Quantum spin Hall effect: a brief introduction

Quantum spin Hall effect: a brief introduction

Topological phases of matter

Are 2D topological phases possible without an applied magnetic field?

Topological phases of matter

Duncan Haldane

Well..., at least one doesn’t need a net magnetic field!

F. D. M. Haldane Model for a Quantum Hall Effect without Landau Levels:Condensed-Matter Realization of the Parity AnomalyPhysical Review Letters, 61, 2015, 1988.

Are 2D topological phases possible without an applied magnetic field?

1988…

2005… next breakthrough: ”There are many more topological phases of matter”

Charlie Kane Gene Mele

New topological invariant for a 2D time-reversal invariant system (no magnetic field!)

C. L. Kane and E. J. Mele,Phys. Rev. Lett. 95, 146802 (2005)




Breakthrough Prize In Fundamental PhysicsCharles Kane and Eugene Mele – University of Pennsylvania Citation: For new ideas about topology and symmetry in physics, leading to the prediction of a new class of materials that conduct electricity only on their surface.Description: Since the days of Ben Franklin, we've come to distinguish between electrical forms of matter that are either conducting or insulating. But that concept has been turned inside-out by Charles Kane and Gene Mele who have predicted a new class of materials – “topological insulators” – that are inviolable conductors of electricity on the boundary but insulators in the interior. Their discovery has important implications for the “space-race” in quantum computing and could lead to new generations of electronic devices that promise enormous energy efficiencies in computation. Topological insulators also offer a window into deep questions about the fundamental nature of matter and energy, since they exhibit particle-like excitations similar to the fundamental particles of physics (electrons and photons) but can be controlled in the laboratory in ways that electrons and photons cannot. These connections offer a new conceptual framework for controlling the flow of charge, light and even of mechanical waves in various states of matter. Unanticipated applications, too, seem inevitable: when the transistor was invented in 1947, no one could realistically predict that it would lead to information technologies that would allow terabytes of data to be crammed onto a tiny silicon chip.

“Kane and Mele introduced new ideas of topology in quantum physics in a quite remarkable way,” said Edward Witten, chair of the selection committee. “It is beautiful how this story has unfolded.”




C. Kane and E.J. Mele,Phys. Rev. Lett. 95, 146802 (2005)

Prediction: new topological phase of matter in HgTe quantum wells!

B. A. Bernevig, T. L. Hughes, andS.-C. Zhang, Science 314, 1757 (2006)

Soucheng Zhang




C. Kane and E.J. Mele,Phys. Rev. Lett. 95, 146802 (2005)

Prediction: new topological phase of matter in HgTe quantum wells!

B. A. Bernevig, T. L. Hughes, andS.-C. Zhang, Science 314, 1757 (2006)

Soucheng Zhang

Laurens Molenkamp

Confirmed experimentally!

M. König et al., Science 318, 766 (2007)

Laurens Molenkamp

Observed in HgTe quantum wells!



2D ”quantum spin Hall insulator” from strong spin-orbit interactions

Laurens Molenkamp

Observed in HgTe quantum wells!


2D ”quantum spin Hall insulator” from strong spin-orbit interactions


d<6.3 nm normal band order conventional insulator

d>6.3 nm inverted band order topological insulator

A quantum spin Hall insulator looks like two copies of an integer quantum Hall system stacked on top of each other. How does a a spin-orbit interaction achieve this?

Consider a Gedanken experiment...

uniformly charged cylinder with electric field

spin-orbit interaction

1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)

A · k ⌃ eB(kyx� kxy)

G = ⌅e2

h

Mc ⌥ 100 meV

D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)

W. Hausler, L. Kecke, and A. H. MacDonald, Phys.Rev. B 65, 085104 (2002)

��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)

with vi and Ki functions of g1⇤ , g2⇤ and g4⇤

⇥ ⌃ bandwidth (2)

c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†

⇤ create excitations at the Fermi points of theright- and left-moving branches with spin projection ⌥

(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥

�2�R cos(Qx)�e�2ik0

F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃

yµ⇥cn+1,⇥�H.c.

⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)

Hint = g1� :R†⇤L⇤L†

�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)

with vi and K functions of g1⇤ , g2⇤ , g3⇤ (11)

1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


B. A. Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)

time-reversal

invariant




cf. with the IQHE in a symmetric gauge

Lorentz force

1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

B = �⇤A

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)





Lorentz force

1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

B = �⇤A

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)

| >





Lorentz force

1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)


1

B = �⇤A

E = E(x, y, 0)

(E ⇤ k) · � = E⌃z(kyx� kxy)

A =B

2(y,�x, 0)


G = ⌅e2

h

Mc ⌥ 100 meV



��0 2⇤ 10�11 eV m⇥ 0.5 eV

vF 1⇤ 106 m/sKc + Ks 1.8

HR

H =⌃

dx [Hc +Hs]

Hi =vi

2[(�x i)2+(�x�i)2]�

mi

⇧acos(

⌥2⇧Ki i), (1)

�R = ⇥1 sin(q0a)



c ⇧ ( + + �)/↵

2 (3)

s ⇧ ( + � �)/↵

2 (4)

R†⇤ and L†


(5)

L+

vF = 2a�

t2 + ⇥20 and �R = ⇥1 sin(q0a)

H⇤ =�ivF

�:R†

⇤ (x)�xR⇤ (x) ::L†⇤ (x)�xL⇤ (x) :

⇥


F (x+a/2)R†⇤ (x)L⇤ (x)+H.c.

