topological quantum matter, 7.5 ectsphysics.gu.se/~tfkhj/topo/topologicalquantummatter.pdf ·...

Topological Quantum Matter, 7.5 ECTSphysics.gu.se/~tfkhj/topomatter.html

Part 1: 10 double hour lectures 12/9 - 3/10, 27/10 - 31/10 (tentatively)Part 2: 5 guest lectures 10/11 - 12/12

Examination: homework problems on part 1, project based on one of the guest lectures

Literature: Lecture notes downloadable from the course homepage.Additional text/references will be made available during the course.

Masters students: To get credit for the course, please contactBengt-Erik Mellander, [email protected], after completion ofthe examination.

mailto:[email protected]


Concepts in Topological Quantum Matter, 4.5 ECTSphysics.gu.se/~tfkhj/topomatter/

Part 1: 10 double hour lectures 12/9 - 3/10, 27/10 - 31/10 (tentatively)Part 2: 5 guest lectures 10/11 - 12/12

Examination: homework problems on part 1, project based on one of the guest lectures

Literature: Lecture notes downloadable from the course homepage.Additional text/references will be made available during the course.

Masters students: To get credit for the course, please contactBengt-Erik Mellander, [email protected], after completion ofthe examination.



What is topological quantum matter & why do we care?

Before the era of topology in physics...

Standard paradigm of emergent order:spontaneous symmetry breaking


Some examples:

Crystals break the translational and rotational symmetry of free space


Crystals

Liquid crystals break rotational but not translational symmetry


Crystals

Liquid crystals

Magnets break time-reversal symmetry and the rotational symmetry of spin space


Crystals

Liquid crystals

Magnets

Superconductors break a gauge symmetry


Crystals

Liquid crystals

Magnets

Superconductors

... and many more examples


Crystals

Liquid crystals

Magnets

Superconductors

... and many more examples

At high temperature, entropy dominates and leads to a disordered state.

At low temperature, energy dominates and leads to an ordered state.

Lev Landau http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Landau_Le...

1 av 1 06-01-16 14.40

The ordered state breaks one or more symmetries ”spontaneously”.

Lev Landau


In 1980, the ”Landau paradigm” was challenged by the discovery of the (integer) quantum Hall effect (von Klitzing et al.)

Electrons confined to a plane and in a strong magnetic field show, at low enough temperature, plateaus in the “Hall conductance”:

to a precision of at least 9 decimal places!

There is no symmetry breaking. What type of ordercauses this precise quantization?

TOPOLOGICAL ORDER!

Topological quantum matter

”Symmetry-protected topological matter”

Unique groundstate protected by atopological invariant (Chern number,Z2-index,...), ordinary electron excitations,short-range quantum entanglement:

integer quantum Hall effect,topological insulators,Chern insulators,topological superconductors,...

Groundstate degeneracies on higher-genus manifolds, fractionalized excitations,long-range quantum entanglement:

fractional quantum Hall effect,quantum spin liquids,...

”Topologically ordered matter” (proper)

Topological quantum matter

”Topologically protected matter”

Unique groundstate protected by atopological invariant (Chern number,Z2-index,...), ordinary electron excitations,short-range quantum entanglement:

integer quantum Hall effect,

topological insulators,Chern insulators,topological superconductors,...

Groundstate degeneracies on higher-genus manifolds, fractionalized excitations,long-range quantum entanglement:

fractional quantum Hall effect,quantum spin liquids,...

”Topologically ordered matter” (proper)

The topological insulatorsprovide the best ”port of entry” to the study of topological quantum matter.This is how we shall go about it in the course!

”An electrical insulator is a material whose internal electric charges do not flow freely... examples include glass, paper, Teflon, plastics, ceramic materials,... ”

?

http://en.wikipedia.org/wiki/Electric_charge

http://en.wikipedia.org/wiki/Electric_charge

Ordinary insulators have nothing to do with topology, but topological insulators do!?

