Andreas P. Schnyder

Max-Planck-Institut für Festkörperforschung, Stuttgart

Andreas P. Schnyder

June 11-13, 2014!

!

Université de Lorraine

Introduction to topological aspects in !

condensed matter physics

1st lecture: !

- Topological band theory!

- Topological insulators in 1D (polyacetylene)!

- Topological insulators in 2D (IQHE, QSHE)

2nd lecture:!

- Topological insulators w/ TRS in 2D & 3D (Z

2

invariant)!

- BdG theory for superconductors!

- Topological superconductors in 1D and 2D!

- Majorana bound states

3rd lecture:!

- Topological superconductors in 2D and 3D w/ TRS!

- Periodic table of topological insulators and superconductors

Ten-fold classification of !

topological insulators and superconductors

4th lecture:!

- Topological crystalline insulators!

- Gapless topological materials

Books and review articles

Review articles:!

- M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)!

- X.L. Qi and S.C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)!

- S. Ryu, A. P. Schnyder, A. Furusaki, A. Ludwig, New J. Phys. 12, 065010 (2010)!

- C. Beenakker, Annual Review of Cond. Mat. Phys. 4, 113 (2013)!

- J. Alicea, Rep. Prog. Phys. 75, 076501 (2012)!

- Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013)!

Books:!

- Shun-Qing Shen, “Topological insulators”, Springer Series in Solid-State ! Sciences, Volume 174 (2012)!

- B. Andrei Bernevig, "Topological Insulators and Topological Superconductors”, ! Princeton University Press (2013)!

- Mikio Nakahara, "Geometry, Topology and Physics", Taylor & Francis (2003) ! - A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger, “The geometric 
 phase in quantum systems”, Springer (2003)!

- M. Franz and L. Molenkamp, “Topological Insulators”, Contemporary Concepts 
 of Condensed Matter Science, Elsevier (2013)!

1st lecture: Topological band theory

1. Introduction!

- What is topology?!

- Topological band theory

3. Topological insulators in 2D!

- Integer quantum Hall effect!

- Bulk boundary correspondence!

- Chern insulator on square lattice!

2. Topological insulators in 1D!

- Berry phase !

- Simple example: Two-level system!

- Polyacetylene (Su-Schrieffer-Heeger model)!

- Domain wall states

For example, consider two-dimensional surfaces in three-dimensional space

Genus can be expressed in terms of an integral of the Gauss curvature over the surface

What is topology?

The study of geometric properties that are insensitive to smooth deformations

Closed surface is characterized by its genus g = # holes

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Gauss:

!

S

κ dA = 4π (1 − g ) (1)

Thermal Halll

κ

_xy

T = π

²

k

_B²

6h n (2)

start labels

4s 3p 3s E

_gap

− π/a + π /a (3)

end labels

H( k ) (4)

and

W ( k

_∥

) = (5)

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

⎞

⎟

⎟

⎠

, (6)

and

λ

_L

≫ ξ

₀

ξ

₀

= ! v

_F

/(π ∆

₀

) (7) (8)

λ

_L

> L ≫ ξ

₀

(9)

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→iω_n

,

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

(9) (10)

topological invariant

P er io dic Ta ble o f T op olo gic al In su la to rs an d S u pe rco n du ct or s

Anti-Unitary Symmetries : -Time Reversal : -Particle -Hole : Unitary (chiral) symmetry :

1 ()()12 ; HHkk

1 ()()12 ; HHkk 1 ()()HHkk ; Real K-theory

Complex K-theory Bott Periodicity d

Altland- Zirnbauer Random Matrix Classes Kitaev, 2008 Schnyder, Ryu, Furusaki, Ludwig 2008

8 antiunitary symmetry classes

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

Chern number g = 0, g = 1

n = !

bands

i 2π

"

F dk ² (1)

γ _C =

#

C A · d k (2)

First Chern number n = 0

n = !

bands

i 2π

"

dk ²

$% ∂ u

∂ k ₁

&

&

&

&

∂ u

∂ k ₂

'

−

% ∂ u

∂ k ₂

&

&

&

&

∂ u

∂ k ₁

'(

(3)

