theory of topological insulators - ifw-website · · 2013-03-08pre hist qhe cs bw kubo red z2 sum...

Pre Hist QHE CS BW Kubo Red Z2 Sum

Helmut EschrigLeibniz-Institut für Festkörper- und Werkstofforschung DresdenLeibniz-Institute for Solid State and Materials Research Dresden

Theory of Topological Insulators

Helmut Eschrig IFW Dresden Theory of Topological Insulators


1 Preliminaries

2 Historical Remark

3 Quantum Hall Effect

4 (2r + 1)-Dimensional Chern-Simons Field Theory

5 Bloch and Wannier functions

6 Kubo’s formula

7 Dimension Reduction

8 Z2-Symmetry

9 Summary



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



Local coordinates on a real manifold M of dimension m:U ⊂ M is an open set small enough to be trivialized:

M ⊃ U ∋ x ↔ x=x = (x1, . . . , xm) ∈ Rm

Élie Cartan’s external calculus on M does not need a metric:

external 0-form: f : x 7→ f (x) = f (x) ∈ R,

external 1-form: ω = df =(∂ f/∂x i)dx i =∑

j ωj dx j ,

external 2-form: σ =∑

i<j σij dx i ∧ dx j = (2!)−1σijdx idx j ,· · ·external r -form: ρ =

∑

i1<...<ir ρi1...ir dx i1 ∧ · · · ∧ dx ir =,

= (r !)−1ρi1...ir dx1 · · ·dx r

alternating tensor ρi1...ir .

y = y(x) : df = (∂ f/∂x i)dx i = (∂ f/∂y j)(∂y j/∂x i)dx i = (∂ f/∂y j)dy j .



dρ =∑

i1<...<ir

dρi1...ir ∧ dx i1 ∧ · · · ∧ dx ir =

=∑

i1<...<ir

∑

j

∂ρi1...ir

∂x j dx j ∧ dx i1 ∧ · · · ∧ dx ir .

∫

Uρ =

∫

U

∑

i1<...<ir

ρi1...ir dx i1 ∧ · · · ∧ dx ir ,

dx i1 ∧ · · · ∧ dx ir =D(x1, . . . , x r )

D(y1, . . . , y r )dy j1 ∧ · · · ∧ dy jr .

Stokes:∫

Mdρ =

∫

∂Mρ,

∂M: boundary of M.



An action integral over M which is an integral of an externalform depends only on the topology of M, not on a metric; it iscalled a topological term.

For instance the action integral of a Maxwell field,∫

d4xFµν Fµν =

∫

d4xFστ gσµ gτν Fµν

depends on the pseudo-Riemannian metric gµν ,while ∫

d4xδµνστ0123 FµνFστ =

∫

d4xE · B

is a topological term.



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



Theoretical prediction of quantum oscillations in σxy :

σxy ≈ −necB

+1ωcτ

σxx .

T. Ando, Y. Matsumoto and Y. Uemura, J. Phys. Soc. Japan 39,279 (1975).

Discovery of QHE:

K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45,494 (1980).



Kubo’s formula as a Chern number: (TKNN-number)D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. denNijs, Phys. Rev. Lett. 49, 405 (1982).Edge currents: B. I. Halperin, Phys. Rev. B 25, 2185(1982).Berry’s phase: M. V. Berry, Les Houches, 1983; Proc. R.Soc. A 392, 45 (1984).Identification of Berry’s phase and TKNN-number as Chernnumbers: Barry Simon, Phys. Rev. Lett. 51, 2167 (1983).

A case of combinatorial topology:Shiing-Shen Chern and James Simons, Ann. Math. 99, 48(1974).

Relevance in gauge-field theory and string theory:Edward Witten, Commun. Math. Phys. 121, 351 (1989).In this paper Witten coined the namesChern-Simons field theory and Chern-Simons action.



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



(2 + 1)-dimensional Chern-Simons field theory:

X.-L. Qi, T. L. Hughes and S.-C. Zhang, Phys. Rev. B 78,195424 (2008).

Sint =

∫

d3x Aµ jµ =σH

2!

