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Topology and solid state physics

Ming-Che Chang

Department of Physics

11/10/2011 @ NSYSU

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A brief review of topology

K≠0

G≠0

K≠0

G=0

extrinsic curvature K vs intrinsic (Gaussian) curvature G

G>0

G=0

G<0

Positive and negative Gaussian curvature

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2 ( ), 2(1 )Mda G M g

• Gauss-Bonnet theorem (for a 2-dim closed surface)

0g 1g 2g

• Gauss-Bonnet theorem (for a surface with boundary)

,2gM MM Mda G ds k

Marder, Phys Today, Feb 2007

Euler characteristic

The most beautiful theorem in differential topology

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• After a cyclic evolution

0

, ( ) , (

'

0

(

)

' )T

ni

dt E

n

t

n Te

dynamical phase

Adiabatic evolution of a quantum system

0 λ(t)

E(λ(t))( ) (0)T

xx

n

n+1

n-1

• Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned

(

, ( , ( )

)

)n

n t n t

ie

• Do we need to worry about this phase?

• Energy spectrum:

( , ; )H r p

A brief review of Berry phase

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Fock, Z. Phys 1928Schiff, Quantum Mechanics (3rd ed.) p.290 No!

Pf :

( ) ( )H t i tt

0( )

,

' ( ')( )

t

nn

it t

n

Ei dt e e

Consider the n-th level,

Stationary, snapshot state

, ,nn nH E

, ,0n n n

i

≡An(λ)

( )

, ,' n

n n

ie

n

Choose a (λ) such that,

redefine the phase,

Thus removing the extra phase!

An’(λ) An(λ)

An’(λ)=0

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One problem:

( )A does not always have a

well-defined (global) solution

Vector flow A

Contour of

Vector flow A

is not defined here

Contour of

( ) (0) 0C

T A d

0CA d

CC

M. Berry, 1984 : Parameter-dependent phase NOT always removable!

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0' ( ')

( ) (0)C

Ti dti E t

Te e

( )C CA d

• Berry phase (path dependent, but gauge indep.)

Index n neglected

• Berry connection

( )A i

• Berry curvature

( ) ( )A i

C C SA d da

• Stokes theorem (3-dim here, can be higher) 2

3

1

( )t

C

S

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• Berry phase (1984)

y

z

x( )B t

CS

Independent of the speed (as long as it’s slow)

1 = ( )

2C C

• Aharonov-Bohm (AB) phase (1959)

Independent of precise path

Φmagnetic fluxsolenoid

AB0

=2

solid angle

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→ Persistent current

0

2kL kL

Φ

AB phase

• AB phase and persistent charge current (in a metal ring)

confirmed in 1990 (Levy et al, PRL)

After one circle, the electron gets a phase

textured B field

S

Ω(C)

• Berry phase and persistent spin current (Loss et al, PRL 1990)

B

S

e-

After circling once, an electron gets

0

2 ( )2zkL CkL

→ persistent charge and spin current

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Berry curvature and Bloch state

• Space inversion symmetry

• Time reversal symmetry

( ) ( )n nk k

( ) ( )n nk k

both symmetries

( ) 0,n k k

For scalar Bloch state (non-degenerate band):

When do we expect to have it?

• SI symmetry is broken

• TR symmetry is broken

• spinor Bloch state (degenerate band)

• band crossing

← electric polarization

← QHE

← SHE

← monopoleAlso,

2

( ) ( ) ( )2

( )n k knknk nk nk nkk i

p kU r u r u r

mu u

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2

1H

eCh

→ Quantization of Hall conductance (Thouless et al 1982)

Remains quantized even with

disorder, e-e interaction

(Niu, Thouless, Wu, PRB, 1985)

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1

2)

BZ

z kd k C

~ Gauss-Bonnet theorem

=

Brillouin zone

1st Chern ~ Euler characteristic number

Topology and Bloch state

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Bulk-edge correspondence

Different topological classes

Semiclassical (adiabatic) picture: energy levels must cross (otherwise topology won’t change).

