Topological insulators 
 and superconductors

Max-Planck-Institut für Festkörperforschung, Stuttgart

Andreas P. Schnyder

August 10-14, 2015

25th Jyväskylä Summer School

Topological insulators 


and superconductors

2nd lecture

1. Chern insulator and IQHE!

- Integer quantum Hall effect!

- Chern insulator on square lattice!

- Topological invariant

[p ic tu re c o u rte s y S. Z h a n g e t a l. ]

Edge states

• There is a gapless chiral edge mode along the sample boundary.

B r

k x

E

!

!

"

Number of edge modes C h

e

xy =

= !

2 /

"

Effective field theory

(

^{x x y y}

) ^{m ( ) y}

^z

iv

H = # ! " + ! " + !

( ) ^y

m

y

domain wall fermion

Robust against disorder (chiral fermions cannot be backscattered)

First example of 2D topological material

- 2D electron gas in large magnetic field, at low T

- There is an energy gap, but it is not an insulator

2D cyclotron motion ! Landau levels

Quantized Hall conductivity:

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h

²

ρ xy = _n ¹ _e ^h

²

n ∈

J y = σ xy E x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3) Θ ² Ξ ² Π ² (4) Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D (x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s _i (x)D _ij (x − x ^′ )s _j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward

[von Klitzing ‘80]

The Integer Quantum Hall State

Integer Quantum Hall State:

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k

^′

) k

F

> 1/ξ

0

sgn(∆

⁺_K

) = − sgn(∆

⁻_K

) and l

_k

antiparallel to l

_ek

sgn(∆

⁺_k

) = − sgn(∆

⁻_k

)

Symmetry Operations: E

gap

= ! ω

c

Θ H ( k )Θ

⁻¹

= + H ( − k ); Θ

²

= ± 1 (1) Ξ H ( k )Ξ

⁻¹

= −H ( − k ); Ξ

²

= ± 1 (2) Π H ( k )Π

⁻¹

= −H ( k ); Π ∝ ΘΞ (3)

Θ

²

Ξ

²

Π

²

(4)

Interaction Hamiltonian

S

int

= g

_ph²

2

!

dx dx

^′

ρ(x)D(x − x

^′

)ρ(x

^′

) + g

_sf²

2

!

dx dx

^′

s

i

(x)D

ij

(x − x

^′

)s

j

(x

^′

) (5)

1. f -Summenregel Modifications:

In passing, let us also comment on the dependence of ∆

_∞

on the integrated pump pulse intensity A

²₀

τ

_p

, which is shown in Fig. ?? (c) for nine different pulse widths τ

_p

. The asym- ptotic gap value ∆

_∞

is linear for A

²₀

τ

_p

→ 0 for τ

_p

≤ 2τ

_l

; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ

_l

exhibit a downward bend, while those with longer pulse widths an upward one. The curve with τ

p

= τ

l

lies in between these two regimes and marks the reach of full effectiveness of the single-photon processes. At relatively high integrated intensity, all downward bending curves (τ

p

≤ τ

l

) show a more or less sharp upward bend before reaching zero. Instead, upward bending curves (τ

p

> τ

l

) tend to flatten before reaching zero and to saturate for pulse widths larger than 4τ

l

with increasing A

²₀

τ

p

. This occurs because long pump pulses create sharp

En e rg y

- Plateaus in resistivity

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ xy = n ^e _h

²

ρ xy = _n ¹ _e ^h

²

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3)

Θ ² Ξ ² Π ² (4)

Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D(x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s i (x)D ij (x − x ^′ )s j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward bend, while those with longer pulse widths an upward one. The curve with τ _p = τ _l lies in between these two regimes and marks the reach of full effectiveness of the single-photon processes. At relatively high integrated intensity, all downward bending curves (τ _p ≤ τ _l )

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ xy = n ^e _h

²

ρ _xy = _n ¹ _e ^h

²

n ∈

J _y = σ _xy E _x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3) Θ ² Ξ ² Π ² (4) Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D (x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s i (x)D ij (x − x ^′ )s j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h

²

ρ _xy = _n ¹ _e ^h

²

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3)

Θ ² Ξ ² Π ² (4)

Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D (x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s _i (x)D _ij (x − x ^′ )s _j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward bend, while those with longer pulse widths an upward one. The curve with τ _p = τ _l lies in between these two regimes and marks the reach of full effectiveness of the single-photon processes. At relatively high integrated intensity, all downward bending curves (τ _p ≤ τ _l )

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h

²

ρ _xy = _n ¹ _e ^h

²

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3)

Θ ² Ξ ² Π ² (4)

Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D (x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s _i (x)D _ij (x − x ^′ )s _j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward bend, while those with longer pulse widths an upward one. The curve with τ _p = τ _l lies in between these two regimes and marks the reach of full effectiveness of the single-photon processes. At relatively high integrated intensity, all downward bending curves (τ _p ≤ τ _l )

B J y

E x

The Integer Quantum Hall State

— chiral edge state cannot be localized by disorder (no backscattering)

— charge cannot flow in bulk; only along 1D channels at edges (chiral edge states)

Edge states

• There is a gapless chiral edge mode along the sample boundary.

B r

k

x

E

!

!

" Number of edge modes C

h e

xy

=

= !

2

/

"

Effective field theory

(

^{x x y y}

) ^{m ( ) y}

^z

iv

H = # ! " + ! " + !

( ) ^y

m

y

domain wall fermion

Robust against disorder (chiral fermions cannot be backscattered)

IQHE has an energy gap in the bulk:

Explanation One:

Explanation Two:

Edge state transport

Topological band theory

Distinction between the integer quantum Hall state and a conventional insulator is a topological property of the band structure

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k ) : H ( k , k ^′ ) k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h

²

ρ xy = _n ¹ _e ^h

²

n ∈

J _y = σ _xy E _x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3) Θ ² Ξ ² Π ² (4) Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D(x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s i (x)D ij (x − x ^′ )s j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump

Hamiltonians with energy gap

What causes the precise quantization in IQHE?

Brillouin zone

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

Ξ H _BdG ( k ) Ξ ⁻¹ = − H _BdG ( − k ) "−→ (1)

∆n

Chern number g = 0, g = 1

n = !

bands

i 2π

"

F dk ² (2)

γ _C =

#

C A · d k (3)

First Chern number n = 0 n = !

bands

i 2π

"

dk ²

$% ∂ u

∂ k ₁

&

&

&

&

∂ u

∂ k ₂

'

−

% ∂ u

∂ k ₂

&

&

&

&

∂ u

∂ k ₁

'(

(4) H ( k ) :

H ( k , k ^′ )

k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h ² ρ _xy = _n ¹ _e ^{h 2} n ∈

J _y = σ _xy E _x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (5) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (6) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (7)

Θ ² Ξ ² Π ² (8)

Classified by Chern number: (= topological invariant)

[Thouless et al, 84]

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{e k} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ xy = n ^e _h

²

ρ _xy = _n ¹ _e ^h

²

n ∈

J _y = σ _xy E _x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3) Θ ² Ξ ² Π ² (4) Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D(x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s i (x)D ij (x − x ^′ )s j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward

does not change under smooth deformations, as long as bulk energy gap is not closed Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Chern number

n = i 2π

!

F dk

^{filled states}

(1)

Gauss:

!

S

κ dA = 4π(1 − g ) (2)

Thermal Halll

κ

_xy

T = π

²

k

²_B

6h n (3)

start labels

4s 3p 3s E

_gap

− π/a + π/a (4)

end labels

H(k) (5)

and

W (k

_∥

) = (6)

H

_BdG

=

⎛

⎜

⎜

⎝

ε

^k

− g

k^z

+∆

_s,^k

+ ∆

_t,^k

ε

^∗_⊥k

0 +∆

_s,^k

+ ∆

_t,^k

−ε

^k

+ g

k^z

0 −ε

^∗_⊥k

ε

_⊥k

0 ε

^k

+ g

k^z

−∆

_s,^k

+ ∆

_t,^k

0 −ε

_⊥k

−∆

_s,^k

+ ∆

_t,^k

−ε

^k

− g

k^z

⎞

⎟

⎟

⎠

, (7)

and

λ

_L

≫ ξ

₀

ξ

₀

= ! v

_F

/(π∆

₀

) (8) (9)

λ

_L

> L ≫ ξ

₀

(10)

charge current operator j

_y

(x) = iek

_F

/β

2π ! (

λ ˜

²

+ 1 )

iωn,ν

+π/2

!

