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)

•  φ,θ).

θ

φ φ θ

φ θ φ

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)

κ

Ε8 κ

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π

π

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1

D

∑a

d2aθa = ei2πc−/8

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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

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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