→ gapless states bound to the interface, which are protected by topology.

Eg

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Edge states in quantum Hall system

Gapless excitations at the edge

number of edge modes = Hall conductance

(Semiclassical picture)

LLs

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2D TI is also called QSHI

1930 1940 1950 1960 1970 1980 1990 2000 2010

Band insulator (Wilson, Bloch)

Mott insulator

Anderson insulator

Quantum Hall insulator

Topological insulator

First, a brief history of insulators

Peierls transition

Hubbard model

Scaling theory of localization

The topology in “topological insulator”

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• Without B field, Chern number C1= 0

• Bloch states at k, -k are not independent

Lattice fermion with time reversal symmetry

• C1 of closed surface may depend on caps

• C1 of the EBZ (mod 2) is independent of caps

2 types of insulator, the “0-type”, and the “1-type”

• EBZ is a cylinder, not a closed torus.

∴ No obvious quantization.

Moore and Balents PRB 07

(topological insulator, TI)

2D Brillouin zone

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• 2D TI characterized by a Z2 number (Fu and Kane 2006 )

0-type 1-type

How can one get a TI?

: band inversion due to SO coupling

2

( )

1mod 2

2 EBZ EBZd k dk A

~ Gauss-Bonnet theorem with edge

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helical edge states

Bulk-edge correspondence in TI

backscattering by non-magnetic impurity forbidden

robust

TRIM

Dirac point

Fermi level

Bulk states

Edge states

(2-fold degeneracy at TRIM due to Kramer’s degeneracy)

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2D TI A stack of 2D TI

Helical edge state

z

x

y

3 TI indices

Helical SS fragile

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3 weak TI indices: (ν1, ν2, ν3 )

Fu, Kane, and Mele PRL 07 Moore and Balents PRB 07 Roy, PRB 09

ν0 = z+-z0

(= y+-y0 = x+-x0 )

z0 x0

y0

z+

x+ y+

Eg., (x0, y0, z0 )

1 strong TI index: ν0

difference between two 2D TI indices

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Topological insulators in real life

• Graphene (Kane and Mele, PRLs 2005)

• HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 2006)

• Bi bilayer (Murakami, PRL 2006)

• …

• Bi1-xSbx, α-Sn ... (Fu, Kane, Mele, PRL, PRB 2007)

• Bi2Te3 (0.165 eV), Bi2Se3 (0.3 eV) … (Zhang, Nature Phys 2009)

• The half Heusler compounds (LuPtBi, YPtBi …) (Lin, Nature Material 2010)

• thallium-based III-V-VI2 chalcogenides (TlBiSe2 …) (Lin, PRL 2010)

• GenBi2mTe3m+n family (GeBi2Te4 …)

• …

SO coupling only 10-3 meV

2D

3D

• strong spin-orbit coupling

• band inversion

22S.Y. Xu et al Science 2011

Band inversion, parity change, and spin-momentum locking (helical Dirac cone)

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Physics related to the Z2 invariant

Topological insulator

• (Generalization of) Gauss-Bonnett theorem

4D QHE

Spin pump

Helical surface state

Magneto-electric response

Majorana fermion in TI-SC interface

Time reversal inv, spin-orbit coupling, band inversion

Interplay with SC, magnetism

QC

Quantum SHE

graphene

…

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Berry phase and topology in condensed matter physics

1982 Quantized Hall conductance (Thouless et al)

1983 Quantized charge transport (Thouless)

1990 Persistent spin current in one-dimensional ring (Loss et al)

1992 Quantum tunneling in magnetic cluster (Loss et al)

1993 Modern theory of electric polarization (King-Smith et al)

1996 Semiclassical dynamics in Bloch band (Chang et al)

2001 Anomalous Hall effect (Taguchi et al)

2003 Spin Hall effect (Murakami et al)

2006 Topological insulator (Kane et al)

…

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Thank you!