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

P er io dic Ta ble o f T op olo gic al In su la to rs an d S u pe rco n du ct or s

Anti-Unitary Symmetries : -Time Reversal : -Particle -Hole : Unitary (chiral) symmetry :

1 ()()12 ; HHkk

1 ()()12 ; HHkk 1 ()()HHkk ; Real K-theory

Complex K-theory Bott Periodicity d

Altland- Zirnbauer Random Matrix Classes Kitaev, 2008 Schnyder, Ryu, Furusaki, Ludwig 2008

8 antiunitary symmetry classes

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

1 frist chapter

Chern number g = 0, g = 1

n = !

bands

i 2π

"

F dk ² (1)

γ _C =

#

C A · d k (2)

First Chern number n = 0

n = !

bands

i 2π

"

dk ²

$% ∂ u

∂ k ₁

&

&

&

&

∂ u

∂ k ₂

'

−

% ∂ u

∂ k ₂

&

&

&

&

∂ u

∂ k ₁

'(

(3)

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

g is an integer topological invariant Gauss-Bonnet Theorem

In condensed matter physics:

Topology of insulating materials, topology of band structures

Band theory of solids and topology

Bloch’s theorem:

Topological equivalence:

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k

_x

k

_y

π /a − π /a (1)

majoranas

γ

₁

= ψ + ψ

^†

(2)

γ

₂

= −i !

ψ − ψ

^†

"

(3) and

ψ = γ

₁

+ iγ

₂

(4)

ψ

^†

= γ

₁

− iγ

₂

(5)

and

γ

_i²

= 1 (6)

{γ

_i

, γ

_j

} = 2δ

_ij

(7)

mean field

γ

_E^† ₌₀

= γ

_E=0

(8)

⇒ γ

_k^†_,E

= γ

_−k,−E

(9) Ξ ψ

₊k,+E

= τ

_x

ψ

₋^∗ k,−E

(10) Ξ

²

= +1 Ξ = τ

_x

K (11)

τ

_x

=

# 0 1

1 0

$

(12) c

^†

c c

^†

c ⇒ ⟨c

^†

c

^†

⟩c c = ∆

^∗

c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m ( k ) = m ( k )

| m ( k )| m ˆ ( k ) : m ˆ ( k ) ∈ S

²

π

₂

(S

²

) = (16) H

_BdG

= (2t [cos k

_x

+ cos k

_y

] − µ) τ

_z

+ ∆

₀

(τ

_x

sin k

_x

+ τ

_y

sin k

_y

) = m ( k ) · τ (17)

m

_x

m

_y

m

_z

(18)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k

_x

k

_y

π /a − π/a (1)

majoranas

γ

₁

= ψ + ψ

^†

(2)

γ

₂

= −i !

ψ − ψ

^†

"

(3) and

ψ = γ

₁

+ iγ

₂

(4)

ψ

^†

= γ

₁

− iγ

₂

(5)

and

γ

_i²

= 1 (6)

{γ

_i

, γ

_j

} = 2δ

_ij

(7)

mean field

γ

_E^† ₌₀

= γ

_E=0

(8)

⇒ γ

_k^†_,E

= γ

_−k,−E

(9) Ξ ψ

₊k,+E

= τ

_x

ψ

₋^∗ k,−E

(10) Ξ

²

= +1 Ξ = τ

_x

K (11)

τ

_x

=

# 0 1

1 0

$

(12) c

^†

c c

^†

c ⇒ ⟨c

^†

c

^†

⟩c c = ∆

^∗

c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m ( k )

|m(k)| m(k) : ˆ m(k) ˆ ∈ S

²

π

₂

(S

²

) = (16) H

_BdG

= (2t [cos k

_x

+ cos k

_y

] − µ) τ

_z

+ ∆

₀

(τ

_x

sin k

_x

+ τ

_y

sin k

_y

) = m ( k ) · τ (17)

m

_x

m

_y

m

_z

(18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

kx ky π/a − π/a (1)

majoranas

γ1 = ψ + ψ^† (2)

γ2 = −i!