−π/2

dθ

_ν

sin θ

_ν

×

* E

Ω

_ν

u

_ν

v

_ν

+

a

^he_ν,ν

+ a

^eh_ν,ν

,

e

^−2iqνx

-.

. .

.

E→iωn

,

j

_l,y

= + e

! t )

ky,σ

sin k

_y

c

^†_lkyσ

c

_lkyσ

− e

! α )

ky

cos k

_y

/

c

^†_lky↓

c

_lky↑

+ c

^†_lky↑

c

_lky↓

0

(10) (11)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Chern number

n = i 2π

!

F d k

^{filled states}

(1)

Gauss:

!

S

κ dA = 4π(1 − g ) (2)

Thermal Halll

κ

_xy

T = π

²

k

_B²

6h n (3)

start labels

4s 3p 3s E

_gap

− π/a + π/a (4)

end labels

H(k) (5)

and

W (k

_∥

) = (6)

H

BdG

=

⎛

⎜

⎜

⎝

ε

^k

− g

k^z

+∆

s,k

+ ∆

t,k

ε

^∗_⊥k

0 +∆

_s,^k

+ ∆

_t,^k

−ε

^k

+ g

k^z

0 −ε

^∗_⊥k

ε

_⊥k

0 ε

^k

+ g

k^z

−∆

s,k

+ ∆

t,k

0 −ε

_⊥k

−∆

s,k

+ ∆

t,k

−ε

^k

− g

k^z

⎞

⎟

⎟

⎠

, (7)

and

λ

L

≫ ξ

0

ξ

0

= ! v

F

/(π∆

0

) (8) (9)

λ

_L

> L ≫ ξ

₀

(10)

charge current operator j

_y

(x) = iek

_F

/β

2π ! (

λ ˜

²

+ 1

)

iωn,ν

+π/2

!

−π/2

dθ

_ν

sin θ

_ν

×

* E

Ω

_ν

u

_ν

v

_ν

+

a

^he_ν,ν

+ a

^eh_ν,ν

,

e

^−2iqνx

-.

. .

.

E→iωn

, j

_l,y

= + e

! t )

ky,σ

sin k

_y

c

^†_lkyσ

c

_lkyσ

− e

! α )

ky

cos k

_y

/

c

^†_lky↓

c

_lky↑

+ c

^†_lky↑

c

_lky↓

0

(10) (11)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

and

n = i 2π

! "

F d

²

k (1)

|u( k )⟩ → e

^iφk

|u( k )⟩ (2)

A → A + ∇

^k

φ

^k

(3)

F = ∇

^k

× A (4)

γ

_C

=

#

C

A · d k (5)

γ

_C

=

"

S

F d

²

k (6)

= ⇒ (7)

Bloch theorem

[T ( R ), H ] = 0 k |ψ

_n

⟩ = e

^ikr

|u

_n

( k )⟩ (8) (9) H ( k ) = e

^−ikr

He

^+ikr

(10) (11) H (k) |u

_n

(k)⟩ = E

_n

(k) |u

_n

(k)⟩ (12) we have

H ( k ) k

_x

k

_y

π/a − π /a k ∈ Brillouin Zone (13) majoranas

γ

₁

= ψ + ψ

^†

(14)

γ

₂

= −i $

ψ − ψ

^†

%

(15) and

ψ = γ

₁

+ iγ

₂

(16)

ψ

^†

= γ

₁

− iγ

₂

(17)

and

γ

_i²

= 1 (18)

{γ

_i

, γ

_j

} = 2δ

_ij

(19)

— edge states are perfect charge conductor!

⇥ _xy = e ² h

i 2

X Z F d ² k

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Chern number

n = i 2π

!

F dk

^{filled states}

(1)

Gauss:

!

