quantum spin hall effect: a brief introduction
TRANSCRIPT
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Quantum spin Hall effect: a brief introduction
![Page 2: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/2.jpg)
Topological phases of matter
Are 2D topological phases possible without an applied magnetic field?
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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…
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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)
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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!)
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.”
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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. 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
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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. 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)
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Laurens Molenkamp
Observed in HgTe quantum wells!
M. König et al., Science 318, 766 (2007)
2005… next breakthrough: ”There are many more topological phases of matter”
2D ”quantum spin Hall insulator” from strong spin-orbit interactions
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Laurens Molenkamp
Observed in HgTe quantum wells!
M. König et al., Science 318, 766 (2007)
2D ”quantum spin Hall insulator” from strong spin-orbit interactions
2005… next breakthrough: ”There are many more topological phases of matter”
d<6.3 nm normal band order conventional insulator
d>6.3 nm inverted band order topological insulator
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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?
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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)
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)
B. A. Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
time-reversal
invariant
![Page 13: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/13.jpg)
Consider a Gedanken experiment...
uniformly charged cylinder with electric field
spin-orbit interaction
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)
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)
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)
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)
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
B = �⇤A
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)
![Page 14: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/14.jpg)
Consider a Gedanken experiment...
uniformly charged cylinder with electric field
spin-orbit interaction
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)
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)
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)
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)
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
B = �⇤A
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)
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![Page 15: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/15.jpg)
Consider a Gedanken experiment...
uniformly charged cylinder with electric field
spin-orbit interaction
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)
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)
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)
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)
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
B = �⇤A
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)
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Two copies of an IQH system, bulk insulator with helical edge states
Quantum spin Hall (QSH) insulator single Kramers pair
![Page 17: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/17.jpg)
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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
?
![Page 18: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/18.jpg)
| >
| >
| >
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
–
+| > | >– new Kramers pair
| >+
![Page 19: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/19.jpg)
Why ”topological”?
![Page 20: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/20.jpg)
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
Why ”topological”?
![Page 21: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/21.jpg)
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
band index
Why ”topological”?
![Page 22: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/22.jpg)
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
band index
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
Brillouin zone (BZ)
Why ”topological”?
![Page 23: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/23.jpg)
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
Bloch wave function
Why ”topological”?
![Page 24: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/24.jpg)
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
⇥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
”Berry connection”
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
”Berry curvature”
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
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)
Why ”topological”?
Bloch wave function
![Page 25: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/25.jpg)
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
BZ
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
⇥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
⇥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
⇥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
”effective” BZ
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
⇥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
⇥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
identify T-conjugatepoints in the BZ
Why ”topological”?
![Page 26: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/26.jpg)
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
BZ
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
⇥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
⇥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
⇥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
”effective” BZ
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
⇥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
⇥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
identify T-conjugatepoints in the BZ
open manifold:NO quantization from the Berry curvature
=
Why ”topological”?
![Page 27: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/27.jpg)
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
⇥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
⇥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
⇥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
⇥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
⇥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
⇥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
⇥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
=
”close” the cylinder!
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
ZFn(k) · dk = C
C =
⇢0 mod 2 ordinary insulator1 mod 2 topological insulator
Why ”topological”?
![Page 28: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/28.jpg)
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
ZFn(k) · dk = C
C =
⇢0 mod 2 ordinary insulator1 mod 2 topological insulator
The parity of C is a Z2 invariant, independent of the ”closure”!
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
ZFn(k) · dk = C
C =
⇢0 mod 2 ordinary insulator1 mod 2 topological insulator
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)
Why ”topological”?
![Page 29: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/29.jpg)
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.
![Page 30: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/30.jpg)
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)
![Page 31: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/31.jpg)
Some cool stuff exploiting the helical edge states in quantum spin Hall insulators:
”On-demand” spin entangler
Phys. Rev. B 91, 245406 (2015)
![Page 32: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/32.jpg)
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.
![Page 33: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/33.jpg)
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)
![Page 34: Quantum spin Hall effect: a brief introduction](https://reader030.vdocuments.net/reader030/viewer/2022012806/61bd414661276e740b10ee24/html5/thumbnails/34.jpg)
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…