ψ − ψ^†"

(3) and

ψ = γ1 + iγ2 (4)

ψ^† = γ1 − iγ2 (5)

and

γ_i² = 1 (6)

{γi,γj} = 2δij (7)

mean field

γ_E^† ₌₀ = γ_E=0 (8)

⇒ γk^†,E = γ₋k,−E (9) Ξψ₊k,+E = τxψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τx =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| m(k) :ˆ m(k)ˆ ∈ S² π2(S²) = (16) HBdG = (2t[coskx + cosky] − µ)τz + ∆0 (τx sinkx + τy sinky) = m(k) · τ (17)

mx my mz (18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

kx ky π/a − π/a (1)

majoranas

γ1 = ψ + ψ^† (2)

γ2 = −i !

ψ − ψ^†"

(3) and

ψ = γ1 + iγ2 (4)

ψ^† = γ1 − iγ2 (5)

and

γ_i² = 1 (6)

{γi,γj} = 2δij (7)

mean field

γ_E=0^† = γ_E=0 (8)

⇒ γk^†,E = γ₋k,−E (9) Ξ ψ₊k,+E = τxψ₋^∗k,−E (10) Ξ² = +1 Ξ = τxK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| mˆ (k) : mˆ (k) ∈ S² π2(S²) = (16) HBdG = (2t[coskx + cosky] − µ)τz + ∆0 (τx sinkx + τy sinky) = m(k) · τ (17)

mx my mz (18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ₂ = −i!

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i,γ_j} = 2δ_ij (7)

mean field

γ_E^† ₌₀ = γ_E=0 (8)

⇒ γ_k^†_,E = γ_−k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| mˆ (k) : mˆ (k) ∈ S² π₂(S²) = (16) H_BdG = (2t [cosk_x + cosk_y] − µ)τ_z + ∆0 (τx sink_x + τ_y sink_y) = m(k) · τ (17)

m_x m_y m_z (18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ2 = −i!

ψ − ψ^†"

(3) and

ψ = γ1 + iγ2 (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i,γ_j} = 2δ_ij (7)

mean field

γ_E=0^† = γ_E=0 (8)

⇒ γk^†,E = γ₋k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| m(k) :ˆ m(k)ˆ ∈ S² π₂(S²) = (16) HBdG = (2t[coskx + cos ky] − µ)τz + ∆0 (τxsinkx + τy sinky) = m(k) · τ (17)

mx my mz (18)

=

Bloch Hamiltonian

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k

_x

k

_y

π /a − π/a k ∈ Brillouin Zone (1)

majoranas

γ

₁

= ψ + ψ

^†

(2)

γ

₂

= −i !

ψ − ψ

^†

"

(3) and

ψ = γ

₁

+ iγ

₂

(4)

ψ

^†

= γ

₁

− iγ

₂

(5)

and

γ

_i²

= 1 (6)

{γ

_i

, γ

_j

} = 2δ

_ij

(7)

mean field

γ

_E^† ₌₀

= γ

_E=0

(8)

⇒ γ

_k^†_,E

= γ

_−k,−E

(9) Ξ ψ

₊k,+E

= τ

_x

ψ

₋^∗ k,−E

(10) Ξ

²

= +1 Ξ = τ

_x

K (11)

τ

_x

=

# 0 1

1 0

$

(12) c

^†

c c

^†

c ⇒ ⟨c

^†

c

^†

⟩c c = ∆

^∗

c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

| m ( k )| m(k) : ˆ m(k) ˆ ∈ S

²

π

₂

(S

²

) = (16) H

_BdG

= (2t [cos k

_x

+ cos k

_y

] − µ) τ

_z

+ ∆

₀

(τ

_x

sin k

_x

+ τ

_y

sin k

_y

) = m ( k ) · τ (17)

m

_x

m

_y

m

_z

(18)

Band structure defines a mapping:

Hamiltonians!

with energy gap 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)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

H ( k ) k

_x

k

_y

π/a − π/a k ∈ Brillouin Zone (1) majoranas

γ

₁

= ψ + ψ

^†

(2)

γ

₂

= −i !