∫

d3x δµνσ012 Aµ∂νAσ, ∂µ =

∂

∂xµ, µ = 0,1,2;

jµ =δSint

δAµ=σH δ

µνσ012 ∂νAσ,

↑

Hall conductance

δcycl{012}012 = +1,

δcycl{210}012 = −1,

δµνσ012 = 0 else.



jµ = σH Fµ, Fµ = δµνσ012 ∂νAσ

j0 = cρ = σH B,

j1 = σH E2, j2 = −σH E1.

F 0 = ∂1A2−∂2A1 = B (= Bz),

F 1 = ∂2A0 − ∂0A2 = E2,

F 2 = ∂0A1−∂1A0 = −E1.

0 = ∂µ jµ ⇒ 1c

∂Bz

∂t = −(∇× E)z : Faraday’s law of induction.



Kubo:

σH =e2

h

(1

2π

∫

T2κ

occ.∑

n

(−2i)(∂1unκ | ∂2unκ) dκ1 ∧ dκ2

︸︷︷︸

integer (first Chern number)

)

, ∂ i =∂

∂κi

σH =e2

~

1(2π)2

∫

T2κ

d2κ δ12ij tr(∂ iAj), Stokes:

∫

MdA =

∫

∂MA

Ajmn = (−i)(umκ | ∂

junκ)

A =∑

j

Aj dκj : Berry-Simon connection form

F = DA = dA + i[A,A] = (tr([A,A]) = 0)

=∑

ij

∂ iAj dκi ∧ dκj : Berry-Simon curvature form



q = (qA) = (ct , x1, x2, κ1, κ2), A = 0,1,2,3,4, ∂A =∂

∂qA,

(AA) = (A0,A1,A2,0,0, ), (AA) = (0,0,0,A3,A4),

Sint =e2

~

12!(2π)2

∫

d5q δABCDE01234 AA∂BAC tr(∂DAE),



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



integer r th Cern number Cr of r th Chern character:

Cr =1r !

∫

tr(

F

2π∧ · · · ∧

F

2π

)

= tr

((2

2π

∫

d2k F12)r)

=

=1

(2π)r

∫

tr(

F12F34 · · · F2r−1,2r)

dk1 ∧ dk2 ∧ · · · ∧ dk2r =

=1

r !2r (2π)r

∫

d2r k δ12···2ri1i2···i2r

tr(F i1i2F i3i4 · · · F i2r−1i2r

).

S(2r+1)int =

e2

~

Cr

(2π)r

∫

A0∂1A2 · · · ∂2r−1A2r cdt ∧ dx1 ∧ · · · ∧ dx2r =

=e2

~

Cr

(r + 1)!(2π)r

∫

d2r+1x δµν···στ012···2r Aµ∂ν · · · ∂σAτ .



r th order response:

jµ =e2

~

Cr

r !(2π)r δµν···στ012···2r∂νA. · · · ∂σAτ ,

S(2r+1)int =

e2

~

1r !(r + 1)!(2π)2r

∫

d4r+1q δA0A1···A4r01···4r ·

· AA0∂A1 · · · ∂A2r−1AA2r tr(DA2r+1AA2r+2 · · ·DA4r−1AA4r

).



Example: 4D QHES.-C. Zhang and J. P. Hu, Science 294, 823 (2001).

r = 2, A0 = A1 = A4 = 0, A2 = B3x1, A3 = −E3ct :

j4 =e2

~

C2

(2π)2 B3E3 = σHE3,

∫

dx1dx2 j4 =e2

~

C2

(2π)2

(∫

dx1dx2 B3

︸︷︷︸

NΦ0=N2π~/e

)

E3 =e

2πC2N E3,

12-plane

34-plane�B3

�E36

j4



Simple (2r + 1)D model, topologically equivalent tomost conceivable 2rD band structures:

(2r + 1)D Dirac model:

H =

∫

d2r x ψ†(

−i∑

j

Γj∂j + Γ0m)

ψ, [Γµ, Γν ]+ = 2δµν I,

j = 1, . . . ,2r , µ, ν = 0,1, . . . ,2r ,

The Γµ are (r × r)-matrices of a faithful representation of theso(2r + 1) Lie algebra (Clifford algebra).

