T
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•
– –
•
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s× s = s̃× s̃ = 1 + s+ s̃s× s̃ = e+ s+ s̃
d1 = de = 1, ds = ds̃ = 1 +√2
θ1 = 1, θe = −1, θs = i, θs̃ = −i
Fusion rules : Quantum dimensions and topological spins :
i -i
•
• • •
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i i
-i -i
• σ κ θ=π• •
• •
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Ψ[ ]
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•
•
Ψ[ ]
i
-i
T 2 = −1
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•
–
•
–
• –