⇥,(6)

⌥ = ±

q0

HR =�i⇧

n,µ,⇥

(⇥0 + ⇥1 cos (Qna))⇤c†n,µ⌃


⌅

⇥j = �ja�1 (j = 0, 1)

c ⇧ ( + + �)/↵

2 (7)

s ⇧ ( + � �)/↵

2 (8)


�⇤R�⇤ : + g2⇤ :R†+R+L†

⇤L⇤ :

+g4⇤

2(:R†

+R+R†⇤R⇤ : +R � L) (9)

g2⇤ ⇧g2⇤ � ⇤⇤+g1⇤ (10)

| >

| >

| >

| >

| >

Two copies of an IQH system, bulk insulator with helical edge states

Quantum spin Hall (QSH) insulator single Kramers pair

| >

| >



perturb with a time-reversal invariant spin-nonconserving interaction

?

| >

| >

| >



perturb with a time-reversal invariant spin-nonconserving interaction

–

+| > | >– new Kramers pair

| >+

Why ”topological”?

1

⇥n,k(r) = un,k(r)eik·r

� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧

Fn(k) = ⌃k ⇥An(k)

1

2�

X

n

Z

BZFn(k) · dk = C, C ⇤ Z

1

2�

X

n

Z

EBZFn(k) · dk = C

C =

⇢0 mod 2 ordinary insulator1 mod 2 topological insulator


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


band index


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


band index

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


Brillouin zone (BZ)


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


Bloch wave function


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


”Berry connection”

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


”Berry curvature”

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


Chern number

The Chern number C measures the quantized Hallconductance in an integer quantum Hall system.

C vanishes for a time-reversal invariant system.However, there is still a topological structure present!

C. L. Kane and E. J. Mele, PRL 95, 226801 (2005)


Bloch wave function

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


BZ

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


”effective” BZ

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


identify T-conjugatepoints in the BZ


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


BZ

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


”effective” BZ

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


identify T-conjugatepoints in the BZ

open manifold:NO quantization from the Berry curvature

=


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

Z

EBZFn(k) · dk = C

C =


=

”close” the cylinder!

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

ZFn(k) · dk = C

C =



1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

ZFn(k) · dk = C

C =


The parity of C is a Z2 invariant, independent of the ”closure”!

1


� A B C

An(k) = �i⌅un,k | ⌃k | un,k⇧


1

2�

X

n

Z


1

2�

X

n

ZFn(k) · dk = C

C =


J. E. Moore and L. Balents, PRB 75, 121306(R) (2007)

signals the presence of robust Kramers pairs on the edge

bulk-edge correspondence L. Fu and C. L. Kane, PRB 74, 195312 (2006)


3D (”strong”) topological insulators have also robustspin-momentum locked edge (= surface) states.Theory: L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007) Experiment on Bi1-xSbx: Hsieh et al., Science 323, 919 (2008) 106803.

3D (”strong”) topological insulators have also robustspin-momentum locked edge (= surface) states.Theory: L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007) Experiment on Bi1-xSbx: Hsieh et al., Science 323, 919 (2008) 106803.

Electron spin polarization in photoemission experiments determined by photon polarization. C. Jozwiak et al., Phys. Rev B 84, 165113 (2011)

Some cool stuff exploiting the helical edge states in quantum spin Hall insulators:

”On-demand” spin entangler

Phys. Rev. B 91, 245406 (2015)

Bad news: Experimental realizations of 2D topological insulators are tricky to handle! Since its discovery in 2006, the topological phase of the HgTe/CdTe quantum well has still only been probed experimentally in Laurens Molenkamp’s lab in Würzburg.

Bad news: Experimental realizations of 2D topological insulators are tricky to handle! Since its discovery in 2006, the topological phase of the HgTe/CdTe quantum well has still only been probed experimentally in Laurens Molenkamp’s lab in Würzburg.

Candidate 2D topological insulators (a.k.a. quantum spin Hall insulators):

”Stanene” (single atomic layer of tin)Xu et al., PRL (2013)

InAs/GaSb quantum wellsSuzuki et al., PRB (2013)

SiliceneC.-C. Liu et al., PRL (2011)

Alternative realizations of helical electron liquids* in high demand!

* … this is the most interesting feature of 2D topological insulators!

More on this on Thursday when discussing topological superconductivity…

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Quantum spin Hall effect: a brief introduction

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