2rD Dirac lattice model (tb):

H =∑

k

ψ†k

(∑

j

Γj sin kj + Γ0(m + c∑

j

cos kj))

ψk =

=∑

k

ψ†k

(∑

µ

Γµdµ(k))

ψk , c : NN hopping integral.



r = 2 :

dµ(k) =

([

m + c4∑

j=1

cos kj], sin k1, sin k2, sin k3, sin k4

)

,

εk = ±

√∑

µ

d2µ(k),

C2 =3

8π2

∫

d4k δabcde01234 da∂1db∂2dc∂3dd∂4de =

=

0 |m| > 4c,

1 −4c < m < −2c,

−3 −2c < m < 0,

3 0 < m < 2c,

−1 2c < m < 4c.



gapless surface states on 3D surface (open x4 boundary) for

εk = 0 for some k .

1 Dirac cone at k = 0 for |C2| = 1,3 Dirac cones at k = (π,0,0), (0, π,0), (0,0, π) for |C2| = 3.

For more details see Qi at al. (2008).



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



d-Tori Tdr (unit cell) and T

dk Brillouin zone.

R · G = 2π · integer.

Crystal Hamiltonian:

H = −~

2

2m∇2+V , Hψnk = ψnkεnk , (ψnk |ψn′k ′)∞ = δnn′δ(k−k ′),

ψnk (r) =eik ·r

|Tdk |

1/2unk (r), unk (r+R) = unk (r), unk+G(r) = unk (r),

Hk = −~

2

2m(−i∇+k)2+V , Hk unk = unkεnk , (unk |un′k )

Tdr

= δnn′ ,

anR(r) =1

|Tdk |

∫

Tdk

ddk eik ·(r−R)occ.∑

n′

Unn′(k)un′k (r), U†(k) = U−1(k),

(anR |an′R′) = δnn′δRR′ ,∑

R

occ.∑

n

|anR(r)|2 = ρ(r).



Change of polarization:

∆P = −e

|Tdr |

occ.∑

n

∫ ∞

dd r r ∆|an0(r)|2 =

=e

|Tdr |

∆

(

−i|Td

k |

∫

Tdk

ddkocc.∑

n

∫

Tdr

dd r(Uu)†nk∇k (Uu)nk

)

.

occ.∑

n

((Uu)nk | ∇k (Uu)nk

)=

occ.∑

n

(unk | ∇k unk )+tr(U−1(k)∇k U(k)

),

trU−1∇U = tr∇ ln U = ∇tr ln U = ∇ ln det U = i∇θ, det U = eiθ,

det U(k+G) = det U(k) ⇒ θ(k) = α(k)+∑

R

′k ·R, α(k+G) = α(k),

∆P is only determined modulo (e/|Tdr |)∑

R′R.



Consider a 1D crystal:

R = Na, G =2πa2 a, |T1

r | = |a|, |T1k | =

2π|a|.

θ(k) = α(k) + Nk ,

∆P =e

|T1r |

∆

(a

2π(−i)

∫ 1

0dk

occ.∑

n

((Uu)nk | ∂

k (Uu)nk))

.



Adiabatic parameter t :

H = Ht , 0 ≤ t ≤ 1, P(0) = 0,P(1) = P,

unk → untk , U(k) → U(k , t).

6

-

t

k

M

�

U

∂M

T1k



P = P(1) − P(0) =e

|T1k |

a2π

∫

∂MtrA =

=

∫ 1

0dt∂P∂t

=e

|T1k |

a2π

∫

MtrF ,

Amn = −i((Uu)mtk | ∂

k (Uu)ntk)dk , (Uu)m,ntk occupied,

Fmn = −i(∂t(Uu)mtk | ∂

k (Uu)ntk)dt∧dk , (Uu)m,ntk occupied,

trA : J. Zak’s Berry phase due to the k -torus,

A : J. Zak’s Berry connection,F : J. Zak’s Berry curvature.