S

κ dA = 4π(1 − g) (2)

Thermal Halll

κ

_xy

T = π

²

k

_B²

6h n (3)

start labels

4s 3p 3s E

_gap

− π/a + π/a (4)

end labels

H(k) (5)

and

W (k

_∥

) = (6)

H

_BdG

=

⎛

⎜

⎜

⎝

ε

^k

− g

k^z

+∆

_s,^k

+ ∆

_t,^k

ε

^∗_⊥k

0 +∆

_s,^k

+ ∆

_t,^k

−ε

^k

+ g

k^z

0 −ε

^∗_⊥k

ε

_⊥k

0 ε

^k

+ g

k^z

−∆

_s,^k

+ ∆

_t,^k

0 −ε

_⊥k

−∆

_s,^k

+ ∆

_t,^k

−ε

^k

− g

k^z

⎞

⎟

⎟

⎠

, (7)

and

λ

_L

≫ ξ

₀

ξ

₀

= ! v

_F

/(π∆

₀

) (8) (9)

λ

_L

> L ≫ ξ

₀

(10)

charge current operator j

_y

(x) = iek

_F

/β

2π ! (

λ ˜

²

+ 1 )

iωn,ν

+π/2

!

−π/2

dθ

_ν

sin θ

_ν

×

* E

Ω

_ν

u

_ν

v

_ν

+

a

^he_ν,ν

+ a

^eh_ν,ν

,

e

^−2iqνx

-.

. .

.

E→iωn

,

j

_l,y

= + e

! t )

ky,σ

sin k

_y

c

^†_lkyσ

c

_lkyσ

− e

! α )

ky

cos k

_y

/

c

^†_lky↓

c

_lky↑

+ c

^†_lky↑

c

_lky↓

0

(10) (11)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Chern number

n = i 2π

!

F d k

^{filled states}

(1)

Gauss:

!

S

κ dA = 4π(1 − g ) (2)

Thermal Halll

κ

_xy

T = π

²

k

_B²

6h n (3)

start labels

4s 3p 3s E

_gap

− π/a + π/a (4)

end labels

H(k) (5)

and

W (k

_∥

) = (6)

H

BdG

=

⎛

⎜

⎜

⎝

ε

^k

− g

k^z

+∆

s,k

+ ∆

t,k

ε

^∗_⊥k

0 +∆

_s,^k

+ ∆

_t,^k

−ε

^k

+ g

k^z

0 −ε

^∗_⊥k

ε

_⊥k

0 ε

^k

+ g

k^z

−∆

s,k

+ ∆

t,k

0 −ε

_⊥k

−∆

s,k

+ ∆

t,k

−ε

^k

− g

k^z

⎞

⎟

⎟

⎠

, (7)

and

λ

L

≫ ξ

0

ξ

0

= ! v

F

/(π∆

0

) (8) (9)

λ

_L

> L ≫ ξ

₀

(10)

charge current operator j

_y

(x) = iek

_F

/β

2π ! (

λ ˜

²

+ 1

)

iωn,ν

+π/2

!

−π/2

dθ

_ν

sin θ

_ν

×

* E

Ω

_ν

u

_ν

v

_ν

+

a

^he_ν,ν

+ a

^eh_ν,ν

,

e

^−2iqνx

-.

. .

.

E→iωn

, j

_l,y

= + e

! t )

ky,σ

sin k

_y

c

^†_lkyσ

c

_lkyσ

− e

! α )

ky

cos k

_y

/

c

^†_lky↓

c

_lky↑

+ c

^†_lky↑

c

_lky↓

0

(10) (11)

Kubo formula:

Follows from the quantization of the topological invariant.

Zero-energy states must exist at the interface between two different topological phases

Bulk-boundary correspondence

smooth transition

n=1 n=0

x y

Zero-energy state at interface

Bulk-boundary correspondence:

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Chern number

n = i 2π

!

F dk

^{filled states}

(1)

Gauss:

!

