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Topology in solid state physics

Ming-Che Chang

Department of Physics

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Curvature

K≠0

G≠0

K≠0

G=0

extrinsic curvature K vs

intrinsic curvature G

(Gaussian curvature)

G>0

G=0

G<0

Positive and negative

Gaussian curvature

外在

內在

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Euler characteristic 2 ( ), 2(1 )M

da G M gπχ χ= = −∫

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

0g = 1g = 2g =

The most beautiful theorem

in differential topology

• Gauss-Bonnet theorem (for a 2D surface with boundary)

( ),2gM M

M Mda G ds k π χ∂

∂+ =∫ ∫

Marder, Phys Today, Feb 2007

• Can be generalized to higher (even) dimension.

歐拉特徵數

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dirty clean

• Flux quantization

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h

eφ =

• Quantum Hall effect

• …

Often protected by topology

Condensed matter physics is physics of dirt - Pauli

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2-dim electron gas (2DEG)

Quantum Hall effect (von Klitzing, PRL 1980)

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1985

ρxy deviates from (h/e2)/n by less than 3 ppm on the very first report.

• This result is independent of the shape/size of sample.

• Different materials lead to the same effect (Si MOSFET, GaAs heterojunction…)

→ a very accurate way to measure α-1 = h/e2c = 137.036 (no unit)

→ a very convenient resistance standard (1990).

Quantum Hall effect (von Klitzing, 1980)

h/e2 = 25.81202 k-ohms

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Gauss-Bonnet theorem

Quantization of Hall conductanceThouless et al’s argument (1982), bulk description

( )2

21( )

2nzH n

BZ

ke

d kh

σπ

Ω

=

∫r

• From linear response theory, the Hall conductivity for the n-th band is

1st Chern number

(an integer for a filled band)

(

( )

)

n nk k

k

n

n

i u u

A k

kΩ = ∇ × ∇

= ∇ ×

r r

r

rr

rr

( )n n nkA k i u u≡ ∇ r

rr

• Berry curvature (for n-th band)

• Berry connection

2

1( ) 2

BZ

nz kd k CπΩ =∫r

2M

da G πχ=∫

C1 is not changed as long as the band does not touch the other bands

Cell-periodic part of the Bloch state

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• Robust against disorder

(no back-scattering)

• Classical picture

Chiral edge state

(skipping orbit)

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

Bulk-edge correspondence Edge states in quantum Hall system

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• classical Hall effect

B

+++++++

B↑ ↑ ↑ ↑

↑↑↑↑↑↑↑

↑↑↑↑

↑ ↑

↑↑↑↑↑↑↑

↑↑↑↑↑↑↑

↑↑↑↑

↑↑↑↑

• spin Hall effect (transverse spin current)

• anomalous Hall effect (in magnetic materials)

No magnetic field required !

Lorentz force

Berry curvature (intrinsic)

Skew scattering (extrinsic)

Berry curvature (intrinsic)

Skew scattering (extrinsic)

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• skew scattering by spinless impurities

• no magnetic field required

(J.E. Hirsch, PRL 1999, Dyakonov and Perel, JETP 1971. Kato et al, Science 2004)

(extrinsic) Spin Hall effect

Local Kerr effect in strained n-type

bulk InGaAs, 0.03% polarization

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From Murakami’s talk

Spin current is no longer quantized

when we turn on the spin-orbit coupling.

Quantum version of spin Hall effect?

• 2006: Kane and Mele, S.C. Zhang et al found Quantum spin Hall insulator

(aka 2D topological insulator)

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Topological insulators in real life

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

• Bi bilayer (Murakami, PRL 2006, Drozdov et al Nat Phys 2014 [exp])

• Silicine (Liu et al, PRL 2011)

• Tin thin film (Xu et al, PRL 2013)

• BiX/SbX monolayer (X=H,F,Cl,Br) (Song et al 2014)

• 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; BiTeCl

[strong SIA], Chen Nat Phys 2014)

• GenBi2mTe3m+n family (GeBi2Te4 …)

• BaBiO3 [SC] (0.7 eV, Yan, Nat Phys 2013 [theo])

• …

2D

3D

band inversion induced by spin-orbit coupling

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• Insulators are characterized by (Fu and Kane 2006)

2

( )

01mod 2

12 EBZ EBZd k dk Aν

π ∂

= Ω − = ∫ ∫

～ Gauss-Bonnet theorem for a 2D surface with boundary

( ),2g

M MM Mda G ds k π χ

∂∂+ =∫ ∫

2χ = 1χ =0χ =

Ordinary insulator

Quantum spin Hall insulator

The topology behind the topological insulator

ν does not changed as long as the band gap remains open

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Oh, Science 2013

Hall effects, their quantum companions, and edge states

3D Topological insulator

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Robust

chiral edge state

Robust

helical edge state

Edge state: Quantum Hall insulator vs topological insulator

Real space Energy spectrum

2 robust edge channels, but

spin transport is not quantized.•Dirac point at TRIM•Dirac cone with spin texture

22S.Y. Xu et al Science 2011

Angle-Resolved Photo-Emission Spectroscopy (ARPES)

Band inversion, parity change, and

spin-momentum locking (Dirac cone with spin texture)