J. Zak, Phys. Rev. B 62, 2747 (1989).



Let Ht be periodic, H1 = H0, then, ∂M = ∅. Is P(1) = P = 0?

Recall that ∆P was via A only determined modulo (e/|T1k |)Na.

12π

∫

MtrF = N = integer (= first Chern number C1).

While trA dependts on U(k , t), trF depends only on U(k)(for smooth U(t)).

trF = d trA holds locally, its continuation to all M is obstructed,if C1 6= 0.

This is a case of the general Chern-Weil theorem:While trF is uniquely defined in such cases, it is independent ofthe local gauge (U(t)) of trA.



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



L. Smrcka and P. Streda, J. Phys. C 10, (1977); TKNN, 1982.

conductivity ∼ current-current correlation function;for non-interacting Bloch electrons at zero temperature:

σH =i~

(2π)2

∫

d2κocc.∑

n

unocc.∑

m

(umκ |jx |unκ)(unκ |jy |umκ) − c.c.(εmκ − εnκ)2



Ax = 0, Ay = Bx − Ect , B = Bez , E = Eey ,

H =~

2

2m

((

−i∂

∂x

)

)2 +(

−i∂

∂y+

e~c

Bx −e~

Et)2)

+ V (x , y)

T (R) = exp(iR ·

(−i∇− (e/~c)eyBx

), R = n1a1 + n2a2,

T (R)T (R′) = exp((ie/~c)B·R×R′

)T (R′)T (R), [H, T (R)] = 0,

choose (e/~c)B · R1 × R2 = 2π· integer, then

Hκunκ = unκεnκ, Hκ =~

2(κ + κ)2

2m+V , κ = −i∇+(e/~c)ey (Bx−Ect).

j = −ep

m|T2r |

= −e~

2∑

i=1

R i

|T2r |

[ ∂

∂κi, Hκ

]

.

σH =i~

(2π)2

∫

d2κocc.∑

n

unocc.∑

m

(umκ |jx |unκ)(unκ |jy |umκ) − c.c.(εmκ − εnκ)2



σH =e2

2π~

(

12π

∫

T2r

occ.∑

n

(−2i)(∂1unκ | ∂2unκ) dκ1 ∧ dκ2

)

=

=e2

2π~C1, C1 ∈ Z ∂ i =

∂

∂κi.

G =∑

n

u∗n (ω − εn)

−1 un,

C1 ∼

∫

tr(

Pocc(G∂ωG−1)(G∂1G−1)((G∂2G−1))

dω∧dκ1∧dκ2.



Remark regarding the literature:

G−1G = 1 ⇒ (∂G−1)G + G−1∂G = 0;

tr(G−1∂G · · · ) = ∓tr(G ∂G−1 · · · )

.



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



D. J. Thouless, Phys. Rev. B 27, 6083 (1983);

Consider a QHE setup with a strip long in y -direction (furtheron with periodic boundary conditions with macroscopic lengthLy ) but constraint in x-direction by width Lx , and let themagnetic field adiabatically change.

A0 = 0, Ax = 0, Ay = B(t)x ,

Bz(t) =∂Ay

∂x= B(t), Ey (t) = −

∂Ay

c∂t= −

xc∂B∂t.

κ2 = κ(x) = −i∂

∂y+

e~c

B(t)x , Hκ → Hλ(x ,t),κ(x).

λ(x , t) ≈ const. within a unit cell: adiabatic and quasiclassic.



[ ∂

∂t, Hλκ

]

= −xc∂B∂t

epy

m=

xc∂B∂t

jy .

jx =∂Px

∂t,

∂jx

∂x= −

∂ρ

∂t=∂2Px

∂t∂x:

(t , x , κ) = (q0,q1,q2), ρ = j0, j i = −δij

01∂Px

∂q j , i , j = 0,1.

AA =

occ.∑

n

(−i)(unq | ∂Aunq), F = dA, ∂i =∂

∂q i , i = 0,1,2 :

j i = −1

2πδ012

i2j

∫

dq2dq jF2j .

i = 0 : quantized charge flow across the QHE strip (Laughlin,1981).i = 1 and B(t) periodic: quantized charge pumping through a1D conductor (Thouless, 1983).