2D topological insulators... taking off from the quantum Hall effect

no channel for backscattering

B“skipping current”

quantization

chiral edge states

perfect conductance along the edge

2D topological insulators... taking off from quantum Hall effect

B

perfect conductance along the edge von Klitzing et al., PRL (1980)

2D topological insulators... taking off from quantum Hall effect

Is this kind of physics possible without a magnetic field?

a bulk insulator with perfectlyconducting edge states

Duncan Haldane

Well..., at least one doesn’t need anet magnetic field... PRL, 1988

Charlie Kane Gene Mele

In fact, one can do away with themagnetic field altogether! PRLs, 2005

To see how this is possible, consider a Gedanken experiment... Bernevig & Zhang, PRL (2006)

spin-orbit interaction

1

E = E(x, y, 0)

(E × k) · σ = Eσz(kyx− kxy)

A =B

2(y,−x, 0)

A · k ∼ eB(kyx− kxy)

G = νe2

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)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

H =

�dx [Hc +Hs]

Hi =vi

2[(∂xϕi)

2+(∂xϑi)

2]−

mi

πacos(

�2πKiϕi), (1)

∆R = γ1 sin(q0a)

with vi and Ki functions of g1τ , g2τ and g4τ

Λ ∼ bandwidth (2)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (3)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (4)

R†τ and L

†τ create excitations at the Fermi points of the

right- and left-moving branches with spin projection τ

(5)

L+

vF = 2a

�t2 + γ

20 and ∆R = γ1 sin(q0a)

Hτ =−ivF

�:R

†τ (x)∂xRτ (x) ::L

†τ (x)∂xLτ (x) :

�

−2∆R cos(Qx)�e−2ik0

F (x+a/2)R

†τ (x)Lτ (x)+H.c.

�,(6)

τ = ±

q0

HR =−i

�

n,µ,ν

(γ0 + γ1 cos (Qna))

�c†n,µσ

yµνcn+1,ν−H.c.

�

γj = αja−1

(j = 0, 1)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (7)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (8)

Hint = g1− :R†τLτL

†−τR−τ : + g2τ :R

†+R+L

†τLτ :

+g4τ

2(:R

†+R+R

†τRτ : +R↔ L) (9)

g2τ ≡g2τ − δτ+g1τ (10)

with vi and K functions of g1τ , g2τ , g3τ (11)

compare with an integer quantum Hall system

Lorentz force

1

E = E(x, y, 0)


A =B

2(y,−x, 0)


G = νe2

h

Mc ≈ 100 meV



Rev. B 65, 085104 (2002)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

H =

�dx [Hc +Hs]

Hi =vi

2[(∂xϕi)

2+(∂xϑi)

2]−

mi

πacos(

�2πKiϕi), (1)

∆R = γ1 sin(q0a)



ϕc ≡ (ϕ+ + ϕ−)/

√2 (3)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (4)

R†τ and L



(5)

L+

vF = 2a

�t2 + γ


Hτ =−ivF

�:R

†τ (x)∂xRτ (x) ::L

†τ (x)∂xLτ (x) :

�


F (x+a/2)R


�,(6)

τ = ±

q0

HR =−i

�

n,µ,ν


�c†n,µσ

yµνcn+1,ν−H.c.

�

γj = αja−1

(j = 0, 1)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (7)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (8)


†−τR−τ : + g2τ :R

†+R+L

†τLτ :

+g4τ

2(:R

†+R+R

†τRτ : +R↔ L) (9)

g2τ ≡g2τ − δτ+g1τ (10)


1

B = ∇×A

E = E(x, y, 0)


A =B

2(y,−x, 0)


G = νe2

h

Mc ≈ 100 meV



Rev. B 65, 085104 (2002)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

H =

�dx [Hc +Hs]

Hi =vi

2[(∂xϕi)

2+(∂xϑi)

2]−

mi

πacos(

�2πKiϕi), (1)

∆R = γ1 sin(q0a)



ϕc ≡ (ϕ+ + ϕ−)/

√2 (3)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (4)

R†τ and L



(5)

L+

vF = 2a

�t2 + γ


Hτ =−ivF

�:R

†τ (x)∂xRτ (x) ::L

†τ (x)∂xLτ (x) :

�


F (x+a/2)R


�,(6)

τ = ±

q0

HR =−i

�

n,µ,ν


�c†n,µσ

yµνcn+1,ν−H.c.