ψ − ψ

^†

"

(3) and

ψ = γ

₁

+ iγ

₂

(4)

ψ

^†

= γ

₁

− iγ

₂

(5)

and

γ

_i²

= 1 (6)

{γ

_i

, γ

_j

} = 2δ

_ij

(7)

mean field

γ

_E^† ₌₀

= γ

_E=0

(8)

⇒ γ

k^†,E

= γ

₋k,−E

(9) Ξ ψ

₊k,+E

= τ

_x

ψ

₋^∗ k,−E

(10) Ξ

²

= +1 Ξ = τ

_x

K (11)

τ

_x

=

# 0 1

1 0

$

(12) c

^†

c c

^†

c ⇒ ⟨c

^†

c

^†

⟩c c = ∆

^∗

c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m ( k ) = m ( k )

| m ( k )| m ˆ ( k ) : m ˆ ( k ) ∈ S

²

π

₂

(S

²

) = (16) H

_BdG

= (2t [cos k

_x

+ cos k

_y

] − µ) τ

_z

+ ∆

₀

(τ

_x

sin k

_x

+ τ

_y

sin k

_y

) = m ( k ) · τ (17)

m

_x

m

_y

m

_z

(18)

Band structures are equivalent if they can be continuously ! deformed into one another without closing the energy gap Electron wavefunction in crystal

crystal momentum

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

kx ky π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ2 = −i!

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i,γ_j} = 2δ_ij (7)

mean field

γ_E=0^† = γ_E=0 (8)

⇒ γk^†,E = γ₋k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τx =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| m(k) :ˆ m(k)ˆ ∈ S² π₂(S²) = (16) HBdG = (2t[coskx + cosky] − µ)τz + ∆0 (τx sinkx + τy sinky) = m(k) · τ (17)

mx my mz (18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ₂ = −i!

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i,γ_j} = 2δ_ij (7)

mean field

γ_E=0^† = γ_E=0 (8)

⇒ γ_k^†_,E = γ_−k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| m(k) :ˆ m(k)ˆ ∈ S² π₂(S²) = (16) H_BdG = (2t[cosk_x + cosk_y] − µ)τ_z + ∆₀ (τ_x sin k_x + τ_y sink_y) = m(k) · τ (17)

m_x m_y m_z (18)

Energy

Momentum

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

homotopy

ν = # k_x (1)

∆^±k = ∆s ± ∆t |d^k| (2)

∆_s > ∆_t ∆_s ∼ ∆_t ν = ±1 for ∆_t > ∆_s (3) and

π₃[U(2)] = q(k) : ∈ U(2) (4)

Lattice BdG H_BdG

h(k) = ε^kσ₀ + αg^k · σ (5)

∆(k) = (∆_sσ₀ + ∆_td^k · σ) iσ_y (6)

h_ex I_y ≃ e

!

! k_F,−

k_F,+

dky

2π sgn

"

#

µ

H_ex^µ ρ^µ₁(0, k_y)

$%

− t sin k_y + λ

Lx/2

#

n=1

ρ^x_n(0, k_y) cos k_y

& . (7) and

j_n,ky = −t sin k_y '

c^†_nky↑c_nky↑ + c^†_nky↓c_nky↓(

(8) + λ cos k_y '

c^†_nky↓c_nky↑ + c^†_nky↑c_nky↓(

(9) The contribution j_n,k⁽¹⁾_y corresponds to nearest-neighbor hopping, whereas j_n,k⁽²⁾_y is due to SOC. We calculate the expectation value of the edge current at zero temperature from the spectrum E_l,ky and the wavefunctions )

)ψ_l,ky*

of H_k⁽¹⁰⁾_y , I_y = −e

! 1 N_y

#

k_y

L_x/2

#

n=1

#

l,E_l<0

⟨ψ_l,ky|j_n,ky|ψ_l,ky⟩ (10) We observe that the current operators presence of the superconducting gaps or the edge;

these only enter through the eigenstates |ψ_l,ky⟩.