S

κ dA = 4π(1 − g) (2)

Thermal Halll

κ

_xy

T = π

²

k

_B²

6h n (3)

start labels

4s 3p 3s E

_gap

− π/a + π/a (4)

end labels

H(k) (5)

and

W (k

_∥

) = (6)

H

_BdG

=

⎛

⎜

⎜

⎝

ε

^k

− g

k^z

+∆

_s,^k

+ ∆

_t,^k

ε

^∗_⊥k

0 +∆

_s,^k

+ ∆

_t,^k

−ε

^k

+ g

k^z

0 −ε

^∗_⊥k

ε

_⊥k

0 ε

^k

+ g

k^z

−∆

_s,^k

+ ∆

_t,^k

0 −ε

_⊥k

−∆

_s,^k

+ ∆

_t,^k

−ε

^k

− g

k^z

⎞

⎟

⎟

⎠

, (7)

and

λ

_L

≫ ξ

₀

ξ

₀

= ! v

_F

/(π∆

₀

) (8) (9)

λ

_L

> L ≫ ξ

₀

(10)

charge current operator j

_y

(x) = iek

_F

/β

2π ! (

λ ˜

²

+ 1 )

iωn,ν

+π/2

!

−π/2

dθ

_ν

sin θ

_ν

×

* E

Ω

_ν

u

_ν

v

_ν

+

a

^he_ν,ν

+ a

^eh_ν,ν

,

e

^−2iqνx

-.

. .

.

E→iωn

,

j

_l,y

= + e

! t )

ky,σ

sin k

_y

c

^†_lkyσ

c

_lkyσ

− e

! α )

ky

cos k

_y

/

c

^†_lky↓

c

_lky↑

+ c

^†_lky↑

c

_lky↓

0

(10) (11)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Chern number

n = i 2π

!

F d k

^{filled states}

(1)

Gauss:

!

S

κ dA = 4π(1 − g ) (2)

Thermal Halll

κ

_xy

T = π

²

k

_B²

6h n (3)

start labels

4s 3p 3s E

_gap

− π/a + π/a (4)

end labels

H(k) (5)

and

W (k

_∥

) = (6)

H

BdG

=

⎛

⎜

⎜

⎝

ε

^k

− g

k^z

+∆

s,k

+ ∆

t,k

ε

^∗_⊥k

0 +∆

_s,^k

+ ∆

_t,^k

−ε

^k

+ g

k^z

0 −ε

^∗_⊥k

ε

_⊥k

0 ε

^k

+ g

k^z

−∆

s,k

+ ∆

t,k

0 −ε

_⊥k

−∆

s,k

+ ∆

t,k

−ε

^k

− g

k^z

⎞

⎟

⎟

⎠

, (7)

and

λ

L

≫ ξ

0

ξ

0

= ! v

F

/(π∆

0

) (8) (9)

λ

_L

> L ≫ ξ

₀

(10)

charge current operator j

_y

(x) = iek

_F

/β

2π ! (

λ ˜

²

+ 1

)

iωn,ν

+π/2

!

−π/2

dθ

_ν

sin θ

_ν

×

* E

Ω

_ν

u

_ν

v

_ν

+

a

^he_ν,ν

+ a

^eh_ν,ν

,

e

^−2iqνx

-.

. .

.

E→iωn

, j

_l,y

= + e

! t )

ky,σ

sin k

_y

c

^†_lkyσ

c

_lkyσ

− e

! α )

ky

cos k

_y

/

c

^†_lky↓

c

_lky↑

+ c

^†_lky↑

c

_lky↓

0

(10) (11)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

and

n = i 2π

! "

F d

²

k (1)

|u( k )⟩ → e

^iφk

|u( k )⟩ (2)

A → A + ∇

^k

φ

^k

(3)

F = ∇

^k

× A (4)

γ

_C

=

#

C

A · d k (5)

γ

_C

=

"

S

F d

²

k (6)

= ⇒ (7)

Bloch theorem

[T (R), H ] = 0 k |ψ

_n

⟩ = e

^ikr

|u

_n

(k)⟩ (8) (9) H ( k ) = e

^−ikr

He

^+ikr

(10) (11) H ( k ) |u

_n

( k )⟩ = E

_n

( k ) |u

_n

( k )⟩ (12) we have

H ( k ) k

_x

k

_y

π/a − π /a k ∈ Brillouin Zone (13) majoranas

γ

₁

= ψ + ψ

^†

(14)

γ

₂

= −i $

ψ − ψ

^†

%

(15) and

ψ = γ

₁

+ iγ

₂

(16)