Now the magnetic state is not any more in a closed positionspace without boundary (unit cell torus or Born-van Karmantorus T

2r ); it is bounded in x-direction by edges.

Since outside the edges the vacuum state has C1 = 0 and theChern number is topological prodected as long as theelectronic spectrum is gapped, C1 6= 0 in the QHE strip canonly happen, if there are gapless edge states.



X.-L. Qi, T. L. Hughes and S.-C. Zhang, Phys. Rev. B 78,195424 (2008).

gave a general prescription how automatically an analogousdimension reduction form a fundamental (2r + 1)DChern-Simons field theory to an adiabatic and quasi-classical((2r + 1) − s)D Chern-Simons field theory is obtained.



Start with the (2r + 1)D Chern-Simons action

S(2r+1)int =

e2

~

1r !(r + 1)!(2π)2r

∫

d4r+1q δA0A1···A4r01···4r ·

· AA0∂A1 · · · ∂A2r−1AA2r tr(DA2r+1AA2r+2 · · ·DA4r−1AA4r

),

remove a term ∂A2r−1AA2r ,remove the integrations over dx2r dk2r/(2π),multiply by the original power r of the external potential A, afters reduction steps find

S(2r+1−s)(s) int =

e2

~

(2r+1−sr

)

(2r + 1 − s)!(2π)2r−s

∫

d4r+1−2sq δA0A1···A4r−2s01···(4r−2s) ·

· AA0∂A1 · · · ∂A2r−2s−1AA2r−2s ·

· tr(DA2r−2s+1AA2r−2s+2 · · ·DA4r−2s−1AA4r−2s

).



There are at most r descendants from a fundamental (2r + 1)DChern-Simons theory, then there is no external gauge potentialA left any more for reduction.

To arrive in our D ≤ 3-dimensional physical position spaces,the following cases are possible:

S(0+1)int ,

S(2+1)int → S(1+1)

1 int → S(0+1)2 int ,

S(4+1)int → S(3+1)

1 int → S(2+1)2 int ,

S(3+1)3 int → S(2+1)

4 int ,

S(3+1)5 int

dHvA, QHE,TRI, QSHE



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



In order to have a topological protection of a non-trivial Chernnumber state in a dimension reduced Chern-Simons theory, anadditional symmetry is needed which enforces the boundarystates to be gapless.

Such a symmetry may be one of the ubiquitous discretesymmetries of the kinetic terms in the action:

charge conjugation: C†H−k C = −H tk , C†C = C∗C = 1,

time reversal: T †H−k T = H tk , T †T = −T ∗T = 1.

C : Aµ → −Aµ, T : Aµ →

{

A0

−Ai

Chern-Simons Lagrangian L(2r+1)int :

C : L(2r+1)int → (−1)r+1L(2r+1)

int , T : L(2r+1)int → (−1)r L(2r+1)

int .



A Chern-Simons theory, depending on the properties of thekinetic terms, may be C-invariant for r odd and T -invariant for reven.

This is why non-trivial S(3+1)1 int and S(2+1)

2 int must be time-reversalinvariant. (In the first (second) descendant of a fundamental(2r + 1)-dimensional case an adiabatic quasiclassicalpolarization (gauge curvature) appears in the action instead ofone (two) external potential factor(s) with the sametransformation properties as the latter.)



1 Preliminaries

2 Historical Remark




6 Kubo’s formula


8 Z2-Symmetry

9 Summary



Chern-Simons action integrals are integrals of externalforms and hence independent of a metric: topologicalterms.

The QHE is the prototype of such a theory in solid statephysics.

A Chern number is the 2r -dimensional integral of a 2r -form(Chern form) which is an invariant polynomial of theBerry-Simon curvature form. It locally derives from aChern-Simons form obtained from the Berry-Simonconnection form, but does not depend on the choice of theconnection (Berry gauge invariance).

The theory combines a Berry phase gauge theory with agauge field theory of an external field.



Thank you for your attention!


theory of topological insulators - ifw-website · · 2013-03-08pre hist qhe cs bw kubo red z2 sum...

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