�

γj = αja−1

(j = 0, 1)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (7)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (8)


†−τR−τ : + g2τ :R

†+R+L

†τLτ :

+g4τ

2(:R

†+R+R

†τRτ : +R↔ L) (9)

g2τ ≡g2τ − δτ+g1τ (10)

To see how this is possible, consider a Gedanken experiment...

spin-orbit interaction

1

E = E(x, y, 0)


A =B

2(y,−x, 0)


G = νe2

h

Mc ≈ 100 meV



Rev. B 65, 085104 (2002)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

H =

�dx [Hc +Hs]

Hi =vi

2[(∂xϕi)

2+(∂xϑi)

2]−

mi

πacos(

�2πKiϕi), (1)

∆R = γ1 sin(q0a)



ϕc ≡ (ϕ+ + ϕ−)/

√2 (3)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (4)

R†τ and L



(5)

L+

vF = 2a

�t2 + γ


Hτ =−ivF

�:R

†τ (x)∂xRτ (x) ::L

†τ (x)∂xLτ (x) :

�


F (x+a/2)R


�,(6)

τ = ±

q0

HR =−i

�

n,µ,ν


�c†n,µσ

yµνcn+1,ν−H.c.

�

γj = αja−1

(j = 0, 1)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (7)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (8)


†−τR−τ : + g2τ :R

†+R+L

†τLτ :

+g4τ

2(:R

†+R+R

†τRτ : +R↔ L) (9)

g2τ ≡g2τ − δτ+g1τ (10)


compare with an integer quantum Hall system

Lorentz force

1

E = E(x, y, 0)


A =B

2(y,−x, 0)


G = νe2

h

Mc ≈ 100 meV



Rev. B 65, 085104 (2002)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

H =

�dx [Hc +Hs]

Hi =vi

2[(∂xϕi)

2+(∂xϑi)

2]−

mi

πacos(

�2πKiϕi), (1)

∆R = γ1 sin(q0a)



ϕc ≡ (ϕ+ + ϕ−)/

√2 (3)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (4)

R†τ and L



(5)

L+

vF = 2a

�t2 + γ


Hτ =−ivF

�:R

†τ (x)∂xRτ (x) ::L

†τ (x)∂xLτ (x) :

�


F (x+a/2)R


�,(6)

τ = ±

q0

HR =−i

�

n,µ,ν


�c†n,µσ

yµνcn+1,ν−H.c.

�

γj = αja−1

(j = 0, 1)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (7)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (8)


†−τR−τ : + g2τ :R

†+R+L

†τLτ :

+g4τ

2(:R

†+R+R

†τRτ : +R↔ L) (9)

g2τ ≡g2τ − δτ+g1τ (10)


1

B = ∇×A

E = E(x, y, 0)


A =B

2(y,−x, 0)


G = νe2

h

Mc ≈ 100 meV



Rev. B 65, 085104 (2002)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

H =

�dx [Hc +Hs]

Hi =vi

2[(∂xϕi)

2+(∂xϑi)

2]−

mi

πacos(

�2πKiϕi), (1)

∆R = γ1 sin(q0a)



ϕc ≡ (ϕ+ + ϕ−)/

√2 (3)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (4)

R†τ and L



(5)

L+

vF = 2a

�t2 + γ


Hτ =−ivF

�:R

†τ (x)∂xRτ (x) ::L

†τ (x)∂xLτ (x) :

�


F (x+a/2)R


�,(6)

τ = ±

q0

HR =−i

�

n,µ,ν


�c†n,µσ

yµνcn+1,ν−H.c.

�

γj = αja−1

(j = 0, 1)

ϕc ≡ (ϕ+ + ϕ−)/

√2 (7)

ϕs ≡ (ϕ+ − ϕ−)/

√2 (8)


†−τR−τ : + g2τ :R

†+R+L

†τLτ :

+g4τ

2(:R

†+R+R

†τRτ : +R↔ L) (9)

g2τ ≡g2τ − δτ+g1τ (10)

| >

| >

To see how this is possible, consider a Gedanken experiment...

| >

| >

Two copies of a quantum Hall system, bulk insulator with helical edge states

2D topological insulator

single Kramers pair

x

How does Nature do it?Also by spin-orbit interactions!

x

”s-band”

”p-band”