Momentum dependent topological number:

∝

3

#

µ=1

H_ex^µ ρ^µ₁(E, ky) ρ^x₁ (11) N_QPI(ω, q) = − 1

πIm +

#

k

G₀(k, ω)T(ω)G₀(k + q, ω) ,

∝ -

S⃗_f) )

) T(ω) ) )

)S⃗_i.

(12)

a (13)

ξk^± = ε^k ± α |(14)g^k|

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Bloch theorem

[T(R), H] = 0 |ψ_n⟩ = e^ikr |u_n(k)⟩ (1) (2) H(k) = e^−ikrHe^+ikr (3) (4) H(k) |u_n(k)⟩ = E_n(k) |u_n(k)⟩ (5) we have

H(k) k_x k_y π/a − π/a k ∈ Brillouin Zone (6) majoranas

γ₁ = ψ + ψ^† (7)

γ₂ = −i !

ψ − ψ^†"

(8) and

ψ = γ₁ + iγ₂ (9)

ψ^† = γ₁ − iγ₂ (10)

and

γ_i² = 1 (11)

{γ_i, γ_j} = 2δ_ij (12)

mean field

γ_E^† ₌₀ = γ_E=0 (13)

⇒ γ_k^†_,E = γ_−k,−E (14) Ξ ψ₊k,+E = τ_xψ₋^∗ k,−E (15) Ξ² = +1 Ξ = τ_xK (16)

τ_x =

#0 1

1 0

$

(17) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (18) weak vs strong

|µ| < 4t (19)

n = 1 (20)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Bloch theorem

[T(R), H] = 0 |ψ_n⟩ = e^ikr |u_n(k)⟩ (1) (2) H(k) = e^−ikrHe^+ikr (3) (4) H(k) |u_n(k)⟩ = E_n(k) |u_n(k)⟩ (5) we have

H(k) k_x k_y π/a − π/a k ∈ Brillouin Zone (6) majoranas

γ₁ = ψ + ψ^† (7)

γ₂ = −i !

ψ − ψ^†"

(8) and

ψ = γ₁ + iγ₂ (9)

ψ^† = γ₁ − iγ₂ (10)

and

γ_i² = 1 (11)

{γ_i, γ_j} = 2δ_ij (12)

mean field

γ_E^† ₌₀ = γ_E=0 (13)

⇒ γ_k^†_,E = γ_−k,−E (14) Ξ ψ₊k,+E = τ_xψ₋^∗ k,−E (15) Ξ² = +1 Ξ = τ_xK (16)

τ_x =

#0 1

1 0

$

(17) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (18) weak vs strong

|µ| < 4t (19)

n = 1 (20)

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Bloch theorem

[T ( R ), H ] = 0 |ψ

_n

⟩ = e

^ikr

|u

_n

( k )⟩ (1) (2) H ( k ) = e

^−ikr

He

^+ikr

(3) (4) H (k) |u

_n

(k)⟩ = E

_n

(k) |u

_n

(k)⟩ (5) we have

H ( k ) k

_x

k

_y

π/a − π /a k ∈ Brillouin Zone (6) majoranas

γ

₁

= ψ + ψ

^†

(7)

γ

₂

= −i !

ψ − ψ

^†

"

(8) and

ψ = γ

₁

+ iγ

₂

(9)

ψ

^†

= γ

₁

− iγ

₂

(10)

and

γ

_i²

= 1 (11)

{γ

_i

, γ

_j

} = 2δ

_ij

(12)

mean field

γ

_E^† ₌₀

= γ

_E=0

(13)

⇒ γ

k^†,E

= γ

₋k,−E

(14) Ξ ψ

₊k,+E

= τ

_x

ψ

₋^∗ k,−E

(15) Ξ

²

= +1 Ξ = τ

_x

K (16)

τ

_x

=

# 0 1

1 0

$

(17) c

^†

c c

^†

c ⇒ ⟨c

^†

c

^†

⟩c c = ∆

^∗

c c (18) weak vs strong

|µ| < 4t (19)

n = 1 (20)

Bloch wavefunction!

has periodicity of potential

gap

consider electron wavefunction in periodic crystal potential