ψ

^†

= γ

₁

− iγ

₂

(17)

and

γ

_i²

= 1 (18)

{γ

_i

, γ

_j

} = 2δ

_ij

(19)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

H ( k , k ^′ ) k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h

²

ρ xy = _n ¹ _e ^h

²

n ∈

J _y = σ _xy E _x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (1) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (2) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (3) Θ ² Ξ ² Π ² (4) Interaction Hamiltonian

S _int = g _ph ² 2

!

dx dx ^′ ρ(x)D (x − x ^′ )ρ(x ^′ ) + g _sf ²

2

!

dx dx ^′ s _i (x)D _ij (x − x ^′ )s _j (x ^′ ) (5) 1. f -Summenregel

Modifications:

In passing, let us also comment on the dependence of ∆ _∞ on the integrated pump pulse intensity A ² ₀ τ _p , which is shown in Fig. ?? (c) for nine different pulse widths τ _p . The asym- ptotic gap value ∆ _∞ is linear for A ² ₀ τ _p → 0 for τ _p ≤ 2τ _l ; for larger values of the pump pulse, it shows instead an upward bend because of the full effectiveness of the two-photon processes. At higher, but still not so large, integrated intensity, the single-phonon proces- ses dominate and the curves corresponding to pulses shorter than τ _l exhibit a downward

topological invariant

Stable gapless edge states:

= number or edge modes

IQHE: chiral Dirac Fermion

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

and

E ₀ k _y (1)

2γ _C = solid angle swept out by ˆ d ( k ) (2)

H (k) = d(k) · σ d ˆ (3)

n = i 2π

! "

F d ² k (4)

|u(k)⟩ → e ^iφ

^k

|u(k)⟩ (5)

A → A + ∇ ^k φ ^k (6)

F = ∇ ^k × A (7)

γ _C =

#

C

A · dk (8)

γ _C =

"

S

F d ² k (9)

= ⇒ (10)

Bloch theorem

[T ( R ), H ] = 0 k |ψ _n ⟩ = e ^{i kr} |u _n ( k )⟩ (11) (12) H ( k ) = e ^{−i kr} He ^{+i kr} (13) (14) H ( k ) |u _n ( k )⟩ = E _n ( k ) |u _n ( k )⟩ (15) we have

H (k) k _x k _y π/a − π/a k ∈ Brillouin Zone (16) majoranas

γ ₁ = ψ + ψ ^† (17)

γ ₂ = −i $

ψ − ψ ^† %

(18) and

ψ = γ ₁ + iγ ₂ (19)

ψ ^† = γ ₁ − iγ ₂ (20)

and

γ _i ² = 1 (21)

{γ _i , γ _j } = 2δ _ij (22)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

and

E ₀ k _y (1)

2γ _C = solid angle swept out by ˆ d ( k ) (2) H ( k ) = d ( k ) · σ d ˆ (3)

n = i 2π

! "

F d ² k (4)

|u( k )⟩ → e ^iφ

^k

|u( k )⟩ (5)

A → A + ∇ ^k φ ^k (6)

F = ∇ ^k × A (7)

γ _C =

#

C

A · d k (8)

γ _C =

"

S

F d ² k (9)

= ⇒ (10)

Bloch theorem

[T ( R ), H ] = 0 k |ψ _n ⟩ = e ^{i kr} |u _n ( k )⟩ (11) (12) H ( k ) = e ^{−i kr} He ^{+i kr} (13) (14) H ( k ) |u _n ( k )⟩ = E _n ( k ) |u _n ( k )⟩ (15) we have

H ( k ) k _x k _y π /a − π /a k ∈ Brillouin Zone (16) majoranas

γ ₁ = ψ + ψ ^† (17)

γ ₂ = −i $

ψ − ψ ^† %

(18) and

ψ = γ ₁ + iγ ₂ (19)

ψ ^† = γ ₁ − iγ ₂ (20)

and

γ _i ² = 1 (21)

{γ _i , γ _j } = 2δ _ij (22)

• robust to smooth deformations !

(respect symmetries of the system)!

• ! insensitive to disorder, impossible to localize!