EF

ordinary insulator

local band structure

x

How does Nature do it?Strong atomic spin-orbit interactions!

x

topological insulatorordinary insulator ordinary insulator

EF

”s-band”

”p-band”

conducting edge stateslocal band structure

x

How does Nature do it?Strong atomic spin-orbit interactions!

x

topological insulator

EFconducting edge states

local band structure

First proposed by Kane and Mele for graphene (2005)C. L. Kane and E. J. Mele, PRL 95, 226801 (2005)

Bernevig et al. proposal for HgTe quantum wells (2006)B. A. Bernevig, T. A. Hughes, and S. C. Zhang, Science 314, 1757 (2006)

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

Experimental realizations...

The helical edge states are stable against time-reversal invariant perturbations,no disorder backscattering

edge states

HgTe quantum well | >–

+| >

1

G =2e

2

h

L ∼ 1µm

L ∼ 20µm

g

g ∼ 1/2v

ξ < L

α(±2kF )

±2kF

z⊥ ≡ 2g

√CK

z� ≡ 4K − 2

γ = (α(2kF ) + α(−2kF ))/2

∂xθ

ψ↑ = η↑ exp�i√

π[φ + θ]�/

√2πκ

ψ↓ = η↓ exp�−i√

π[φ− θ]�/

√2πκ

H0 + Hd + Hf + HR

O(α2)

α(x) = �α(x)�+

�

n

α(kn)eikxn

Ψ↑↓(x)=ψ↑↓(x)e±ikF x

Ψ↓(x)=ψ↓(x)e−ikF x

HR = α(x)(z × k) · σ

B = ∇×A

E = E(x, y, 0)


A =B

2(y,−x, 0)


G = νe2

h

Mc ≈ 100 meV



Rev. B 65, 085104 (2002)

�α0 � 2× 10−11

eV m

Λ � 0.5 eV

vF � 1× 106

m/s

Kc + Ks � 1.8

HR

ballistic transport

”symmetry

-protecte

d”

topologic

al phase

König et al. (2007)

normal band gap

large samples(inelastic scatteringfrom the bulk)

Fermi level notinside the inverted gap

What is ”topological” about a topological insulator?

Warm up... Gauss-Bonnet theorem

1

1

2π

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

Pierre Ossian Bonnet, 1819-1892

1

�

MKdS = 2(1− g)

K = κmin · κmax

1

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M


Warm up... Gauss-Bonnet theorem

1

1

2π

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

1

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

1

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

1

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

1

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

1

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

What is ”topological” about a topological insulator? 1

1

2π

�

MKdS = 2(1− g)

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

Generalized Gauss-Bonnet theorem

”Chern theorem”

Berry curvature

Chern numberShiing-Shen Chern, 1911-2004


1

H = −t

�

n,α

c†n,αcn+1,α +

µ

2

�

n,α

c†n,αcn,α −i

�

n,α,β

c†n,α

�γD σx

αβ+γR σyαβ

�cn+1,β (1)

− iγ�R

�

n,α,β

cos(Qna)c†n,ασyαβcn+1,β +

µ�

2

�

n,α

cos(Qna)c†n,αcn,α (2)

+�

n,n�;α,β

V (n− n�)c†n,αc

†n�,βcn�,βcn,α + h. c. (3)

SA(n,N)− SA(n− 1, N − 1) = Simp

whereJK = J1 = J2 = Kondo couplingn = number of sites in A incl. the impurity siteN = total number of sites in A+B

HRashba = αkxσy

EF

HZeeman = (µBgB/2)σx

SA(JK , n,NL, NR)− SA(1, n− 1, NL, NR) = Simp

whereNL and NR are the number of sites in the left and right blue block, respectively

1

2π

�

MF · dS = C

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

Berry curvature

Chern number

un,k(r) = � r |un,k �

1

ψn,k(r) = un,k(r)eik·r

Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = ∇k ×An(k)

1

2π

�

n

�

BZFn(k) · dk = C, C ∈ Z

1

2π

�

n

�

EBZFn(k) · dk = C

C =

�0 mod 2 ordinary insulator1 mod 2 topological insulator

band index

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�

EBZFn(k) · dk = C

C =


Brillouin zone (BZ)

Electron wave function in a crystal

1

H = −t

�

n,α

c†n,αcn+1,α +

µ

2

�

n,α

c†n,αcn,α −i

�

n,α,β

c†n,α

�γD σx

αβ+γR σyαβ

�cn+1,β (1)

− iγ�R

�

n,α,β


µ�

2

�

n,α


+�

n,n�;α,β


†n�,βcn�,βcn,α + h. c. (3)

SA(n,N)− SA(n− 1, N − 1) = Simp


HRashba = αkxσy

EF




1

2π

�

MF · dS = C

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

Berry curvature

Chern number

un,k(r) = � r |un,k �

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�

EBZFn(k) · dk = C

C =



Michael Berry

1

H = −t

�

n,α

c†n,αcn+1,α +

µ

2

�

n,α

c†n,αcn,α −i

�

n,α,β

c†n,α

�γD σx

αβ+γR σyαβ

�cn+1,β (1)

− iγ�R

�

n,α,β


µ�

2

�

n,α


+�

n,n�;α,β


†n�,βcn�,βcn,α + h. c. (3)

SA(n,N)− SA(n− 1, N − 1) = Simp


HRashba = αkxσy

EF




1

2π

�

MF · dS = C

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

Berry curvature

Chern number

un,k(r) = � r |un,k �

Berry curvature Berry, RSPSA (1984)

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�

EBZFn(k) · dk = C

C =



1

H = −t

�

n,α

c†n,αcn+1,α +

µ

2

�

n,α

c†n,αcn,α −i

�

n,α,β

c†n,α

�γD σx

αβ+γR σyαβ

�cn+1,β (1)

− iγ�R

�

n,α,β


µ�

2

�

n,α


+�

n,n�;α,β


†n�,βcn�,βcn,α + h. c. (3)

SA(n,N)− SA(n− 1, N − 1) = Simp


HRashba = αkxσy

EF




1

2π

�

MF · dS = C

K = κmin · κmax

radius =1

κ

g = 0 g = 1 g = 2 g = 3

⊂ M

Berry curvature

Chern number

un,k(r) = � r |un,k �

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n=1

� filled bands

�

M=BZFn(k) · dk = C, C ∈ Z

1

2π

� filled bands�

n=1

�

M=BZFn,z(k) dkx dky = C

1

2π

�

n

�Fn(k) · dk = C

C =



vanishing Chern number

odd undertime reversal

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�

EBZFn(k) · dk = C

C =



1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n=1

� filled bands

�


1

2π

� filled bands�

n=1

�


1

2π

�

n

�Fn(k) · dk = C

C =




1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�

EBZFn(k) · dk = C

C =



1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n=1

� filled bands

�


1

2π

� filled bands�

n=1

�


1

2π

�

n

�Fn(k) · dk = C

C =


broken time-reversal

symmetry (like in the

quantum Hall effect)

David Thouless

integer quantum Hall effect: ”TKNN invariant”Thouless et al., PRL (1982)



1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�

EBZFn(k) · dk = C

C =



1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n=1

� filled bands

�


1

2π

� filled bands�

n=1

�


1

2π

�

n

�Fn(k) · dk = C

C =



What if we identify time-reversed points in the BZ?

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�


1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


The parity of the Chern number is unique!

1


Γ A B C

An(k) = −i�un,k | ∇k | un,k�

Fn(k) = −i∇k × �un,k | ∇k | un,k�

1

2π

�

n=1

�


1

2π

�

n

�


1

2π

�

n

�Fn(k) · dk = C

C =


”Z2 topological invariant”,counts the number of Kramers pairs at the edge of the topological insulator(bulk-boundary correspondence)

Kane & Mele, PRL (2005)

Why all the hoopla?

https://www.youtube.com/watch?v=HBuLMrzgbgM



• Future electronics/spintronics:

• New physics in hybrid structures: Majorana fermions, magnetic monopoles, ”dyons”,...

• An inroad to the general study of topological quantum matter

Why all the hoopla?

An inroad to the general study of topological quantum matter...

2D topological insulator

”Periodic tables” for symmetry-protected topological matter:

Now for some serious work on the blackboard!

topological quantum matter, 7.5 ectsphysics.gu.se/~tfkhj/topo/topologicalquantummatter.pdf ·...

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