! • cannot exist in a purely 1D system ! (Fermion doubling theorem)

n = | n _L n _R |

Topology and Two Bands Model

1 Empty Flat Band 1 Filled Flat Band

k

h(k)

h

2⇡

It is not possible to have a coherent phase convention for all points of the sphere

‣ if does not cover the whole sphere : single phase convention possible. «Standard trivial case»

‣ If spreads over the whole sphere :

we need 2 independent phase conventions

! signals a topological property : the wavefunction phase winds by around the sphere

h(k) h(k)

single phase convention possible

Trivial Band Twisted Band

Chern number ⟺ winding of electronic phase

Topological Property

| u

_k

| u ˜

_k

e

^{i (k)}

vendredi 13 septembre 13

Topology and Two Bands Model

1 Empty Flat Band

1 Filled Flat Band

k

h(k)

h

2⇡

It is not possible to have a coherent phase convention for all points of the sphere

‣ if does not cover the whole sphere : single phase convention possible. «Standard trivial case»

‣ If spreads over the whole sphere :

we need 2 independent phase conventions

! signals a topological property : the wavefunction phase winds by around the sphere

h(k) h(k)

single phase convention possible

Trivial Band Twisted Band

Chern number ⟺ winding of electronic phase

Topological Property

| u

_k

| u ˜

_k

e

^{i (k)}

vendredi 13 septembre 13

Chern insulator on square lattice

[D. Haldane PRL ’88]

Tight-binding model:

d _x (k) = sin k _x d _y (k) = sin k _y d _z (k) = (2 + M cos k _x cos k _y )

Chern number:

Spectrum flattening:

E _± = ± | d(k) | ^{d(k) = ˆ d(k)}

| d(k) |

no edge state trivial phase

d _y

d _x d _z

chiral edge state non-trivial phase

d _z

d _x d _y

quantized Hall effect _xy = e ² h n H ^CI = d(k) · ~ + ✏ ₀ (k) ₀

Chern insulator (“integer quantum Hall state on a lattice”)

E

k

4 < M < 0

n _Z = ± 1 chiral edge state

n _Z = 0

n _Z = 1 8⇡

Z

BZ

d ² k ✏ ^µ⌫ d ˆ · h

@ _k

_µ

d ˆ ⇥ @ _k

_⌫

d ˆ i

H

_CI

= ⇣

c

^†_s,k

c

^†_p,k

⌘

H

^CI

✓ c

_s,k

c

_p,k

Experimental realization: Cr-doped (Bi,Sb)

2

Te

3

◆ [Chang et al. Science ’13]

M > 0

Brillouin zone

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

Ξ H _BdG ( k ) Ξ ⁻¹ = − H _BdG ( − k ) "−→ (1)

∆n

Chern number g = 0, g = 1

n = !

bands

i 2π

"

F dk ² (2)

γ _C =

#

C A · d k (3)

First Chern number n = 0 n = !

bands

i 2π

"

dk ²

$% ∂ u

∂ k ₁

&

&

&

&

∂ u

∂ k ₂

'

−

% ∂ u

∂ k ₂

&

&

&

&

∂ u

∂ k ₁

'(

(4) H ( k ) :

H ( k , k ^′ )

k _F > 1/ξ ₀

sgn(∆ ⁺ _K ) = − sgn(∆ ⁻ _K ) and l _k antiparallel to l _{k e} sgn(∆ ⁺ _k ) = − sgn(∆ ⁻ _k )

σ _xy = n ^e _h ² ρ _xy = _n ¹ _e ^{h 2} n ∈

J _y = σ _xy E _x

Symmetry Operations: E _gap = ! ω _c

Θ H ( k )Θ ⁻¹ = + H ( − k ); Θ ² = ± 1 (5) Ξ H ( k )Ξ ⁻¹ = −H ( − k ); Ξ ² = ± 1 (6) Π H ( k )Π ⁻¹ = −H ( k ); Π ∝ ΘΞ (7)

Θ ² Ξ ² Π ² (8)

Mapping d(k) : ˆ d(k) ˆ 2 S ²

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

energy spectrum Simple example Polyacethylene:

“⌅ ₂ (S ² ) = ” H (k) = d(k) · =

⌅ 0 h(k) h ^† (k ) 0

⇧

(1) and

[C ₂ H ₂ ] _n (2)

F = 1 2

k ˆ

k ² (3)

C =

⌃

S F ^⌅ d⇤d⇧ = (4)

F ^µ⇥ = ⇥ _µ⇥⇤ F ^⇤ (5)

F = ⇤ ^k ⇥ A (6)

d(k) = k (7)

C =

⌃

S F · dk (8)

Berry curvature tensor

F ^µ⇥ (k) = ⌃

⌃ k _µ A ^⇥ (k) ⌃

⌃k _⇥ A ^µ (k) (9)

Berry curvature

F ^k

i

,k

_j

= sin ⇤ 2

⌃ (⇤, ⇧)

⌃ (k _i , k _j ) (10)

k d(k) (11)

F ^⌅ = ⌃ A _⌅ ⌃ _⌅ A = sin ⇤

2 (12)

Berry vector potential

A = i u _k ⇤ ⇤ ⌃ ⇤ ⇤ u _k ⇥

= 0 (13)

A _⌅ = i u _k ⇤ ⇤ ⌃ _⌅ ⇤ ⇤ u _k ⇥

= sin ² (⇤/2) (14)

and

A = (15)

⇤ ⇤ u ⁺ _k ⇥

=

⌅ cos(⇤ /2)e ^i⌅

sin(⇤ /2)

⇧

(16)

⇤ ⇤ u _k ⇥

=

⌅ sin(⇤ /2)e ^i⌅

cos(⇤/2)

⇧

(17) (18)

E _± = ± | d | (19)

Chern insulator on square lattice

[D. Haldane PRL ’88]

Tight-binding model:

d _x (k) = sin k _x d _y (k) = sin k _y d _z (k) = (2 + M cos k _x cos k _y )

Spectrum flattening:

E _± = ± | d(k) | ^{d(k) = ˆ d(k)}

| d(k) |

H ^CI = d(k) · ~ + ✏ ₀ (k) ₀

Chern insulator (“integer quantum Hall state on a lattice”)

E

k

chiral edge state

H

_CI

= ⇣

c

^†_s,k

c

^†_p,k

⌘

H

^CI

✓ c

_s,k

c

_p,k

Experimental realization: Cr-doped (Bi,Sb)

2

Te

3

◆ [Chang et al. Science ’13]

n _Z = 0

M > 0 ⁴ < M < 0

n _Z = ± 1

trivial phase non-trivial phase

Chern number: n _Z = 1 8⇡

Z

BZ

d ² k ✏ ^µ⌫ d ˆ · h

@ _k

_µ

d ˆ ⇥ @ _k

_⌫

d ˆ i

Texture of unit vector d(k) ˆ

Chern insulator on square lattice

Chern insulator on square lattice:

d _x (k) = sin k _x d _y (k) = sin k _y d _z (k) = (2 + M cos k _x cos k _y )

Effective low-energy continuum theory for M=0: (expand around ; term can be neglected) k = 0

E _± = ± = ± p

k ² + M ² two eigenfunctions with energies:

u

⁺_k

↵

= 1

p 2 ( M )

✓ k

_x

ik

_y

M

◆

u _k ↵

= 1

p 2 ( + M )

✓ k _x + ik _y + M

◆

Berry curvature:

zero-energy state ! at boundary

F _xy = ⇥ _k

_x

A _k

_y

⇥ _k

_y

A _k

_x

= + M 2 ³ gives nonzero Chern number !

(= Hall conductance ) _xy n = 1 2

Z

d ² k F _xy = 1

2 sgn(M )

NB: Chern number must be integer for integrals over compact manifolds. ! Proper regularization of Dirac Hamiltonian will lead to

Chiral edge state at boundary between two Chern insulators with different n

n = 0 n = 1

0 = 1

p 2

✓ 1 1

◆

e ^ik

^y

^y e ^R

^0x

^{M (x}

⁰

^)dx

⁰

H _CI = k _{x x} + k _{y y} + M _z

H ^CI = d(k) · ~ + ✏ ₀ (k) ₀

0

n 2 Z